A New Algorithm for Computing the Square Root of a Matrix
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Extremely Accurate Solutions Using Block Decomposition and Extended Precision for Solving Very Ill-Conditioned Equations
Extremely accurate solutions using block decomposition and extended precision for solving very ill-conditioned equations †Kansa, E.J.1, Skala V.2, and Holoborodko, P.3 1Convergent Solutions, Livermore, CA 94550 USA 2Computer Science & Engineering Dept., Faculty of Applied Sciences, University of West Bohemia, University 8, CZ 301 00 Plzen, Czech Republic 3Advanpix LLC, Maison Takashima Bldg. 2F, Daimachi 10-15-201, Yokohama, Japan 221-0834 †Corresponding author: [email protected] Abstract Many authors have complained about the ill-conditioning associated the numerical solution of partial differential equations (PDEs) and integral equations (IEs) using as the continuously differentiable Gaussian and multiquadric continuously differentiable (C ) radial basis functions (RBFs). Unlike finite elements, finite difference, or finite volume methods that lave compact local support that give rise to sparse equations, the C -RBFs with simple collocation methods give rise to full, asymmetric systems of equations. Since C RBFs have adjustable constent or variable shape parameters, the resulting systems of equations that are solve on single or double precision computers can suffer from “ill-conditioning”. Condition numbers can be either the absolute or relative condition number, but in the context of linear equations, the absolute condition number will be understood. Results will be presented that demonstrates the combination of Block Gaussian elimination, arbitrary arithmetic precision, and iterative refinement can give remarkably accurate numerical salutations to large Hilbert and van der Monde equation systems. 1. Introduction An accurate definition of the condition number, , of the matrix, A, is the ratio of the largest to smallest absolute value of the singular values, { i}, obtained from the singular value decomposition (SVD) method, see [1]: (A) = maxjjminjj (1) This definition of condition number will be used in this study. -
Phi)Ve Is a Magic Number
(Phi)ve is a Magic Number James J. Solberg Professor Emeritus, Purdue University Written for the 13th Gathering for Gardner, 2018 Martin Gardner would surely have noted the centennial significance of January 6 of this year. On that day, the digits of the date 1/6/18 matched 1.618 – the leading digits of the famous golden ratio number Phi. If you wanted three more digits, you could have set your alarm to celebrate at 3:39 am or at a more reasonable 3:39 pm when the eight digits of that moment aligned to 1.6180339. I am sure that Martin would have also noticed the puns in my title. The first word refers to both Phi and five, and the fourth word refers to magic squares. My goal is to reveal several surprising connections between the two values and their powers using both Fibonacci numbers and five-by-five magic squares. A warning: if ordinary word puns tend to make you numb, then my mathematical puns will make you number. As you presumably know, the golden ratio number Phi (or �) shows up in nature, in art, in architecture, and even beauty salons, as well as mathematics. If you are not familiar with �, you are in for a treat. There are plenty of books and websites that explain the fascinating properties and ubiquitous nature of this mysterious number. I want to use a few of the many identities, so here is a short summary. One of the ways to determine the golden ratio is to find the place to divide a line so that the ratio of the length of the longer segment to the shorter is the same as the ratio of the whole line to the longer segment. -
A Study of Convergence of the PMARC Matrices Applicable to WICS Calculations
