Generalized Vandermonde Matrices and Determinants in Electromagnetic Compatibility

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Mälardalen University Press Licentiate Theses No. 253

Mälardalen University Press Licentiate Theses No. 253 GENERALIZED VANDERMONDE MATRICES AND DETERMINANTS IN ELECTROMAGNETIC COMPATIBILITY

GENERALIZED VANDERMONDE MATRICES AND DETERMINANTS IN ELECTROMAGNETIC COMPATIBILITY

Karl Lundengård

Karl2017 Lundengård

2017

School of Education, Culture and Communication

School of Education, Culture and Communication Abstract

Matrices whose rows (or columns) consists of monomials of sequential powers are called Vandermonde matrices and can be used to describe several useful concepts and have properties that can be helpful for solving many kinds of problems. In this thesis we will discuss this matrix and some of its properties as well as a generalization of it and how it can be applied to curve fitting discharge current for the purpose of ensuring electromagnetic compatibility. In the first chapter the basic theory for later chapters is introduced. This includes the Vandermonde matrix and some of its properties, history, applications and generalizations, interpolation and regression problems, optimal experiment design and modelling of electrostatic discharge currents with the purpose to ensure electromagnetic compatibility. The second chapter focuses on finding the extreme points for the determinant for the Vandermonde matrix on various surfaces including spheres, ellipsoids, cylinders and tori. The extreme points are analysed in three dimensions or more. The third chapter discusses fitting a particular model called the p-peaked Analytically Extended Function (AEF) to data taken either from a standard for electromagnetic compatibility or experimental measurements. More specifically the AEF will be fitted to discharge currents from the IEC 62305-1 and IEC 61000-4-2 standards for lightning protection and electrostatic discharge immunity as well as some experimentally measured data of similar phenomena.

1 Abstract

1 Acknowledgements

First I am very grateful to my closest colleagues. My main supervisor Pro- fessor Sergei Silvestrov introduced the Vandermonde matrix to me and suggested the research pursued in this thesis. From him and my co-supervisor Professor Anatoliy Malyarenko I have learned invaluable lessons about mathematics research that will be very important in my future career. Optimising the Vandermonde determinant would have been less fruitful and engaging without fellow research student Jonas Osterberg¨ whose skills and ideas com- plement my own very well. I am very glad to have worked with Dr. Milica Ranˇcićwhose conscientiousness, work ethic and patience when introducing me to the world of electromagnetic compatibility and developing and evaluating the ideas used in this thesis makes me consider her a true role model for an interdisciplinary researcher. The research related to electromagnetic compatibility owes a lot to the regular input from Assistant Professor Vesna Javor to ensure the relevance and quality of the work. I am also grateful to the other research students at Mälardalen University that I have worked alongside with. I have found the environment at the School for Education, Culture and Communication at Mälardalen University excellent and full of friendly, helpful and skilled co-workers. Last but not least a very heartfelt thank you to my family for all the support, encouragement and assistance you have given me. A special men- tion to my sister for help with translating from 18th century French. I am also very sad that I will not get to spend time explaining the contents of this thesis to my father whose entire mathematics career consisted of unsuc- cessfully solving a single problem on the blackboard in 9th grade. On the other hand, my mother’s modest academic credentials have never stopped her from engaging, discussing and enjoying my work, interests and other new knowledge so I am sure she will keep me busy. Without the ideas, requests, remarks, questions, encouragements and patience of those around me this work would not have been completed.

Karl Lundeng˚ard V¨aster˚as, March, 2017

3 Acknowledgements

Karl Lundeng˚ard V¨aster˚as, March, 2017

3 Generalized Vandermonde matrices and determinants in electromagnetic compatibility

Popul¨arvetenskaplig sammanfattning Popular-science summary

Denna licentiatuppsats behandlar tv˚aolika ämnen, optimering av determi- This licentiate thesis discusses two different topics, optimisation of the Van- nanten av Vandermonde-matrisen över olika volymer i olika dimensioner och dermonde determinant over different volumes in various dimensions and how hur en viss klass av funktioner kan användas för att approximera strömmen a certain class of functions can be used to approximate the current in elec- i elektrostatiska urladdningar som används för att säkerställa elektromag- trostatic discharges used in ensuring electromagnetic compatibility. An ex- netisk kompatibilitet. En exempel p˚akopplingen mellan de tv˚aomr˚adena ample of how the two topics can be connected is given in the final part of ges i den sista delen av uppsatsen. the thesis. En Vandermonde-matris ären matris därraderna (eller kolonnerna) ges A Vandermonde matrix is a matrix with rows (or column) given by av stigande potenser och s˚adana matriser förekommer i m˚anga olika sam- increasing powers and such matrices appear in many different circumstances, manhang, b˚ade inom abstrakt matematik och tillämpningar inom andra both in abstract mathematics and various applications. In the thesis a brief omr˚aden. I uppsatsen ges en kort genomg˚ang av Vandermonde-matrisens history of the Vandermonde matrix is given as well as a discussion of some historia och tillämpningar av den och n˚agra besläktade matriser. Fokus applications of the Vandermonde matrix and some related matrices. The ligger p˚ainterpolation or regression vilket kortfattat kan beskrivas som main topics will be interpolation and regression which can be described as metoder för att anpassa en matematisk beskrivning till given data, t.ex. methods for fitting a mathematical description to collected data from for fr˚anexperimentella mätningar. example experimental measurements. Determinanten av en matris ärett tal som beräknas fr˚anmatrisens el- The determinant of a matrix is a number calculated from the elements of ement och som p˚aett kompakt sätt kan beskriva flera olika egenskaper av the matrix in a particular way and it can describe properties of the matrix matrisen eller systemet som matrisen beskriver. I denna uppsats diskuteras or the system it describes in a compact way. In this thesis it is discussed hur man skall välja element i Vandermonde-matrisen föratt maximera de- how to choose the elements of the Vandermonde matrix to maximise the terminanten under förutsättningen att elementen i Vandermonde-matrisen determinant under the constraint that the elements that define the Vander- tolkas som en punkt i en volym (som kan ha fler än tre dimensioner). Flera monde determinant are interpreted as points in a certain volume (that can volymer undersöks, bland annat klot, kuber, ellipsoider och torus. have a dimension higher than three). Examined volumes include spheres, En motivering till varför det är användbart att veta hur Vandermonde- cubes, ellipsoids and tori. matrisens determinant kan maximeras äratt det kan användas till optimal One way to motivate the usefulness of knowing how to maximise the experiment design, det vill säga att avgöra hur man skall välja mätpunkter Vandermonde determinant is that it can be used in optimal experiment de- föratt kunna bygga en s˚abra matematisk modell som möjligt. Ett exempel sign, that is determining how to choose the data points in an experiment to p˚ahur detta kan g˚atill ges i sista delen av uppsatsen. construct the best possible mathematical model. An example of how to do Ett omr˚ade där det är användbart att kunna bygga matematiska mod- this can be found in the final section of the thesis. eller fr˚an experimentella data är elektromagnetisk kompatibilitet. Detta One area where it is useful to construct mathematical model from ex- omr˚ade handlar om att säkerställa att system som inneh˚aller elektronik inte perimental data is electromagnetic compatibility. This is the study if how to p˚averkas förmycket av externa elektromagnetiska störningar eller störan- ensure that a system that contains electronics is not disturbed too much by dra system d˚ade används. En viktig del av detta omr˚ade är att undersöka external electromagnetic disturbances or disturbs other systems when it is hur systemet reagerar p˚aolika externa störningar s˚asomde beskrivs i olika used. An important aspect of this field is examining how systems respond konstruktionsstandarder. I uppsatsen diskuterar vi hur man med hjälp av to different external disturbances described in different construction stan- en specifik klass av funktioner kan konstruera matematiska modeller baser- dards. In this thesis we discuss how a specific class of functions can be used ade p˚aspecifikationer i standarder eller experimentella data. Välbekanta to construct mathematical models based on specifications in standards or fenomen s˚asom blixtnedslag och urladdningar av statisk elektricitet mellan experimental data. Well-known phenomenon such as lightning strikes and människa och metallförem˚al diskuteras. electrostatic discharges from a human being to a metal object are discussed.

4 5 Generalized Vandermonde matrices and determinants in electromagnetic compatibility

Popul¨arvetenskaplig sammanfattning Popular-science summary

4 5 Notation Matrix and vector notation v, M - Bold, roman lower- and uppercase letters denote vectors and matrices respectively.

Mi,j - Element on the ith row and jth column of M.

M ,j, Mi, - Column (row) vector containing all elements · · from the jth column (ith row) of M. [a ]nm - n m matrix with element a in ij ij × ij the ith row and jth column. V , V = V - n m Vandermonde matrix. nm n nn × G , G = G - n m generalized Vandermonde matrix. nm n nn × Standard sets Z, N, R, C - Sets of all integers, natural numbers (including 0), real numbers and complex numbers. n n n Sp , S = S2 - The n-dimensional sphere deﬁned by the p - norm, n+1 n n+1 p S (r)= x R xk = r . p ∈ | | k=1 k [K] - All functions on K with continuous kth derivative. C Special functions Deﬁnitions can be found in standard texts. Suggested sources use notation consistent with thesis. (α) (α,β) Hn, Cn , Pn - Hermite, Gegenbauer and Jacobi polynomials, see [2]. Γ(x), ψ(x) - The Gamma- and Digamma functions, see [2].

2F2(a, b; c; x) - The hypergeometric function, see [2]. a Gm,n z - The Meijer G-function, see [139]. p,q b Other df dx = f (x) - Derivative of the function f with respect to x. dkf (k) dxk = f (x)-kth derivative of the function f with respect to x. ∂f ∂x = f (x) - Partial derivative of the function f with respect to x. a¯b - Rising factorial a¯b = a(a + 1) (a + b 1). ··· −

7 Notation Matrix and vector notation v, M - Bold, roman lower- and uppercase letters denote vectors and matrices respectively.

Mi,j - Element on the ith row and jth column of M.

7 Contents

List of Papers 13

1 Introduction 15 1.1TheVandermondematrix...... 19 1.1.1 WhowasVandermonde?...... 19 1.1.2 The Vandermonde determinant ...... 21 1.1.3 Inverse of the Vandermonde matrix ...... 26 1.1.4 Thealternantmatrix...... 27 1.1.5 The generalized Vandermonde matrix ...... 30 1.2Interpolation...... 32 1.2.1 Polynomialinterpolation...... 33 1.3Regression...... 37 1.3.1 Linearregressionmodels...... 38 1.3.2 Non-linear regression models ...... 39 1.3.3 The Marquardt least-squares method ...... 39 1.3.4 D-optimalexperimentdesign...... 43 1.4 Electromagnetic compatibility and electrostaticdischargecurrents...... 46 1.4.1 Electrostatic discharge modelling ...... 48 1.5Summariesofpapers...... 51

9 Contents

List of Papers 13

9 Generalized Vandermonde matrices and determinants in electromagnetic compatibility CONTENTS

2 Extreme points of the Vandermonde determinant 53 3.2.2 Estimating parameters for underdetermined systems . 111 2.1 Extreme points of the Vandermonde determinant and related 3.2.3 Fitting with data points as well as charge flow and determinants on various surfaces in three dimensions . . . . . 55 specificenergyconditions...... 112 2.1.1 Optimization of the generalized Vandermonde deter- 3.2.4 Calculating the η-parameters from the β-parameters . 115 minantinthreedimensions...... 55 3.2.5 Explicit formulas for a single-peak AEF ...... 116 2.1.2 Extreme points of the Vandermonde determinant on 3.2.6 Fitting to lightning discharge currents ...... 117 the three-dimensional unit sphere ...... 59 3.3 Approximation of electrostatic discharge currents ...... 121 2.1.3 Optimisation of the Vandermonde determinant on the three-dimensionaltorus...... 60 3.3.1 IEC 61000-4-2 Standard current waveshape ...... 122 2.1.4 Optimisation using Gröbner bases ...... 64 3.3.2 D-Optimal approximation for exponents given by a classofarithmeticsequences...... 125 2.1.5 Extreme points on the ellipsoid in three dimensions . 66 3.3.3 Examples of models from applications and experiments 129 2.1.6 Extreme points on the cylinder in three dimensions . . 68 3.3.4 SummaryofESDmodelling...... 131 2.1.7 Optimizing the Vandermonde determinant on a surface defined by a homogeneous polynomial ...... 70 References 135 2.2 Optimization of the Vandermonde determinant on some n-dimensional surfaces ...... 72 Index 151 2.2.1 The extreme points on the sphere given by roots of a polynomial...... 73 List of Figures 153 2.2.2 Further visual exploration on the sphere ...... 81 List of Tables 156 2.2.3 The extreme points on spheres defined by the p-norms givenbyrootsofapolynomial...... 88 List of Definitions 156 2.2.4 The Vandermonde determinant on spheres defined by the4-norm...... 96 List of Theorems 157

3 Approximation of electrostatic discharge currents using the List of Lemmas 157 analytically extended function 99 3.1 The analytically extended function (AEF) ...... 101 3.1.1 The p-peak analytically extended function ...... 102 3.2 Approximation of lightning discharge current functions . . . . 109 3.2.1 FittingtheAEF...... 110

10 11 Generalized Vandermonde matrices and determinants in electromagnetic compatibility CONTENTS

10 11 List of Papers

Paper A. Karl Lundeng˚ard, Jonas Osterberg¨ and Sergei Silvestrov. Extreme points of the Vandermonde determinant on the sphere and some limits involving the generalized Vandermonde determinant. Preprint: arXiv:1312.6193 [math.ca], 2013.

Paper B. Karl Lundeng˚ard, Jonas Osterberg,¨ and Sergei Silvestrov. Optimization of the determinant of the Vandermonde matrix and related matrices. In AIP Conference Proceedings 1637, ICNPAA, Narvik, Norway, pages 627– 636, 2014.

Paper C. Karl Lundeng˚ard, Jonas Osterberg,¨ and Sergei Silvestrov. Optimization of the determinant of the Vandermonde matrix on the sphere and related surfaces. In Christos H Skiadas, editor, ASMDA 2015 Proceedings: 16th Applied Stochastic Models and Data Analysis International Conference with 4th De- mographics 2015 Workshop, pages 637–648. ISAST: International Society for the Advancement of Science and Technology, 2015.

Paper D. Karl Lundeng˚ard, Milica Ranˇci´c, Vesna Javor and Sergei Silvestrov. On some properties of the multi-peaked analytically extended function for approximation of lightning discharge currents. Sergei Silvestrov and Milica Ranˇci´c, editors, Engineering Mathematics I: Electromagnetics, Fluid Mechanics, Ma- terial Physics and Financial Engineering, volume 178 of Springer Proceedings in Mathematics & Statistics. Springer International Publishing, 2016.

Paper E. Karl Lundeng˚ard, Milica Ranˇci´c, Vesna Javor and Sergei Silvestrov. Estima- tion of parameters for the multi-peaked AEF current functions. Methodology and Computing in Applied Probability, pages 1–15, 2016.

Paper F. Karl Lundeng˚ard, Milica Ranˇci´c, Vesna Javor and Sergei Silvestrov. Electro- static discharge currents representation using the multi-peaked analytically extended function by interpolation on a D-optimal design. Preprint: arXiv:1701.03728 [physics.comp-ph], 2017.

13 List of Papers

13 Generalized Vandermonde matrices and determinants in electromagnetic compatibility

Parts of the thesis have been presented at the following international conferences:

1. 10th International Conference on Mathematical Problems in Engineering, Aerospace and Sciences - ICNPAA 2014, Narvik, Norway, July 15-18, 2014. 2. 16th Applied Stochastic Models and Data Analysis International Conference with 4th Demographics 2015 Workshop - ASMDA 2015, Piraeus, Greece, June 30 - July 4, 2015. Chapter 1 Summaries of papers A-F with a brief description of the thesis authors contributions to each paper can be found in Section 1.5. Introduction

This chapter is partially based on Paper D:

Paper D. Karl Lundeng˚ard, Milica Ranˇci´c, Vesna Javor and Sergei Silvestrov. On some properties of the multi-peaked analytically extended function for approximation of lightning discharge currents. Sergei Silvestrov and Milica Ranˇci´c,editors, Engineering Mathematics I: Electromag- netics, Fluid Mechanics, Material Physics and Financial Engineer- ing, volume 178 of Springer Proceedings in Mathematics & Statistics. Springer International Publishing, 2016.

14 Generalized Vandermonde matrices and determinants in electromagnetic compatibility

Parts of the thesis have been presented at the following international conferences:

This chapter is partially based on Paper D:

14 INTRODUCTION

The Vandermonde matrix is a well-known type of matrix that appears in many different areas. In this thesis we will discuss this matrix and some of its properties, specifically the extreme points of the determinant on various surfaces and we will use a generalized Vandermonde matrix for fitting a certain type of curve to data taken from sources that are important when analysing electromagnetic compatibility. This thesis is based on the six papers listed on page 13. The contents have been rearranged to clarify the relations between the material in the different papers. If a section is based on a paper this is specified at the beginning of the section and unless otherwise specified any subsection is from the same source. A section that is based on a paper consists of text from the paper unchanged except for modifications to correct misprints and preserve consistency within the thesis. Parts of several papers have also been omitted to avoid repetition and improve cohesion. The relations between contents of the the sections are of many kinds, common definitions and dependent results, conceptual connections as well as similarities in proof techniques and problem formulations. This is illustrated in Figure 1.1. A reader only interested in a particular section or in a hurry can consult Figure 1.2 to find a short route to the desired content. In chapter 1 the basic theory for later chapters is introduced. The Van- dermonde matrix and some of its properties, history, applications and generalizations are briefly introduced in Section 1.1. In Section 1.2 interpolation problems and their relations to alternant- and Vandermonde matrices is described. In Section 1.3 various regression models and the Marquardt least-squares method for non-linear regression problems are discussed. The optimal design of experiments with respect to regression is discussed as well. Section 1.4 introduces electromagnetic compatibility and certain curve- fitting problems that appear in modelling of electrostatic discharge currents. Chapter 2 discusses the optimisation of the Vandermonde determinant over various surfaces. First the extreme points on the sphere in three dimensions are examined, Section 2.1.1 and 2.1.2. Further discussion includes the torus, cylinder and ellipsoid, Section 2.1.3–2.1.7. In Section 2.2 the determinant is optimised on the sphere and related surfaces in higher dimensions. Chapter 3 discusses fitting a piecewise non-linear regression model to data. The particular model is introduced in Section 3.1 and a general framework for fitting it to data using the Marquardt least-squares method is described in Section 3.2.1–3.2.5. The framework is then applied to lightning discharge currents in Section 3.2.6. An alternate curve-fitting method based on D-optimal interpolation (found analogously to the results in Section 2.2) is described and applied to electrostatic discharge currents in Section 3.3.

17 INTRODUCTION

17 Generalized Vandermonde matrices and determinants in electromagnetic compatibility 1.1. THE VANDERMONDE MATRIX

Section 1.1.3 Section 1.1 Section 1.1.1 1.1 The Vandermonde matrix

The Vandermonde matrix is a well-known matrix with a very special form Section 1.1.4 Section 1.1.2 Section 1.1.5 that appears in many different circumstances, a few examples are polynomial interpolation (see Section 1.2.1), least square regression (see Sec- Section 2.2 Section 2.1.1 tion 1.3), optimal experiment design (see Section 1.3.4), construction of Section 1.2 Paper A–C Paper A error-detecting and error-correcting codes (see [18, 69, 143] as well as more recent work such as [17]), determining if a market with a finite set of Sections 2.1.2-2.1.7 traded assets is complete [32], calculation of the discrete Fourier trans- Section 1.2.1 Paper A–C form [142] and related transforms such as the fractional discrete Fourier transform [122], the quantum Fourier transform [36], and the Vandermonde Section 1.3.2 Section 1.3 Section 1.3.1 transform [6, 7], solving systems of differential equations with constant coefficients [120], various problems in mathematical- [168], nuclear- [29], and Section 1.3.3 quantum physics [148,159] and describing properties of the Fisher informa- Section 1.4 Section 1.3.4 Paper D tion matrix of stationary stochastic processes [93]. In this section we will review some of the basic properties of the Van- Section 3.2 Section 3.1 Section 3.3 dermonde matrix, starting with its definition. Paper E Paper D Paper F Definition 1.1. A Vandermonde matrix is an n m matrix of the form × 11 1 Figure 1.1: Relations between sections of the thesis. Arrows indicate that the target ··· m,n x x x section uses some definition or theorem from the source section. Dashed i 1  1 2 ··· n  Vmn(xn)= xj− = . . . . (1) lines indicates a tangential or conceptual relation. i,j . . .. .  m 1 m 1 m 1 x − x − x   1 2 ··· n −  Sections Section 1.1 Sections 1.1.4, 1.2.1   2.1.2-2.1.7, 2.2 where xi C, i =1,...,n. If the matrix is square, n = m, the notation ∈ Paper A–C Vn = Vnm will be used.

Section 1.1.2 Section 1.3 Sections 1.3.1, 1.3.4 Remark 1.1. Note that in the literature the term Vandermonde matrix is often used for the transpose of the matrix given in expression (1). Section 3.3 Section 1.1.5 Section 1.3.2 Paper F 1.1.1 Who was Vandermonde?

Section 2.1.1 Section 1.3.3 Sections 1.4, 3.1 The matrix is named after Alexandre Théophile Vandermonde (1735–1796) Paper A Paper D Paper D who had a varied career that began with law studies and some success as a concert violinist, transitioned into work in science and mathematics in the beginning of the 1770s that gradually turned into administrative and Section 3.2 leadership positions at various Parisian institutions as well as work in politics Paper E and economics in the end of the 1780s [42]. His entire mathematical career consisted of four published papers, first presented to the French Academy Figure 1.2: Reference that demonstrates short routes to the different chapters. of Sciences in 1770 and 1771 and published a few years later.

18 19 Generalized Vandermonde matrices and determinants in electromagnetic compatibility 1.1. THE VANDERMONDE MATRIX

Section 1.1.3 Section 1.1 Section 1.1.1 1.1 The Vandermonde matrix

18 19 Generalized Vandermonde matrices and determinants in electromagnetic compatibility 1.1. THE VANDERMONDE MATRIX

n k n k n 1.1.2 The Vandermonde determinant r +1 j − s +1 j r + s +1 j − − = − , Often it is not the Vandermonde matrix itself that is useful, instead it is the  j   j   j  k=1 j=1 j=1 j=1 multivariate polynomial given by its determinant that is examined and used.       The determinant of the Vandermonde matrix is usually called the Vander- where r, s R and n Z. The identity was first found by Chu Shih-Chieh ∈ ∈ ca 1260 – ca 1320, traditional chinese: 朱世傑 in 1303 in The precious monde determinant (or Vandermonde polynomial or Vandermondian [168]) mirror of the four elements 四元玉 and was rediscovered (apparently and can be written using an exceptionally simple formula. But before we independently) by Vandermonde [4,127]. discuss the Vandermonde determinant we will disuss the general determi- In the fourth paper Mémoire sur l’élimination [167] Vandermonde dis- nant. cusses some ideas for what we today call determinants, which is functions Definition 1.2. The determinant is a function of square matrices over a that can tell us if a linear equation system has a unique solution or not. field F to the field F, det : n n(F) F such that if we consider the M × → The paper predates the modern definitions of determinants but Vander- determinant as a function of the columns monde discusses a general method for solving linear equation systems using det(M) = det(M ,1, M ,2,...,M ,n) alternating functions, which has strong relation to determinants. He also · · · notices that exchanging exponents for indices in a class of expressions from of the matrix the determinant must have the following properties his first paper will give a class of expressions that he discusses in his fourth paper [179]. This relation is mirrored in the relationship between the deter- The determinant must be multilinear • minant of the Vandermonde matrix and the determinant of a general matrix det(M ,1,...,aM ,k + bN ,k,...,M ,n) described in Theorem 1.3. · · · · While Vandermonde’s papers can be said to contain many important = a det(M ,1,...,M ,k,...,M ,n)+b det(M ,1,...,N ,k,...,M ,n). · · · · · · ideas they do not bring any of them to maturity and he is therefore usu- The determinant must be alternating, that is if M ,i = M ,j for some • · · ally considered a minor scientist and mathematician, especially compared i =j then det(M)=0.

20 21 Generalized Vandermonde matrices and determinants in electromagnetic compatibility 1.1. THE VANDERMONDE MATRIX

The first paper, Mémoire sur la résolution des équations [164], discusses to well-known mathematicians such as Etienne´ Bézout (1730 – 1783) and some properties of the roots of polynomial equations, more specifically for- Pierre-Simon de Laplace (1749 – 1827) as well as the chemist Antoine mulas for the sum of the roots and a sum of symmetric functions of the pow- Lavoisier (1743 – 1794) that he worked with for some time after his math- ers of the roots. This paper has been mentioned as important since it con- ematical career. The Vandermonde matrix does not appear in any of Van- tains some of the fundamental ideas of group theory (see for instance [99]), dermonde’s published works, which is not surprising considering that the but generally this work is overshadowed by the works of the contempo- modern matrix concept did not really take shape until almost a hundred rary Joseph Louis Lagrange (1736–1813) [97]. He also notices the equality years later in the works of Sylvester and Cayley [25, 157]. It is therefore a2b+b2c+ac2 a2c ab2 bc2 =(a b)(a c)(b c), which is a special case strange that the Vandermonde matrix was named after him, a thorough − − − − − − of the formula for the determinant of the Vandermonde matrix. It seems discussion on this can be found in [179], but a possible reason is the simple that Vandermonde did not understand the significance of the expression. formula for the determinant that Vandermonde briefly discusses in his fourth The second paper, Remarques sur des problèmes de situation [165], dis- paper can be generalized to a Vandermonde matrix of any size. One of the cusses the problem of the knight’s tour (what sequence of moves allows a main reasons that the Vandermonde matrix has become known is that it knight to visit all squares on a chessboard exactly once). This paper is con- has an exceptionally simple expression for its determinant that in turn has sidered the first mathematical paper that uses the basic ideas of what is now a surprisingly fundamental relation to the determinant of a general matrix. called knot theory [140]. We will be taking a closer look at the determinant of the Vandermonde ma- The third paper, Mémoire sur des irrationnelles de différents ordres avec trix and related matrices several times in this thesis so the next section will une application au cercle [166], is a paper on combinatorics and the most introduce it and some of its properties. well-known result from the paper is the Chu-Vandermonde identity, n k n k n 1.1.2 The Vandermonde determinant r +1 j − s +1 j r + s +1 j − − = − , Often it is not the Vandermonde matrix itself that is useful, instead it is the  j   j   j  k=1 j=1 j=1 j=1 multivariate polynomial given by its determinant that is examined and used.       The determinant of the Vandermonde matrix is usually called the Vander- where r, s R and n Z. The identity was first found by Chu Shih-Chieh ∈ ∈ ca 1260 – ca 1320, traditional chinese: 朱世傑 in 1303 in The precious monde determinant (or Vandermonde polynomial or Vandermondian [168]) mirror of the four elements 四元玉 and was rediscovered (apparently and can be written using an exceptionally simple formula. But before we independently) by Vandermonde [4,127]. discuss the Vandermonde determinant we will disuss the general determi- In the fourth paper Mémoire sur l’élimination [167] Vandermonde dis- nant. cusses some ideas for what we today call determinants, which is functions Definition 1.2. The determinant is a function of square matrices over a that can tell us if a linear equation system has a unique solution or not. field F to the field F, det : n n(F) F such that if we consider the M × → The paper predates the modern definitions of determinants but Vander- determinant as a function of the columns monde discusses a general method for solving linear equation systems using det(M) = det(M ,1, M ,2,...,M ,n) alternating functions, which has strong relation to determinants. He also · · · notices that exchanging exponents for indices in a class of expressions from of the matrix the determinant must have the following properties his first paper will give a class of expressions that he discusses in his fourth paper [179]. This relation is mirrored in the relationship between the deter- The determinant must be multilinear • minant of the Vandermonde matrix and the determinant of a general matrix det(M ,1,...,aM ,k + bN ,k,...,M ,n) described in Theorem 1.3. · · · · While Vandermonde’s papers can be said to contain many important = a det(M ,1,...,M ,k,...,M ,n)+b det(M ,1,...,N ,k,...,M ,n). · · · · · · ideas they do not bring any of them to maturity and he is therefore usu- The determinant must be alternating, that is if M ,i = M ,j for some • · · ally considered a minor scientist and mathematician, especially compared i =j then det(M)=0.

