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Algorithms for Matrix Functions and Their Frechet Derivatives and Condition Numbers

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Algorithms for Matrix Functions and Their Frechet Derivatives and Condition Numbers Algorithms for Matrix Functions and their Frechet Derivatives and Condition Numbers Relton, S. D. 2014 MIMS EPrint: 2014.55 Manchester Institute for Mathematical Sciences School of Mathematics The University of Manchester Reports available from: http://eprints.maths.manchester.ac.uk/ And by contacting: The MIMS Secretary School of Mathematics The University of Manchester Manchester, M13 9PL, UK ISSN 1749-9097 THE UNIVERSITY OF MANCHESTER - APPROVED ELECTRONICALLY GENERATED THESIS/DISSERTATION COVER-PAGE Electronic identifier: 13526 Date of electronic submission: 24/10/2014 The University of Manchester makes unrestricted examined electronic theses and dissertations freely available for download and reading online via Manchester eScholar at http://www.manchester.ac.uk/escholar. This print version of my thesis/dissertation is a TRUE and ACCURATE REPRESENTATION of the electronic version submitted to the University of Manchester's institutional repository, Manchester eScholar. Approved electronically generated cover-page version 1.0 ALGORITHMS FOR MATRIX FUNCTIONS AND THEIR FRECHET´ DERIVATIVES AND CONDITION NUMBERS A thesis submitted to the University of Manchester for the degree of Doctor of Philosophy in the Faculty of Engineering and Physical Sciences 2014 Samuel David Relton School of Mathematics Contents List of Tables5 List of Figures6 Abstract9 Declaration 10 Copyright Statement 11 Acknowledgements 12 Publications 13 1 Introduction 14 1.1 Matrix functions.............................. 17 1.2 Fr´echet derivatives............................. 20 1.3 Condition numbers............................. 21 1.4 Forwards and backward errors....................... 23 2 New Algorithms for Computing the Matrix Cosine and Sine Sepa- rately or Simultaneously 25 2.1 Introduction................................. 25 2.2 Backward error analysis for the sine and cosine............. 28 2.2.1 Pad´eapproximants of matrix sine................. 28 2.2.2 Pad´eapproximants of matrix cosine................ 30 2.2.3 Exploiting Pad´eapproximants of the exponential........ 31 2.2.4 Backward error propagation in multiple angle formulas..... 33 2 2.3 Recomputing the cosine and sine of 2 × 2 matrices........... 34 2.4 Algorithm for the matrix cosine...................... 35 2.5 Algorithm for the matrix sine....................... 40 2.6 Algorithm for simultaneous computation of the matrix cosine and sine. 44 2.7 Numerical experiments........................... 48 2.7.1 Matrix cosine............................ 50 2.7.2 Matrix sine............................. 51 2.7.3 Matrix cosine and sine....................... 52 2.7.4 Triangular matrices......................... 52 2.7.5 Wave discretization problem.................... 53 2.8 Evaluating rational approximants with the Paterson{Stockmeyer method 54 3 Algorithms for the Matrix Logarithm, its Fr´echet Derivative, and Condition Number 65 3.1 Introduction................................. 65 3.2 Basic algorithm............................... 66 3.3 Backward error analysis.......................... 67 3.4 Condition number estimation....................... 71 3.5 Complex algorithm............................. 71 3.6 Real algorithm............................... 73 3.7 Comparison with existing methods.................... 75 3.7.1 Methods to compute the logarithm................ 75 3.7.2 Methods to compute the Fr´echet derivative............ 76 3.8 Numerical experiments........................... 77 3.8.1 Real versus complex arithmetic for evaluating the logarithm.. 78 3.8.2 Fr´echet derivative evaluation.................... 80 3.8.3 Condition number estimation................... 83 4 Higher Order Fr´echet Derivatives of Matrix Functions and the Level- 2 Condition Number 85 4.1 Introduction................................. 85 4.2 Higher order derivatives.......................... 86 4.3 Existence and computation of higher Fr´echet derivatives........ 88 3 4.4 Kronecker forms of higher Fr´echet derivatives.............. 92 4.5 The level-2 condition number of a matrix function............ 95 4.5.1 Matrix exponential......................... 97 4.5.2 Matrix inverse............................ 100 4.5.3 Hermitian matrices......................... 102 4.5.4 Numerical experiments....................... 103 4.6 The necessity of the conditions in Theorem 4.3.5............. 105 5 Estimating the Condition Number of the Fr´echet Derivative of a Matrix Function 107 5.1 Introduction................................. 107 5.2 The condition number of the Fr´echet derivative............. 109 5.3 Maximizing the second Fr´echet derivative................. 113 5.4 An algorithm for estimating the relative condition number....... 