A Study of Convergence of the PMARC Matrices applicable to WICS Calculations /- by Amitabha Ghosh Department of Mechanical Engineering Rochester Institute of Technology Rochester, NY 14623 Final Report NASA Cooperative Agreement No: NCC 2-937 Presented to NASA Ames Research Center Moffett Field, CA 94035 August 31, 1997 Table of Contents Abstract ............................................ 3 Introduction .......................................... 3 Solution of Linear Systems ................................... 4 Direct Solvers .......................................... 5 Gaussian Elimination: .................................. 5 Gauss-Jordan Elimination: ................................ 5 L-U Decompostion: ................................... 6 Iterative Solvers ........................................ 6 Jacobi Method: ..................................... 7 Gauss-Seidel Method: .............................. .... 7 Successive Over-relaxation Method: .......................... 8 Conjugate Gradient Method: .............................. 9 Recent Developments: ......................... ....... 9 Computational Efficiency .................................... 10 Graphical Interpretation of Residual Correction Schemes .................... 11 Defective Matrices ............................. , ......... 15 Results and Discussion ............ ......................... 16 Concluding Remarks ...................................... 19 Acknowledgements ....................................... 19 References .......................................... -
Applied Linear Algebra
APPLIED LINEAR ALGEBRA Giorgio Picci November 24, 2015 1 Contents 1 LINEAR VECTOR SPACES AND LINEAR MAPS 10 1.1 Linear Maps and Matrices . 11 1.2 Inverse of a Linear Map . 12 1.3 Inner products and norms . 13 1.4 Inner products in coordinate spaces (1) . 14 1.5 Inner products in coordinate spaces (2) . 15 1.6 Adjoints. 16 1.7 Subspaces . 18 1.8 Image and kernel of a linear map . 19 1.9 Invariant subspaces in Rn ........................... 22 1.10 Invariant subspaces and block-diagonalization . 23 2 1.11 Eigenvalues and Eigenvectors . 24 2 SYMMETRIC MATRICES 25 2.1 Generalizations: Normal, Hermitian and Unitary matrices . 26 2.2 Change of Basis . 27 2.3 Similarity . 29 2.4 Similarity again . 30 2.5 Problems . 31 2.6 Skew-Hermitian matrices (1) . 32 2.7 Skew-Symmetric matrices (2) . 33 2.8 Square roots of positive semidefinite matrices . 36 2.9 Projections in Rn ............................... 38 2.10 Projections on general inner product spaces . 40 3 2.11 Gramians. 41 2.12 Example: Polynomial vector spaces . 42 3 LINEAR LEAST SQUARES PROBLEMS 43 3.1 Weighted Least Squares . 44 3.2 Solution by the Orthogonality Principle . 46 3.3 Matrix least-Squares Problems . 48 3.4 A problem from subspace identification . 50 3.5 Relation with Left- and Right- Inverses . 51 3.6 The Pseudoinverse . 54 3.7 The Euclidean pseudoinverse . 63 3.8 The Pseudoinverse and Orthogonal Projections . 64 3.9 Linear equations . 66 4 3.10 Unfeasible linear equations and Least Squares . 68 3.11 The Singular value decomposition (SVD) . -
Radical Expressions and Equations”, Chapter 8 from the Book Beginning Algebra (Index.Html) (V
This is “Radical Expressions and Equations”, chapter 8 from the book Beginning Algebra (index.html) (v. 1.0). This book is licensed under a Creative Commons by-nc-sa 3.0 (http://creativecommons.org/licenses/by-nc-sa/ 3.0/) license. See the license for more details, but that basically means you can share this book as long as you credit the author (but see below), don't make money from it, and do make it available to everyone else under the same terms. This content was accessible as of December 29, 2012, and it was downloaded then by Andy Schmitz (http://lardbucket.org) in an effort to preserve the availability of this book. Normally, the author and publisher would be credited here. However, the publisher has asked for the customary Creative Commons attribution to the original publisher, authors, title, and book URI to be removed. Additionally, per the publisher's request, their name has been removed in some passages. More information is available on this project's attribution page (http://2012books.lardbucket.org/attribution.html?utm_source=header). For more information on the source of this book, or why it is available for free, please see the project's home page (http://2012books.lardbucket.org/). You can browse or download additional books there. i Chapter 8 Radical Expressions and Equations 1256 Chapter 8 Radical Expressions and Equations 8.1 Radicals LEARNING OBJECTIVES 1. Find square roots. 2. Find cube roots. 3. Find nth roots. 4. Simplify expressions using the product and quotient rules for radicals. Square Roots The square root1 of a number is that number that when multiplied by itself yields the original number. -