20 21 Generalized Vandermonde matrices and determinants in electromagnetic compatibility 1.1. THE VANDERMONDE MATRIX

If I is the identity matrix then det(I) = 1. We will now discuss the Vandermonde determinant specifically. • Remark 1.2. Defining the multilinear and alternating properties from the Theorem 1.2. The Vandermonde determinant, vn(x1,...,xn), is given by rows of the matrix will give the same determinant. The name of the alternating property comes from the fact that it combined with multilinearity v (x ,...,x ) = det(V (x ,...,x )) = (x x ). n 1 n n 1 n j − i implies that switching places between two columns changes the sign of the 1 ilinear algebra says that the determinant is unique and that it is given by the duction [95, 117] but there are also several proofs using combinatorial [11] following formula or graph-based techniques [60, 141]. Here we will present a simple proof, n (σ) that dates back to some very early results on determinants [24] and has an det(M)= ( 1)I m (2) − i,σ(i) interesting connection to the general concept of a determinant. σ Sn i=1 ∈ where S is the set of all permutations of the set 1, 2,...,n , that is all lists Proof of Theorem 1.2. There are many versions of this proof, see for exam- n { } that contain the numbers 1, 2,...,n exactly once, and if σ is a permutation ple [10,21,24,71], with focus on different aspects of the proof. Here we will then σ(i) is the ith element of that permutation. provide a fairly concise version that still makes all the steps of the proof clear. We start by only considering one of the variables xk, which gives a Remark 1.3. Often formula (2) is used immediately as the definition of single variable function vn(xk). From the general expression for the deter- the determinant of a matrix, see for instance [5]. The formula is usually minant (Theorem 1.1) it is clear that vn(xk) must be a polynomial of degree attributed to Gottfried Wilhem Leibniz (1646–1716), probably due to a n in x . We also know that if we let x = x for any 1 i n, i = k, the k k i ≤ ≤ letter that he wrote to Guillaume de l’Hôpital (1661–1704) in 1693 where he determinant will be equal to zero since the corresponding matrix will have describes a method of solving linear equation systems that is closely related two identical columns. Thus if vn(xi) = 0 we can write to Cramer’s rule [123], the particular letter was published in [100] and a n translation can be found in [153]. v (x )=P (x ) (x x ) n k k k − i The determinant has several uses and interpretation, the two most well- i=1 i =k known ones are where P (x ) is a polynomial. If we repeat this argument for all the variables, If det(M) = 0 then the vectors corresponding to the columns (or k • and ensure that no roots appear twice in the factorization, we get rows) are linearly independent. Compare this to the properties of the n 1 Wronskian matrix described on page 28. − vn(x1,...,xn)=Pn(x1,...,xn) (xn xi) If the columns (or rows) of M are interpreted as sides defining an − • i=1 n-dimensional parallelepiped the absolute value of det(M) will give the n 2 n 1 − − volume of this parallelepiped. Compare this to the interpretation of = Pn 1(x1,...,xn) (xn 1 xi) (xn xi) D-optimality on page 43. The sign of the determinant is also important − − − − i=1 i=1 when considering the orientation of the surface which is highly relevant n 1 − in geometric algebra and integration over several variables, see for = P (x ,...,x )(x x )(x x )(x x ) (x x ) 0 1 n 2 − 1 3 − 2 3 − 1 ··· n − i instance [112, 146] for examples. i=1

22 23 Generalized Vandermonde matrices and determinants in electromagnetic compatibility 1.1. THE VANDERMONDE MATRIX

22 23 Generalized Vandermonde matrices and determinants in electromagnetic compatibility 1.1. THE VANDERMONDE MATRIX and since this factorization has each x appear as a root n times we can Multilinearity: If we denote the left hand side in (3) with w k • conclude that n w = xk vn(x1,...,xn) vn(x1,...,xn)=C det(Vn(x1,...,xn)) = (xj xi) − k=1 1 i

24 25 Generalized Vandermonde matrices and determinants in electromagnetic compatibility 1.1. THE VANDERMONDE MATRIX and since this factorization has each x appear as a root n times we can Multilinearity: If we denote the left hand side in (3) with w k • conclude that n w = xk vn(x1,...,xn) vn(x1,...,xn)=C det(Vn(x1,...,xn)) = (xj xi) − k=1 1 i

24 25 Generalized Vandermonde matrices and determinants in electromagnetic compatibility 1.1. THE VANDERMONDE MATRIX

k 1 where P (I), I Z>0, does not contain any terms of the form x − In the literature there are many cases where the inverse is instead written ⊂ k for all k I. Thus applying the transformation corresponding to the as a product of several simpler matrices, usually triangular or diagonal [121, ∈ identity matrix we get 129,130,162]. There is also a lot of literature that takes a more algorithmic approach and tries to find fast ways of computing the elements, classical n n n k examples include the Parker-Traub algorithm [160] and the Björck-Pereyra xi vn(x1,...,xn)= xk + P (n,...,1) 1+0=1. → algorithm [13], and more recent results can be found in [40]. i=1 k=1 k=1 Thus if we take the right hand side in equation (3) and exchange exponents 1.1.4 The alternant matrix for indices we get a determinant be Definition 1.2 and since the determinant Many generalizations of the Vandermonde matrix have been proposed and is unique by Theorem 1.1 and x = xj in the Vandermonde matrix this i,j i studied in the literature. An early generalization is the alternant matrix must be equal to which is a matrix that exchanges the powers in the Vandermonde matrix n n with other functions [124]. (σ) i x v (x ,...,x )= ( 1)I x . i n 1 n − σ(i) Definition 1.3. An alternant matrix is a matrix of the form i=1 σ Sn i=1 ∈ f1(x1) f1(x2) f1(xn) 1.1.3 Inverse of the Vandermonde matrix ··· m,n f2(x1) f2(x2) f2(xn) Amn(fm; xn)=[fi(xj)] =  . . ···. .  (6) The inverse for the Vandermonde matrix has been known for a long time, es- i,j . . .. .   pecially since the solution to a Lagrange interpolation problems (see Section f (x ) f (x ) f (x )  m 1 m 2 ··· m n  1.2.1) gives the inverse indirectly. Here we will only give a short overview   of the work on expressing the inverse as an explicit matrix. where fi : F F where F is a field. → An explicit expression for the inverse matrix has been known since at Remark 1.4. In some literature the alternant matrix is used as an al- least the end of the 1950s, see [113]. ternative name for the Vandermonde matrix or the Vandermonde matrix Theorem 1.4. The elements of the inverse of an n-dimensional Vander- multiplied by a diagonal matrix [161]. monde matrix V can be calculated by There are several special cases of alternant matrices that are useful or interesting in various mathematical fields: j 1 1 ( 1) − σn j,i − Vn− ij = −n (4) (x x ) Interpolation and regression k − i k=1 k=i Just like the Vandermonde matrix can be used for polynomial interpolation the alternant matrix can be used to describe interpolation with other sets where σj,i is the j:th elementary symmetric polynomial with variable xi set of function, see Section 1.2 and 1.3. to zero.

j 1 ,a= b Alternant codes σj,i = xm (1 δm ,i) ,δa,b = (5) k − k 0 ,a =b As mentioned on page 19 there are several diﬀerent error-detecting and 1 m1

26 27 Generalized Vandermonde matrices and determinants in electromagnetic compatibility 1.1. THE VANDERMONDE MATRIX

i 1 d − n 1 systems. The literature on the subject is vast but a systematic introduction If fi = i 1 and gi C − [C], i =1,...,n the alternant matrix An(fn; gn) dx − ∈ is offered in [96]. will be the Wronskian matrix. The Wronskian matrix has a long history [70] and is commonly used to test if a set of functions g are linearly indepen- { i} dent as well as finding solutions to ordinary differential equations [52]. If Moore matrix the determinant of the Wronskian matrix is non-zero then the functions are When working in a finite field with prime characteristic p an analogue of linearly independent, see [16, 19], but proving linear dependence requires the Vandermonde and Wronskian matrix can be constructed by taking an further conditions, see [14, 15, 133, 134, 174]. alternant matrix where the rows are given by power of the Frobenius au- A classical application of the Wronskian is confirming that a set of so- tomorphism, F (ω)=ωp. This matrix is called the Moore matrix and is lutions to a linear differential equation are linearly independent or if n 1 named after its originator E. H. Moore who also calculated its determinant, − linearly independent solutions are know construct the remaining linearly independent solution using Abel’s identity (for n = 2) or a generalisation of ω1 ωn p ··· p n 1 p 1 p 1 ω ωn it [20]. 1 ··· − − − . . . = (ωi + ki 1ωi 1 + ...+ k1ω1)(mod p), If Li is a linear partial differential operator of order i then the alter- . .. . ··· − − i=1 k k =0 n 1 n 1 i 1 1 nant matrix An(Ln; gn), where Ln =(L1,...,Ln), is the generalized Wron- ωp − ωp − − 1 n skian matrix [131], that has been used in for example diophantine geome- ··· try [39, 145] and for solving Korteweg-de Vries equations, see [111] and the and showed that if this determinant was not equal to zero then ω1, ..., ωn references therein. The generalized Wronskian matrix has similar properties are linearly independent [118]. There are several uses for the determinant of with respect to the linear dependence of the functions it is created from as the Moore matrix in function field arithmetic, see for instance [62], a classical the standard Wronskian [175]. example is finding the modular invariants of the general linear group over

28 29 Generalized Vandermonde matrices and determinants in electromagnetic compatibility 1.1. THE VANDERMONDE MATRIX

Jacobian matrix Both the Wronskian and generalized Wronskian is also useful in algebraic geometry, see [52] for several examples. One of the most well-known examples of an alternant matrix is the Jaco- bian matrix. Let f : Fn Fn be a vector-valued function that is n times → Bell matrix differentiable with respect to each variable, then the Jacobian matrix is the matrix J, given by Alternating matrices can be used to convert function composition into ma- i 1 ∂y1 ∂y2 ∂yn d − j trix multiplication. By letting Di = i 1 and gj(x)=(f(x)) , where f is ∂x1 ∂x1 ··· ∂x1 dx − ∂y1 ∂y2 ∂yn infinitely differentiable, the alternant matrix B[f]=An(Dn, gn) is called  ∂x2 ∂x2 ··· ∂x2  . . .. . a Bell matrix (its transpose is known as the Carleman matrix). Some au-  . . . .  thors, for instance [94], refer to Bell matrices as Jabotinsky matrices due to  ∂y1 ∂y2 ∂yn     ∂xn ∂xn ··· ∂xn  a special case of Bell matrices considered in [77].   where y = f(x). The most common application of the Jacobian matrix is That Bell matrices converts function composition into matrix multiplica- to use its determinant to describe how volume elements are deformed when tion can be seen by noting that the power series expansion of the jth power changing variables in multivariate calculus [146]. The numerous applications of f can be written as and generalizations that follow from this alone are too numerous to list j ∞ i so here we will only note that it holds a central role in many methods (f(x)) = B[f]ijx for multivariate optimizations, such as the Marquardt least-square method i=1 described in Section 1.3.3. and from this equality follows that B[f g]=B[g]B[f]. This is the basic ◦ property behind a popular technique called Carleman linearisation or Carle- Wronskian matrix man embedding that has seen wide use in the theory of non-linear dynamical i 1 d − n 1 systems. The literature on the subject is vast but a systematic introduction If fi = i 1 and gi C − [C], i =1,...,n the alternant matrix An(fn; gn) dx − ∈ is offered in [96]. will be the Wronskian matrix. The Wronskian matrix has a long history [70] and is commonly used to test if a set of functions g are linearly indepen- { i} dent as well as finding solutions to ordinary differential equations [52]. If Moore matrix the determinant of the Wronskian matrix is non-zero then the functions are When working in a finite field with prime characteristic p an analogue of linearly independent, see [16, 19], but proving linear dependence requires the Vandermonde and Wronskian matrix can be constructed by taking an further conditions, see [14, 15, 133, 134, 174]. alternant matrix where the rows are given by power of the Frobenius au- A classical application of the Wronskian is confirming that a set of so- tomorphism, F (ω)=ωp. This matrix is called the Moore matrix and is lutions to a linear differential equation are linearly independent or if n 1 named after its originator E. H. Moore who also calculated its determinant, − linearly independent solutions are know construct the remaining linearly independent solution using Abel’s identity (for n = 2) or a generalisation of ω1 ωn p ··· p n 1 p 1 p 1 ω ωn it [20]. 1 ··· − − − . . . = (ωi + ki 1ωi 1 + ...+ k1ω1)(mod p), If Li is a linear partial differential operator of order i then the alter- . .. . ··· − − i=1 k k =0 n 1 n 1 i 1 1 nant matrix An(Ln; gn), where Ln =(L1,...,Ln), is the generalized Wron- ωp − ωp − − 1 n skian matrix [131], that has been used in for example diophantine geome- ··· try [39, 145] and for solving Korteweg-de Vries equations, see [111] and the and showed that if this determinant was not equal to zero then ω1, ..., ωn references therein. The generalized Wronskian matrix has similar properties are linearly independent [118]. There are several uses for the determinant of with respect to the linear dependence of the functions it is created from as the Moore matrix in function field arithmetic, see for instance [62], a classical the standard Wronskian [175]. example is finding the modular invariants of the general linear group over

28 29 Generalized Vandermonde matrices and determinants in electromagnetic compatibility 1.1. THE VANDERMONDE MATRIX a finite field [37, 128]. The determinant also plays an important role in the are the Schur functions that were introduced by Cauchy [24] but named theory of Drinfeld modules [125]. after Issai Schur (1875 – 1941) that showed that they were highly useful in invariant theory and representation theory. For instance they can be used 1.1.5 The generalized Vandermonde matrix to determine the character of conjugacy classes of representations of the symmetric group [50]. They have also been used in other areas, for instance There are several types of matrices (or determinants) that have been re- to describe the generating function of many classes of plane partitions, see ferred to as generalized Vandermonde matrices, for example the confluent for instance [21] for several examples. The literature on Schur polynomials Vandermonde matrix is sometimes referred to as the generalized Vander- is vast and so are the applications so there will be no attempt to summarise monde matrix [85,86,102,109,155], this matrix and its role in interpolation them here. problems is briefly described on page 36. Other examples include modified versions of confluent Vandermonde matrices [45], as well as matrices with Integration of an exponential function over a unitary group elements given by multivariate monomials of increasing multidegree [23], or similarly over the algebraic closure of a field [31], matrices with elements If we let U(n) be the n-dimensional unitary group and dU a Haar mea- given by multivariate polynomials with univariate terms [168]. sure normalised to 1 then the Harish-Chandra–Itzykson-Zuber integral for- α1 αn In this thesis we call the alternant matrix Amn(x ,...,x ; x1,...,xn) mula [64,76], says that if A and B are Hermitian matrices with eigenvalues the generalized Vandermonde matrix. λ (A) ... λ (A) and λ (B) ... λ (B) then 1 ≤ ≤ n 1 ≤ ≤ n Definition 1.4. A generalized Vandermonde matrix is an n m matrix of nn n 1 × det [exp(tλj(A)λk(B))]j,k − the form t tr(AUBU ∗) e dU = n(n 1) i! (8) U(n) t 2− v (λ(A))v (λ(B)) α1 α1 α1 n n i=1 x1 x2 xn α2 α2 ··· α2 m,n x1 x2 xn where vn is the determinant of the Vandermonde matrix. If t = 1 and A αi  ···  Gmn(xn)= xj = . . . . (7) i,j . . .. . and B are chosen as diagonal matrices  αm αm αm  x1 x2 xn   ···  ai if i = j, bi if i = j,   Aij = Bij = where xi C, αi C, i =1,...,n. 0 if i =j, 0 if i =j, ∈ ∈ This name has been used for quite some time, see [67] for instance. then formula (8) reduces to an expression involving determinants of a gen- The main reason to study this matrix seems to be its connection to Schur eralized Vandermonde matrix and two Vandermonde matrices, polynomials, see below, and thus the research on the matrix is primarily focused on its determinant. Many of the results are algorithmic in nature ea1b1 ea1b2 ... ea1bn [28,33–35,92] but there are also more algebraic examinations [41,49,149,177]. ea2b1 ea2b2 ... ea2bn There are a couple of examples where the determinant of generalized . . . . . . .. . Vandermonde matrices are interesting or useful: eanb1 eanb2 ... eanbn tr(AUBU ∗) e dU = . Schur polynomials U(n) vn(a1,...,an)vn(b1,...,bn) If λ =(λ ,...,λ ) is an integer partition, that is 0 <λ λ ...λ and 1 n 1 ≤ 2 ≤ n each λi N, then ∈

a(λ1+n 1,λ2+n 2,...,λn)(x1,...,xn) − − = det(G (λ + n 1,λ + n 2,...,λ ; x ,...,x )) n 1 − 2 − n 1 n

30 31 Generalized Vandermonde matrices and determinants in electromagnetic compatibility 1.1. THE VANDERMONDE MATRIX a finite field [37, 128]. The determinant also plays an important role in the are the Schur functions that were introduced by Cauchy [24] but named theory of Drinfeld modules [125]. after Issai Schur (1875 – 1941) that showed that they were highly useful in invariant theory and representation theory. For instance they can be used 1.1.5 The generalized Vandermonde matrix to determine the character of conjugacy classes of representations of the symmetric group [50]. They have also been used in other areas, for instance There are several types of matrices (or determinants) that have been re- to describe the generating function of many classes of plane partitions, see ferred to as generalized Vandermonde matrices, for example the confluent for instance [21] for several examples. The literature on Schur polynomials Vandermonde matrix is sometimes referred to as the generalized Vander- is vast and so are the applications so there will be no attempt to summarise monde matrix [85,86,102,109,155], this matrix and its role in interpolation them here. problems is briefly described on page 36. Other examples include modified versions of confluent Vandermonde matrices [45], as well as matrices with Integration of an exponential function over a unitary group elements given by multivariate monomials of increasing multidegree [23], or similarly over the algebraic closure of a field [31], matrices with elements If we let U(n) be the n-dimensional unitary group and dU a Haar mea- given by multivariate polynomials with univariate terms [168]. sure normalised to 1 then the Harish-Chandra–Itzykson-Zuber integral for- α1 αn In this thesis we call the alternant matrix Amn(x ,...,x ; x1,...,xn) mula [64,76], says that if A and B are Hermitian matrices with eigenvalues the generalized Vandermonde matrix. λ (A) ... λ (A) and λ (B) ... λ (B) then 1 ≤ ≤ n 1 ≤ ≤ n Definition 1.4. A generalized Vandermonde matrix is an n m matrix of nn n 1 × det [exp(tλj(A)λk(B))]j,k − the form t tr(AUBU ∗) e dU = n(n 1) i! (8) U(n) t 2− v (λ(A))v (λ(B)) α1 α1 α1 n n i=1 x1 x2 xn α2 α2 ··· α2 m,n x1 x2 xn where vn is the determinant of the Vandermonde matrix. If t = 1 and A αi  ···  Gmn(xn)= xj = . . . . (7) i,j . . .. . and B are chosen as diagonal matrices  αm αm αm  x1 x2 xn   ···  ai if i = j, bi if i = j,   Aij = Bij = where xi C, αi C, i =1,...,n. 0 if i =j, 0 if i =j, ∈ ∈ This name has been used for quite some time, see [67] for instance. then formula (8) reduces to an expression involving determinants of a gen- The main reason to study this matrix seems to be its connection to Schur eralized Vandermonde matrix and two Vandermonde matrices, polynomials, see below, and thus the research on the matrix is primarily focused on its determinant. Many of the results are algorithmic in nature ea1b1 ea1b2 ... ea1bn [28,33–35,92] but there are also more algebraic examinations [41,49,149,177]. ea2b1 ea2b2 ... ea2bn There are a couple of examples where the determinant of generalized . . . . . . .. . Vandermonde matrices are interesting or useful: eanb1 eanb2 ... eanbn tr(AUBU ∗) e dU = . Schur polynomials U(n) vn(a1,...,an)vn(b1,...,bn) If λ =(λ ,...,λ ) is an integer partition, that is 0 <λ λ ...λ and 1 n 1 ≤ 2 ≤ n each λi N, then ∈ a(λ1+n 1,λ2+n 2,...,λn)(x1,...,xn) − − = det(G (λ + n 1,λ + n 2,...,λ ; x ,...,x )) n 1 − 2 − n 1 n

30 31 Generalized Vandermonde matrices and determinants in electromagnetic compatibility 1.2. INTERPOLATION

1.2 Interpolation appropriate alternant matrix,

The problem of ﬁnding a function that generates a given set of points is g1(x1) g2(x1) ... gm(x1) a1 y1 usually referred to as an interpolation problem and the function generating g1(x2) g2(x2) ... gm(x2) a2 y2 A =  . . . .  , a =  .  , y =  .  . the points is called an interpolating function. A common type of inter- ......       polation problem is to ﬁnd a continuous function, f, such that the given g (x ) g (x ) ... g (x ) a  y   1 n 2 n m n   n  n set of points (x ,y ), (x ,y ),... can be generated by calculating the set       { 1 1 2 2 } (x ,f(x )), (x ,f(x )),... . Often the interpolating function is also a lin- { 1 1 2 2 } 1.2.1 Polynomial interpolation ear combination of elementary functions, but interpolation can also be done in other ways, for instance with fractals (the classical texts on this is [8, 9]) A classical form of interpolation is polynomial interpolation where n data or parametrised curves. For some examples, see Figure 1.3. points are interpolated by a polynomial of at most degree n 1. − The Vandermonde matrix can be used to describe this type of interpolation problem simply by rewriting the equation system given by p(xk)=yk as a matrix equation

n 1 1 x1 x1− a1 y1 ··· n 1 1 x2 x2− a2 y2 . . ···. .   .  =  .  ......  n 1     1 x x  an yn  n ··· n−            That the polynomial is unique (if it exists) is easy to see when considering Figure 1.3: Some examples of diﬀerent interpolating curves. The set of red points the determinant of the Vandermonde matrix are interpolated by a polynomial (left), a self-aﬃne fractal (middle) and a Lissajous curve (right). det(Vn(x1,...,xn)) = (xj xi). − 1 i

In the case of the interpolating function being a linear combination of Clearly this determinant is non-zero whenever all xi are disctinct which other functions and the interpolation is achieved by changing the coefficients means that he matrix is invertible whenever all xi are distinct. If not all xi of the linear combination this is said to be a linear model (not to be confused are distinct there is no function of the x coordinate that can interpolate all with linear interpolation that is interpolation with piecewise straight lines). the points. For linear models the interpolation problem can be described using al- There are several ways to construct the interpolating polynomial without ternant matrices. Suppose we want to find a function explicitly inverting the Vandermonde matrix. The most straight-forward is probably Lagrange interpolation, named after Joseph-Louis Lagrange (1736 m – 1813) [98] who independently discovered it a few years after Edward War- f(x)= aigi(x) (9) ing (1736 – 1798) [172]. i=1 The idea behind Lagrange interpolation is simple, construct a set of n polynomials p ,p ,...,p such that { 1 2 n} that fits as well as possible to the data points (xi,yi), i =1,...,n. We then get an interpolation problem described by the linear equation system 0,i=j pi(xj)= Xa = y where a are the coefficients of f, y are the data values and A is the 1,i= j

32 33 Generalized Vandermonde matrices and determinants in electromagnetic compatibility 1.2. INTERPOLATION

1.2 Interpolation appropriate alternant matrix,

32 33 Generalized Vandermonde matrices and determinants in electromagnetic compatibility 1.2. INTERPOLATION and then construct the final interpolating polynomial by the sum of these 1.5 1.5 1.5 pi weighted by the corresponding yi. The pi polynomials are called Lagrange basis polynomials and can easily be constructed by placing the roots appropriately and then normalizing the 1 1 1 result such that pi(xi) = 1, this gives the expression 0.5 0.5 0.5 (x x1) (x xi 1)(x xi+1) (x xn) pi(x)= − ··· − − − ··· − . (xi x1) (xi xk 1)(xi xi+1) (xi xn) − ··· − − − ··· − 0 0 0 40 20 0 20 40 40 20 0 20 40 40 20 0 20 40 − − − − − − p1(x) p2(x) p3(x) p4(x) p(x) (x, y) Figure 1.5: Illustration of Runge’s phenomenon. Here we attempt to approximate a function (dashed line) by polynomial interpolation (solid line). With 7 2 equidistant sample points (left figure) the approximation is poor near the edges of the interval and increasing the number of sample points to 14 (center) and 19 (right) clearly reduces accuracy at the edges further. 0 fitted to equidistantly sampled points will sometimes lose precision when the number of interpolating points is increased, see Figure 1.5 for an example. 2 One way to predict this instability of polynomial interpolation is that − the conditional number of the Vandermonde matrix can be very large for 0 1 2 3 4 5 6 7 8 equidistant points [59]. There are different ways to mitigate the issue of stability, for example Figure 1.4: Illustration of Lagrange interpolation of 4 data points. The red dots are choosing data points that minimize the conditional number of the relevant 4 matrix [57, 59] or by choosing a polynomial basis that is more stable for the data set and p(x)= ykp(xk) is the interpolating polynomial. the given set of data points such as Bernstein polynomials in the case of k =1 equidistant points [126]. Other polynomial schemes can also be considered, The explicit formula for the full interpolating polynomial is for instance by interpolating with different basis functions in different inter- vals, for example using polynomial splines. n (x x1) (x xk 1)(x xk+1) (x xn) Naturally another choice is to not choose polynomials as basis functions p(x)= yk − ··· − − − ··· − (xk x1) (xk xk 1)(xk xk+1) (xk xn) but instead choose some other functions that are more suitable. For an k=1 − ··· − − − ··· − example of this see Section 3.3. and from this formula the expression for the inverse of the Vandermonde While the instability of polynomial interpolation does not prevent it from matrix can be found by noting that the jth row of the inverse will consist being useful for analytical examinations it is generally considered imprac- of the coefficients of pj, the resulting expression for the elements is given in tical when there is noise present or when calculations are performed with Theorem 1.4. limited precision. Often interpolating polynomials are not constructed by Polynomial interpolation is mostly used when the data set we wish to inverting the Vandermonde matrix or calculating the Lagrange basis poly- interpolate is small. The main reason for this is the instability of the inter- nomials, instead a more computationally efficient method such as Newton polation method. One example of this is Runge’s phenomenon that shows interpolation or Neville’s algorithm are used [138]. There are some variants that when certain functions are approximated by polynomial interpolation of Lagrange interpolation, such as barycentric Lagrange interpolation, that

34 35 Generalized Vandermonde matrices and determinants in electromagnetic compatibility 1.2. INTERPOLATION and then construct the final interpolating polynomial by the sum of these 1.5 1.5 1.5 pi weighted by the corresponding yi. The pi polynomials are called Lagrange basis polynomials and can easily be constructed by placing the roots appropriately and then normalizing the 1 1 1 result such that pi(xi) = 1, this gives the expression 0.5 0.5 0.5 (x x1) (x xi 1)(x xi+1) (x xn) pi(x)= − ··· − − − ··· − . (xi x1) (xi xk 1)(xi xi+1) (xi xn) − ··· − − − ··· − 0 0 0 40 20 0 20 40 40 20 0 20 40 40 20 0 20 40 − − − − − − p1(x) p2(x) p3(x) p4(x) p(x) (x, y) Figure 1.5: Illustration of Runge’s phenomenon. Here we attempt to approximate a function (dashed line) by polynomial interpolation (solid line). With 7 2 equidistant sample points (left figure) the approximation is poor near the edges of the interval and increasing the number of sample points to 14 (center) and 19 (right) clearly reduces accuracy at the edges further. 0 fitted to equidistantly sampled points will sometimes lose precision when the number of interpolating points is increased, see Figure 1.5 for an example. 2 One way to predict this instability of polynomial interpolation is that − the conditional number of the Vandermonde matrix can be very large for 0 1 2 3 4 5 6 7 8 equidistant points [59]. There are different ways to mitigate the issue of stability, for example Figure 1.4: Illustration of Lagrange interpolation of 4 data points. The red dots are choosing data points that minimize the conditional number of the relevant 4 matrix [57, 59] or by choosing a polynomial basis that is more stable for the data set and p(x)= ykp(xk) is the interpolating polynomial. the given set of data points such as Bernstein polynomials in the case of k =1 equidistant points [126]. Other polynomial schemes can also be considered, The explicit formula for the full interpolating polynomial is for instance by interpolating with different basis functions in different inter- vals, for example using polynomial splines. n (x x1) (x xk 1)(x xk+1) (x xn) Naturally another choice is to not choose polynomials as basis functions p(x)= yk − ··· − − − ··· − (xk x1) (xk xk 1)(xk xk+1) (xk xn) but instead choose some other functions that are more suitable. For an k=1 − ··· − − − ··· − example of this see Section 3.3. and from this formula the expression for the inverse of the Vandermonde While the instability of polynomial interpolation does not prevent it from matrix can be found by noting that the jth row of the inverse will consist being useful for analytical examinations it is generally considered imprac- of the coefficients of pj, the resulting expression for the elements is given in tical when there is noise present or when calculations are performed with Theorem 1.4. limited precision. Often interpolating polynomials are not constructed by Polynomial interpolation is mostly used when the data set we wish to inverting the Vandermonde matrix or calculating the Lagrange basis poly- interpolate is small. The main reason for this is the instability of the inter- nomials, instead a more computationally efficient method such as Newton polation method. One example of this is Runge’s phenomenon that shows interpolation or Neville’s algorithm are used [138]. There are some variants that when certain functions are approximated by polynomial interpolation of Lagrange interpolation, such as barycentric Lagrange interpolation, that

34 35 Generalized Vandermonde matrices and determinants in electromagnetic compatibility 1.3. REGRESSION have good computational performance [12]. 1.3 Regression In applications where the data is noisy it is often suitable to use regression instead of interpolation, which will be discussed in the next section. Regression is similar to interpolation except that the presence of noise in the data is taken into consideration. The typical regression problem assumes that the data points (x ,y ),i=1,...,n can be described by Before moving on to regression we will discuss an interesting and im- { i i } portant (but for the rest of the thesis irrelevant) form of polynomial interpolation called Hermite interpolation where it is not only required that yi = f(β; xi)+i p(xk)=yk but also that the derivatives up to a certain order (sometimes allowed vary per point) are also given. This requires a higher degree poly- where f(β; x) is a given function with a ﬁxed number of undetermined pa- nomial that can be found by solving the equation system rameters β and i for i =1,...,n are random variables with expected ∈B value zero, usually referred to as the errors or the noise for the data set.