117 5.5 Numerical experiments........................... 119 5.5.1 Condition number of Fr´echet derivative of matrix logarithm.. 121 5.5.2 Condition number of Fr´echet derivative of matrix power.... 122 5.5.3 An ill conditioned Fr´echet derivative............... 123 5.6 Continued proof of Theorem 5.3.3..................... 124 6 Conclusions 129 Bibliography 132 Index 143 4 List of Tables 2.4.1 The number of matrix multiplications π(cm(X)) required to evaluate cm(X), values of θm, values of p to be considered, and an upper bound κm for the condition number of the denominator polynomial of cm (and sm) for αp(X) ≤ θm.......................... 36 2.5.1 The number of matrix multiplications π(·) required to evaluate rm(x), values of βm in (2.2.6), values of p to be considered, and an upper bound κm for the condition number of the denominator polynomial of rm for αp(X) ≤ βm.................................. 41 2.6.1 Number of matrix multiplications πm required to evaluate both cm and sm....................................... 44 2.8.1 The number of matrix multiplications required to form cm(X) by ex- plicit computation of the powers and by the Paterson{Stockmeyer scheme. 55 3.3.1 Maximal values θm of αp(X) such that the bound in (3.3.2) for k∆Xk=kXk does not exceed u.............................. 68 3.3.2 Maximal values βm of kXk such that the bound in (3.3.5) for k∆Ek=kEk does not exceed u.............................. 70 3.8.1 Run times in seconds for Fortran implementations of iss schur real and iss schur complex on random, full n × n matrices......... 80 5 List of Figures 1.3.1 Illustration showing the sensitivity of a matrix function f........ 21 2.2.1 The boundary of the region within which jrm(x)j ≤ 1, where rm(x) is the [m=m] Pad´eapproximant to the sine function, for m = 1; 3; 5; 7.. 29 2.2.2 The boundary of the region within which jrm(x)j ≤ 1, where rm(x) is the [m=m] Pad´eapproximant to the cosine function, for m = 2; 4; 6; 8. 31 2.7.1 The forward and backward errors of competing algorithms for the ma- trix cosine, for full matrices. The first four plots are for the transformation- free version of Algorithm 2.4.2, whereas for the remaining four plots an initial Schur decomposition is used. The results are ordered by decreas- ing condition number............................ 56 2.7.2 The forward and backward errors of competing algorithms for the ma- trix sine, for full matrices. The first four plots are for the transformation- free version of Algorithm 2.5.2, whereas for the remaining four plots an initial Schur decomposition is used. The results are ordered by decreas- ing condition number............................ 57 2.7.3 The maximum forward and backward errors of competing algorithms for the matrix cosine and sine, for full matrices. The first four plots are for the transformation-free version of Algorithm 2.6.2, while an initial Schur decomposition is used for the lower four plots. The results are ordered by decreasing condition number.................. 58 6 2.7.4 Cost plots for the experiments in sections 2.7.1, 2.7.2, and 2.7.3. The first row shows the cost of computing the matrix cosine whilst the second and third rows show the cost of computing the matrix sine and both functions together. The left column corresponds to the transformation- free versions of our new algorithms, whilst the right corresponds to an initial Schur decomposition. All plots are ordered by decreasing cost of our new algorithms.............................. 59 2.7.5 Forward and backward errors of competing algorithms for the matrix cosine for triangular matrices. The results are ordered by decreasing condition number............................... 60 2.7.6 The forward and backward errors of competing algorithms for the ma- trix sine for triangular matrices. The results are ordered by decreasing condition number............................... 61 2.7.7 The maximum forward and backward errors of competing algorithms for the matrix cosine and sine, for triangular matrices. The results are ordered by decreasing condition number.................. 62 2.7.8 Cost plots for all the algorithms run on triangular matrices. All plots are ordered by decreasing cost of our new algorithms........... 63 2.7.9 Forward errors of cosm new and costay for the matrix (2.7.7) that arises from the semidiscretization of a nonlinear wave equation with parameter α. The matrix becomes increasingly nonnormal as n increases...... 64 3.8.1 Normwise relative errors in computing the logarithm........... 78 3.8.2 Performance profile for the data in Figure 3.8.1.............. 78 3.8.3 Performance profile for the accuracy of the logarithm algorithms on full matrices.................................... 79 3.8.4 Normwise relative errors in computing Llog(A; E)............. 81 3.8.5 Performance profile for the data in Figure 3.8.4.............. 82 3.8.6 Performance profile
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