Surface Fitting and Some Associated Confidence Procedures
SURFACE FITTING AND SOME ASSOCIATED CONFIDENCE PROCEDURES by ARNOLD LEON SCHROEDER, JR . A THESIS submitted to OREGON STATE UNIVERSITY in partial fulfillment of the requirements for the degree of MASTER OF SCIENCE June 1962 APPROVED: Redacted for Privacy Associate Professor of Statistics In Charge of Major Redacted for Privacy Chaii n of the Departrnent of Statistics Redacted for Privacy Chairrnan Cradua 'e Cornrnittee Redacted for Privacy of Graduate Date thesis is presented May 15, 1962 Typed by Jolene 'Wuest TABLE OF CONTENTS CHAPTER PAGE 1 INTRODUCTION .. ••.•... • . 1 2 REGRESSION ANALYSIS APPLIED TO ORE DISTRIBUTION ..• .... 4 3 FITTING DATA TO REGRESSION MODELS . ....•.. • 7 4 CONFIDENCE LIMITS OF REGRESSION COEFFICIENTS .... • •..... ... 13 5 CONFIDENCE PROCEDURES FOR THE DISCRIMINANT . 16 6 CONFIDENCE LIMITS FOR LOCATION OF MAXIMUM . .. 19 7 EXAMPLE ANALYSES 25 BIBLIOGRAPHY 32 APPENDIX . ...... 34 SURFACE FITTING AND SOME ASSOCIATED CONFIDENCE PROCEDURES CHAPTER 1 INTRODUCTION The use of statistics in geology has expanded rapidly, until today we have statistical analysis being applied in many branches of geology. One of the main reasons for this expansion is the enormous increase in the amount of available numerical data, especially in the field ofmining where thousands of sample assays are often taken in the course of mining operations. This thesis is one approach to the problem ofhow to analyze these vast amounts of sample data and is specifically a des crip tion of the analysis of the distribution of ore in the veins of two large mixed-metal ore deposits . However, the statistical techniques pre sented are applicable to geological data in general, as well as to the specific problem of analyzing ore distribution in metalliferous veins. -
Matrix Computations
Chapter 5 Matrix Computations §5.1 Setting Up Matrix Problems §5.2 Matrix Operations §5.3 Once Again, Setting Up Matrix Problems §5.4 Recursive Matrix Operations §5.5 Distributed Memory Matrix Multiplication The next item on our agenda is the linear equation problem Ax = b. However, before we get into algorithmic details, it is important to study two simpler calculations: matrix-vector mul- tiplication and matrix-matrix multiplication. Both operations are supported in Matlab so in that sense there is “nothing to do.” However, there is much to learn by studying how these com- putations can be implemented. Matrix-vector multiplications arise during the course of solving Ax = b problems. Moreover, it is good to uplift our ability to think at the matrix-vector level before embarking on a presentation of Ax = b solvers. The act of setting up a matrix problem also deserves attention. Often, the amount of work that is required to initialize an n-by-n matrix is as much as the work required to solve for x. We pay particular attention to the common setting when each matrix entry aij is an evaluation of a continuous function f(x, y). A theme throughout this chapter is the exploitation of structure. It is frequently the case that there is a pattern among the entries in A which can be used to reduce the amount of work. The fast Fourier transform and the fast Strassen matrix multiply algorithm are presented as examples of recursion in the matrix computations. The organization of matrix-matrix multiplication on a ring of processors is also studied and gives us a nice snapshot of what algorithm development is like in a distributed memory environment. -
Newton's Method for the Matrix Square Root*