p(xk)=yk0 There are many different classes of regression problems defined by the type of function f(β1,...,βm; x) and the probability distribution of the error  p (xk)=yk1 variables.  .  .  Here we will only consider the situation when the i variables are in- (i) dependent and normally distributed with identical variance and that the p (xk)=yk  i k  parameter space is a compact subset of R and that for all xi the function  B  f(β; x ) is a continuous function of β . It is well known in statistics that for all k =1, 2,...,n where ki are integers that defines the order of the i ∈B under these conditions the parameter set β that minimizes the sum of the derivative that needs to match at the point given by xk. ∗ When this equation system is written as a matrix equation the resulting squares of the residuals n matrix, C, will have dimension m m with m = k with rows given by n i 2 × S(β)= (yi f(β; xi)) i=1 − i=1 j 0,bkj is the maximum-likelihood estimator of the true parameter set β [150]. Ca,b = (b 1)! b c 1 ≤ with c = a ki and ck − ≤ (b c 1)! k− − j i=1 When choosing the parameters β∗ the function is usually said to be the − − best possible fit the data in the least-square sense. The matrix C is called a confluent Vandermonde matrix and has been The most wide-spread form of regression is linear regression where, anal- studied extensively since Hermite interpolation is important both for nu- ogously to linear interpolation, the function f(β; x) depends linearly on β. merical and analytical purposes. For example the confluent Vandermonde This is a common type of regression that has a unique solution that is simple matrix also has a very elegant formula for the determinant [3] to find. It is commonly known as the least-squares method and we describe this method in the next section. det(C)= (x x )(ki+1)(kj +1). With a non-linear f(β; x) it is usually much more difficult to solve the j − i 1 i

36 37 Generalized Vandermonde matrices and determinants in electromagnetic compatibility 1.3. REGRESSION have good computational performance [12]. 1.3 Regression In applications where the data is noisy it is often suitable to use regression instead of interpolation, which will be discussed in the next section. Regression is similar to interpolation except that the presence of noise in the data is taken into consideration. The typical regression problem assumes that the data points (x ,y ),i=1,...,n can be described by Before moving on to regression we will discuss an interesting and im- { i i } portant (but for the rest of the thesis irrelevant) form of polynomial interpolation called Hermite interpolation where it is not only required that yi = f(β; xi)+i p(xk)=yk but also that the derivatives up to a certain order (sometimes allowed vary per point) are also given. This requires a higher degree poly- where f(β; x) is a given function with a fixed number of undetermined pa- nomial that can be found by solving the equation system rameters β and i for i =1,...,n are random variables with expected ∈B value zero, usually referred to as the errors or the noise for the data set. p(xk)=yk0 There are many different classes of regression problems defined by the type of function f(β1,...,βm; x) and the probability distribution of the error  p (xk)=yk1 variables.  .  .  Here we will only consider the situation when the i variables are in- (i) dependent and normally distributed with identical variance and that the p (xk)=yk  i k  parameter space is a compact subset of R and that for all xi the function  B  f(β; x ) is a continuous function of β . It is well known in statistics that for all k =1, 2,...,n where ki are integers that defines the order of the i ∈B under these conditions the parameter set β that minimizes the sum of the derivative that needs to match at the point given by xk. ∗ When this equation system is written as a matrix equation the resulting squares of the residuals n matrix, C, will have dimension m m with m = k with rows given by n i 2 × S(β)= (yi f(β; xi)) i=1 − i=1 j 0,bkj is the maximum-likelihood estimator of the true parameter set β [150]. Ca,b = (b 1)! b c 1 ≤ with c = a ki and c k − ≤ (b c 1)! k− − j i=1 When choosing the parameters β∗ the function is usually said to be the − − best possible fit the data in the least-square sense. The matrix C is called a confluent Vandermonde matrix and has been The most wide-spread form of regression is linear regression where, anal- studied extensively since Hermite interpolation is important both for nu- ogously to linear interpolation, the function f(β; x) depends linearly on β. merical and analytical purposes. For example the confluent Vandermonde This is a common type of regression that has a unique solution that is simple matrix also has a very elegant formula for the determinant [3] to find. It is commonly known as the least-squares method and we describe this method in the next section. det(C)= (x x )(ki+1)(kj +1). With a non-linear f(β; x) it is usually much more difficult to solve the j − i 1 i

36 37 Generalized Vandermonde matrices and determinants in electromagnetic compatibility 1.3. REGRESSION

g1(x1) g2(x1) ... gm(x1) β1 y1 points it is usually referred to as a non-linear regression model. g1(x2) g2(x2) ... gm(x2) β2 y2 There is no general analogue to the Gauss-Markov theorem for non-linear A =  . . . .  , β =  .  , y =  .  . regression models and therefore finding the appropriate estimator requires ......       more knowledge about the specifics of the model. In practice non-linear g (x ) g (x ) ... g (x ) β  y   1 n 2 n m n   n  n regression problems are often solved using some numerical method for non-       This is an overdetermined version of the linear interpolation problem linear optimization of which there are many (see for instance [147] for an described in Section 1.2. overview). How can we actually find the coefficients that minimize the sum of the In this thesis we will use a standard method called the Marquardt least- squares of the residuals? First we can define the square of the length of the squares method that the next section will give an overview of. In Section residual vector, e = Aβ y, as a function 3.2.2 we will use a combination of the Marquardt least-squares method and − methods for fitting linear regression models to fit a non-linear regression n 2 model described by s(e)=e e = e =(Aβ y) (Aβ y) | i| − − G (β; t) η = i i=1 where β, η are vectors of parameters to be fitted, i is the data we wish to This kind of function is a positive second degree polynomial with no mixed fit the model to and G (β; t) is the generalized Vandermonde matrix terms and thus has a global minima where ∂s = 0 for all 1 i n. We ∂ei ≤ ≤ can find the global minima by looking at the derivative of the function, e 1 t1 β1 1 t1 β2 1 t1 βn i (t1e − ) (t1e − ) (t1e − ) is determined by β and 1 t2 β1 1 t2 β2 ··· 1 t2 βn i (t2e − ) (t2e − ) (t2e − ) ∂ei G (β; t)= . . ···. .  . = Ai,j . . .. . ∂βj   (t e1 tm )β1 (t e1 tm )β2 (t e1 tm )βn  thus  m − m − ··· m −    n n ∂s ∂ei 1.3.3 The Marquardt least-squares method = 2ei = 2(Ai, β yi)Ai,j =0 A Aβ = A y ∂β ∂β · − ⇔ i i=1 j i=1 This section is based on Section 3.1 of Paper D This gives 1 The Marquardt least-squares method, also known as the Levenberg-Marquardt A Aβ = A y β =(A A)− A y ⇔ algorithm or damped least-squares, is an efficient method for least-squares and by the Gauss-Markov theorem ( [53, 54, 114], see for instance [116] for estimation for functions with non-linear parameters that was developed in 1 a more modern description), if (A A)− exists then (10) gives the linear, the middle of the 20th century (see [101], [115]).

38 39 Generalized Vandermonde matrices and determinants in electromagnetic compatibility 1.3. REGRESSION

1.3.1 Linear regression models unbiased estimator that gives the lowest variance possible for any linear, unbiased estimator. The matrix given by (A A) 1A is often referred to Suppose we want to find a function − as the Moore-Penrose pseudoinverse of A. m i 1 Clearly a linear regression model with gi(x)=x − gives a regression f(x)= βigi(x) (10) model described by a rectangular Vandermonde matrix. i=1 that fits as well as possible in the least-square sense to the data points 1.3.2 Non-linear regression models (x ,y ), i =1,...,n, n>m. We then get a regression problem described by i i So far we have only considered models that are linear with respect to the the linear equation system Aβ = y where β are the coefficients of f, y is parameters that specify them. If we relax the linearity condition and simply the vector of data values and A is the appropriate alternant matrix, consider fitting a function with m parameters, f(β1,...,βm; x), to n data g1(x1) g2(x1) ... gm(x1) β1 y1 points it is usually referred to as a non-linear regression model. g1(x2) g2(x2) ... gm(x2) β2 y2 There is no general analogue to the Gauss-Markov theorem for non-linear A =  . . . .  , β =  .  , y =  .  . regression models and therefore finding the appropriate estimator requires ......       more knowledge about the specifics of the model. In practice non-linear g (x ) g (x ) ... g (x ) β  y   1 n 2 n m n   n  n regression problems are often solved using some numerical method for non-       This is an overdetermined version of the linear interpolation problem linear optimization of which there are many (see for instance [147] for an described in Section 1.2. overview). How can we actually find the coefficients that minimize the sum of the In this thesis we will use a standard method called the Marquardt least- squares of the residuals? First we can define the square of the length of the squares method that the next section will give an overview of. In Section residual vector, e = Aβ y, as a function 3.2.2 we will use a combination of the Marquardt least-squares method and − methods for fitting linear regression models to fit a non-linear regression n 2 model described by s(e)=e e = e =(Aβ y) (Aβ y) | i| − − G (β; t) η = i i=1 where β, η are vectors of parameters to be fitted, i is the data we wish to This kind of function is a positive second degree polynomial with no mixed fit the model to and G (β; t) is the generalized Vandermonde matrix terms and thus has a global minima where ∂s = 0 for all 1 i n. We ∂ei ≤ ≤ can find the global minima by looking at the derivative of the function, e 1 t1 β1 1 t1 β2 1 t1 βn i (t1e − ) (t1e − ) (t1e − ) is determined by β and 1 t2 β1 1 t2 β2 ··· 1 t2 βn i (t2e − ) (t2e − ) (t2e − ) ∂ei G (β; t)= . . ···. .  . = Ai,j . . .. . ∂βj   (t e1 tm )β1 (t e1 tm )β2 (t e1 tm )βn  thus  m − m − ··· m −    n n ∂s ∂ei 1.3.3 The Marquardt least-squares method = 2ei = 2(Ai, β yi)Ai,j =0 A Aβ = A y ∂β ∂β · − ⇔ i i=1 j i=1 This section is based on Section 3.1 of Paper D This gives 1 The Marquardt least-squares method, also known as the Levenberg-Marquardt A Aβ = A y β =(A A)− A y ⇔ algorithm or damped least-squares, is an efficient method for least-squares and by the Gauss-Markov theorem ( [53, 54, 114], see for instance [116] for estimation for functions with non-linear parameters that was developed in 1 a more modern description), if (A A)− exists then (10) gives the linear, the middle of the 20th century (see [101], [115]).

38 39 Generalized Vandermonde matrices and determinants in electromagnetic compatibility 1.3. REGRESSION

The least-squares estimation problem for functions with non-linear pa- It is obvious from (11) that δ(r) depends on the value of the fudge factor rameters arises when a function of m independent variables and described λ. Note that if λ = 0, then (11) reduces to the regular Gauss-Newton by k unknown parameters needs to be fitted to a set of n data points such method [115], and if λ the method will converge towards the steepest →∞ that the sum of squares of residuals is minimized. descent method [115]. The reason that the two methods are combined is that The vector containing the independent variables is x =(x , ,x ), the the Gauss-Newton method often has faster convergence than the steepest 1 ··· n vector containing the parameters β =(β , ,β ) and the data points descent method, but is also an unstable method [115]. Therefore, λ must be 1 ··· k chosen appropriately in each step. In the Marquardt least-squares method (Y ,X ,X , ,X )=(Y , X ) ,i=1, 2, , n. i 1i 2i ··· mi i i ··· this amounts to increasing λ with a chosen factor v whenever an iteration Let the residuals be denoted by E = f(X ; β) Y and the sum of increases S, and if an iteration reduces S then λ is reduced by a factor v as i i − i squares of Ei is then written as many times as possible. Below follows a detailed description of the method n using the following notation: S = (f(X ; β) Y )2 , i i n − 2 i=1 (r) (r) S = Yi f(Xi, b , c) , (13) which is the function to be minimized with respect to β. − i=1 The Marquardt least-square method is an iterative method that gives ap- n proximate values of β by combining the Gauss-Newton method (also known 2 S λ(r) = Y f(X , b(r) + δ(r), c) . (14) as the inverse Hessian method) and the steepest descent (also known as the i − i gradient) method to minimize S. The method is based around solving the i=1 linear equation system The iteration step of the Marquardt least-squares method can be described as follows: (r) (r) (r) (r) A∗ + λ I δ∗ = g∗ , (11) Input: v>1 and b(r), λ(r). (r) (r) • where A∗ is a modified Hessian matrix of E(b) (or f(Xi; b)), g∗ is a (r) rescaled version of the gradient of S, r is the number of the current iteration Compute S λ . of the method, and λ is a real positive number sometimes referred to as the (r) If λ(r) 1 then compute S λ , else go to . fudge factor [138]. The Hessian, the gradient and their modifications are • v defined as follows: (r) (r) A = J J, If S λ S(r) let λ(r+1) = λ . • v ≤ v ∂fi ∂Ei Jij = = ,i=1, 2, ,m; j =1, 2, , k, If S λ(r) S(r) let λ(r+1) = λ(r). ∂bj ∂bj ··· ··· ≤ and (r) (r) aij If S λ >S then find the smallest integer ω>0 such that (A∗)ij = , • (r) ω (r) (r+1) (r) ω √aii√ajj S λ v S , and set λ = λ v . ≤ while (r+1) (r) (r) (r) gi Output: b = b + δ , δ . g = J(Y f0), f0i = f(Xi, b, c), gi∗ = . • − aii This iteration procedure is also described in figure 1.6. Naturally, some Solving (11) gives a vector which, after some scaling, describes how the condition for what constitutes an acceptable fit for the function must also parameters b should be changed in order to get a new approximation of β, be chosen. If this condition is not satisfied the new values for b(r+1) and (r) (r+1) δ∗ λ will be used as input for the next iteration and if the condition is b(r+1) = b(r) + δ(r), δ(r) = i . (12) √aii satisfied the algorithm terminates. The quality of the fitting, in other words

40 41 Generalized Vandermonde matrices and determinants in electromagnetic compatibility 1.3. REGRESSION

40 41 Generalized Vandermonde matrices and determinants in electromagnetic compatibility 1.3. REGRESSION the value of S, is determined by the stopping condition and the initial values 1.3.4 D-optimal experiment design for b(0). The initial value of λ(0) affects the performance of the algorithm For the class of linear non-weighted regression problems described in Sec- to some extent since after the first iteration λ(r) will be self-regulating. tion 1.3.1 minimizing the square of the sum of residuals gives the maximum- Suitable values for b(0) are challenging to find for many functions f and likelihood estimation of the parameters that specify the fitted function. This they are often, together with λ(0), found using heuristic methods. estimation naturally has a variance as well and minimizing this variance can Input: be interpreted as improving the reliability of the fitted function by minimiz- Compute S λ(r) b(r), λ(r) and v>1 ing its sensitivity to noise in measurements. This minimization is usually done by choosing where to sample the data carefully, in other words, given the regression problem defined by

λ(r) yi = f(β; xi)+i ω = ω +1 ω =1 λ(r) 1 Compute S YES v for i =1,...,n with the same conditions on f(β; x) and i as in Section 1.3 NO NO NO we want to choose a design x ,i=1,...,n that minimizes the variance of { i } the values predicted by the regression model. This is usually referred to as (r) ω (r) (r) (r) λ(r) (r) S λ v S S λ S S v S G-optimality. ≤ ≤ NO ≤ To give a proper definition of G-optimality we will need the concept of YES YES YES the Fischer information matrix. Definition 1.5. For a finite design x Rn the Fischer information (r+1) (r) ω (r+1) (r+1) λ(r) ∈X ⊆ λ = λ v λ = λ(r) λ = v matrix is the matrix defined by n M(x)= f(xi)f(xi) Output: i=1 (r+1) (r) (r) (r) b = b + δ , δ where f(x)= f (x) f (x) f (x) . 1 2 ··· n Definition 1.6 (The G-optimality criterion). A design ξ is said to be Figure 1.6: The basic iteration step of the Marquardt least-squares method, definitions of computed quantities are given in (12), (13) and (14). G-optimal if it minimizes the maximum variance of any predicted value

Var(y(ξ)) = min max Var(y(x)) = min max f(x) M(z)f(x). xi,i=1,2,...,n x z x In Section 3.2 the Marquardt least-squares method will be used for re- ∈X ∈X ∈X gression with power-exponential functions. The G-optimality condition was first introduced in [154] (the name G- optimality comes from later work by Kiefer and Wolfowitz where they describe several different types of optimal design using alphabetical letters [89], [90]) and is an example of a minimax criterion, since it minimizes the maximum variance of the values given by the regression model [116]. There are many kinds of optimality conditions related to G-optimality. One which is suitable for us to consider is D-optimality. This type of optimality was first introduced in [169] and instead of focusing on the variance of the predicted values of the model it instead minimizes the volume of the confidence ellipsoid for the parameters (for a given confidence level).

42 43 Generalized Vandermonde matrices and determinants in electromagnetic compatibility 1.3. REGRESSION the value of S, is determined by the stopping condition and the initial values 1.3.4 D-optimal experiment design for b(0). The initial value of λ(0) affects the performance of the algorithm For the class of linear non-weighted regression problems described in Sec- to some extent since after the first iteration λ(r) will be self-regulating. tion 1.3.1 minimizing the square of the sum of residuals gives the maximum- Suitable values for b(0) are challenging to find for many functions f and likelihood estimation of the parameters that specify the fitted function. This they are often, together with λ(0), found using heuristic methods. estimation naturally has a variance as well and minimizing this variance can Input: be interpreted as improving the reliability of the fitted function by minimiz- Compute S λ(r) b(r), λ(r) and v>1 ing its sensitivity to noise in measurements. This minimization is usually done by choosing where to sample the data carefully, in other words, given the regression problem defined by

42 43 Generalized Vandermonde matrices and determinants in electromagnetic compatibility 1.3. REGRESSION

Definition 1.7 (The D-optimality criterion). A design ξ is said to be will demonstrate one way to optimize the Vandermonde determinant over a D-optimal if it maximizes the determinant of the Fischer information matrix cube. The shape of the volume to optimize the determinant in is given by con- det(M(ξ)) = max det(M(x)). straints on the data points. For example, if there is a cost associated with x ∈X each data point that increases quadratically with x and there is a total bud- The D-optimal designs are often good design with respect to other get, C, for the experiment that cannot be exceeded the constraint on the x-values becomes x2 + x2 + ...+ x2 C and the determinant needs to be types of criterion (see for example [61] for a brief discussion on this) and 1 2 n ≤ is often practical to consider due to being invariant with respect to lin- optimized in a ball. In Chapter 2 we examine the optimization of the Van- ear transformations of the design matrix. A well-known theorem called the dermonde determinant over several different surfaces in several dimensions. Kiefer-Wolfowitz equivalence theorem shows that under certain conditions In Section 3.3 we use a D-optimal design to improve the stability of an G-optimality is equivalent to D-optimality. interpolation problem as an alternative to the non-linear regression done in Section 3.2. Note that while choosing a D-optimal design can give an Theorem 1.5 (Kiefer-Wolfowitz equivalence theorem). For any linear re- approximation method that is more stable since it minimizes the variance gression model with independent, uncorrelated errors and continuous and of the parameters the function used in the approximation can still be highly linearly independent basis functions fi(x) defined on a fixed compact topo- sensitive to changes in parameters (the variance of the predicted values can logical space there exists a D-optimal design and any D-optimal design is be minimized but still high) so it does necessarily maximize stability or X also G-optimal. stop instability phenomenons similar to Runge’s phenomenon for polynomial interpolation. This equivalence theorem was originally proven in [91] but the formulation above is taken from [116]. Thus maximizing the determinant of the Fischer information matrix corresponds to minimizing the variance of the estimated β. Interpolation can be considered a special case of regression when the sum of the square of the residuals can be reduced to zero. Thus we can speak of D-optimal design for interpolation as well, in fact optimal experiment design is often used to find the minimum number of points needed for a certain model. For a linear interpolation problem defined by the alternant matrix A(f; x) the Fis- cher information matrix is M(x)=A(f; x)A(f; x) and since A(f; x) is an n n matrix det(M(x)) = det(A(f; x) ) det(A(f; x)) = det(f; x))2. Thus × the maximization of the determinant of the Fischer information matrix is equivalent to finding the extreme points of the determinant of an alternant matrix in some volume given by the set of possible designs. A standard case of this is polynomial interpolation where the x-values are in a limited interval, for instance 1 x 1 for i =1, 2,...,n. In this − ≤ i ≤ case the regression problem can be written as Vn(x)β = y where Vn(x) is a Vandermonde matrix as defined in equation (1) and the constraints on the elements of β means that the volume we want to optimize over is a cube in n dimensions. There is a number of classical results that describe how to find the D-optimal designs for weighted univariate polynomials with various efficiency functions, see for instance [43], and in Section 2.2.3 we

44 45 Generalized Vandermonde matrices and determinants in electromagnetic compatibility 1.3. REGRESSION

44 45 Generalized Vandermonde matrices and determinants in 1.4. ELECTROMAGNETIC COMPATIBILITY AND electromagnetic compatibility ELECTROSTATIC DISCHARGE CURRENTS

1.4 Electromagnetic compatibility and tions of these currents should give an accurate description of the observed electrostatic discharge currents behaviour that the standard is based on as well being computationally efficient (since computer simulations replacing construction of prototypes can save both time and resources) and be compatible with the mathematical There are many examples of electromagnetic phenomena that involve two tools that are commonly used in electromagnetic calculations, for instance objects influencing each other without touching. Almost everyone is familiar Laplace and Fourier transforms. with magnets that attract or repel other object, sparks that bridge physical In this thesis we will discuss approximations of electrostatic discharge gaps and radio waves that send messages across the globe. While this action- currents, either from a standard or based on experimental data. In Section at-a-distance can be very useful it can also cause unintended interactions 1.4.1 a review of models in the literature can be found and in Chapter between different systems. This is usually referred to as electromagnetic 3 we propose a new function, the analytically extended function (AEF), disturbance or electromagnetic interference and the field of electromagnetic for modelling these currents that has some advantages compared to the compatibility (EMC) is the study and design of systems that are not sus- commonly used models and can be applied to many different cases, typically ceptible to disturbances from other systems and does not cause interference at the cost of some extra manual work in fitting the model. with other systems or themselves [132, 173]. Electrostatic discharge (ESD) is a common phenomenon where a sud- There are many possible causes of electromagnetic disturbance including den flow of electricity between two charged object occurs, examples include a multitude of sources. Some examples are man-made sources such as broad- sparks and lightning strikes. The main mechanism behind is usually said casting and receiving devices, power generators and converters, power con- to be contact electrification, this phenomena is due to all materials occa- version and ignition systems for combustion engines, manufacturing equip- sionally emitting electrons, usually at a higher rate when they are heated. ment like ovens, saws, mills, welders, blenders and mixers, other equipment Typically the emission and absorption balances out but since the rate of such as fans, heaters, coolers, lights, computers, instruments for measure- emission varies between different materials an imbalance can occur when ments and control, examples of natural sources are atmospheric-, solar- and two materials come sufficiently close to each other. When the materials are cosmic noise, static discharges and lightning [119]. separated this charge imbalance might remain for some time, it can be re- Mathematical modelling is an important tool for EMC [119]. Using com- stored by the charged objects slowly emitting electrons to the surrounding puters for electromagnetic analysis have been done since the 1950s [63] and objects but in the right conditions, for example if the charged object comes it rapidly became more and more useful and important over time [137]. In near a conductive material with an opposite charge, the restoration of the practice many different types of models and methods are used, all with their charge balance can be very rapid resulting in an electrostatic discharge. The own advantages and disadvantages, and the design process often involves reader is likely to be familiar with the case of two materials rubbing against a combination of analytical and numerical techniques [46]. The sources of each other building up a charge imbalance and one of the objects generating electromagnetic disturbances are not always well understood or cannot be a spark when moved close to a metal object. This case is common since fric- well described which means that it is not always feasible to derive all parts tion between objects typically means a larger contact area where charges can of the model using first principles, especially since the behaviour of many transfer and movement is necessary for charge separation. For this reason systems contain some degree of randomness which means that it is some- this mechanism is often referred to as friction charging or the triboelectric times most reasonable to use models that are based on typical behaviours effect. Contact charging can happen between any material, including liq- based on statistical data [27, 84]. uids and gases, and can also be affected by many other types of phenomena, Requirements for a product or system to be considered electromagneti- such as ion transfer or energetic charged particles colliding with other ob- cally compatible can be found in standards such as the IEC 61000-4-2 [73] jects [51]. Therefore the exact mechanisms behind electrostatic discharges and IEC 62305-1 [74]. In several of these standard approximations of typical can be difficult to understand and describe, even when the circumstances currents for various phenomena are given and electromagnetic compatibility where the electrostatic discharge are likely are well known [110]. requirements are based on the effects of the system being exposed to these In this thesis we focus on two types of electrostatic discharge, lightning currents, such as the radiated electromagnetical fields. Ideally the descrip- discharge and human-to-object (human-to-metal or human-to-human).

46 47 Generalized Vandermonde matrices and determinants in 1.4. ELECTROMAGNETIC COMPATIBILITY AND electromagnetic compatibility ELECTROSTATIC DISCHARGE CURRENTS

Lightning discharges can cause electromagnetic disturbances in three this type of function is also applied in other types of engineering, see sec- ways, by passing through an system directly, by passing through a nearby tion 3.1 for some examples. object which then radiates electrical fields that disturbs the system, or by This model was also extended with a four-exponential version by Keenan indirectly inducing transient currents in systems when the electrical field and Rossi [88]: associated with a thundercloud disappears when the lightning discharge re- t t t t moves the charge imbalance between cloud and ground [27]. We discuss i(t)=I e− τ1 e− τ2 I e− τ3 e− τ4 . (15) 1 − − 2 − modelling of some lightning discharges from standards and experimental data in Section 3.2. The Pulse function was proposed in [44], Electrostatical discharges from humans are very common and are typ- t p t τ τ ically just a nuisance, but they can damage sensitive electronics and can i(t)=I0 1 e− 1 e− 2 , cause severe accidents, either by the shock from the discharge causing a − human error or by directly causing gas or dust explosions [84,110]. We dis- and has been used for representation of lightning discharge currents both in cuss modelling of a simulated human-to-object electrostatical discharge in its single term form [105] as well as linear combinations of two [156], three Section 3.3. or four Pulse functions [180]. The Heidler function [65] is one of the most commonly used functions 1.4.1 Electrostatic discharge modelling for lightning discharge modelling n Well-defined representation of real electrostatic discharge currents is needed t I0 τ1 t in order to establish realistic requirements for ESD generators used in testing i(t)= e− τ2 , n the equipment and devices, as well as to provide and improve the repeata- η 1+ t τ1 bility of tests. It should be able to approximate the current for various test levels, test set-ups and procedures, and also for various ESD conditions Wang et al. [170] proposed an ESD model in the form of a sum of two Heidler such as approach speeds, types of electrodes, relative arc length, humidity, functions: n n etc. A mathematical function is necessary for computer simulation of such t t I1 τ1 t I2 τ3 t phenomena, for verification of test generators and for improving standard i(t)= e− τ2 + e− τ4 , (16) n n η1 1+ t η2 1+ t waveshape definitions. τ1 τ3 A number of current functions, mostly based on exponential functions, 1 /n 1/n have been proposed in the literature to model the ESD currents, [26,47,48, with η = exp τ1 nτ2 and η = exp τ3 nτ4 being the 1 τ2 τ1 2 τ4 τ3 80, 82, 88, 156, 163, 170, 171, 180, 181]. Here we will give a brief presentation − − peak correction factors. The function has been used to fit different electro- of some of them and in Section 3.1 we will propose an alternative function static discharge currents using different methods [47,170,181]. and a scheme for fitting it to a waveshape. Berghe and Zutter [163] proposed an ESD current model constructed as A number of mathematical expressions have been introduced in the liter- a sum of two Gaussian functions in the form: ature for the purpose of representation of the ESD currents, either the IEC 61000-4-2 Standard one [73], or experimentally measured ones, e.g. [47]. In 2 2 t τ1 t τ2 this section we give an overview of most commonly applied ESD current i(t)=A exp − + Btexp − . (17) − σ1 − σ2 approximations. A double-exponential function has been proposed by Cerri et al. [26] for The following approximation using exponential polynomials is presented representation of ESD currents for commercial simulators in the form in [171] by Wang et al.

t t Ct Dt i(t)=I e− τ1 I e− τ2 , i(t)=Ate− + Bte− , (18) 1 − 2

48 49 Generalized Vandermonde matrices and determinants in 1.4. ELECTROMAGNETIC COMPATIBILITY AND electromagnetic compatibility ELECTROSTATIC DISCHARGE CURRENTS

48 49 Generalized Vandermonde matrices and determinants in electromagnetic compatibility 1.5. SUMMARIES OF PAPERS and has been used for design of simple electric circuits which can be used 1.5 Summaries of papers to simulate ESD currents. One of the most commonly used ESD standard currents is the IEC 61000- Paper A 4-2 current that represent a typical electrostatic discharge generated by the In this paper we examine the extreme points of the Vandermonde determi- human body [73]. In the IEC 61000-4-2 standard [73] this current is given by nant in three or more dimensions. The paper discusses the three-dimensional a graphical representation, see Figure 3.10, together with some constraints, case, see Section 2.1.2, and gives a more detailed description of the method see page 122. In Figure 1.7 the models discussed in this section have been used to solve the n-dimensional problem from [158], see Section 2.2.1. The fitted to the graph given in the standard. The data from the standard is not extreme points are given in terms of roots of rescaled Hermite polynomials included in this figure since some features, notably the initial delay visible and explicit expressions are given for dimensions three to seven. The results in the standard is not reproduced in either model. The different models are also visualized in three to seven dimensions by using symmetries of the also give quite different quantitative behaviour in the region 2.5 25 ns. In − answers to project all the extreme points onto a two-dimensional plane, see Section 3.1 we propose a new scheme for modelling this type of functions Section 2.2.2. There is also a brief discussion on optimising the generalised and in Section 3.3 we fit this model to the IEC 61000-4-2 standard current Vandermonde determinant in three dimensions, see Section 2.1.1. The thesis and some experimental data. author contributed primarily to the derivation of some of the recursive properties of the Vandermonde determinant and its derivatives and to a lesser Two Heidler, [47] Two Heidler, [181] extent to the visualisation aspects of the problem. 14 Pulse binomial, [156] Exponential polynomial, [171] 12 Two Gaussians, [163] Four exponential, [88] Paper B

10 Here the Vandermonde determinant is optimised over the three-dimensional torus, see Section 2.1.3, and the sphere deﬁned by the p - norm in n di- 8 mensions, see Section 2.2.3. Main focus is on optimisation over the cube

i(t) [A] that corresponds to p = . The thesis author contributed primarily to the ∞ 6 examination of the torus. 4 Paper C 2 The value of the Vandermonde determinant is optimized over the ellipsoid 0 and cylinder in three dimensions, see Section 2.1.5 and 2.1.6. Lagrange mul- 0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 5.5 tipliers are used to find a system of polynomial equations which give the local t [s] 10 8 extreme points as its solutions. Using Gröbner basis and other techniques · − the extreme points are given either explicitly or as roots of polynomials in Figure 1.7: Functions representing the Standard ESD current waveshape for 4kV. one variable. The behaviour of the Vandermonde determinant is also presented visually in some interesting cases. The method is also extended to The model given in Section 3.1 is also fitted to both lightning discharge surfaces defined by homogeneous polynomials, see Section 2.1.7. Finally the current from the standard and from measured data in Section 3.2. paper discusses the extreme points on sphere defined by the p - norm (primarily p = 4). The thesis author primarily contributed to the examination of the ellipsoid, cylinder and surfaces defined by homogenous polynomials.

