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, appli- cations 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 deter- minant 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 stan- dard 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 dis- charge immunity as well as some experimentally measured data of similar phenomena. Copyright © Karl Lundengård, 2017 ISBN 978-91-7485-312-4 ISSN 1651-9256 Printed by E-Print AB, Stockholm, Sweden 1 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, appli- cations 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 deter- minant 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 stan- dard 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 dis- charge immunity as well as some experimentally measured data of similar phenomena. 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 sug- gested the research pursued in this thesis. From him and my co-supervisor Professor Anatoliy Malyarenko I have learned invaluable lessons about math- ematics 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 Ranˇci´cwhose conscientiousness, work ethic and patience when introducing me to the world of electromagnetic compatibility and developing and evalu- ating 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 M¨alardalen University that I have worked alongside with. I have found the environment at the School for Education, Culture and Communication at M¨alardalen 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 First I am very grateful to my closest colleagues. My main supervisor Pro- fessor Sergei Silvestrov introduced the Vandermonde matrix to me and sug- gested the research pursued in this thesis. From him and my co-supervisor Professor Anatoliy Malyarenko I have learned invaluable lessons about math- ematics 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 Ranˇci´cwhose conscientiousness, work ethic and patience when introducing me to the world of electromagnetic compatibility and developing and evalu- ating 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 M¨alardalen University that I have worked alongside with. I have found the environment at the School for Education, Culture and Communication at M¨alardalen 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 Generalized Vandermonde matrices and determinants in electromagnetic compatibility Popul¨arvetenskaplig sammanfattning Popular-science summary Denna licentiatuppsats behandlar tv˚aolika ¨amnen, optimering av determi- This licentiate thesis discusses two different topics, optimisation of the Van- nanten av Vandermonde-matrisen ¨over olika volymer i olika dimensioner och dermonde determinant over different volumes in various dimensions and how hur en viss klass av funktioner kan anv¨andas f¨or att approximera str¨ommen a certain class of functions can be used to approximate the current in elec- i elektrostatiska urladdningar som anv¨ands f¨or att s¨akerst¨alla 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 ¨aren matris d¨arraderna (eller kolonnerna) ges A Vandermonde matrix is a matrix with rows (or column) given by av stigande potenser och s˚adana matriser f¨orekommer i m˚anga olika sam- increasing powers and such matrices appear in many different circumstances, manhang, b˚ade inom abstrakt matematik och till¨ampningar 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 till¨ampningar av den och n˚agra besl¨aktade 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 f¨or att anpassa en matematisk beskrivning till given data, t.ex. methods for fitting a mathematical description to collected data from for fr˚anexperimentella m¨atningar. example experimental measurements. Determinanten av en matris ¨arett tal som ber¨aknas fr˚anmatrisens el- The determinant of a matrix is a number calculated from the elements of ement och som p˚aett kompakt s¨att 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.