MATHEMATICS OF COMPUTATION VOLUME 46, NUMBER 174 APRIL 1986, PAGES 537-549 Newton's Method for the Matrix Square Root* By Nicholas J. Higham Abstract. One approach to computing a square root of a matrix A is to apply Newton's method to the quadratic matrix equation F( X) = X2 - A =0. Two widely-quoted matrix square root iterations obtained by rewriting this Newton iteration are shown to have excellent mathematical convergence properties. However, by means of a perturbation analysis and supportive numerical examples, it is shown that these simplified iterations are numerically unstable. A further variant of Newton's method for the matrix square root, recently proposed in the literature, is shown to be, for practical purposes, numerically stable. 1. Introduction. A square root of an n X n matrix A with complex elements, A e C"x", is a solution X e C"*" of the quadratic matrix equation (1.1) F(X) = X2-A=0. A natural approach to computing a square root of A is to apply Newton's method to (1.1). For a general function G: CXn -* Cx", Newton's method for the solution of G(X) = 0 is specified by an initial approximation X0 and the recurrence (see [14, p. 140], for example) (1.2) Xk+l = Xk-G'{XkylG{Xk), fc = 0,1,2,..., where G' denotes the Fréchet derivative of G. Identifying F(X+ H) = X2 - A +(XH + HX) + H2 with the Taylor series for F we see that F'(X) is a linear operator, F'(X): Cx" ^ C"x", defined by F'(X)H= XH+ HX. -
The Collage of Backgrounds and the Golden Ratio
The Collage of Backgrounds and the Golden Ratio Supplemental Information about Sacred Geometry and the Art of Doug Craft All Materials and Art © 2008 Doug Craft Doug Craft [email protected] http://www.DougCraftFineArt.com The Collage of Backgrounds and the Golden Ratio © 2008 Doug Craft Introduction While my earlier collages were figurative with a clear visual distinction between foreground objects and background, my work since 1997 has been exploring a new concept that I call The Collage of Backgrounds. This technique basically arranges natural abstract images - what would be considered backgrounds in my figurative collages - according to simple geometric rules based on Sacred Geometry and the Golden Ratio. This idea can be seen in the collage to the right, AIR Elements Proportional Offset Square 2004-003. The following is an explanation and visual guide to the Golden Ratio and the shapes and geometric forms that I use in my collages, montages, and artwork in other media. Sacred Geometry and The Golden Ratio, Φ When I refer to Sacred Geometry, I am talking about geometry that is derived from or directly related to the structure of nature. Our universe is structured in a highly complex yet sublimely ordered manner. This is a truth that is readily felt by sensitive people, and has also been demonstrated by science and mathematics. Structural forms seen at the microscopic level are repeated at other scales, and the laws of fractional symmetry appear to apply throughout. So, geometry that refers to the structural unity of nature is a powerful metaphor for the mystery of life, and thus sacred. -
Sensitivity and Stability Analysis of Nonlinear Kalman Filters with Application to Aircraft Attitude Estimation
Graduate Theses, Dissertations, and Problem Reports 2013 Sensitivity and stability analysis of nonlinear Kalman filters with application to aircraft attitude estimation Matthew Brandon Rhudy West Virginia University Follow this and additional works at: https://researchrepository.wvu.edu/etd Recommended Citation Rhudy, Matthew Brandon, "Sensitivity and stability analysis of nonlinear Kalman filters with application ot aircraft attitude estimation" (2013). Graduate Theses, Dissertations, and Problem Reports. 3659. https://researchrepository.wvu.edu/etd/3659 This Dissertation is protected by copyright and/or related rights. It has been brought to you by the The Research Repository @ WVU with permission from the rights-holder(s). You are free to use this Dissertation in any way that is permitted by the copyright and related rights legislation that applies to your use. For other uses you must obtain permission from the rights-holder(s) directly, unless additional rights are indicated by a Creative Commons license in the record and/ or on the work itself. This Dissertation has been accepted for inclusion in WVU Graduate Theses, Dissertations, and Problem Reports collection by an authorized administrator of The Research Repository @ WVU. For more information, please contact [email protected]. SENSITIVITY AND STABILITY ANALYSIS OF NONLINEAR KALMAN FILTERS WITH APPLICATION TO AIRCRAFT ATTITUDE ESTIMATION by Matthew Brandon Rhudy Dissertation submitted to the Benjamin M. Statler College of Engineering and Mineral Resources at West Virginia University in partial fulfillment of the requirements for the degree of Doctor of Philosophy in Aerospace Engineering Approved by Dr. Yu Gu, Committee Chairperson Dr. John Christian Dr. Gary Morris Dr. Marcello Napolitano Dr. Powsiri Klinkhachorn Department of Mechanical and Aerospace Engineering Morgantown, West Virginia 2013 Keywords: Attitude Estimation, Extended Kalman Filter, GPS/INS Sensor Fusion, Stochastic Stability Copyright 2013, Matthew B. -