50 51 Generalized Vandermonde matrices and determinants in electromagnetic compatibility 1.5. SUMMARIES OF PAPERS and has been used for design of simple electric circuits which can be used 1.5 Summaries of papers to simulate ESD currents. One of the most commonly used ESD standard currents is the IEC 61000- Paper A 4-2 current that represent a typical electrostatic discharge generated by the In this paper we examine the extreme points of the Vandermonde determi- human body [73]. In the IEC 61000-4-2 standard [73] this current is given by nant in three or more dimensions. The paper discusses the three-dimensional a graphical representation, see Figure 3.10, together with some constraints, case, see Section 2.1.2, and gives a more detailed description of the method see page 122. In Figure 1.7 the models discussed in this section have been used to solve the n-dimensional problem from [158], see Section 2.2.1. The fitted to the graph given in the standard. The data from the standard is not extreme points are given in terms of roots of rescaled Hermite polynomials included in this figure since some features, notably the initial delay visible and explicit expressions are given for dimensions three to seven. The results in the standard is not reproduced in either model. The different models are also visualized in three to seven dimensions by using symmetries of the also give quite different quantitative behaviour in the region 2.5 25 ns. In − answers to project all the extreme points onto a two-dimensional plane, see Section 3.1 we propose a new scheme for modelling this type of functions Section 2.2.2. There is also a brief discussion on optimising the generalised and in Section 3.3 we fit this model to the IEC 61000-4-2 standard current Vandermonde determinant in three dimensions, see Section 2.1.1. The thesis and some experimental data. author contributed primarily to the derivation of some of the recursive properties of the Vandermonde determinant and its derivatives and to a lesser Two Heidler, [47] Two Heidler, [181] extent to the visualisation aspects of the problem. 14 Pulse binomial, [156] Exponential polynomial, [171] 12 Two Gaussians, [163] Four exponential, [88] Paper B

50 51 Generalized Vandermonde matrices and determinants in electromagnetic compatibility

Paper D This paper is a detailed description and derivation of some properties of the analytically extended function (AEF) and a scheme for how it can be used in approximation of lightning discharge currents, see sections 3.1.1 and 3.2.2. Lightning discharge currents are classified in the IEC 62305-1 Stan- dard into waveshapes representing important observed phenomena. These Chapter 2 waveshapes are approximated with mathematical functions in order to be used in lightning discharge models for ensuring electromagnetic compatibility. A general framework for estimating the parameters of the AEF using the Marquardt least-squares method (MLSM) for a waveform with an arbi- Extreme points of the trary (finite) number of peaks as well as for the given charge transfer and specific energy is described, see sections 1.3.3, 3.2 and 3.2.3. This framework Vandermonde determinant is used to find parameters for some single-peak waveshapes and advantages and disadvantages of the approach are discussed, see Section 3.2.6. The thesis author contributed with the p-peak formulation of the AEF, modification to the MLSM and basic software for fitting the AEF to data.

Paper E In this paper is an examination of how the analytically extended function (AEF) can be used to approximate multi-peaked lightning current waveforms. A general framework for estimating the parameters of the AEF using This chapter is based on Papers A, B and C: the Marquardt least-squares method (MLSM) for a waveform with an arbi- trary (finite) number of peaks is presented, see Section 3.2. This framework Paper A. Karl Lundeng˚ard, Jonas Osterberg¨ and Sergei Silvestrov. Extreme is used to find parameters for some waveforms, such as lightning currents points of the Vandermonde determinant on the sphere and some limits from the IEC 62305-1 Standard and recorded lightning current data, see Sec- involving the generalized Vandermonde determinant. tion 3.2.6. The thesis author contributed with improved software for fitting Preprint: arXiv:1312.6193 [math.ca], 2013. the AEF to the more complicated waveforms (compared to Paper D). Paper B. Karl Lundeng˚ard, Jonas Osterberg,¨ and Sergei Silvestrov. Optimiza- Paper F tion of the determinant of the Vandermonde matrix and related matrices. In AIP Conference Proceedings 1637, ICNPAA, Narvik, Norway, The multi-peaked analytically extended function (AEF) is used in this pa- pages 627–636, 2014. per for representation of electrostatic discharge (ESD) currents. In order to minimize unstable behaviour and the number of free parameters the expo- Paper C. Karl Lundeng˚ard, Jonas Osterberg,¨ and Sergei Silvestrov. Optimiza- nents of the AEF are chosen from an arithmetic sequence. The function is tion of the determinant of the Vandermonde matrix on the sphere fitted by interpolating data chosen according to a D-optimal design. ESD and related surfaces. In Christos H Skiadas, editor, ASMDA 2015 current modelling is illustrated through two examples: an approximation of Proceedings: 16th Applied Stochastic Models and Data Analysis In- the IEC Standard 61000-4-2 waveshape, and a representation of some mea- ternational Conference with 4th Demographics 2015 Workshop, pages sured ESD current. The contents of this paper is in Section 3.3. The thesis 637–648. ISAST: International Society for the Advancement of Science author contributed with the derivation of the D-optimal design, motivating and Technology, 2015. its use as well as software for fitting the AEF according to the design.

52 Generalized Vandermonde matrices and determinants in electromagnetic compatibility

52 2.1. EXTREME POINTS OF THE VANDERMONDE DETERMINANT AND RELATED DETERMINANTS ON VARIOUS SURFACES IN THREE DIMENSIONS

2.1 Extreme points of the Vandermonde determinant and related determinants on various surfaces in three dimensions

In this chapter we will discuss how to optimize the determinant of the Van- dermonde matrix and some related determinants over various surfaces in three dimensions and the results will be visualized.

2.1.1 Optimization of the generalized Vandermonde determinant in three dimensions This section is based on Section 1.1 of Paper A

In this section we plot the values of the determinant

v (x )=(x x )(x x )(x x ), 3 3 3 − 2 3 − 1 2 − 1 and also the generalized Vandermonde determinant g3(x3, a3) for three dif- 2 2 2 3 ferent choices of a3 over the unit sphere x1 +x2 +x3 = 1 in R . Our plots are over the unit sphere but the determinant exhibits the same general behavior over centered spheres of any radius. This follows directly from (1.4) and that exactly one element from each row appears in the determinant. For any scalar c we get

n ai gn(cxn, an)= c gn(xn, an), i=1 which for vn becomes

n(n 1) − vn(cxn)=c 2 vn(xn), (19) and so the values over diﬀerent radii diﬀer only by a constant factor. In Figure 2.1 value of v3(x3) has been plotted over the unit sphere and the curves where the determinant vanishes are traced as black lines. The coordinates in Figure 2.1 (b) are related to x3 by

2 01 1/√60 0 x = 111 01/√20t, (20) 3 −    1 11 0 01/√3 − −     55 2.1. EXTREME POINTS OF THE VANDERMONDE DETERMINANT AND RELATED DETERMINANTS ON VARIOUS SURFACES IN THREE DIMENSIONS

2.1 Extreme points of the Vandermonde determinant and related determinants on various surfaces in three dimensions

In this chapter we will discuss how to optimize the determinant of the Van- dermonde matrix and some related determinants over various surfaces in three dimensions and the results will be visualized.

2.1.1 Optimization of the generalized Vandermonde determinant in three dimensions This section is based on Section 1.1 of Paper A

In this section we plot the values of the determinant

n ai gn(cxn, an)= c gn(xn, an), i=1 which for vn becomes

2 01 1/√60 0 x = 111 01/√20t, (20) 3 −    1 11 0 01/√3 − −     55 2.1. EXTREME POINTS OF THE VANDERMONDE Generalized Vandermonde matrices and determinants in DETERMINANT AND RELATED DETERMINANTS ON electromagnetic compatibility VARIOUS SURFACES IN THREE DIMENSIONS

For reasons that will become clear in Section 2.2.1 it is also useful to think about these coordinates as the roots of the polynomial 1 P (x)=x3 x. 3 − 2

So far we have only considered the behavior of v3(x3), that is g3(x3, a3) with a3 = (0, 1, 2). We now consider three generalized Vandermonde determinants, namely g3 with a3 = (0, 1, 3), a3 = (0, 2, 3) and a3 = (1, 2, 3). (a) Plot with respect to (b) Plot with respect to (c) Plot with respect These three determinants show increasingly more structure and they all have the regular x-basis. the t-basis, see (20). to parametrization a neat formula in terms of v3 and the elementary symmetric polynomials (21).

ekn = ek(x1, ,xn)= xi1 xi2 xik , Figure 2.1: Plot of v3(x3) over the unit sphere. ··· ··· 1 i1

cos(φ) sin(θ) t(θ, φ)= sin(φ) . (21)   cos(φ) cos(θ)   We will use this t-basis and spherical parametrization throughout this section. From the plots in Figure 2.1 it can be seen that the number of extreme (a) Plot with respect to (b) Plot with respect to (c) Plot with respect to points for v3 over the unit sphere seem to be 6 = 3!. It can also been seen the regular x-basis. the t-basis, see (20). angles given in (21). that all extreme points seem to lie in the plane through the origin that is orthogonal to an apparent symmetry axis in the direction (1, 1, 1), the Figure 2.2: Plot of g3(x3, (0, 1, 3)) over the unit sphere. direction of t3. We will see later that the extreme points for vn indeed lie in n In Figure 2.2 we see the determinant the hyperplane xi = 0 for all n, see Theorem 2.2, and the total number 111 i=1 of extreme points for vn equals n!, see Remark 2.1. g3(x3, (0, 1, 3)) = x1 x2 x3 = v3(x3)e1, 3 3 3 The black lines where v3(x3) vanishes are actually the intersections be- x x x 1 2 3 tween the sphere and the three planes x3 x1 = 0, x3 x2 = 0 and − − x x = 0, as these diﬀerences appear as factors in v (x ). plotted over the unit sphere. The expression v 3(x3)e1 is easy to derive, the 2 − 1 3 3 We will see later on that the extreme points are the six points acquired v3(x3) is there since the determinant must vanish whenever any two columns from permuting the coordinates in are equal, which is exactly what the Vandermonde determinant expresses. The e1 follows by a simple polynomial division. As can be seen in the plots 1 we have an extra black circle where the determinant vanishes compared to x3 = ( 1, 0, 1) . √2 − Figure 2.1. This circle lies in the plane e1 = x1 + x2 + x3 = 0 where we

56 57 2.1. EXTREME POINTS OF THE VANDERMONDE Generalized Vandermonde matrices and determinants in DETERMINANT AND RELATED DETERMINANTS ON electromagnetic compatibility VARIOUS SURFACES IN THREE DIMENSIONS

For reasons that will become clear in Section 2.2.1 it is also useful to think about these coordinates as the roots of the polynomial 1 P (x)=x3 x. 3 − 2

ekn = ek(x1, ,xn)= xi1 xi2 xik , Figure 2.1: Plot of v3(x3) over the unit sphere. ··· ··· 1 i1

previously found the extreme points of v3(x3) and thus doubles the number a3 g3(x3, a3) of extreme points to 2 3!. · (0, 1, 2) v (x )e =(x x )(x x )(x x ) A similar treatment can be made of the remaining two generalized de- 3 3 0 3 − 2 3 − 1 2 − 1 (0, 1, 3) v (x )e =(x x )(x x )(x x )(x + x + x ) terminants that we are interested in, plotted in the following two ﬁgures. 3 3 1 3 − 2 3 − 1 2 − 1 1 2 3 (0, 2, 3) v (x )e =(x x )(x x )(x x )(x x + x x + x x ) 3 3 2 3 − 2 3 − 1 2 − 1 1 2 1 3 2 3 (1, 2, 3) v (x )e =(x x )(x x )(x x )x x x 3 3 3 3 − 2 3 − 1 2 − 1 1 2 3 Table 2.1: Table of some determinants of generalized Vandermonde matrices.

and so g3(x3, (0, 2, 3)) vanish on the sphere on two circles lying on the planes x + x + x + 1 = 0 and x + x + x 1 = 0. These can be seen in Figure 1 2 3 1 2 3 − 2.3 as the two black circles perpendicular to the direction (1, 1, 1).

Note also that while v3 and g3(x3, (0, 1, 3)) have the same absolute value (a) Plot with respect to (b) Plot with respect to (c) Plot with respect to on all their respective local extreme points (by symmetry) we have that both the regular x-basis. the t-basis, see (20). angles given in (21). g3(x3, (0, 2, 3)) and g3(x3, (1, 2, 3)) have diﬀerent absolute values for some of their respective extreme points. Figure 2.3: Plot of g3(x3, (0, 2, 3)) over the unit sphere.

2.1.2 Extreme points of the Vandermonde determinant on the three-dimensional unit sphere

This section is based on Section 2.2 of Paper A

It is fairly simple to describe v3(x3) on the circle that is formed by the intersection of the unit sphere and the plane x1 + x2 + x3 = 0. Using Rodrigues’ rotation formula to rotate a point, x, around the axis 1 (1, 1, 1) √3 (a) Plot with respect to (b) Plot with respect to (c) Plot with respect to with the angle θ will give the rotation matrix the regular x-basis. the t-basis, see (20). angles given in (21). 2 cos(θ)+1 1 cos(θ) √3 sin(θ)1 cos(θ)+√3 sin(θ) Figure 2.4: Plot of g3(x3, (1, 2, 3)) over the unit sphere. 1 − − − R = 1 cos(θ)+√3 sin(θ) 2 cos(θ)+1 1 cos(θ) √3 sin(θ) . θ 3 − − −  The four determinants treated so far are collected in Table 2.1. Deriva- 1 cos(θ) √3 sin(θ)1 cos(θ)+√3 sin(θ) 2 cos(θ)+1 − − − tion of these determinants is straight forward. We note that all but one of   them vanish on a set of planes through the origin. For a = (0, 2, 3) we have A point which already lies on S2 can then be rotated to any other point the usual Vandermonde planes but the intersection of e2 = 0 and the unit on S2 by letting R act on the point. Choosing the point x = 1 ( 1, 0, 1) θ √2 − sphere occur at two circles. gives the Vandermonde determinant a convenient form on the circle since: 1 x x + x x + x x = (x + x + x )2 (x2 + x2 + x2) 1 2 1 3 2 3 2 1 2 3 − 1 2 3 √3 cos(θ) sin(θ) 1 − − 1 2 1 Rθx = 2 sin(θ) , = (x1 + x2 + x3) 1 = (x1 + x2 + x3 + 1) (x1 + x2 + x3 1) , √  −  2 − 2 − 6 √3 cos(θ) + sin(θ)   58 59 2.1. EXTREME POINTS OF THE VANDERMONDE Generalized Vandermonde matrices and determinants in DETERMINANT AND RELATED DETERMINANTS ON electromagnetic compatibility VARIOUS SURFACES IN THREE DIMENSIONS

2.1.2 Extreme points of the Vandermonde determinant on the three-dimensional unit sphere

This section is based on Section 2.2 of Paper A

2v3(Rθx)=2 √3 cos(θ) + sin(θ) ∂h ∂h ∂h f(φ, θ)= + + =0, ∂x ∂x ∂x √ 1 x=x(φ,θ) 2 x=x(φ,θ) 3 x=x(φ,θ) 3 cos(θ) + sin(θ) + 2 sin(θ) 2 2 sin(θ)+√3 cos(θ) + sin(θ) where h is the previously given implicit equation for T , − 1 3 1 2 2 2 2 2 2 2 2 = 4 cos(θ) 3 cos(θ) = cos(3θ). h(x)= x + x + x + r r +4r2 x + x , √2 − √2 1 2 3 2 − 1 1 2 Note that the ﬁnal equality follows from cos(nθ)=Tn(cos(θ)) where Tn is then f(φ + π, θ)=0as well unless x1 = x2 =0. the nth Chebyshev polynomial of the ﬁrst kind. From formula (48) if follows Proof. Calculate and parametrise the partial derivatives of h: that P3(x)=T3(x) but for higher dimensions the relationship between the Chebyshev polynomials and P is not as simple. n ∂h Finding the maximum points for v3(x3) on this form is simple. The Van- =8r2r1(r2 + r1 cos(φ)) cos(φ) cos(θ), dermonde determinant will be maximal when 3θ =2nπ where n is some ∂x1 2π ∂h integer. This gives three local maxima corresponding to θ1 = 0, θ2 = 3 =8r2r1(r2 + r1 cos(φ)) cos(φ) sin(θ), 4π ∂x2 and θ3 = 3 . These points correspond to cyclic permutation of the coordinates of x = 1 ( 1, 0, 1). Analogously the minimas for v (x ) can be shown ∂h √2 − 3 3 =8r2r1(r2 + r1 cos(φ)) sin(φ). to be a transposition followed by cyclic permutation of the coordinates of x. ∂x3 Thus any permutation of the coordinates of x correspond to a local extreme This gives point just like it was stated on page 56.

f(φ, θ)=8r2r1(r2 + r1 cos(φ))(cos(φ)(cos(θ) + sin(θ)) + sin(φ)) 2.1.3 Optimisation of the Vandermonde determinant on the

three-dimensional torus unless cos(φ) = 0 or r2 + r1 cos(φ) = 0 the condition f(φ, θ) = 0 can be This section is based on page 627–630 in Paper B rewritten as sin(φ) cos(θ) + sin(θ)= . (25) There are two equivalent conditions that describe the three-dimensional −cos(φ) torus with radii r and r , T2 = S1(r ) S1(r ): 2 1 1 × 2 Substituting (25) into the explicit expression for f(φ + π, θ) gives 2 2 3 2 2 2 2 T = x R g(x)= r2 x + y + z r =0 , (22) ∈ − − 1 sin(φ) f(φ + π, θ)=8r2r1(r2 r1 cos(φ)) cos(φ) sin(φ) =0. − − −cos(φ) − 2 3 2 2 2 2 2 2 2 2 T = x R h(x)= x + y + z + r r +4r2(x + y )=0 . ∈ 2 − 1 If cos(θ) = 0 then sin(θ)= 1 which gives f(φ, θ)= 8r2r1 thus f(φ, θ) =0 (23) ± ± if cos(θ) = 0. It remains to see what happens when r +r cos(φ) = 0. From 2 1 The surface of the torus can also be parametrised as follows: the parametrization (24) it is clear that this corresponds to x1 = x2 =0 which means that v3 at any such point will be zero. x =(r2 + r1 cos(φ)) cos(θ)  y =(r2 + r1 cos(φ)) sin(θ) (24) Using the method of Lagrange multipliers and the constraint g(x) = 0 in  z = r1 sin(θ) (23) gives the following conditions on stationary points for the Vandermonde  60  61 2.1. EXTREME POINTS OF THE VANDERMONDE Generalized Vandermonde matrices and determinants in DETERMINANT AND RELATED DETERMINANTS ON electromagnetic compatibility VARIOUS SURFACES IN THREE DIMENSIONS which gives Lemma 2.1. For any x(φ, θ) such that

f(φ, θ)=8r2r1(r2 + r1 cos(φ))(cos(φ)(cos(θ) + sin(θ)) + sin(φ)) 2.1.3 Optimisation of the Vandermonde determinant on the three-dimensional torus unless cos(φ) = 0 or r2 + r1 cos(φ) = 0 the condition f(φ, θ) = 0 can be This section is based on page 627–630 in Paper B rewritten as sin(φ) cos(θ) + sin(θ)= . (25) There are two equivalent conditions that describe the three-dimensional −cos(φ) torus with radii r and r , T2 = S1(r ) S1(r ): 2 1 1 × 2 Substituting (25) into the explicit expression for f(φ + π, θ) gives 2 2 3 2 2 2 2 T = x R g(x)= r2 x + y + z r =0 , (22) ∈ − − 1 sin(φ) f(φ + π, θ)=8r2r1(r2 r1 cos(φ)) cos(φ) sin(φ) =0. − − −cos(φ) − 2 3 2 2 2 2 2 2 2 2 T = x R h(x)= x + y + z + r r +4r2(x + y )=0 . ∈ 2 − 1 If cos(θ) = 0 then sin(θ)= 1 which gives f(φ, θ)= 8r2r1 thus f(φ, θ) =0 (23) ± ± if cos(θ) = 0. It remains to see what happens when r +r cos(φ) = 0. From 2 1 The surface of the torus can also be parametrised as follows: the parametrization (24) it is clear that this corresponds to x1 = x2 =0 which means that v3 at any such point will be zero. x =(r2 + r1 cos(φ)) cos(θ)  y =(r2 + r1 cos(φ)) sin(θ) (24) Using the method of Lagrange multipliers and the constraint g(x) = 0 in  z = r1 sin(θ) (23) gives the following conditions on stationary points for the Vandermonde  60  61 2.1. EXTREME POINTS OF THE VANDERMONDE Generalized Vandermonde matrices and determinants in DETERMINANT AND RELATED DETERMINANTS ON electromagnetic compatibility VARIOUS SURFACES IN THREE DIMENSIONS determinant on the surface of the three-dimensional torus x , given by ±

r x2 + y2 ∂v3 ∂g 2 − = λ = λ x = λ 2 r1 cos(φ) cos(θ), 1 ∂x ∂x 2 2 x˜ r2 z˜ − x + y x = 1 cos arcsin , ± 2 2 2 ± r1 2r1 ∂v ∂g r2 x + y (26) 3 − 1 = λ = λ y = λ 2 r1 cos(φ) sin(θ), y˜ r z˜ − ∂y ∂y x2 + y2 y = 1 2 cos arcsin , ± 2 ± r1 2r1 ∂v3 ∂g = λ = λ 2 z = λ 2 r1 sin(φ), z˜ ∂z ∂z z = . ± 2 where g(x) is deﬁned by (22). From equation (26) it is clear that if the partial derivatives of v3 at a 2 2 particular point on T corresponds to the coordinates a point on S (2r1). Figures 2.5-2.7 illustrates this result for a regular, horn and spindle torus. It is also known (see Section 2.1.2) that the sum of all partial derivatives of v3 vanishes,

∂v3 ∂v3 ∂v3 3 + + =0, (x, y, z) R . (27) ∂x ∂y ∂z ∈

Thus all points where the condition given in (26) is satisfied for λ = 0 can 2 be found in the intersection between S (2r1) and the plane given by (27). A simple way of finding this intersection is to finds some points that belongs to the intersection and rotate the point around the planes normal vector. Using Rodrigues’ rotation formula to rotate a point, x, around the axis Figure 2.5: Plot of v3(x3) over a proper torus (r1 =1, r2 =3), 3D-plot with curve 1 (1, 1, 1) with the angle θ will give the rotation matrix marked (left), parametrised plot with curve marked (center), values of √3 v3(x(α)) along the curve (right).

2 cos(α)+1 1 cos(α) √3 sin(α)1 cos(α)+√3 sin(α) 1 − − − Rα = 1 cos(α)+√3 sin(α) 2 cos(α)+1 1 cos(α) √3 sin(α) . 3 − − −  1 cos(α) √3 sin(α)1 cos(α)+√3 sin(α) 2 cos(α)+1  − − −  A point which already lies on S2 can then be rotated to any other 2 π point on S by letting Rθ act on the point. Rotating the point x 0, 4 = r1+r2 (1, 1, 0) gives the new point: √2 −

3 cos(α)+√3 sin(α) π √2r1 x˜(α)=Rαx 0, = 3 cos(α)+√3 sin(α) . Figure 2.6: Plot of v3(x3) over a horn torus (r1 =1, r2 =1), 3D-plot with curve 4 3 −  2√3 sin(α) marked (left), parametrised plot with curve marked (center), values of v (x(α)) along the curve (right).   3 A rotated point,x ˜(α)=(˜x, y,˜ z˜), will correspond to two points on the torus,

62 63 2.1. EXTREME POINTS OF THE VANDERMONDE Generalized Vandermonde matrices and determinants in DETERMINANT AND RELATED DETERMINANTS ON electromagnetic compatibility VARIOUS SURFACES IN THREE DIMENSIONS determinant on the surface of the three-dimensional torus x , given by ± r x2 + y2 ∂v3 ∂g 2 − = λ = λ x = λ 2 r1 cos(φ) cos(θ), 1 ∂x ∂x 2 2 x˜ r2 z˜ − x + y x = 1 cos arcsin , ± 2 2 2 ± r1 2r1 ∂v ∂g r2 x + y (26) 3 − 1 = λ = λ y = λ 2 r1 cos(φ) sin(θ), y˜ r z˜ − ∂y ∂y x2 + y2 y = 1 2 cos arcsin , ± 2 ± r1 2r1 ∂v3 ∂g = λ = λ 2 z = λ 2 r1 sin(φ), z˜ ∂z ∂z z = . ± 2 where g(x) is deﬁned by (22). From equation (26) it is clear that if the partial derivatives of v3 at a 2 2 particular point on T corresponds to the coordinates a point on S (2r1). Figures 2.5-2.7 illustrates this result for a regular, horn and spindle torus. It is also known (see Section 2.1.2) that the sum of all partial derivatives of v3 vanishes,

∂v3 ∂v3 ∂v3 3 + + =0, (x, y, z) R . (27) ∂x ∂y ∂z ∈

Proof. Considering the expression for the Vandermonde determinant in The- n n(n 1) orem 1.2 the number of factor of v (x) is i 1= − . Thus n − 2 i=1 n(n 1) − vn(cx)=c 2 vn(x). (28)

Figure 2.7: Plot of v3(x3) over a spindle torus (r1 =3, r2 =1), 3D-plot with curve marked (left), parametrised plot with curve marked (center), values of Gröbner bases together with algorithms to find them, and algorithms v3(x(α)) along the curve (right). for solving a polynomial equation is an important tool that arises in many applications. One such application is the optimization of polynomials over The explicit expression for the curves are: affine varieties through the method of Lagrange multipliers. We will here give some main points and informal discussion on these methods as an in- √2 cos(α) troduction and to describe some notation. v3 (x(α)) = ± 2(1 + 2 cos(α)2) 3 + 6 cos(α)2 Definition 2.1. ( [30]) Let f1, ,fm be polynomials in R[x1, ,xn]. The ··· ··· 2 3 2 3 2 affine variety V (f1, ,fm) defined by f1, ,fm is the set of all points 3 3 + 6 cos(α) r1 2 3 + 6 cos(α) r1 cos(α) n ··· ··· − − (x1, ,xn) R such that fi(x1, ,xn) = 0 for all 1 i m. ··· ∈ ··· ≤ ≤ 2 3 4 2 2 2 +8 3 + 6 cos(α) r cos(α) 15r r2 6 cos(α) r r2 When n = 3 we will sometimes use the variables x, y, z instead of 1 ∓ 1 ∓ 1 4 2 2 2 x1,x2,x3. Affine varieties are this way the common zeros of a set of multi- 48 cos(α) r1r2 7 3 + 6 cos(α) r1r2 ± − variate polynomials. Such sets of polynomials will generate a greater set of + 16 cos(α)2r2 3 + 6 cos(α)2)r 12 cos(α)2r3 3r3 . polynomials [30] by 2 1 ± 2 ∓ 2 m 1 Note that as r2 = 0 the expression simplifies to v3 (x(α)) = √ cos(3α) f1, ,fm hifi : h1, ,hm R[x1, ,xn] , ± 2 ··· ≡ ··· ∈ ··· which is the same expression that you would get if the same method was i=1 used on the regular unit sphere. and this larger set will define the same variety. But it will also define an ideal (a set of polynomials that contains the zero-polynomial and is closed 2.1.4 Optimisation using Gröbner bases under addition, and absorbs multiplication by any other polynomial) by I(f , ,f )= f , ,f . A Gröbner basis for this ideal is then a finite This section is based on Section 4 of Paper C 1 ··· m 1 ··· m set of polynomials g , ,g such that the ideal generated by the leading { 1 ··· k} terms of the polynomials g , ,g is the same ideal as that generated by In this section we will find the extreme points of the Vandermonde de- 1 ··· k all the leading terms of polynomials in I = f , ,f . terminant on a few different surfaces. This will be done using Lagrange 1 ··· m multipliers and Gröbner bases but first we will make an observation about In this paper we consider the optimization of the Vandermonde deter- the Vandermonde determinant that will be useful later. minant vn(x) over surfaces defined by a polynomial equation on the form n Lemma 2.2. The Vandermonde determinant is a homogeneous polynomial p n(n 1) sn(x1, ,xn ; p; a1, ,an) ai xi =1, (29) of degree 2− . ··· ··· ≡ | | i=1 where we will select the constants ai and p to get ellipsoids in three dimensions, cylinders in three dimensions, and spheres under the p-norm in

Proof. Considering the expression for the Vandermonde determinant in The- n n(n 1) orem 1.2 the number of factor of v (x) is i 1= − . Thus n − 2 i=1 n(n 1) − vn(cx)=c 2 vn(x). (28)

64 65 2.1. EXTREME POINTS OF THE VANDERMONDE Generalized Vandermonde matrices and determinants in DETERMINANT AND RELATED DETERMINANTS ON electromagnetic compatibility VARIOUS SURFACES IN THREE DIMENSIONS n dimensions. The cases of the ellipsoids and the cylinders are suitable for give the following three basis polynomials: solution by Gröbner basis methods, but due to the existing symmetries for 2 the spheres other methods are more suitable, as provided in Section 2.2.4. g1(z) =(a + b)(a b) − From (28) and the convexity of the interior of the sets defined by (29), 4(a + b)2(a + c)(b + c)+3c2(a2 + ab + b2)+3c(a3 + b3) z2 − under a suitable choice of the constant p and non-negative ai, it is easy 2 2 2 4 n +3 c(a + b + c) 4(a + b)(a + c)(b + c)+(a + b )c +(a + b)c z to see that the optimal value of v on a x p 1 will be attained on c2(b + c)(a + c)(a + b + c)2z6, (31) n i| i| ≤ − i=1 2 2 2 2 n p g2(y, z)= 2(a + b) (a + c)(b + c)+c(a +2b )(a + b + c)+2bc (a + b) z ai xi = 1. And so, by the method of Lagrange multipliers we have i=1 | | 5 3 that the minimal/maximal values of vn(x1, ,xn) on sn(x1, ,xn) 1 + q1z q2z b(a b)(a + b)(a + b +3c)y, (32) ··· ··· ≤ − − − will be attained at points such that ∂vn/∂xi λ∂sn/∂xi = 0 for 1 i n g (x, z)= 2(a + b)2(a + c)(b + c)+c(2a2 + b2)(a + b + c)+2ac2(a + b) z − ≤ ≤ 3 and some constant λ and sn(x1, ,xn) 1 = 0, [144]. 5 3 ··· − q1z + q2z a(a b)(a + b)(a + b +3c)x, (33) − − − For p = 2 the resulting set of equations will form a set of polynomials in 2 2 q1 =9c (b + c)(a + c)(a + b + c) , λ, x1, ,xn. These polynomials will define an ideal over R[λ, x1, ,xn], ··· ··· 2 2 2 2 2 2 and by finding a Gröbner basis for this ideal we can use the especially nice q2 =3c(a + b + c)(3a b +4a c +3ab +6abc +4ac +4b c +4bc ). properties of Gröbner bases to find analytical solutions to these problems, that is, to find roots for the polynomials in the computed basis. This basis was calculated using software for symbolic computation [1]. Since g1 only depends on z and g2 and g3 are first degree polynomial in y and x respectively the stationary points can be found by finding the roots of g and then calculate the corresponding x and y coordinates. A general 2.1.5 Extreme points on the ellipsoid in three dimensions 1 formula can be found in this case (since g1 only contains even powers of z it can be treated as a third degree polynomial) but it is quite cumbersome This section is based on Section 5 of Paper C and we will therefore not give it explicitly.

Lemma 2.3. The extreme points of v3 on an ellipsoid will have real coor- In this section we will find the extreme points of the Vandermonde determi- dinates. nant on the three dimensional ellipsoid given by Proof. The discriminant is a useful tool for determining how many real roots ax2 + by2 + cz2 = 1 (30) low-level polynomials have. Following Irving [75] the discriminant, ∆(p), of 2 3 a third degree polynomial p(x)=c0 + c1x + c2x + c3x is where a>0, b>0, c>0. ∆ = 18c c c c 4c3c + c2c2 4c c3 27c2c2 1 2 3 4 − 2 4 2 3 − 1 3 − 1 4 Using the method of Lagrange multipliers together with (30) and some rewriting gives that all stationary points of the Vandermonde determinant and if p(x) is non-negative then all roots will be real (but not necessarily lie in the variety distinct). Since the first basis polynomial g1 only contains terms with even 2 exponents and is of degree 6 the polynomialg ˜1 defined byg ˜1(z )=g1(z) will be a polynomial of degree 3 whose roots are the square roots of g1. V = V ax2 + by2 + cz2 1, ax + by + cz, − Calculating the discriminant ofg ˜1 gives ax(z x)(y x) by(z y)(y x)+cz(z y)(z x) . − − − − − − − ∆(˜g ) = 9(a b)2(a + b +3c)2(a + b + c)4abc3 1 − 3 2 3 2 2 3 2 3 3 2 2 3 2 Computing a Gröbner basis for V using the lexicographic order x>y>z 32(a b + a c + a b + a c + b c + b c )+61abc(a + b + c) . 66 67 2.1. EXTREME POINTS OF THE VANDERMONDE Generalized Vandermonde matrices and determinants in DETERMINANT AND RELATED DETERMINANTS ON electromagnetic compatibility VARIOUS SURFACES IN THREE DIMENSIONS n dimensions. The cases of the ellipsoids and the cylinders are suitable for give the following three basis polynomials: solution by Gröbner basis methods, but due to the existing symmetries for 2 the spheres other methods are more suitable, as provided in Section 2.2.4. g1(z) =(a + b)(a b) − From (28) and the convexity of the interior of the sets defined by (29), 4(a + b)2(a + c)(b + c)+3c2(a2 + ab + b2)+3c(a3 + b3) z2 − under a suitable choice of the constant p and non-negative ai, it is easy 2 2 2 4 n +3 c(a + b + c) 4(a + b)(a + c)(b + c)+(a + b )c +(a + b)c z to see that the optimal value of v on a x p 1 will be attained on c2(b + c)(a + c)(a + b + c)2z6, (31) n i| i| ≤ − i=1 2 2 2 2 n p g2(y, z)= 2(a + b) (a + c)(b + c)+c(a +2b )(a + b + c)+2bc (a + b) z ai xi = 1. And so, by the method of Lagrange multipliers we have i=1 | | 5 3 that the minimal/maximal values of vn(x1, ,xn) on sn(x1, ,xn) 1 + q1z q2z b(a b)(a + b)(a + b +3c)y, (32) ··· ··· ≤ − − − will be attained at points such that ∂vn/∂xi λ∂sn/∂xi = 0 for 1 i n g (x, z)= 2(a + b)2(a + c)(b + c)+c(2a2 + b2)(a + b + c)+2ac2(a + b) z − ≤ ≤ 3 and some constant λ and sn(x1, ,xn) 1 = 0, [144]. 5 3 ··· − q1z + q2z a(a b)(a + b)(a + b +3c)x, (33) − − − For p = 2 the resulting set of equations will form a set of polynomials in 2 2 q1 =9c (b + c)(a + c)(a + b + c) , λ, x1, ,xn. These polynomials will define an ideal over R[λ, x1, ,xn], ··· ··· 2 2 2 2 2 2 and by finding a Gröbner basis for this ideal we can use the especially nice q2 =3c(a + b + c)(3a b +4a c +3ab +6abc +4ac +4b c +4bc ). properties of Gröbner bases to find analytical solutions to these problems, that is, to find roots for the polynomials in the computed basis. This basis was calculated using software for symbolic computation [1]. Since g1 only depends on z and g2 and g3 are first degree polynomial in y and x respectively the stationary points can be found by finding the roots of g and then calculate the corresponding x and y coordinates. A general 2.1.5 Extreme points on the ellipsoid in three dimensions 1 formula can be found in this case (since g1 only contains even powers of z it can be treated as a third degree polynomial) but it is quite cumbersome This section is based on Section 5 of Paper C and we will therefore not give it explicitly.

Using the method of Lagrange multipliers gives the equation system ∂v 3 =0, ∂x ∂v 3 =2λby, ∂y ∂v 3 =2λcz. ∂z Taking the sum of each expression gives c by + cz =0 y = z. (35) ⇔ −b x2 y2 Figure 2.8: Illustration of the ellipsoid defined by + + z2 =0with the extreme Combining (34) and (35) gives 9 4 points of the Vandermonde determinant marked. Displayed in Cartesian c b 1 c 1 coordinates on the right and in ellipsoidal coordinates on the left. +1 cz2 =1 z = y = . b ⇒ ± c √b + c ⇒ ∓ b √b + c Thus the plane defined by (35) intersects with the cylinder along the Since a, b and c are all positive numbers it is clear that ∆(g1) is non- lines negative. Furthermore, since a, b and c are positive numbers all terms ing ˜ 1 c 1 b 1 with odd powers have negative coefficients and all terms with even powers 1 = x, , x R = (x, r, s) x R , b √b + c − c √b + c ∈ { − | ∈ } have positive coefficients. Thus if w<0 theng ˜ (w) > 0 and thus all roots 1 must be positive. c 1 b 1 2 = x, , x R = (x, r, s) x R . − b √ c √ ∈ { − | ∈ } b + c b + c Finding the stationary points for v3 along 1: An illustration of an ellipsoid and the extreme points of the Vandermonde 1 b c 1 determinant on its surface is shown in Figure 2.8. v (x, r, s)= x2 + x + (r + s) , 3 − √ c − b b + c b + c ∂v 1 b c 3 (x, r, s)= 2x + (r + s) . 2.1.6 Extreme points on the cylinder in three dimensions ∂x − √ c − b b + c From this it follows that This section is based on Section 6 of Paper C ∂v 1 c b 3 (x, r, s)=0 x = . ∂x − ⇔ √ b − c 2 b + c In this section we will examine the local extreme points on an infinitely long Thus cylinder aligned with the x-axis in 3 dimensions. In this case we do not need 1 1 c b c b to use Gröbner basis techniques since the problem can be reduced to a one x1 = , , (36) √b + c 2 b − c b − c dimensional polynomial equation. is the only stationary point on . An analogous argument shows that x = The cylinder is defined by 1 2 x is the only stationary point on . − 1 2 An illustration of where these points are placed on the cylinder is shown by2 + cz2 =1, where b>0, c>0. (34) in Figure 2.9.

68 69 2.1. EXTREME POINTS OF THE VANDERMONDE Generalized Vandermonde matrices and determinants in DETERMINANT AND RELATED DETERMINANTS ON electromagnetic compatibility VARIOUS SURFACES IN THREE DIMENSIONS

The stationary points on the surface given by g(cx)=ck will be given by n(n 1) − ∂vn k ∂g c 2 = c λ ,k 1, 2,...,n ∂x ∂x ∈ k x=y k x=y n(1 n) − k and if c is chosen such that λ = c 2 c then the stationary points are defined by ∂v ∂g n = ,k 1, 2,...,n. ∂xk ∂xk ∈ n k Suppose that y x R g(x)=c is a stationary point for vn then 16 ∈{ ∈ | } 1 Figure 2.9: Illustration of the cylinder defined by y2 + z2 =1with the extreme the point given by z = cy where c = g(y)− k will be a stationary point 25 points of the Vandermonde determinant marked. Displayed in Cartesian for the Vandermonde determinant and will lie on the surface defined by coordinates on the right and in cylindrical coordinates on the left. g(x)=1. Lemma 2.5. If z is a stationary point for the Vandermonde determinant 2.1.7 Optimizing the Vandermonde determinant on a surface on the surface g(x)=1where g(x) is a homogeneous polynomial then z is − defined by a homogeneous polynomial either a stationary point or does not lie on the surface. Proof. Since g( x)=( 1)kg(x) is either 1 or 1 then v (x) = v ( x) This section is based on Section 7 of Paper C − − − | n | | n − | for any point, including z and the points in a neighbourhood around it which means that if g( x)=g(x) then the stationary points are preserved and When using Lagrange multipliers it can be desirable to not have to consider − otherwise the point will lie on the surface defined by g(x)= 1 instead of the λ-parameter (the scaling between the gradient and direction given by − the constraint). We demonstrate a simple way to remove this parameter g(x)=1. when the surface is defined by an homogeneous polynomial. A well-known example of homogeneous polynomials are quadratic forms. If we let Lemma 2.4. Let g : R R be a homogeneous polynomial such that k n→(n 1) n g(x)=xaSx g(cx)=c g(x) with k = − . If g(x)=1, x C defines a continu- 2 ∈ ous bounded surface then any point on the surface that is a stationary point then g(x) is a quadratic form which in turn is a homogeneous polynomial for the Vandermonde determinant, z Cn, can be written as z = cy where with k = 2. If S is a positive definite matrix then g(x) = 1 defines an ∈ ellipsoid. Here we will demonstrate the use of Lemma 2.4 to find the extreme ∂v ∂g n = ,i 1, 2,...,n (37) points on a rotated ellipsoid. ∂x ∂x ∈ i x=y i x=y Consider the ellipsoid defined by 1 2 5 2 3 5 2 1 x + y + yz + z = 1 (38) and c = g(y)− k . 9 8 4 8 then by Lemma 2 we can instead consider the points in the variety Proof. By the method of Lagrange multipliers the point y x Rn g(x)= ∈{ ∈ | 2 1 is a stationary point for the Vandermonde determinant if V = V 2xy +2xz + y2 z2 x, } − − − 9 ∂vn ∂g 2 2 5 3 = λ ,k 1, 2,...,n x +2xy 2yz + z y z, ∂x ∂x ∈ − − − 4 − 4 k x=y k x=y 2 2 3 5 2xz y +2yz + x y z . for some λ R. − − − 4 − 4 ∈ 70 71 2.1. EXTREME POINTS OF THE VANDERMONDE Generalized Vandermonde matrices and determinants in DETERMINANT AND RELATED DETERMINANTS ON electromagnetic compatibility VARIOUS SURFACES IN THREE DIMENSIONS

The stationary points on the surface given by g(cx)=ck will be given by n(n 1) − ∂vn k ∂g c 2 = c λ ,k 1, 2,...,n ∂x ∂x ∈ k x=y k x=y n(1 n) − k and if c is chosen such that λ = c 2 c then the stationary points are defined by ∂v ∂g n = ,k 1, 2,...,n. ∂xk ∂xk ∈ n k Suppose that y x R g(x)=c is a stationary point for vn then 16 ∈{ ∈ | } 1 Figure 2.9: Illustration of the cylinder defined by y2 + z2 =1with the extreme the point given by z = cy where c = g(y)− k will be a stationary point 25 points of the Vandermonde determinant marked. Displayed in Cartesian for the Vandermonde determinant and will lie on the surface defined by coordinates on the right and in cylindrical coordinates on the left. g(x)=1. Lemma 2.5. If z is a stationary point for the Vandermonde determinant 2.1.7 Optimizing the Vandermonde determinant on a surface on the surface g(x)=1where g(x) is a homogeneous polynomial then z is − defined by a homogeneous polynomial either a stationary point or does not lie on the surface. Proof. Since g( x)=( 1)kg(x) is either 1 or 1 then v (x) = v ( x) This section is based on Section 7 of Paper C − − − | n | | n − | for any point, including z and the points in a neighbourhood around it which means that if g( x)=g(x) then the stationary points are preserved and When using Lagrange multipliers it can be desirable to not have to consider − otherwise the point will lie on the surface defined by g(x)= 1 instead of the λ-parameter (the scaling between the gradient and direction given by − the constraint). We demonstrate a simple way to remove this parameter g(x)=1. when the surface is defined by an homogeneous polynomial. A well-known example of homogeneous polynomials are quadratic forms. If we let Lemma 2.4. Let g : R R be a homogeneous polynomial such that k n→(n 1) n g(x)=xaSx g(cx)=c g(x) with k = − . If g(x)=1, x C defines a continu- 2 ∈ ous bounded surface then any point on the surface that is a stationary point then g(x) is a quadratic form which in turn is a homogeneous polynomial for the Vandermonde determinant, z Cn, can be written as z = cy where with k = 2. If S is a positive definite matrix then g(x) = 1 defines an ∈ ellipsoid. Here we will demonstrate the use of Lemma 2.4 to find the extreme ∂v ∂g n = ,i 1, 2,...,n (37) points on a rotated ellipsoid. ∂x ∂x ∈ i x=y i x=y Consider the ellipsoid defined by 1 2 5 2 3 5 2 1 x + y + yz + z = 1 (38) and c = g(y)− k . 9 8 4 8 then by Lemma 2 we can instead consider the points in the variety Proof. By the method of Lagrange multipliers the point y x Rn g(x)= ∈{ ∈ | 2 1 is a stationary point for the Vandermonde determinant if V = V 2xy +2xz + y2 z2 x, } − − − 9 ∂vn ∂g 2 2 5 3 = λ ,k 1, 2,...,n x +2xy 2yz + z y z, ∂x ∂x ∈ − − − 4 − 4 k x=y k x=y 2 2 3 5 2xz y +2yz + x y z . for some λ R. − − − 4 − 4 ∈ 70 71 Generalized Vandermonde matrices and determinants in 2.2. OPTIMIZATION OF THE VANDERMONDE electromagnetic compatibility DETERMINANT ON SOME N -DIMENSIONAL SURFACES

Finding the Gr¨obner basis of V gives

g (z)=z(6z + 1)(260642z2 27436z + 697), 1 − g (y, z)= 1138484256z3 127275604z2 + 16689841z + 6277879y, 2 − − g (x, z) = 10246358304z3 + 1145480436z2 93707658z + 6277879x. 3 − This system is not diﬃcult to solve and the resulting points are:

p0 = (0, 0, 0), 1 1 p = 0, , , 1 6 −6 Figure 2.10: Illustration of the ellipsoid deﬁned by (38) with the extreme points of the 45√2 1 5√2 1 5√2 Vandermonde determinant marked. Displayed in Cartesian coordinates p2 = , , , on the right and in ellipsoidal coordinates on the left. 361 −19 − 722 19 − 722 45√2 1 5√2 1 5√2 p3 = , + , + . 2.2.1 The extreme points on the sphere given by roots of a 361 −19 722 19 722 polynomial

The point p0 is an artifact of the rewrite and does not lie on any ellipsoid This section is based on Section 2.1 of Paper A and can therefore be discarded. By Lemma 4 there are also three more stationary points p4 = p1, p5 = p2 and p6 = p3. Rescaling each of The extreme points of the Vandermonde determinant on the unit sphere in − − − n these points according to Lemma 2 gives qi = g(pi) which are all points R are known and given by Theorem 2.3 where we present a special case of on the ellipsoid defined by g(x) = 1. The result is illustrated in Figure 2.10. Theorem 6.7.3 in [158]. We will also provide a proof that is more explicit Note that this example gives a simple case with a Gröbner basis that is than the one in [158] and that exposes more of the rich symmetric prop- small and easy to find. Using this technique for other polynomials and in erties of the Vandermonde determinant. For the sake of convenience some higher dimensions can require significant computational resources. properties related to the extreme points of the Vandermonde determinant defined by real vectors xn will be presented before Theorem 2.3.

2.2 Optimization of the Vandermonde Theorem 2.1. For any 1 k n determinant on some n-dimensional surfaces ≤ ≤ n ∂v v (x ) n = n n (39) In this section we will consider the extreme points of the Vandermonde ∂xk xk xi n i=1 − determinant on the n-dimensional unit sphere in R . We want both to ﬁnd i =k an analytical solution and to identify some properties of the determinant that can help us to visualize it in some area around the extreme points in This theorem will be proven after the introduction of the following useful dimensions n>3. lemma:

Finding the Gröbner basis of V gives g (z)=z(6z + 1)(260642z2 27436z + 697), 1 − g (y, z)= 1138484256z3 127275604z2 + 16689841z + 6277879y, 2 − − g (x, z) = 10246358304z3 + 1145480436z2 93707658z + 6277879x. 3 − This system is not difficult to solve and the resulting points are: p0 = (0, 0, 0), 1 1 p = 0, , , 1 6 −6 Figure 2.10: Illustration of the ellipsoid defined by (38) with the extreme points of the 45√2 1 5√2 1 5√2 Vandermonde determinant marked. Displayed in Cartesian coordinates p2 = , , , on the right and in ellipsoidal coordinates on the left. 361 −19 − 722 19 − 722 45√2 1 5√2 1 5√2 p3 = , + , + . 2.2.1 The extreme points on the sphere given by roots of a 361 −19 722 19 722 polynomial

Lemma 2.6. For any 1 k n 1 ≤ ≤ − n 1 ∂vn vn(xn) − ∂vn 1 = + (xn xi) − (40) ∂xk −xn xk − ∂xk − i=1 72 73 Generalized Vandermonde matrices and determinants in 2.2. OPTIMIZATION OF THE VANDERMONDE electromagnetic compatibility DETERMINANT ON SOME N -DIMENSIONAL SURFACES and Supposing that formula (39) is true for n 1 results in − n 1 ∂vn − vn(xn) n 1 n 1 = . (41) ∂vn vn(xn) − − vn 1(xn 1) ∂x x x = + (xn xi) − − n i=1 n i ∂x − x x − x x − k n k i=1 i=1 k i − i =k − n 1 n Proof. Note that the determinant can be described recursively v (x ) − v (x ) v (x ) = n n + n n = n n . x x x x x x k − n i=1 k − i i=1 k − i n 1 i =k i =k − v (x )= (x x ) (x x ) n n n − i j − i Showing that (39) is true for n = 2 completes the proof i=1 1 i

74 75 Generalized Vandermonde matrices and determinants in 2.2. OPTIMIZATION OF THE VANDERMONDE electromagnetic compatibility DETERMINANT ON SOME N -DIMENSIONAL SURFACES and Supposing that formula (39) is true for n 1 results in − n 1 ∂vn − vn(xn) n 1 n 1 = . (41) ∂vn vn(xn) − − vn 1(xn 1) ∂x x x = + (xn xi) − − n i=1 n i ∂x − x x − x x − k n k i=1 i=1 k i − i =k − n 1 n Proof. Note that the determinant can be described recursively v (x ) − v (x ) v (x ) = n n + n n = n n . x x x x x x k − n i=1 k − i i=1 k − i n 1 i =k i =k − v (x )= (x x ) (x x ) n n n − i j − i Showing that (39) is true for n = 2 completes the proof i=1 1 i

74 75 Generalized Vandermonde matrices and determinants in 2.2. OPTIMIZATION OF THE VANDERMONDE electromagnetic compatibility DETERMINANT ON SOME N -DIMENSIONAL SURFACES

n Remark 2.1. Note that if xn =(x1,x2,... xn) is an extreme point of the Λ (x ,λ)=v(x ) λ x2 1 Vandermonde determinant then any other point whose coordinates are a n n n − i − i=1 permutation of the coordinates of xn is also an extreme point. This follows from the determinant function being, by definition, alternating with respect for some λ. Explicitly this requirement becomes to the columns of the matrix and the xis defines the columns of the Vander- monde matrix. Thus any permutation of the xis will give the same value for ∂Λn v (x ) . Since there are n! permutations there will be at least n! extreme = 0 for all 1 k n, (45) | n n | ∂xk ≤ ≤ points. The roots of the polynomial (48) defines the set of xis fully and thus ∂Λ n =0. (46) there are exactly n! extreme points, n!/2 positive and n!/2 negative. ∂λ

n n n Proof of Theorem 2.3. By the method of Lagrange multipliers condition (45) ∂Λn ∂vn = 2λxk = 2λ xk =0. (47) must be satisﬁed for any extreme point. If xn is a ﬁxed extreme point so ∂xk ∂xk − − k=1 k=1 k=1 that n vn(xn)=vmax, There are two ways to satisfy condition (47) either λ = 0 or xk = 0. k=1 then (45) can be written explicitly, using (39), as When λ = 0 equation (45) reduces to n ∂Λn vmax ∂v = 2λx = 0 for all 1 k n, n ∂x x x − k ≤ ≤ = 0 for all 1 k n, k i=1 k i ∂xk ≤ ≤ i =k − and by equation (19) this can only be true if vn(xn) = 0, which is of no n or alternatively by introducing a new multiplier ρ as interest to us, and so all extreme points must lie in the hyperplane xk = n k=1 1 2λ ρ = x = x for all 1 k n. (49) 0. x x v k n k ≤ ≤ i=1 k i max i =k −

76 77 Generalized Vandermonde matrices and determinants in 2.2. OPTIMIZATION OF THE VANDERMONDE electromagnetic compatibility DETERMINANT ON SOME N -DIMENSIONAL SURFACES

n Thus if equation (44) is true for n 1 it is also true for n. Showing that the Theorem 2.3. A point on the unit sphere in R , xn =(x1,x2,... xn), − equation holds for n = 2 is very simple is an extreme point of the Vandermonde determinant if and only if all xi, i 1, 2,... n , are distinct roots of the rescaled Hermite polynomial ∂v ∂v ∈{ } 2 + 2 = 1+1=0. ∂x ∂x − 1 2 n n(n 1) Pn(x) = (2n(n 1))− 2 Hn − x . (48) Proof of Theorem 2.2. Using the method of Lagrange multipliers it follows − 2 that any xn on the unit sphere that is an extreme point of the Vandermonde determinant will also be a stationary point for the Lagrange function n Remark 2.1. Note that if xn =(x1,x2,... xn) is an extreme point of the Λ (x ,λ)=v(x ) λ x2 1 Vandermonde determinant then any other point whose coordinates are a n n n − i − i=1 permutation of the coordinates of xn is also an extreme point. This follows from the determinant function being, by definition, alternating with respect for some λ. Explicitly this requirement becomes to the columns of the matrix and the xis defines the columns of the Vander- monde matrix. Thus any permutation of the xis will give the same value for ∂Λn v (x ) . Since there are n! permutations there will be at least n! extreme = 0 for all 1 k n, (45) | n n | ∂xk ≤ ≤ points. The roots of the polynomial (48) defines the set of xis fully and thus ∂Λ n =0. (46) there are exactly n! extreme points, n!/2 positive and n!/2 negative. ∂λ

Equation (46) corresponds to the restriction to the unit sphere and is there- Remark 2.2. All terms in Pn(x) are of even order if n is even and of odd fore immediately satisfied. Since all the partial derivatives of the Lagrange order when n is odd. This means that the roots of Pn(x) will be symmetrical function should be equal to zero it is obvious that the sum of the partial in the sense that if xi is a root then xi is also a root. derivatives will also be equal to zero. Combining this with Lemma 2.7 gives − n n n Proof of Theorem 2.3. By the method of Lagrange multipliers condition (45) ∂Λn ∂vn = 2λxk = 2λ xk =0. (47) must be satisfied for any extreme point. If xn is a fixed extreme point so ∂xk ∂xk − − k=1 k=1 k=1 that n vn(xn)=vmax, There are two ways to satisfy condition (47) either λ = 0 or xk = 0. k=1 then (45) can be written explicitly, using (39), as When λ = 0 equation (45) reduces to n ∂Λn vmax ∂v = 2λx = 0 for all 1 k n, n ∂x x x − k ≤ ≤ = 0 for all 1 k n, k i=1 k i ∂xk ≤ ≤ i =k − and by equation (19) this can only be true if vn(xn) = 0, which is of no n or alternatively by introducing a new multiplier ρ as interest to us, and so all extreme points must lie in the hyperplane xk = n k=1 1 2λ ρ = x = x for all 1 k n. (49) 0. x x v k n k ≤ ≤ i=1 k i max i =k −

76 77 Generalized Vandermonde matrices and determinants in 2.2. OPTIMIZATION OF THE VANDERMONDE electromagnetic compatibility DETERMINANT ON SOME N -DIMENSIONAL SURFACES

n By forming the polynomial f(x)=(x x1)(x x2) (x xn) and noting By setting g(z)=f(az) and choosing a = a differential equation that − − ··· − ρ that matches the definition for the Hermite polynomials is found: n n n f (x )= (x x ) = (x x ), g (z) 2zg (z)+2ng(z)=0. (52) k − i k − i − j=1 i=1 x=xk i=1 i=j i=k By definition the solution to (52) is g(z)=bH (z) where b is a constant. n n n n n n n n An exact expression for the constant a can be found using Lemma 2.8 (for f (x )= (x x ) = (x x )+ (x x ) k − i k − i k − i the sake of convenience the lemma is stated and proved after this theorem). l=1 j=1 i=1 x=xk j=1 i=1 l=1 i=1 j=l i=j j=k i=j l=k i=l We get i=l i= k i= k n n 2 2 2 2 n(n 1) n n x = a z =1 a − =1, i i ⇒ 2 =2 (x x ), i=1 i=1 k − i j=1 i=1 and so j=k i=j i= k 2 a = . n(n 1) we can rewrite (49) as − Thus condition (45) is satisfied when xi are the roots of 1 f (xk) ρ = xk, 2 f (xk) n n(n 1) Pn(x)=bHn (z)=bHn − x . or 2