Contributions of Issai Schur to Analysis
Contributions of Issai Schur to Analysis Harry Dym and Victor Katsnelson∗ November 1, 2018 The name Schur is associated with many terms and concepts that are widely used in a number of diverse fields of mathematics and engineering. This survey article focuses on Schur’s work in analysis. Here too, Schur’s name is commonplace: The Schur test and Schur-Hadamard multipliers (in the study of estimates for Hermitian forms), Schur convexity, Schur complements, Schur’s results in summation theory for sequences (in particular, the fundamental Kojima-Schur theorem), the Schur-Cohn test, the Schur al- gorithm, Schur parameters and the Schur interpolation problem for functions that are holomorphic and bounded by one in the unit disk. In this survey, we shall discuss all of the above mentioned topics and then some, as well as some of the generalizations that they inspired. There are nine sections of text, each of which is devoted to a separate theme based on Schur’s work. Each of these sections has an independent bibliography. There is very little overlap. A tenth section presents a list of the papers of Schur that focus on topics that are commonly considered to be analysis. We shall begin with a review of Schur’s less familiar papers on the theory of commuting differential operators. Acknowledgement: The authors extend their thanks to Bernd Kirstein for carefully reading the manuscript and spotting a number of misprints. 1 . Permutable differential operators and fractional powers of differential operators. Let arXiv:0706.1868v1 [math.CA] 13 Jun 2007 dny dn−1y P (y)= p (x) + p (x) + + p (x)y (1.1) n dxn n−1 dxn−1 ··· 0 and dmy dm−1y Q(y)= q (x) + q (x) + + q (x)y, (1.2) m dxm m−1 dxm−1 ··· 0 ∗HD thanks Renee and Jay Weiss and VK thanks Ruth and Sylvia Shogam for endowing the chairs that support their respective research. -
Matlib: Matrix Functions for Teaching and Learning Linear Algebra and Multivariate Statistics
Package ‘matlib’ August 21, 2021 Type Package Title Matrix Functions for Teaching and Learning Linear Algebra and Multivariate Statistics Version 0.9.5 Date 2021-08-10 Maintainer Michael Friendly <[email protected]> Description A collection of matrix functions for teaching and learning matrix linear algebra as used in multivariate statistical methods. These functions are mainly for tutorial purposes in learning matrix algebra ideas using R. In some cases, functions are provided for concepts available elsewhere in R, but where the function call or name is not obvious. In other cases, functions are provided to show or demonstrate an algorithm. In addition, a collection of functions are provided for drawing vector diagrams in 2D and 3D. License GPL (>= 2) Language en-US URL https://github.com/friendly/matlib BugReports https://github.com/friendly/matlib/issues LazyData TRUE Suggests knitr, rglwidget, rmarkdown, carData, webshot2, markdown Additional_repositories https://dmurdoch.github.io/drat Imports xtable, MASS, rgl, car, methods VignetteBuilder knitr RoxygenNote 7.1.1 Encoding UTF-8 NeedsCompilation no Author Michael Friendly [aut, cre] (<https://orcid.org/0000-0002-3237-0941>), John Fox [aut], Phil Chalmers [aut], Georges Monette [ctb], Gaston Sanchez [ctb] 1 2 R topics documented: Repository CRAN Date/Publication 2021-08-21 15:40:02 UTC R topics documented: adjoint . .3 angle . .4 arc..............................................5 arrows3d . .6 buildTmat . .8 cholesky . .9 circle3d . 10 class . 11 cofactor . 11 cone3d . 12 corner . 13 Det.............................................. 14 echelon . 15 Eigen . 16 gaussianElimination . 17 Ginv............................................. 18 GramSchmidt . 20 gsorth . 21 Inverse............................................ 22 J............................................... 23 len.............................................. 23 LU.............................................. 24 matlib . 25 matrix2latex . 27 minor . 27 MoorePenrose .