2ρ n f (xk) xkf (xk)=0. Choosing b = (2n(n 1))− 2 gives P (x) with leading coefficient 1. This can − n − n be confirmed by calculating the leading coefficient of P (x) using the explicit And since the last equation must vanish for all k we must have expression for the Hermite polynomial (54). This completes the proof. 2ρ f (x) xf (x)=cf(x), (50) Lemma 2.8. Let x , i =1, 2,...,n be roots of the Hermite polynomial − n i Hn(x). Then n n for some constant c. To find c the x -terms of the right and left part of n(n 1) x2 = − . equation (50) are compared to each other, i 2 i=1 n 2ρ n 1 n c cnx = xncnx − = 2ρ cnx c = 2ρ. Proof. By letting e (x ,... x ) denote the elementary symmetric polynomi- · − n − · ⇒ − k 1 n als Hn(x) can be written as Thus the following differential equation for f(x) must be satisfied Hn(x)=An(x x1) (x xn) 2ρ − ··· − n n 1 n 2 f (x) xf (x)+2ρf(x)=0. (51) = A (x e (x ,...,x )x − + e (x ,...,x )x − + q(x)) − n n − 1 1 n 2 1 n Choosing x = az gives where q(x) is a polynomial of degree n 3. Noting that − n 2ρ 2 f (az) a zf (az)+2ρf(az) x2 =(x + ...+ x )2 2 x x − (n 1) i 1 n − i j 2 − i=1 1 i

78 79 Generalized Vandermonde matrices and determinants in 2.2. OPTIMIZATION OF THE VANDERMONDE electromagnetic compatibility DETERMINANT ON SOME N -DIMENSIONAL SURFACES

78 79 Generalized Vandermonde matrices and determinants in 2.2. OPTIMIZATION OF THE VANDERMONDE electromagnetic compatibility DETERMINANT ON SOME N -DIMENSIONAL SURFACES it is clear the the sum of the square of the roots can be described using the From this follows coeﬃcients for xn, xn 1 and xn 2. The explicit expression for H (x) is [158] 1 1 − − n = − . 2ρ n2(n 1) n − 2 i n 2i ( 1) (2x) − n l H (x)=n! − Comparing the x − terms in (56) gives the following relation n i! (n 2i)! i=0 − n 1 1 2 i n 2i an l = (n l + 2)(n l)an l+2 +(n l)aa l n n n 2 n 2 ( 1) (2x) − − 2ρ − − − − − n =2 x 2 − n(n 1)x − + n! − . (54) − − i! (n 2i)! i=3 − which is equivalent to Comparing the coeﬃcients in the two expressions for Hn(x) gives (n l + 2)(n l + 1) n an l = an l+2 − − − . An =2 , − − ln2(n 1) − Ane1(x1,...,xn)=0, n 2 Letting k = n l gives (55). A e (x ,...,x )= n(n 1)2 − . n 2 1 n − − − Thus by (53) n 2.2.2 Further visual exploration on the sphere n(n 1) x2 = − . i 2 This section is based on Section 2.4 of Paper A i=1

Visualization of the determinant v3(x3) on the unit sphere is straightforward, Theorem 2.4. The coefficients, a , for the term xk in P (x) given by (48) k n as well as visualizations for g3(x3, a) for different a. In three dimensions all are given by the following relations points on the sphere can be mapped to the plane. In higher dimensions 1 we need to reduce the set of visualized points somehow. In this section an =1,an 1 =0,an 2 = , we provide visualizations for v ,...,v by using symmetry properties of the − − 2 4 7 (k + 1)(k + 2) Vandermonde determinant. a = a , 1 k n 3. (55) k −n(n 1)(n k) k+2 ≤ ≤ − − − Four dimensions Proof. equation (51) tells us that 1 1 By Theorem 2.2 we know that the extreme points of v4(x4) on the sphere Pn(x)= Pn(x) xPn (x). (56) all lie in the hyperplane x + x + x + x = 0. The intersection of this 2ρ − n 1 2 3 4 hyperplane with the unit sphere in R4 can be described as a unit sphere in 3 That an = 1 follows from the definition of Pn and an 1 = 0 follows from the R , under a suitable basis, and can then be easily visualized. − Hermite polynomials only having terms of odd powers when n is odd and This can be realized using the transformation 1 even powers when n is even. That an 2 = can be easily shown using the − 2 definition of Pn and the explicit formula for the Hermite polynomials (54). 1 10 n 2 − − 1/√40 0 The value of the ρ can be found by comparing the x − terms in (56) 11 0 x = −  01/√20t. (57) 10 1   1 1 − 0 01/√2 an 2 = n(n 1)an + (n 2)an 2.  101 − 2ρ − n − −       80 81 Generalized Vandermonde matrices and determinants in 2.2. OPTIMIZATION OF THE VANDERMONDE electromagnetic compatibility DETERMINANT ON SOME N -DIMENSIONAL SURFACES it is clear the the sum of the square of the roots can be described using the From this follows coefficients for xn, xn 1 and xn 2. The explicit expression for H (x) is [158] 1 1 − − n = − . 2ρ n2(n 1) n − 2 i n 2i ( 1) (2x) − n l H (x)=n! − Comparing the x − terms in (56) gives the following relation n i! (n 2i)! i=0 − n 1 1 2 i n 2i an l = (n l + 2)(n l)an l+2 +(n l)aa l n n n 2 n 2 ( 1) (2x) − − 2ρ − − − − − n =2 x 2 − n(n 1)x − + n! − . (54) − − i! (n 2i)! i=3 − which is equivalent to Comparing the coefficients in the two expressions for Hn(x) gives (n l + 2)(n l + 1) n an l = an l+2 − − − . An =2 , − − ln2(n 1) − Ane1(x1,...,xn)=0, n 2 Letting k = n l gives (55). A e (x ,...,x )= n(n 1)2 − . n 2 1 n − − − Thus by (53) n 2.2.2 Further visual exploration on the sphere n(n 1) x2 = − . i 2 This section is based on Section 2.4 of Paper A i=1

For reasons that will become clear in Section 2.2.1 it is also useful to think about these coordinates as the roots of the polynomial

1 P (x)=x3 x. 3 − 2

By Theorem 2.3 the extreme points on the unit sphere for v4(x4) is described by the roots of this polynomial

1 1 P (x)=x4 x2 + . 4 − 2 48 (a) Plot with t-basis given by (57). (b) Plot with θ and φ given by (21). The roots of P4(x) are: Figure 2.11: Plot of v4(x4) over points on the unit sphere.

n(n 1) Five dimensions v (( 1)x )=( 1) 2− v (x ), n − n − n n By Theorem 2.3 or 2.4 we see that the polynomials providing the coordinates and so opposite points are both maxima or both minima if n =4k or of the extreme points have all even or all odd powers. From this it is easy n =4k + 1 for some k Z+ and opposite points are of diﬀerent types ∈ to see that all coordinates of the extreme points must come in pairs xi, xi. if n =4k 2 or n =4k 1 for some k Z+. − − − ∈ Furthermore, by Theorem 2.2 we know that the extreme points of v5(x5) on the sphere all lie in the hyperplane x1 + x2 + x3 + x4 + x5 = 0. 5 1 1 We use this to visualize v5(x5) by selecting a subspace of R that contains x2 = , . −√2 √2 all points that have coordinates which are symmetrically placed on the real line, (x ,x , 0, x , x ). 1 2 − 2 − 1 The coordinates in Figure 2.12 (a) are related to x5 by P (x)=x2 1. 2 − 101 We will see later on that the extreme points are the six points acquired − 0 11 1/√20 0 from permuting the coordinates in 2.1.1  −  x = 0 01 01/√20t. (58) 5   1  0 11 0 01/√5 x = ( 1, 0, 1) .   3  1 01   √2 −     82 83 Generalized Vandermonde matrices and determinants in 2.2. OPTIMIZATION OF THE VANDERMONDE electromagnetic compatibility DETERMINANT ON SOME N -DIMENSIONAL SURFACES

For reasons that will become clear in Section 2.2.1 it is also useful to think about these coordinates as the roots of the polynomial

1 P (x)=x3 x. 3 − 2

By Theorem 2.3 the extreme points on the unit sphere for v4(x4) is described by the roots of this polynomial

1 1 P (x)=x4 x2 + . 4 − 2 48 (a) Plot with t-basis given by (57). (b) Plot with θ and φ given by (21). The roots of P4(x) are: Figure 2.11: Plot of v4(x4) over points on the unit sphere.

1 2 1 2 The results of plotting the v4(x4) after performing this transformation x41 = 1+ ,x42 = 1 , can be seen in Figure 2.11. All 24 = 4! extreme points are clearly visible. −2 3 −2 − 3 From Figure 2.11 we see that whenever we have a local maxima we have 1 2 1 2 a local maxima at the opposite side of the sphere as well, and the same for x43 = 1 ,x44 = 1+ . 2 − 3 2 3 minima. This is due to the occurrence of the exponents in the rows of Vn. From equation (19) we have n(n 1) Five dimensions v (( 1)x )=( 1) 2− v (x ), n − n − n n By Theorem 2.3 or 2.4 we see that the polynomials providing the coordinates and so opposite points are both maxima or both minima if n =4k or of the extreme points have all even or all odd powers. From this it is easy n =4k + 1 for some k Z+ and opposite points are of diﬀerent types ∈ to see that all coordinates of the extreme points must come in pairs xi, xi. if n =4k 2 or n =4k 1 for some k Z+. − − − ∈ Furthermore, by Theorem 2.2 we know that the extreme points of v5(x5) on the sphere all lie in the hyperplane x1 + x2 + x3 + x4 + x5 = 0. 5 1 1 We use this to visualize v5(x5) by selecting a subspace of R that contains x2 = , . −√2 √2 all points that have coordinates which are symmetrically placed on the real line, (x ,x , 0, x , x ). 1 2 − 2 − 1 The coordinates in Figure 2.12 (a) are related to x5 by P (x)=x2 1. 2 − 101 We will see later on that the extreme points are the six points acquired − 0 11 1/√20 0 from permuting the coordinates in 2.1.1  −  x = 0 01 01/√20t. (58) 5   1  0 11 0 01/√5 x = ( 1, 0, 1) .   3  1 01   √2 −     82 83 Generalized Vandermonde matrices and determinants in 2.2. OPTIMIZATION OF THE VANDERMONDE electromagnetic compatibility DETERMINANT ON SOME N -DIMENSIONAL SURFACES

Six dimensions

As for v5(x5) we use symmetry to visualize v6(x6). We select a subspace 6 of R that contains all symmetrical points (x1,x2,x3, x3, x2, x1) on the − − − sphere.

The coordinates in Figure 2.13 (a) are related to x6 by

10 0 − 0 10 − 1/√20 0  00 1 x6 = − 01/√20t. (59)  001     0 01/√2  010     (a) Plot with t-basis given by (58). (b) Plot with θ and φ given by (21).  100     Figure 2.12: Plot of v5(x5) over points on the unit sphere.

The result, see Figure 2.12, is a visualization of a subspace containing 8 of the 120 extreme points. Note that to satisfy the condition that the coordinates should be symmetrically distributed pairs can be fulfilled in two other subspaces with points that can be described in the following ways: (x ,x , 0, x , x ) and (x , x , 0,x , x ). This means that a transfor- 1 2 − 1 − 2 2 − 2 1 − 1 mation similar to (58) can be used to describe 3 8 = 24 different extreme · points. The transformation (58) corresponds to choosing x3 = 0. Choosing another coordinate to be zero will give a different subspace of R5 which behaves identically to the visualized one. This multiplies the number of extreme points by five to the expected 5 4! = 120. · By Theorem 2.3 the extreme points on the unit sphere for v5(x5) is described by the roots of this polynomial (a) Plot with t-basis given by (59). (b) Plot with θ and φ given by (21).

1 3 Figure 2.13: Plot of v6(x6) over points on the unit sphere. P (x)=x5 x3 + x. 5 − 2 80 In Figure 2.13 there are 48 visible extreme points. The remaining ex- The roots of P5(x) are: treme points can be found using arguments analogous the ﬁve-dimensional case. x51 = x55,x52 = x54,x53 =0, − − By Theorem 2.3 the extreme points on the unit sphere for v6(x6) is described by the roots of this polynomial 1 2 1 2 x54 = 1 ,x55 = 1+ . 2 − 5 2 5 1 1 1 P (x)=x6 x4 + x2 . 6 − 2 20 − 1800

84 85 Generalized Vandermonde matrices and determinants in 2.2. OPTIMIZATION OF THE VANDERMONDE electromagnetic compatibility DETERMINANT ON SOME N -DIMENSIONAL SURFACES

Six dimensions

As for v5(x5) we use symmetry to visualize v6(x6). We select a subspace 6 of R that contains all symmetrical points (x1,x2,x3, x3, x2, x1) on the − − − sphere.

The coordinates in Figure 2.13 (a) are related to x6 by

84 85 Generalized Vandermonde matrices and determinants in 2.2. OPTIMIZATION OF THE VANDERMONDE electromagnetic compatibility DETERMINANT ON SOME N -DIMENSIONAL SURFACES

The roots of P6(x) are: Seven dimensions As for v (x ) we use symmetry to visualize v (x ). We select a subspace of x61 = x66,x62 = x65,x63 = x64, 6 6 7 7 − − − 7 3 1 R that contains all symmetrical points (x1,x2,x3, 0, x3, x2, x1) on the 4 1 1 2 − − − ( 1) 3 3 3 sphere. x64 = − 10i √10 z6w + z6w 2√15 − 6 6 The coordinates in Figure 2.14 (a) are related to x7 by 1 = 10 2√10 √3l k , (60) 10 0 √ 6 6 − 2 15 − − 0 10 1 1  −  ( 1) 4 1 1 2 00 1 1/√20 0 √3 3 3 − x65 = − 10i 10 z6w6 + z6w6 x =  000 01/√20t. (63) 2√15 − − 7      001 0 01/√2 1   = 10 2√10 √3l + k , (61)  010   √ − 6 6   2 15  100 1   1 1 1 2   x = √3 10 w 3 + w 3 +5 66 30 6 6 1 = 2√10 k +5 , (62) 30 · 6 z6 =√3+i, w6 =2+i√6 1 3 1 3 k6 = cos arctan ,l6 = sin arctan . 3 2 3 2

(a) Plot with t-basis given by (63). (b) Plot with θ and φ given by (21).

Figure 2.14: Plot of v7(x7) over points on the unit sphere.

In Figure 2.14 48 extreme points are visible just like it was for the six- dimensional case. This is expected since the transformation corresponds 7 to choosing x4 = 0 which restricts us to a six-dimensional subspace of R which can then be visualized in the same way as the six-dimensional case. The remaining extreme points can be found using arguments analogous the ﬁve-dimensional case. By Theorem 2.3 the extreme points on the unit sphere for v4 is described by the roots of this polynomial 1 5 5 P (x)=x7 x5 + x3 x. 7 − 2 84 − 3528

86 87 Generalized Vandermonde matrices and determinants in 2.2. OPTIMIZATION OF THE VANDERMONDE electromagnetic compatibility DETERMINANT ON SOME N -DIMENSIONAL SURFACES

The roots of P6(x) are: Seven dimensions As for v (x ) we use symmetry to visualize v (x ). We select a subspace of x61 = x66,x62 = x65,x63 = x64, 6 6 7 7 − − − 7 3 1 R that contains all symmetrical points (x1,x2,x3, 0, x3, x2, x1) on the 4 1 1 2 − − − ( 1) 3 3 3 sphere. x64 = − 10i √10 z6w + z6w 2√15 − 6 6 The coordinates in Figure 2.14 (a) are related to x7 by 1 = 10 2√10 √3l k , (60) 10 0 √ 6 6 − 2 15 − − 0 10 1 1  −  ( 1) 4 1 1 2 00 1 1/√20 0 √3 3 3 − x65 = − 10i 10 z6w6 + z6w6 x =  000 01/√20t. (63) 2√15 − − 7      001 0 01/√2 1   = 10 2√10 √3l + k , (61)  010   √ 6 6   2 15 −  100 1   1 1 1 2   x = √3 10 w 3 + w 3 +5 66 30 6 6 1 = 2√10 k +5 , (62) 30 · 6 z6 =√3+i, w6 =2+i√6 1 3 1 3 k6 = cos arctan ,l6 = sin arctan . 3 2 3 2

(a) Plot with t-basis given by (63). (b) Plot with θ and φ given by (21).

Figure 2.14: Plot of v7(x7) over points on the unit sphere.

86 87 Generalized Vandermonde matrices and determinants in 2.2. OPTIMIZATION OF THE VANDERMONDE electromagnetic compatibility DETERMINANT ON SOME N -DIMENSIONAL SURFACES

The roots of P7(x) are:

x = x ,x = x ,x = x ,x =0, 71 − 77 72 − 76 73 − 75 74 3 1 4 1 1 2 ( 1) 3 3 3 x75 = − 14i √14 z6w + z6w 2√21 − 6 6 2 2 2 Figure 2.15: Value of v3(x3) over: S2 (left), S4 (middle left), S8 (middle right) and 1 2 = 14 2√14 √3l6 k6 , (64) S (right). √ ∞ 2 21 − − 1 1 4 1 1 2 ( 1) 3 3 3 2 x76 = − 14i √14 z6w + z6w regions. As we soon will see the extreme points on S2 are 1/√2, 0, 1/√2 2√21 − − 6 6 − and the vectors constructed by permutating the coordinates of this vector, 1 = 14 2√14 √3l + k , (65) making a total of 6 = 3! extreme points. Similarly, for the cube we have the √ 7 7 2 21 − vectors formed by the six permutations of the coordinates in ( 1, 0, 1). The 1 − 1 1 2 two stated vectors are both maxima and correspond to the top left maxima 1 3 3 3 x77 = √14 w + w +5 in the ﬁgures, odd permutations of these vectors will give minima and even 42 6 6 permutations will again give maxima. This follows directly from the anti- 1 = 2√14k +5 , (66) symmetry of the Vandermonde determinant, given a permutation σ Sn 42 7 ∈ we have vn(xσ1 , ,xσn ) = sgn(σ)vn(x1, ,xn). ··· ··· n 1 z6 =√3+i, w6 =2+i√10 We are thus faced with the problem of maximizing v (x ) over S − . | n n | p By the symmetry of v we do not loose any generality by assuming that 1 5 | n| k7 = cos arctan , the coordinates are ordered and pairwise distinct, x1 <

Consider the unit (n 1)-sphere under the p-norm (p Z,p 1), we deﬁne − ∈ ≥ Proof. Suppose that we have two diﬀerent sequences of coordinates xn and n 1 n p p xn : S − = xn R : x1 + + xn =1 . p { ∈ | | ··· | | } x <

88 89 Generalized Vandermonde matrices and determinants in 2.2. OPTIMIZATION OF THE VANDERMONDE electromagnetic compatibility DETERMINANT ON SOME N -DIMENSIONAL SURFACES

The roots of P7(x) are: x = x ,x = x ,x = x ,x =0, 71 − 77 72 − 76 73 − 75 74 3 1 4 1 1 2 ( 1) 3 3 3 x75 = − 14i √14 z6w + z6w 2√21 − 6 6 2 2 2 Figure 2.15: Value of v3(x3) over: S2 (left), S4 (middle left), S8 (middle right) and 1 2 = 14 2√14 √3l6 k6 , (64) S (right). √ ∞ 2 21 − − 1 1 4 1 1 2 ( 1) 3 3 3 2 x76 = − 14i √14 z6w + z6w regions. As we soon will see the extreme points on S2 are 1/√2, 0, 1/√2 2√21 − − 6 6 − and the vectors constructed by permutating the coordinates of this vector, 1 = 14 2√14 √3l + k , (65) making a total of 6 = 3! extreme points. Similarly, for the cube we have the √ 7 7 2 21 − vectors formed by the six permutations of the coordinates in ( 1, 0, 1). The 1 − 1 1 2 two stated vectors are both maxima and correspond to the top left maxima 1 3 3 3 x77 = √14 w + w +5 in the ﬁgures, odd permutations of these vectors will give minima and even 42 6 6 permutations will again give maxima. This follows directly from the anti- 1 = 2√14k +5 , (66) symmetry of the Vandermonde determinant, given a permutation σ Sn 42 7 ∈ we have vn(xσ1 , ,xσn ) = sgn(σ)vn(x1, ,xn). ··· ··· n 1 z6 =√3+i, w6 =2+i√10 We are thus faced with the problem of maximizing v (x ) over S − . | n n | p By the symmetry of v we do not loose any generality by assuming that 1 5 | n| k7 = cos arctan , the coordinates are ordered and pairwise distinct, x1 <

88 89 Generalized Vandermonde matrices and determinants in 2.2. OPTIMIZATION OF THE VANDERMONDE electromagnetic compatibility DETERMINANT ON SOME N -DIMENSIONAL SURFACES

where σ>0 is a normalization constant to assure that zn lies on the sphere. The condition xi = xn i+1 for the ordered maximum x1 < convex set. This ≤ p the maxima are constructed from different orderings of these, it makes sense follows directly from the absolute homogeneity and the triangle inequality to instead consider the polynomial constructed from these coordinates. Our associated with the p-norm: task now is to find the polynomials that define the optimizing coordinates for different n 2 and p 1, xn + xn xn xn ≥ ≥ + =1. n 2 p ≤ 2 p 2 p n P (x)= (x xi), p − n 1 i=1 We have established that z lies on S − . For each 1 i

xj + xj xi xi xj xi + xj xi 1 1 n n n − − | − | − 2 zj zi = = xj xi 2 x x , n j i P (xk)= (x xi) = (xk xi), | − | 2σ 2σ ≥| − | − p − − j=1 i=1 x=xk i=1 i =j i =k where the last step follows from σ<1 and the general relation n n n n n n n n 2 2 P (x )= (x x ) = (x x )+ (x x ) a + b a b p k − i k − i k − i = − + ab ab. l=1 j=1 i=1 x=xk j=1 i=1 l=1 i=1 2 2 ≥ j = il =j j =k i =j l =k i =l i=l i=k i=k It follows that n n =2 (xk xi), 1 1 − 2 2 max j=1 i=1 vn(zn) vn(xn) vn(xn ) = vn , j =k i =j | |≥| | i=k and to not have a contradiction, we must have that the equality holds, that and it is easy to show that is xn = xn , and so the maximum is unique. n n Now, consider Pp (xk) 1 n(n 1) =2 . (67) − n 2 P (x ) x x vn( xn)=( 1) vn(xn), p k i=1 k i − − i =k − which follows easily from the degree of the expansion of vn, where every n(n 1) term is of degree − . If follows that if x is a maximum of v on Deﬁne 2 n | n| n the sphere then xn is also a maximum. Now, since the maximum with n p − sp (xn)= xi 1, ordered coordinates x1 <

90 91 Generalized Vandermonde matrices and determinants in 2.2. OPTIMIZATION OF THE VANDERMONDE electromagnetic compatibility DETERMINANT ON SOME N -DIMENSIONAL SURFACES

where σ>0 is a normalization constant to assure that zn lies on the sphere. The condition xi = xn i+1 for the ordered maximum x1 <

90 91 Generalized Vandermonde matrices and determinants in 2.2. OPTIMIZATION OF THE VANDERMONDE electromagnetic compatibility DETERMINANT ON SOME N -DIMENSIONAL SURFACES

To maximize v (x ) over the surface sn(x ) = 0 we transform the Proof. By equation (69) we have | n n | p n objective function by applying a (strictly increasing) logarithm: n n P2 (x)+ρxP2 (x) =0, 1 k n. wn(xn) = log ( vn(xn) )= log ( xj xi ) , ≤ ≤ | | | − | x=xk 1 i j ≤ ≤ The left part of this equation represents n evaluations of a polynomial of with partial derivatives degree n that vanishes on x1, ,xn and must thus be a constant multiple n n n ··· P (x ) of P , that we defined as the polynomial that vanishes on x1, ,xn, and ∂wn(xn) 1 1 p k 2 ··· = = n . so ∂x x x 2 P (x ) k i=1 k i p k n n n i=k − P2 (x)+ρxP2 (x)+σP2 (x)=0. By the method of Lagrange multipliers we now have that the maxima of Two find the coefficients σ and ρ we need to adapt this polynomial to n 1 n vn(xn) on Sp− must be stationary points to the Lagrangian | | the sphere. We have xi = 0. The condition cn = 1 is by choice. The n i=1 Λ(xn,λ)=wn(xn) λsp (xn), (68) n − condition cn 1 = 0 follows from the expansion of the coefficients in P2 . which explicitly means − n n n p n k k ∂wn(xn) ∂sp (xn) Pn (x)= (x xi)= ( 1) − en k(x1, ,xn)x , = λ , − − − ··· i=1 k=0 ∂xk ∂xk for some multiplier λ R. We then get where ek is the elementary symmetric polynomial defined by ∈ n 1 Pp (xk) p 1 ek(x1, ,xn)= xi1 xi2 xik . n = λp xk − sgn(xk). ··· ··· 2 P (xk) | | i1< 2 and with a leading co- cn 2 = e2(x1, ,xn)= . n − ··· −2 efficient of 1, whose roots form the coordinates in the points xn R that n 1 ∈ maximize vn(xn) over the Euclidean hypersphere S − satisfy the differen- This establishes equation (71) in the theorem. | | tial equation Now, the coefficients c , ,c for any polynomial solution p(x) of degree n ··· 0 n n 2 n n to a differential equation on the form P2 (x)+n(1 n)xP2 (x)+n (n 1)P2 (x)=0. (70) − − n n n Furthermore, the coefficients for the three terms of highest degree are p (x)+ρxp (x)+σp (x)=0, 1 must satisfy cn =1,cn 1 =0,cn 2 = , (71) − − −2 ρnc + σc =0 and the subsequent coefficients are defined recursively by n n ρ(n 1)cn 1 + σcn 1 =0 (k + 1)(k + 2) − − − ck = ck+2. (72) n(n 1)cn + ρ(n 2)cn 2 + σcn 2 =0. −n(n 1)(n k) − − − − − − 92 93 Generalized Vandermonde matrices and determinants in 2.2. OPTIMIZATION OF THE VANDERMONDE electromagnetic compatibility DETERMINANT ON SOME N -DIMENSIONAL SURFACES

n The second of these equations is trivial since cn 1 = 0. The first and third where m = n 2 is the degree of the polynomial p . Furthermore, the first − − n ∞ equation simplifies to two coefficients for p are cm =1,cm 1 =0and the subsequent coefficients − satisfy ∞ ρn + σ =0, (k + 1)(k + 2) c = c . (75) 1 1 k k(k + 3) m(m + 3) k+2 n(n 1) ρ(n 2) σ =0, − − − 2 − − 2 Proof. It is easy to show that the coordinates 1 and +1 must be present − and so in the maxima points, if they were not then we could rescale the point so 2n(n 1) that the value of vn(xn) is increased, which is not allowed. We may thus ρ = − − = n(n 1) | | n (n 2) − − assume the ordered sequence of coordinates − − 2 σ = n (n 1), 1=x < 2 and with a leading that is ∞ n 2 n n n coefficient of 1, whose roots form the coordinates in the points xn R that (1 x )p (x) 4xp (x)+σp (x)=0. (77) n 1 ∈ − ∞ − ∞ ∞ maximize vn(xn) over the cube S − satisfy m | | ∞ The constant σ is found by considering the coefficient for x : P n (x)=(x 1)(x + 1)pn (x). (74) ∞ − ∞ m(m 1) 4m + σ =0 σ = m(m + 3). − − − ⇔ where pn is defined by the differential equation: ∞ Finally 2 n n n 2 n n n (1 x )p (x) 4xp (x)+m(m + 3)p (x)=0, (1 x )p (x) 4xp (x)+m(m + 3)p (x)=0. − ∞ − ∞ ∞ − ∞ − ∞ ∞

94 95 Generalized Vandermonde matrices and determinants in 2.2. OPTIMIZATION OF THE VANDERMONDE electromagnetic compatibility DETERMINANT ON SOME N -DIMENSIONAL SURFACES

The polynomial that provide all coordinates for the maxima is then From this it is easy to show that (80) can be rewritten by introducing the n n n P (x)=(x 1)(x + 1)p . (78) univariate polynomial Pn(x)= (x xi) as ∞ − ∞ − Equation (75) follows by the same methods as for p = 2, that is, by identi- i=1 fying all coefficients in the differential equation to be identically zero. ∂ ln v 1 P (x ) | n| = k . ∂x 2 P (x ) We have provided the means to describe the coordinates of the extreme k k points of the Vandermonde determinant over the unit spheres under the Now the leftmost equation of (79) can be written Euclidean norm and under the infinity norm. The resulting polynomials can be identified by rescaled Hermite polynomials, H x n(n 1)/2 , 1 P (xk) ∂sn n − = λ , (3/2) 2 P (xk) ∂xk and Gegenbauer polynomials, Cn (x), respectively. In the next section we will discuss the case p = 4. or more succinct ∂sn P (xk) 2λ P (xk)=0, (81) 2.2.4 The Vandermonde determinant on spheres defined by − ∂xk the 4-norm In the case p = 2 we are lucky since (81) becomes, by introducing the new “multiplier” ρ This section is based on Section 8 of Paper C P (x)+ρ xP (x) =0, (82) n |x=xk The optimization of the Vandermonde determinant on the sphere (p = 2) and since the left part of this equation is a polynomial of degree n and has and on the cube (p = ) lends themselves to methods in orthogonal poly- roots x , ,x we must have ∞ 1 n nomials. In fact, as shown by Stieltjes and recaptured by Szeg˝o[158], and ··· presented in more detail in and extended in Section 2.2.1 there is a fairly P (x)+ρnxP (x)+σnP (x)=0, (83) straightforward solution derived from electrostatic considerations, which im- for some ρn,σn that may depend on n. Now if choose P (x) to be monic, plicitly deals with the Vandermonde determinant. n note that ρn = 0, and require us to be on the sphere we get P (x)=x Consider the optimization of vn(x) over the sphere 1 n 2 − x − + , and by identifying coefficients we get ρn and σn: n 2 ··· p 2 sn(x)= xi =1, P (x)+n(1 n)xP (x)+n (n 1)P (x)=0, − − i=1 which is a nice and well known form of differential equation and defines a for suitable choices for p (even). Instead of optimizing vn we are free to sequence of orthogonal polynomials that are rescaled Hermite polynomials optimize ln vn over the sphere to the same effect (vn(x) = 0 is not a solution, | | [158], so we can find a recurrence relation for Pn+1 in terms of Pn and Pn 1, all xi are pair-wise distinct). This leaves us with the set of equations by − Lagrange multipliers. and we can, for a fixed n, construct the coefficients of Pn recursively, without explicitly finding P1, ,Pn 1. ∂ ln vn ∂sn ··· − | | = λ ,sn =1, (79) Now, this is for p = 2. For p = 4 we continue from (81) instead with ∂xk ∂xk 3 where the left-most equation holds for 1 k n. It is easy to show that P (x)+ρnx P (x) x=xk =0. (84) ≤ ≤ | the partial derivatives can be written Now the polynomial in x in the left part of this equation has shared roots n ∂ ln v 1 with P (x) and so by the same method as for p = 2 we get: | n| = . (80) ∂x x x k i=1 k i 3 2 i =k − P (x)+ρnx P (x)+(σnx + τnx + υn)P (x)=0. (85)

96 97 Generalized Vandermonde matrices and determinants in 2.2. OPTIMIZATION OF THE VANDERMONDE electromagnetic compatibility DETERMINANT ON SOME N -DIMENSIONAL SURFACES

96 97 Generalized Vandermonde matrices and determinants in electromagnetic compatibility

It is easy to show for the sphere under any p-norm that the extreme points of ln v (x) where x <

98 Generalized Vandermonde matrices and determinants in electromagnetic compatibility

It is easy to show for the sphere under any p-norm that the extreme points of ln v (x) where x <

98 3.1. THE ANALYTICALLY EXTENDED FUNCTION (AEF)

3.1 The analytically extended function (AEF)

In this section we consider least square approximation using a particular function we call the power exponential function, defined in Definition 3.1, as a basis. Definition 3.1. Here we will refer to the function defined by (87) as the power exponential function,

1 t β x(β; t)= te − , 0 t. (87) ≤ For non-negative values of t and β the power exponential function has a steeply rising initial part followed by a more slowly decaying part, see Figure 3.1. This makes it qualitatively similar to several functions that are popular for approximation of important phenomena in diﬀerent ﬁelds such as approximation of lightning discharge currents and pharmacokinetics. Examples include the biexponential function [22], [151], the Heidler function [65] and the Pulse function [178].

Figure 3.1: An illustration of how the steepness of the power exponential function varies with β.

The power exponential function has been used in applications, for example to model attack rate of predatory ﬁsh, see [135, 136]. Here we examine linear combinations of piecewise power exponential functions that in later sections will be used to approximate electrostatic discharge current functions.

101 3.1. THE ANALYTICALLY EXTENDED FUNCTION (AEF)

3.1 The analytically extended function (AEF)

Figure 3.1: An illustration of how the steepness of the power exponential function varies with β.

101 Generalized Vandermonde matrices and determinants in electromagnetic compatibility 3.1. THE ANALYTICALLY EXTENDED FUNCTION (AEF)

3.1.1 The p-peak analytically extended function Remark 3.1. The p -peak AEF can be written more compactly if we introduce the vectors This section is based on Section 2 of Paper D

ηq =[ηq,1 ηq,2 ... ηq,nq ] , (89)

The p -peaked AEF is constructed using the power exponential function 2 2 2 β +1 β +1 βq,n +1 given in Definition 3.1. In order to get a function with multiple peaks and xq(t) q,1 xq(t) q,2 ... xq(t) q , 1 q p, x (t)= ≤ ≤ (90) q  2 2 2 where the steepness of the rise between each peak as well as the slope of β β βq,n  xq(t) q,1 xq(t) q,2 ... xq(t) q ,q= p +1. the decaying part is not dependent on each other, we define the analyti-  cally extended function (AEF) as a function that consist of piecewise linear The more compact form is combinations of the power exponential function that has been scaled and q 1 translated so that the resulting function is continuous. With a given differ- − Imk + Imq ηq xq(t),tmq 1 t tmq , 1 q p, ence in height between subsequent peaks I , I , ... , I , corresponding − m1 m2 mp  · ≤ ≤ ≤ ≤ i(t)= k=1 (91) times tm1 , tm2 ,...,tmp , integers nq > 0, real values βq,k, ηq,k,1 q p+1,  q ≤ ≤  1 k nq such that the sum over k of ηq,k is equal to one, the p -peaked  I η x (t),tt, q = p +1. ≤ ≤ mk · q q mq ≤ AEF i(t) is given by (88). k=1   Definition 3.2. Given Imq R and tmq R, q =1, 2, ... ,p such that If the AEF is used to model an electrical current, than the derivative ∈ ∈ tm0 =0linear function of elementary functions its derivative The analytically extended function (AEF), i(t), with p peaks is defined as can be found using standard methods.

q 1 nq − 2 Theorem 3.1. The derivative of the p -peak AEF is βq,k+1 Imk +Imq ηq,kxq(t) ,tmq 1 t tmq , 1 q p,  − ≤ ≤ ≤ ≤ t t k=1 k=1 mq xq(t) i(t)= n (88) Imq − ηq Bq xq(t),tmq 1 t tmq , 1 q p,  p p+1 t t ∆t −  2 di(t) mq 1 mq ≤ ≤ ≤ ≤  βp+1,k = − − (92)  Imk ηp+1,kxp+1(t) ,tmp t, ≤ dt  xq(t) tmq t k=1 k=1 I − η B x (t),t t, q = p +1,   mq q q q mq  t tmq ≤    where where  t tmq 1 tmq t  − − exp − , 1 q p, 2 ∆tmq ∆tmq ≤ ≤ βq,1 +1 0 ... 0 xq(t)= 2  t t 0 βq,2 +1 ... 0  B =   ,  exp 1 ,q= p +1, q . . .. . tmq − tmq . . . .  2    00... βq,nq +1 and ∆tmq = tmq tmq1 .   − −  2  Sometimes the notation i(t; β, η) with βp+1,1 0 ... 0 2 0 βp+1,2 ... 0 β = β β ... β ... β , Bp+1 =  . . . .  , 1,1 1,2 q,k p+1,np+1 . . .. .  2  η = η1,1 η1,2 ... ηq,k ... ηp+1,np+1 ,  00... β   p+1,np+1    will be used to clarify what the particular parameters for a certain AEF are. for 1 q p. ≤ ≤

102 103 Generalized Vandermonde matrices and determinants in electromagnetic compatibility 3.1. THE ANALYTICALLY EXTENDED FUNCTION (AEF)

3.1.1 The p-peak analytically extended function Remark 3.1. The p -peak AEF can be written more compactly if we introduce the vectors This section is based on Section 2 of Paper D

ηq =[ηq,1 ηq,2 ... ηq,nq ] , (89)

102 103 Generalized Vandermonde matrices and determinants in electromagnetic compatibility 3.1. THE ANALYTICALLY EXTENDED FUNCTION (AEF)

Proof. From the definition of the AEF (see (88)) and the derivative of the value from both directions unless all η 0, but if all η 0 then q,k ≤ q,k ≤ power exponential function (87) given by nq η = 1. q,k d β 1 β(1 t) k=1 x(β; t)=β(1 t)t − e − , dt − Noting that for any diagonal matrix B the expression expression (92) can easily be derived since differentiation is a linear operation nq β2 +1 and the result can be rewritten in the compact form analogously to (91). q,k ηq Bxq(t)= ηq,kBkkxq(t) , 1 q p, Illustration of the AEF function and its derivative for various values of ≤ ≤ k=1 βq,k-parameters is shown in Fig. 3.2. is well-defined and that the equivalent statement holds for q = p it is easy to see from (92) that the factor (t t) in the derivative ensures that the mq − derivative is zero every time t = tmq .

When interpolating a waveform with p peaks it is natural to require that there will not appear new peaks between the chosen peaks. This corresponds to requiring monotonicity in each interval. One way to achieve this is given in lemma 3.2.

Lemma 3.2. If η 0, k =1,...,n the AEF, i(t), is strictly monotonic q,k ≥ q on the interval tmq 1

line) with diﬀerent βq,k-parameters but the same Imq

and tmq . (a) 0 <βq,k < 1, (b) 4 <βq,k < 5, Lemma 3.3. For any tmq 1 t0 t1 tmq , 1 q p, − ≤ ≤ ≤ ≤ ≤ (c) 12 <βq,k < 13, (d) a mixture of large and small βq,k-parameters t1 β β e t1 tmq t0 tmq xq(t) dt = β+1 ∆γ β +1, − , − (93) Lemma 3.1. The AEF is continuous and at each t the derivative is equal t β β∆tmq β∆tmq mq 0 to zero. with ∆tmq = tmq tmq 1 and − − Proof. Within each interval tmq 1 t tmq the AEF is a linear combination − ≤ ≤ of continuous functions and at each t the function will approach the same ∆γ(β,t ,t )=γ (β +1,βt ) γ (β +1,βt ) , mq 0 1 1 − 0

104 105 Generalized Vandermonde matrices and determinants in electromagnetic compatibility 3.1. THE ANALYTICALLY EXTENDED FUNCTION (AEF)

Lemma 3.2. If η 0, k =1,...,n the AEF, i(t), is strictly monotonic q,k ≥ q on the interval tmq 1

If we allow some of the ηq,k-parameters to be negative, the derivative can change sign and the function might get an extra peak between two other peaks, see Fig. 3.3. The integral of the electric current represents the charge flow. Unlike the Heidler function the integral of the AEF is relatively straightforward to find. How to do this is detailed in lemma 3.3, lemma 3.4, theorem 3.2, and Figure 3.2: Illustration of the AEF (solid line) and its derivative (dashed theorem 3.3. line) with different βq,k-parameters but the same Imq and tmq . (a) 0 <βq,k < 1, (b) 4 <βq,k < 5, Lemma 3.3. For any tmq 1 t0 t1 tmq , 1 q p, − ≤ ≤ ≤ ≤ ≤ (c) 12 <βq,k < 13, (d) a mixture of large and small βq,k-parameters t1 β β e t1 tmq t0 tmq xq(t) dt = β+1 ∆γ β +1, − , − (93) Lemma 3.1. The AEF is continuous and at each t the derivative is equal t β β∆tmq β∆tmq mq 0 to zero. with ∆tmq = tmq tmq 1 and − − Proof. Within each interval tmq 1 t tmq the AEF is a linear combination − ≤ ≤ of continuous functions and at each t the function will approach the same ∆γ(β,t ,t )=γ (β +1,βt ) γ (β +1,βt ) , mq 0 1 1 − 0

104 105 Generalized Vandermonde matrices and determinants in electromagnetic compatibility 3.1. THE ANALYTICALLY EXTENDED FUNCTION (AEF)

t tm Changing variables according to τ = β − q gives ∆tmq

t1 β τ1 β e β τ x (t) dt = τ e− dt = q ββ+1 t0 τ0 eβ = (γ(β +1,τ ) γ(β +1,τ )) ββ+1 1 − 0 eβ = ∆γ(β +1,τ ,τ ) ββ+1 1 0 β e t1 tm t0 tm = ∆γ β +1,β − q ,β − q . ββ+1 ∆t ∆t mq mq

When t0 = tmq 1 and t1 = tmq then Figure 3.3: An example of a two-peaked AEF where some of the ηq,k-parameters are − negative, so that it has points where the ﬁrst derivative changes sign t1 eβ between two peaks. The solid line is the AEF and the dashed lines is the x (t)β dt = ∆γ (β +1,β) q ββ+1 derivative of the AEF. t0 and with γ(β +1, 0) = 0 we get (94). where Lemma 3.4. For any tmq 1 t0 t1 tmq , 1 q p, − ≤ ≤ ≤ ≤ ≤ q 1 nq t t1 − β 1 τ i(t)dt =(t t ) I + I η g (t ,t ), (95) γ(β,t)= τ − e− dτ 1 − 0 mk mq q,k q 1 0 0 t0 k=1 k=1 where 2 is the lower incomplete Gamma function [2]. βq,k e t1 tmq 1 t0 tmq 1 2 − − gq(t1,t0)= 2 ∆γ βq,k +2, − , − If t0 = tmq 1 and t1 = tmq then βq,k+1 ∆t ∆t − 2 mq mq βq,k +1 with ∆γ(β,t0,t1) deﬁned as in (93). tmq eβ x (t)β dt = γ (β +1,β) . (94) q ββ+1 Proof. tmq 1 − t t q 1 nq 1 1 − 2 βq,k+1 i(t)dt = Imk + Imq ηq,kxq(t) dt t0 t0 Proof. k=1 k=1 q 1 nq t − 1 2 βq,k+1 =(t1 t0) Imk + Imq ηq,k xq(t) dt − t0 t1 t1 β k=1 k=1 β t tmq t tmq x (t) dt = − exp 1 − dt q 1 nq q − t0 t0 ∆tmq − ∆tmq =(t1 t0) Imk + Imq ηq,k gq(t0,t1). β 1 t β − e 1 t tm t tm k=1 k=1 = − β − q exp 1 β − q dt. ββ ∆t − ∆t t0 mq mq

106 107 Generalized Vandermonde matrices and determinants in electromagnetic compatibility 3.1. THE ANALYTICALLY EXTENDED FUNCTION (AEF)

t tm Changing variables according to τ = β − q gives ∆tmq

106 107 Generalized Vandermonde matrices and determinants in 3.2. APPROXIMATION OF LIGHTNING DISCHARGE electromagnetic compatibility CURRENT FUNCTIONS

Theorem 3.2. If tma 1 ta tma , tmb 1 tb tmb and 0 ta tb tmp In the next section we will estimate the parameters of the AEF that gives − ≤ ≤ − ≤ ≤ ≤ ≤ ≤ then the best ﬁt with respect to some data and for this the partial derivatives with respect to the βm parameters will be useful. Since the AEF is a linear t a 1 na q b − function of elementary functions these partial derivatives can easily be found i(t)dt =(tma ta) Imk + Ima ηa,k ga(ta,tma ) ta − using standard methods. k=1 k=1 b 1 q 1 nq − − Theorem 3.4. The partial derivatives of the p-peak AEF with respect to + ∆t I + I η gˆ β2 +1 mq mk mq q,k q,k the β parameters are q=a+1 k=1 k=1 b 1 nb − 0, 0 t tmq 1 , − +(tb tmb ) Imk + Imb ηb,k gb(tmb ,tb), (96) ∂i β2 +1 ≤ ≤ − = q,k k=1 k=1 2 Imq ηq,k βq,k hq(t)xq(t) ,tmq 1 t tmq , 1 q p, ∂βq,k − ≤ ≤ ≤ ≤ 0,tt, mq ≤ where gq(t0,t1) is deﬁned as in lemma 3.4 and (99)  β e ∂i 0, 0 t tmp , gˆ(β)= γ (β +1,β) . = β2 ≤ ≤ β+1 ∂β p+1,k β p+1,k 2 Imp+1 ηp+1,k βp+1,k hp+1(t)xp+1(t) ,tmp t, ≤ (100) Proof. This theorem follows from integration being linear and lemma 3.4. where

t tmq 1 t tmq 1 ln − − − − +1, 1 q p, Theorem 3.3. For tmp t0

p np+1 ferentiation of composite functions and products of functions. ∞ 2 i(t)dt = Imk ηp+1,k g˜ βp+1,k , (98) tmp k=1 k=1 3.2 Approximation of lightning discharge current where functions eβ g˜(β)= (Γ(β + 1) γ (β +1,β)) ββ+1 − This section is based on Section 3 of Paper E with ∞ β 1 t Many diﬀerent types of systems, objects and equipment are susceptible Γ(β)= t − e− dt to damage from lightning discharges. Lightning eﬀects are usually anal- 0 is the Gamma function [2]. ysed using lightning discharge models. Most of the engineering and electromagnetic models imply channel-base current functions. Various single and Proof. This theorem follows from integration being linear and Lemma 3.4. multi-peaked functions are proposed in the literature for modelling lightning channel-base currents, examples include [65, 66, 78, 79, 83]. For engineering

108 109 Generalized Vandermonde matrices and determinants in 3.2. APPROXIMATION OF LIGHTNING DISCHARGE electromagnetic compatibility CURRENT FUNCTIONS

t tmq 1 t tmq 1 ln − − − − +1, 1 q p, Theorem 3.3. For tmp t0

108 109 Generalized Vandermonde matrices and determinants in 3.2. APPROXIMATION OF LIGHTNING DISCHARGE electromagnetic compatibility CURRENT FUNCTIONS and electromagnetic models, a general function that would be able to re- The Jacobian matrix, J, can in this case be described as produce desired waveshapes is needed, such that analytical solutions for its derivatives, integrals, and integral transformations, exist. A multi-peaked ∂i ∂i ... ∂i ∂βq,1 ∂βq,2 ∂βq,n channel-base current function has been proposed by Javor [78] as a gen- t=tq,1 t=tq,1 q t=tq,1  ∂i ∂i ... ∂i  eralization of the so-called TRF (two-rise front) function from [79], which ∂βq,1 ∂βq,2 ∂βq,n t=tq,2 t=tq,2 q t=tq,2 possesses such properties. J =   (102)  . . .. .   . . . .  In this paper we analyse a modification of such multi-peaked function,    ∂i ∂i ... ∂i  a so-called p -peak analytically extended function (AEF). The possibility ∂βq,1 ∂βq,2 ∂βq,nq  t=tq,kq t=tq,kq t=tq,kq    of application of the AEF to approximation of various multi-peaked wave-   shapes is investigated. Estimation of its parameters has been performed where the partial derivatives are given by (99) and (100). using the Marquardt least-squares method (MLSM), an efficient method for the estimation of non-linear function parameters, see Section 1.3.3. It has been applied in many fields, including lightning research, e.g. for optimiz- 3.2.2 Estimating parameters for underdetermined systems ing parameters of the Heidler function in Lovric et al. [103], or the Pulse This section is based on Section 3.2 of Paper D function in Lundeng˚ard et al. [105]- [106]. Some numerical results are presented, including those for the Standard For the Marquardt least-squares method to work at least one data point per IEC 62305 [74] current of the first-positive strokes, and an example of a fast- unknown parameter is needed, m k. It can still be possible to estimate ≥ decaying lightning current waveform. Fitting a p-peaked AEF to recorded all unknown parameters if there is insufficient data, m1 and initial values b(0), λ(0). •

kq r =0 2 • Sq = (i(tq,k) iq,k) , (101) − Find c(r) using b(r) together with extra relations. k=1 Find b(r+1) and δ(r) using MLSM. is minimized. One way to estimate these parameters is to use the Marquardt • least-square method described in Section 1.3.3. Check chosen termination condition for MLSM, if it is not satisfied go In order to fit the AEF it is sufficient that k n . Suppose we have some • q ≥ q to . estimate of the β-parameters which is collected in the vector b. It is then fairly simple to calculate an estimate for the η-parameters, see Section 3.2.4, Output: b, c. which we collect in h. We define a residual vector by (e) = i(t ; b, h) i • k q,k − q,k where i(t; b, h) is the AEF with the estimated parameters. The algorithm is illustrated in figure 3.4.

110 111 Generalized Vandermonde matrices and determinants in 3.2. APPROXIMATION OF LIGHTNING DISCHARGE electromagnetic compatibility CURRENT FUNCTIONS and electromagnetic models, a general function that would be able to re- The Jacobian matrix, J, can in this case be described as produce desired waveshapes is needed, such that analytical solutions for its derivatives, integrals, and integral transformations, exist. A multi-peaked ∂i ∂i ... ∂i ∂βq,1 ∂βq,2 ∂βq,n channel-base current function has been proposed by Javor [78] as a gen- t=tq,1 t=tq,1 q t=tq,1  ∂i ∂i ... ∂i  eralization of the so-called TRF (two-rise front) function from [79], which ∂βq,1 ∂βq,2 ∂βq,n t=tq,2 t=tq,2 q t=tq,2 possesses such properties. J =   (102)  . . .. .   . . . .  In this paper we analyse a modification of such multi-peaked function,    ∂i ∂i ... ∂i  a so-called p -peak analytically extended function (AEF). The possibility ∂βq,1 ∂βq,2 ∂βq,nq  t=tq,kq t=tq,kq t=tq,kq    of application of the AEF to approximation of various multi-peaked wave-   shapes is investigated. Estimation of its parameters has been performed where the partial derivatives are given by (99) and (100). using the Marquardt least-squares method (MLSM), an efficient method for the estimation of non-linear function parameters, see Section 1.3.3. It has been applied in many fields, including lightning research, e.g. for optimiz- 3.2.2 Estimating parameters for underdetermined systems ing parameters of the Heidler function in Lovric et al. [103], or the Pulse This section is based on Section 3.2 of Paper D function in Lundeng˚ard et al. [105]- [106]. Some numerical results are presented, including those for the Standard For the Marquardt least-squares method to work at least one data point per IEC 62305 [74] current of the first-positive strokes, and an example of a fast- unknown parameter is needed, m k. It can still be possible to estimate ≥ decaying lightning current waveform. Fitting a p-peaked AEF to recorded all unknown parameters if there is insufficient data, m1 and initial values b(0), λ(0). • kq r =0 2 • Sq = (i(tq,k) iq,k) , (101) − Find c(r) using b(r) together with extra relations. k=1 Find b(r+1) and δ(r) using MLSM. is minimized. One way to estimate these parameters is to use the Marquardt • least-square method described in Section 1.3.3. Check chosen termination condition for MLSM, if it is not satisfied go In order to fit the AEF it is sufficient that k n . Suppose we have some • q ≥ q to . estimate of the β-parameters which is collected in the vector b. It is then fairly simple to calculate an estimate for the η-parameters, see Section 3.2.4, Output: b, c. which we collect in h. We define a residual vector by (e) = i(t ; b, h) i • k q,k − q,k where i(t; b, h) is the AEF with the estimated parameters. The algorithm is illustrated in figure 3.4.

110 111 Generalized Vandermonde matrices and determinants in 3.2. APPROXIMATION OF LIGHTNING DISCHARGE electromagnetic compatibility CURRENT FUNCTIONS

2 p q 1 q 1 nq Input: choose v and − − r =0 W (b, h)= I + I I η gˆ(β2 + 1) initial values for b(0) and λ(0)  mk mk mq q,k q,k q=1 k=1 k=1 k=1  nq + I2 η2 gˆ 2 β2 +2 Find b(r+1) and δ(r) Find h(r) using b(r) mq q,k q,k k=1 using MLSM together with extra relations nq 1 nq − +2I2 η η gˆ β2 + β2 +2 mq q,r q,s q,r q,s  r=1 s=r+1 Termination condition NO r = r +1 2 satisﬁed p np  2 2 + Imk ηp,k g˜ 2 βp,k YES k=1 k=1 np+1 1 np+1 Output: b, h − +2 η η g˜ β2 + β2 p+1,r p+1,s p+1,r p+1,s  r=1 s=r+1 Figure 3.4: Schematic description of the parameter estimation algorithm (106)

p q 1 nq J1 ... 0 − 2 . .. . Q(b, h)= ∆tmq Imk + Imq ηq,k gˆ(βq,k + 1)  . . .  q=1 k=1 k=1 J = 0 ... Jp+1 (107) p np+1  ∂E ∂E ∂E ∂E   Q0 ... Q0 ... Q0 ... Q0  2  ∂β1,1 ∂β1,n1 ∂βp+1,1 ∂βp+1,np+1  + Imk ηp+1,k g˜(βp+1,k), (105)   ∂EW0 ∂EW0 ∂EW0 ∂EW0 k=1 k=1  ......   ∂β1,1 ∂β1,n ∂βp+1,1 ∂βp+1,n   1 p+1    112 113 Generalized Vandermonde matrices and determinants in 3.2. APPROXIMATION OF LIGHTNING DISCHARGE electromagnetic compatibility CURRENT FUNCTIONS

3.2.3 Fitting with data points as well as charge flow and where gˆ(β) and g˜(β) are defined in Theorem 3.2 and 3.3. specific energy conditions Proof. Formula (105) is found by combining (96) and (97). By considering the charge flow at the striking point, Q0, unitary resistance Formula (106) is found by noting that R and the specific energy, W0, we get two further conditions: n 2 n n 1 n ∞ − Q0 = i(t)dt, (103) a = a2 + a a 0 k k r s k=1 k=1 r=1 s=r+1 ∞ 2 W0 = i(t) dt. (104) 0 and then reasoning analogously to the proofs for (96) and (97). First we will define ∞ We can calculate the charge flow and specific energy given by the AEF Q(b, h)= i(t; b, h)dt with formulas (105) and (106), respectively, and get two additional residual 0 terms EQ0 = Q(b, h) Q0 and EW0 = W (b, h) W0. Since these are global ∞ 2 − − W (b, h)= i(t; b, h) dt. conditions this means that the parameters η and β no longer can be fitted 0 separately in each interval. This means that we need to consider all data These two quantities can be calculated as follows. points simultaneously. The resulting J-matrix is Theorem 3.5. p q 1 nq J1 ... 0 − 2 . .. . Q(b, h)= ∆tmq Imk + Imq ηq,k gˆ(βq,k + 1)  . . .  q=1 k=1 k=1 J = 0 ... Jp+1 (107) p np+1  ∂E ∂E ∂E ∂E   Q0 ... Q0 ... Q0 ... Q0  2  ∂β1,1 ∂β1,n1 ∂βp+1,1 ∂βp+1,np+1  + Imk ηp+1,k g˜(βp+1,k), (105)   ∂EW0 ∂EW0 ∂EW0 ∂EW0 k=1 k=1  ......   ∂β1,1 ∂β1,n ∂βp+1,1 ∂βp+1,n   1 p+1    112 113 Generalized Vandermonde matrices and determinants in 3.2. APPROXIMATION OF LIGHTNING DISCHARGE electromagnetic compatibility CURRENT FUNCTIONS where where Γ(β) is the Gamma function, Ψ(β) is the digamma function, see for ∂i ∂i ... ∂i example [2], and G(β) is a special case of the Meijer G-function and can be ∂βq,1 ∂βq,2 ∂βq,n t=tq,1 t=tq,1 q t=tq,1 defined as  ∂i ∂i ... ∂i  ∂βq,1 ∂βq,2 ∂βq,n 3,0 1, 1 t=tq,2 t=tq,2 q t=tq,2 G(β)=G β Jq =  . . .  2,3 0, 0,β+1  . . .. .   . . . .     ∂i ∂i ... ∂i  using the notation from [139]. When evaluating this function it might be  ∂βq,1 ∂βq,2 ∂βq,nq   t=tq,kq t=tq,kq t=tq,kq  more practical to rewrite G using other special functions   and the partial derivatives in the last two rows are given by 1, 1 ββ+1 G(β)=G3,0 β = F (β +1,β+ 1; β +2,β+ 2; β) 2,3 0, 0,β+1 2 2 2 − dˆg (β + 1) 2 Imq ηq,s βq,s , 1 q p, ∂Q dβ 2 ≤ ≤ π csc (πβ)  β=βq,s+1 Ψ(β)+π cot(πβ) + ln(β) = d˜g − Γ( β) ∂βq,s  − 2 Imp ηp+1,s βp+1,s ,q= p +1. dβ β=β2 p+1,s where   For 1 q p  2 ≤ ≤ ∞ k k (β + 1) 2F2(β +1,β+ 1; β +2,β+ 2; β)= ( 1) β 2 q 1 − − (β + k + 1) ∂W − dˆg k=0 =2 Imk Imq ηq,s βq,s ∂βq,s dβ 2 k=1 β=βq,s+1 is a special case of the hypergeometric function. These partial derivatives were found using software foe symbolic computation [1]. nq 2 dˆg dˆg Note that all η-parameters must be recalculated for each step and how +4Imq ηq,sβq,s ηq,s + ηq,k  dβ 2 dβ 2 2 this is done is detailed in Section 3.2.4. β=2βq,s+2 k=1 β=βq,s+βq,k+2  k =s      3.2.4 Calculating the η-parameters from the β-parameters and

p Suppose that we have nq 1 points (tq,k,iq,k) such that ∂W − =4 Imk ηp+1,sβp+1,s ∂βp+1,s tmq 1

nq For an AEF that interpolates these points it must be true that d˜g d˜g ηp+1,s + ηp+1,k  . dβ 2 dβ 2 2 q 1 nq β=2βp+1,s k=1 β=βp+1,s+βp+1,k −  k =s  I + I η x (t )βq,s = i ,k=1, 2,...,n 1. (110)   mk mq q,s q q,k q,k q −   k=1 s=1 The derivatives ofg ˆ(β) andg ˜(β) are

Since ηq,1 + ηq,2 + ...+ ηq,nq = 1 equation (110) can be rewritten as dˆg 1 eβ = 1+ Γ(β + 1) Ψ(β) ln(β) G(β) , (108) β nq 1 q 1 dβ e β − − − − βq,s βq,n βq,n β I η x (t ) x (t ) q = i x (t ) q I d˜g 1 e mq q,s q q,k − q q,k q,k − q q,k − ms = G(β) 1 , (109) s=1 s=1 dβ e ββ − (111)

114 115 Generalized Vandermonde matrices and determinants in 3.2. APPROXIMATION OF LIGHTNING DISCHARGE electromagnetic compatibility CURRENT FUNCTIONS where where Γ(β) is the Gamma function, Ψ(β) is the digamma function, see for ∂i ∂i ... ∂i example [2], and G(β) is a special case of the Meijer G-function and can be ∂βq,1 ∂βq,2 ∂βq,n t=tq,1 t=tq,1 q t=tq,1 deﬁned as  ∂i ∂i ... ∂i  ∂βq,1 ∂βq,2 ∂βq,n 3,0 1, 1 t=tq,2 t=tq,2 q t=tq,2 G(β)=G β Jq =  . . .  2,3 0, 0,β+1  . . .. .   . . . .     ∂i ∂i ... ∂i  using the notation from [139]. When evaluating this function it might be  ∂βq,1 ∂βq,2 ∂βq,nq   t=tq,kq t=tq,kq t=tq,kq  more practical to rewrite G using other special functions   and the partial derivatives in the last two rows are given by 1, 1 ββ+1 G(β)=G3,0 β = F (β +1,β+ 1; β +2,β+ 2; β) 2,3 0, 0,β+1 2 2 2 − dˆg (β + 1) 2 Imq ηq,s βq,s , 1 q p, ∂Q dβ 2 ≤ ≤ π csc (πβ)  β=βq,s+1 Ψ(β)+π cot(πβ) + ln(β) = d˜g − Γ( β) ∂βq,s  − 2 Imp ηp+1,s βp+1,s ,q= p +1. dβ β=β2 p+1,s where   For 1 q p  2 ≤ ≤ ∞ k k (β + 1) 2F2(β +1,β+ 1; β +2,β+ 2; β)= ( 1) β 2 q 1 − − (β + k + 1) ∂W − dˆg k=0 =2 Imk Imq ηq,s βq,s ∂βq,s dβ 2 k=1 β=βq,s+1 is a special case of the hypergeometric function. These partial derivatives were found using software foe symbolic computation [1]. nq 2 dˆg dˆg Note that all η-parameters must be recalculated for each step and how +4Imq ηq,sβq,s ηq,s + ηq,k  dβ 2 dβ 2 2 this is done is detailed in Section 3.2.4. β=2βq,s+2 k=1 β=βq,s+βq,k+2  k =s      3.2.4 Calculating the η-parameters from the β-parameters and p Suppose that we have nq 1 points (tq,k,iq,k) such that ∂W − =4 Imk ηp+1,sβp+1,s ∂βp+1,s tmq 1

114 115 Generalized Vandermonde matrices and determinants in 3.2. APPROXIMATION OF LIGHTNING DISCHARGE electromagnetic compatibility CURRENT FUNCTIONS

for k =1, 2,...,nq 1. This can be written as a matrix equation − β2 +1 (β2 +1)(1 τ) fk(τ)=2η1,k β1,kτ 1,k e 1,k − ln(τ)+1 τ , ˜ ˜ − Imq Xqη˜q = iq, (112) i1 β2 2 1,2 (β1,2+1)(1 τ1) q 1 η1,1 = τ1 e − ,η1,2 =1 η1,1, − Im1 − − βq,n ˜ q 2 2 η˜q = ηq,1 ηq,2 ... ηq,nq 1 , iq = iq,k xq(tq,k) Ims , β β (1 τ) − k − − g (τ)=2η β τ 2,k e 2,k − ln(τ)+1 τ , s=1 k 2,k 2,k − 2 i3 β 2 ˜ βq,s βq,nq 2,2 β1,2(1 τ3) Xq =˜xq(k, s)=xq(tq,k) xq(tq,k) , η2,1 = τ3 e − ,η2,2 =1 η2,1, k,s − Im1 − − 2 2 2 2 β = β1,1 +1 β1,2 +1 β2,1 β2,2 , and xq(t) given by (89). η = η1,1 η1,2 η2,1 η2,2 , When all βq,k, k =1, 2,...,nq are known then ηq,k, k =1, 2,...,nq 1 can 2 nq 1 − 2 β − Q(β, η) e 1,s = η γ β2 +2,β2 +1 be found by solving equation (112) and ηq,n =1 ηq,k. 1,s 2 1,s 2,s q I β1,s+1 − m1 s=1 β2 +1 k=1 1,s If we have k >n 1 data points then the parameters can be estimated q q 2 2 − β2,s with the least-squares solution to (112), more speciﬁcally the solution to e 2 2 2 + η2,s 2 Γ β2,s +1 γ β2,s +1,β2,s , 2β2,s+1 − s=1 β 2 2,s I X˜ X˜ η˜ = X˜ ˜i . mq q q q q q dˆg 2 Im1 η1,s β1,s ,q=1, ∂Q dβ β=β2 +1  1,s = 3.2.5 Explicit formulas for a single-peak AEF ∂βq,s  d˜g 2 Imq ηp,s β2,s ,q=2, dβ 2 t β=β2,s Consider the case where p = 1, n1 = n2 = 2 and τ = t . Then the explicit m1  formula for the AEF is  with derivatives ofg ˆ(β) andg ˜(β) given by (108) and (109),

2 2 2 2 β1,1+1 (β1,1+1)(1 τ) β1,2+1 (β1,2+1)(1 τ) i(τ) η1,1 τ e − + η1,2 τ e − , 0 τ 1, 2 2 2 2 2 2 2 2 = 2 2 2 2 ≤ ≤ (113) β = β1,1 + β1,2 +2 β1,1 + β1,2 +2 (β2,1 + β2,2)(β2,1 + β2,2) , I β2,1 β2,1(1 τ) β2,2 β2,2(1 τ) m1 η2,1 τ e − + η2,2 τ e − , 1 τ. 2 2 2 2 ≤ η = η1,1 η1,2 η2,1 η2,2 , η = (η1,1η1,2)(η1,1η1,2)(η2,1η2,2)(η2,1η2,2) , Assume that four datapoints, (ik,τk), k =1, 2, 3, 4, as well as the charge ∂ ∂ ∂ flow Q0 and specific energy W0, are known. W (β, η)= 2 β Q (2β, η)+β Q β, η . q,s q, (s 1 mod 2)+1 If we want to fit the AEF to this data using MLSM equation (107) gives ∂βq,s ∂βq,s − ∂βq,s f1(τ1) f2(τ1)0 0 3.2.6 Fitting to lightning discharge currents f (τ ) f (τ )0 0  1 2 2 2  00g1(τ3) g2(τ3) This section is based on Section 4 of Paper E  00g (τ ) g (τ )  J =  1 4 2 4  ,  ∂ ∂ ∂ ∂   Q(β, η) Q(β, η) Q(β, η) Q(β, η)  Some results of fitting the AEF to a few waveforms will be given. Some  ∂β ∂β ∂β ∂β   1,1 1,2 2,1 2,2  single-peak waveforms given by Heidler functions in IEC 62305-1 standard  ∂ ∂ ∂ ∂   W (β, η) W (β, η) W (β, η) W (β, η) [74] will be approximated using the AEF, and furthermore, fitting the multi- ∂β ∂β ∂β ∂β   1,1 1,2 2,1 2,2  peaked waveform to experimental data will be presented.   116 117 Generalized Vandermonde matrices and determinants in 3.2. APPROXIMATION OF LIGHTNING DISCHARGE electromagnetic compatibility CURRENT FUNCTIONS

Single-peak waveforms In this section some numerical results of fitting the AEF function to single- peak waveshapes are presented and compared with the corresponding fitting of the Heidler function. The AEF given by (113) is used to model few lightning current waveshapes whose parameters (rise/decay time ratio, T1/T2, peak current value, Im1, time to peak current, tm1, charge flow at the striking point, Q0, specific energy, W0, and time to 0.1Im1, t1) are given in table 1. Data points were chosen as follows:

(i ,τ )=(0.1 I ,t ), 1 1 m1 1 Figure 3.7: Fast-decaying waveshape represented by the AEF function. Here it is (i2,τ2)=(0.9 Im ,t2 = t1 +0.8 T1), fitted with the extra constraint 0 η 1 for all η-parameters. 1 ≤ ≤ (i ,τ )=(0.5 I ,t = t 0.1 T + T ), 3 3 m1 h 1 − 1 2 (i4,τ4)=(i(1.5 th), 1.5 th). The AEF representation of the waveshape denoted as the first positive stroke current in IEC 62305 standard [74], is shown in figure 3.5. Rising and decaying parts of the first negative stroke current from IEC 62305 standard [74] are shown in figure 3.6 - left and right, respectively. β and η parameters of both waveshapes optimized by the MLSM are given in table 3.1. We have also observed a so-called fast-decaying waveshape whose parameters are given in table 3.1. It’s representation using the AEF function is shown in figure 3.7, and corresponding β and η parameter values in table 3.1. Apart from the AEF function (solid line), the Heidler function representation of the same waveshapes (dashed line), and used data points (red solid circles) are also shown in the figures.

Multi-peaked AEF waveforms for measured data Figure 3.5: First-positive stroke represented by the AEF function. Here it is fitted with respect to both the data points as well as Q0 and W0. In this section the AEF will be constructed by fitting to measured data rather than approximation of the Heidler function. We will use data based on the measurements of flash number 23 in [152]. Two AEFs have been constructed, one by choosing peaks corresponding to local maxima, see figure 3.8, and one by choosing peaks corresponding to local maxima and local minima, see figure 3.9. For both AEFs there are two terms in each interval which means that for each peak there are two parameters that are chosen manually (time and current for each peak) and for each interval there are two parameters that are fitted using the MLSM. The AEF in figure 3.8 demonstrates that the AEF can handle cases where the function is not constant or monotonically increasing/decreasing Figure 3.6: First-negative stroke represented by the AEF function. Here it is fitted between peaks. This is only possible if the AEF has more than one term in with the extra constraint 0 η 1 for all η-parameters. ≤ ≤ the interval.

118 119 Generalized Vandermonde matrices and determinants in 3.2. APPROXIMATION OF LIGHTNING DISCHARGE electromagnetic compatibility CURRENT FUNCTIONS

118 119 Generalized Vandermonde matrices and determinants in 3.3. APPROXIMATION OF ELECTROSTATIC DISCHARGE electromagnetic compatibility CURRENTS

First-positive First-negative Fast-decaying stroke stroke T1/T2 10/350 1/200 8/20

tm1 [µs] 31.428 3.552 15.141 Im1 [kA] 200 100 0.001 Q0 [C] 100 // W0 [MJ/Ω] 10 // t1 [µs] 14.51.47 6.34 β1,1 0.114 1.84 7.666 β1,2 2.17 9.99 2.626 Figure 3.8: AEF ﬁtted to measurements from [152]. Here the peaks have been chosen β2,1 0.284 0.099 0.925 to correspond to local maxima in the measured data. β2,2 00.127 2.420 η 0.197 1 0 1,1 − η1,2 1.197 0 1 η2,1 10.401 0.2227 η2,2 00.599 0.7773

Table 3.1: AEF function’s parameters for some current waveshapes

Conclusions Figure 3.9: AEF fitted to measurements from [152]. Here the peaks have been chosen This section investigated the possibility to approximate, in general, multi- to correspond to local maxima and minima in the measured data. peaked lightning currents using an AEF function. Furthermore, existence of the analytical solution for the derivative and the integral of such function has been proven, which is needed in order to perform lightning electromagnetic and [81]. field (LEMF) calculations based on it. Two single-peak Standard IEC 62305-1 waveforms, and a fast-decaying one, have been represented using a variation of the proposed AEF function 3.3 Approximation of electrostatic discharge cur- (113). The estimation of their parameters has been performed applying the rents MLS method using two pairs of data points for each function part (rising and decaying). The results show that there are several factors that need to This section is based on Paper F be taken into consideration to get the best possible approximation of a given waveform. The accuracy of the approximation varies with the chosen data points and the number of terms in the AEF. In several cases the two-term In this paper we analyse the applicability of the generalized multi-peaked sum converged towards a single term sum. This can probably be improved AEF function to representation of ESD currents by interpolation of data by choosing the number of terms and the number and placement of data points chosen according to a D-optimal design. This is illustrated through points in other ways which the authors intend to examine further. Further two examples corresponding to modelling of the IEC Standard 61000-4-2 examples of fitted (single- and multi-peaked) waveforms can be found in [108] waveshape, [72,73] and an experimentally measured ESD current from [87].

120 121 Generalized Vandermonde matrices and determinants in 3.3. APPROXIMATION OF ELECTROSTATIC DISCHARGE electromagnetic compatibility CURRENTS

First-positive First-negative Fast-decaying stroke stroke T1/T2 10/350 1/200 8/20 tm1 [µs] 31.428 3.552 15.141 Im1 [kA] 200 100 0.001 Q0 [C] 100 // W0 [MJ/Ω] 10 // t1 [µs] 14.51.47 6.34 β1,1 0.114 1.84 7.666 β1,2 2.17 9.99 2.626 Figure 3.8: AEF ﬁtted to measurements from [152]. Here the peaks have been chosen β2,1 0.284 0.099 0.925 to correspond to local maxima in the measured data. β2,2 00.127 2.420 η 0.197 1 0 1,1 − η1,2 1.197 0 1 η2,1 10.401 0.2227 η2,2 00.599 0.7773

Table 3.1: AEF function’s parameters for some current waveshapes

120 121 Generalized Vandermonde matrices and determinants in 3.3. APPROXIMATION OF ELECTROSTATIC DISCHARGE electromagnetic compatibility CURRENTS 61000-4-2 © IEC:1995+A1:1998 – 43 – +A2:2000

Voltage [kV] Ipeak [A] tr [ns] I30 [A] I60 [A]

27.5 15% 0.8 25% 4.0 30% 2.0 30% ± ± ± ± 4 15.0 15% 0.8 25% 8.0 30% 4.0 30% ± ± ± ± 6 22.5 15% 0.8 25% 12.0 30% 6.0 30% ± ± ± ± 8 30.0 15% 0.8 25% 16.0 30% 8.0 30% ± ± ± ± Table 3.2: IEC 61000-4-2 Standard ESD Current and its Key Parameters, [73].

3.3.1 IEC 61000-4-2 Standard current waveshape tr ESD generators used in testing of the equipment and devices should be able Figure 3.10: ValuesIllustration are given in table 2. of the IEC 61000-4-2 Standard ESD current and its key to reproduce the same ESD current waveshape each time. This repeata- parameters,Figure 3 –[73]. Typical waveform of the output current of the ESD generator bility feature is ensured if the design is carried out in compliance with the requirements defined in the IEC 61000-4-2 Standard [73]. radiated field generated by the ESD current can change abruptly at Among other relevant issues, the Standard includes graphical represen- that moment. tation of the typical ESD current, figure 3.10, and also defines, for a given the ESD current function must be time-integrable in order to allow test level voltage, required values of ESD current’s key parameters. These • are listed in table 3.2 for the case of the contact discharge, where: numerical calculation of the ESD radiated fields.

Ipeak is the ESD current initial peak; Multi-peaked analytically extended function • t is the rising time defined as the difference between time moments A so-called multi-peaked analytically extended function (AEF) has been • r corresponding to 10% and 90% of the current peak Ipeak, figure 3.10; proposed and applied to lightning discharge current modelling in Section 3.1 and [107]. Initial considerations on applying the function to ESD currents I and I is the ESD current values calculated for time periods of 30 • 30 60 have also been made in [108]. and 60 ns, respectively, starting from the time point corresponding to The AEF consists of scaled and translated power-exponential functions, 10% of Ipeak, figure 3.10. 1 t β that is functions of the form x(β; t)= te − , see Definition 3.1. Here we define the AEF with p peaks as Important Features of ESD Currents q 1 nq − i(t)= I + I η x (t), (114) Various mathematical expressions have been introduced in the literature mk mq q,k q,k k=1 k=1 that can be used for representation of the ESD currents, either the IEC

61000-4-2 Standard one [73], or experimentally measured ones, e.g. [47]. for tmq 1 t tmq ,1 q p, and − ≤ ≤ ≤ ≤ These functions are to certain extent in accordance with the requirements p np+1 given in table 3.2. Furthermore, they have to satisfy the following: Imk ηp+1,kxp+1,k(t), (115) the value of the ESD current and its ﬁrst derivative must be equal to k=1 k=1 • zero at the moment t = 0, since neither the transient current nor the for t t. mp ≤

122 123 Generalized Vandermonde matrices and determinants in 3.3. APPROXIMATION OF ELECTROSTATIC DISCHARGE electromagnetic compatibility CURRENTS 61000-4-2 © IEC:1995+A1:1998 – 43 – +A2:2000

Voltage [kV] Ipeak [A] tr [ns] I30 [A] I60 [A]

122 123 Generalized Vandermonde matrices and determinants in 3.3. APPROXIMATION OF ELECTROSTATIC DISCHARGE electromagnetic compatibility CURRENTS

The current value of the first peak is denoted by Im1 , the difference 3.3.2 D-Optimal approximation for exponents given by a between each pair of subsequent peaks by Im2 ,Im3 ,...,Imp , and their cor- class of arithmetic sequences responding times by t ,t ,...,t . In each time interval q, with 1 q m1 m2 mp ≤ ≤ It can be desirable to minimize the number of points used when constructing p + 1, the number of terms is given by nq,0real number. Note that  tmq in order to guarantee continuity of the AEF derivative the condition is that Remark 3.2. When previously applying the AEF, see Section 3.1.1, all k>c. exponents (β-parameters) of the AEF were set to β2+1 in order to guarantee Then in each interval we want an approximation of the form 1 t 1 that the derivative of the AEF is continuous. Here this condition will be and by setting z(t)=(te − ) c then satisfied in a different manner. n k+i 1 Since the AEF is a linear function of elementary functions its derivative y(t)= ηiz(t) − . and integral can be found using standard methods. For explicit formulae i=1 please refer to Theorem 3.1-3.3. If we have n sample points, ti, i =1,...,n, then the Fischer information Previously, the authors have fitted AEF functions to lightning discharge matrix, M, of this system is M = UU where currents and ESD currents using the Marquardt least square method but k k k z(t1) z(t2) ... z(tn) have noticed that the obtained result varies greatly depending on how the k+1 k+1 k+1 z(t1) z(t2) ... z(tn) waveforms are sampled. This is problematic, especially since the methodol- U =  . . . .  ...... ogy becomes computationally demanding when applied to large amounts of   z(t )k+n 1 z(t )k+n 1 ... z(t )k+n 1 data. Here we will try one way to minimize the data needed but still enough  1 − 2 − n −  to get an as good approximation as possible.   Thus if we want to maximize det(M) = det(U)2 it is sufficient to maximize The method examined here will be based on D-optimality of a regression 1 t 1 or minimize the determinant det(U). Set z(t )=(t e i ) c = x then model. A D-optimal design is found by choosing sample points such that the i i − i determinant of the Fischer information matrix of the model is minimized. un(t1,...,tn) = det(U) For a standard linear regression model this is also equivalent, by the so- n called Kiefer-Wolfowitz equivalence criterion, to G-optimality which means = xk (xj xi) . (117) that the maximum of the prediction variance will be minimized. These are  −  k=1 1 i

124 125 Generalized Vandermonde matrices and determinants in 3.3. APPROXIMATION OF ELECTROSTATIC DISCHARGE electromagnetic compatibility CURRENTS

The current value of the first peak is denoted by Im1 , the difference 3.3.2 D-Optimal approximation for exponents given by a between each pair of subsequent peaks by Im2 ,Im3 ,...,Imp , and their cor- class of arithmetic sequences responding times by t ,t ,...,t . In each time interval q, with 1 q m1 m2 mp ≤ ≤ It can be desirable to minimize the number of points used when constructing p + 1, the number of terms is given by nq,0c. exponents (β-parameters) of the AEF were set to β2+1 in order to guarantee Then in each interval we want an approximation of the form 1 t 1 that the derivative of the AEF is continuous. Here this condition will be and by setting z(t)=(te − ) c then satisfied in a different manner. n k+i 1 Since the AEF is a linear function of elementary functions its derivative y(t)= ηiz(t) − . and integral can be found using standard methods. For explicit formulae i=1 please refer to Theorem 3.1-3.3. If we have n sample points, ti, i =1,...,n, then the Fischer information Previously, the authors have fitted AEF functions to lightning discharge matrix, M, of this system is M = UU where currents and ESD currents using the Marquardt least square method but k k k z(t1) z(t2) ... z(tn) have noticed that the obtained result varies greatly depending on how the k+1 k+1 k+1 z(t1) z(t2) ... z(tn) waveforms are sampled. This is problematic, especially since the methodol- U =  . . . .  ...... ogy becomes computationally demanding when applied to large amounts of   z(t )k+n 1 z(t )k+n 1 ... z(t )k+n 1 data. Here we will try one way to minimize the data needed but still enough  1 − 2 − n −  to get an as good approximation as possible.   Thus if we want to maximize det(M) = det(U)2 it is sufficient to maximize The method examined here will be based on D-optimality of a regression 1 t 1 or minimize the determinant det(U). Set z(t )=(t e i ) c = x then model. A D-optimal design is found by choosing sample points such that the i i − i determinant of the Fischer information matrix of the model is minimized. un(t1,...,tn) = det(U) For a standard linear regression model this is also equivalent, by the so- n called Kiefer-Wolfowitz equivalence criterion, to G-optimality which means = xk (xj xi) . (117) that the maximum of the prediction variance will be minimized. These are  −  k=1 1 i

124 125 Generalized Vandermonde matrices and determinants in 3.3. APPROXIMATION OF ELECTROSTATIC DISCHARGE electromagnetic compatibility CURRENTS

The Lambert W is multivalued but since we are only interested in real- for j =1,...,n 1. Since f(x) is a polynomial of degree n that has x =1 − valued solutions we are restricted to the main branches W0 and W 1. Since as a root then equation (119) implies − W0 1 and W 1 1 the two branches correspond to the rising and ≥− − ≤− f(x) decaying parts of the AEF respectively. We will deal with the details of xf (x)+2kf (x)=c x 1 finding the correct points for the two parts separately. − where c is some constant. Set f(x)=(x 1)g(x) and the resulting differential − equation is D-Optimal interpolation on the rising part x(x 1)g (x) + ((2k + 2)x 2k)g (x)+(2k c)g(x)=0. To find the D-optimal points on the rising part we use Theorem 3.6. − − − The constant c can be found by examining the terms with degree n 1 and − Theorem 3.6. The determinant is given by c =2k +(n 1)(2k + n), thus − n x(1 x)g (x)+(2k (2k + 2)x)g (x) u (k; x ,...,x )= xk (x x ) − − n 1 n i  j − i  +(n 1)(2k + n)g(x)=0. (120) i=1 1 i

126 127 Generalized Vandermonde matrices and determinants in 3.3. APPROXIMATION OF ELECTROSTATIC DISCHARGE electromagnetic compatibility CURRENTS

D-Optimal interpolation on the decaying part guarantees that 1 1) when xi = xmax yi and · In this section some results of applying the described scheme to two dif- xn = xmax, or some permutation thereof. ferent waveforms will be presented. The two waveforms are the Standard Proof. This theorem follows from Theorem 3.6 combined with the fact that ESD current given in IEC 61000-4-2 [73] and a waveform from experimental un(k; x1,...,xn) is a homogeneous polynomial. Since measurements from [87]. The values of n, k and c have been chosen by manual experimentation n(n 1) k+ − un(k; b x1,...,c xn)=b 2 un(k; x1,...,xn) and since both waveforms are given as data rather than explicit functions · · · the D-optimal points have been calculated and then the closest available n if (x1,...,xn) is an extreme point in [0, 1] then (b x1,...,bxn) is an extreme data points have been chosen. n · · point in [0,b] . Thus by Theorem 3.6 the points given by xi = xmax yi will Note that the quality of the results can vary greatly depending on how n · maximize or minimize wn(k; x1,...,xn) on [0,xmax] . the k and m parameters are chosen before this type of approximation scheme is applied, and in practice a strategy for choosing the values eﬀectively should Remark 3.4. It is in many cases possible to ensure the condition 1 < be devised. In many cases increasing the number of interpolation points, n, (2k 1,0) x y without actually calculating the roots of P − (1 2y). In the improves the results but there are many cases where the interpolation is not max · 1 n 1 − literature on orthogonal polynomials there are many− expressions for upper stable. and lower bounds of the roots of the Jacobi polynomials. For instance in [38] an upper bound on the largest root of a Jacobi polynomial is given that in Interpolated AEF representing the IEC 61000-4-2 Standard cur- our case can be rewritten as rent 3 In this section we present the results of ﬁtting 2- and 3-peak AEF to the y1 > 1 − 4k2 +2kn + n2 k 2n +1 Standard ESD current given in IEC 61000-4-2. Data points which are used − − in the optimization procedure are manually sampled from the graphically and thus 3 1 given Standard [73] current function. The peak currents and corresponding 1 > − 4k2 +2kn + n2 k 2n +1 x times are also extracted, and the results of D-optimal interpolation with 2 − − max 128 129 Generalized Vandermonde matrices and determinants in 3.3. APPROXIMATION OF ELECTROSTATIC DISCHARGE electromagnetic compatibility CURRENTS

0 0 Table 3.3: Parameters’ values of the multi-peaked AEFs representing the IEC 61000- 0 2 4 6 8 0 2 4 6 8 4-2 Standard waveshape. t [s] #10-8 t [s] #10-8 (a) (b) with Figure 3.11: AEF representing the IEC 61000-4-2 Standard ESD current waveshape I1 = 31.365 A,I2 =6.854 A,nH =4.036, for 4kV with (a) 2 peaks, (b) 3 peaks. For parameters see Table 3.3. τ1 =1.226 ns,τ2 =1.359 ns,τ3 =3.982 ns,τ4 = 28.817 ns. Note that this function does not reproduce the second local minimum but that all three AEF functions can reproduce all local minima and maxima 3-peaked AEF representing measured data (to a modest degree of accuracy) when suitable values for the n, k and m parameters are chosen. In this section we present the results of fitting a 1-, 2- and a 3-peaked AEF to a waveform from experimental measurements from [87]. The result is also 3.3.4 Summary of ESD modelling compared to a common type of function used for modelling ESD current, Here we examined a mathematical model for representation of ESD currents, also from [87]. either from the IEC 61000-4-2 Standard [73], or experimentally measured In figures 3.12 (a), 3.12 (b) and 3.13 the results of the interpolation ones. The model has been proposed and successfully applied to lightning of D-optimal points for certain parameters are shown together with the current modelling in Section 3.2 and [107] and named the multi-peaked measured data, as well as a sum of two Heidler functions that was fitted to analytically extended function (AEF). the experimental data in [87]. This function is given by It conforms to the requirements for the ESD current and its first derivative, which are imposed by the Standard [73] stating that they must be equal n n t H t H to zero at moment t = 0. Furthermore, the AEF function is time-integrable, τ1 t τ3 t i(t)=I e− τ2 + I e− τ4 , (121) see Section 3.1.1, which is necessary for numerical calculation of radiated 1 nH 2 nH 1+ t 1+ t τ1 τ3 fields originating from the ESD current. 130 131 Generalized Vandermonde matrices and determinants in 3.3. APPROXIMATION OF ELECTROSTATIC DISCHARGE electromagnetic compatibility CURRENTS and 3 peaks are illustrated, see figure 3.11 (a) and 3.11 (b). The parameters Local maxima and minima and corresponding times are listed in table 3.3. In the illustrated examples a fairly good fit is found extracted from the IEC 61000-4-2, [73] but typically areas with steep rise and the decay part are somewhat more Imax1 = 15 [A] Imin1 = 7.1484 [A] Imax2 = 9.0921 [A] difficult to fit with good accuracy than the other parts of the waveform. tmax1 = 6.89 [ns] tmin1 = 12.85 [ns] tmax2 = 25.54 [ns] Parameters of interpolated AEF shown in figure 3.11 (a) 1 1 Interval n kc IEC 61000-4-2 IEC 61000-4-2 0 t tmax1 3 35 1 Peaks Peaks ≤ ≤ 0.8 0.8 tmax1 t tmax2 3 32 Interpolated points Interpolated points ≤ ≤ t

Local maxima and corresponding times extracted from [87, ﬁgure 3] Imax1 = 7.37 [A] Imax2 = 5.02 [A] Imax3 = 3.82 [A] Measured data Measured data tmax1 = 1.23 [ns] tmax2 = 6.39 [ns] tmax3 = 15.5 [ns] Parameters of interpolated AEF shown in ﬁgure 3.12 (a) Two Heidler Two Heidler 6 Peaks 6 Peaks Interval nkc Interpolated points Interpolated points 0 t t 12 8 0.8 1-peak AEF 2-peaked AEF ≤ ≤ max3

Here we consider how the model can be ﬁtted to a waveform using D- optimal interpolation and the resulting methodology is illustrated on the IEC 61000-4-2 Standard waveform [73] and experimental data from [87]. Measured data Two Heidler The resulting methodology can give fairly accurate results even with a 6 Peaks modest number of interpolated points but strategies for choosing some of Interpolated points the involved parameters should be further investigated. 3-peaked AEF ) t 4 ( i

0 2 4 6 8 t [s] #10-8

Figure 3.13: 3-peaked AEF interpolated to D-optimal points chosen from measured ESD current from [87, ﬁgure 3]. Parameters are given in Table 3.4.

132 133 Generalized Vandermonde matrices and determinants in 3.3. APPROXIMATION OF ELECTROSTATIC DISCHARGE electromagnetic compatibility CURRENTS

0 2 4 6 8 t [s] #10-8

Figure 3.13: 3-peaked AEF interpolated to D-optimal points chosen from measured ESD current from [87, ﬁgure 3]. Parameters are given in Table 3.4.

132 133 REFERENCES

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