Verfügbarkeit: Functions of matrices

Functions of matrices: theory and computation

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Bibliographische Detailangaben
1. Verfasser:	Higham, Nicholas J. 1961-2024 (VerfasserIn)
Format:	Buch
Sprache:	English
Veröffentlicht:	Philadelphia SIAM, Soc. for Industrial and Applied Math. 2008
Schlagworte:	Matrices Matrixfuncties Numerieke wiskunde Functions Factorization (Mathematics) Funktion > Mathematik Matrizenzerlegung Matrixfunktion Matrix > Mathematik
Online-Zugang:	Inhaltsverzeichnis
Beschreibung:	Includes bibliographical references and index
Beschreibung:	XX, 425 S. Ill., graph. Darst.
ISBN:	9780898716467

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adam_text	Contents List of Figures xiii List of Tables xv Preface xvii 1 Theory of Matrix Functions 1 1.1 Introduction ................................ 1 1.2 Definitions of f (A) ............................ 2 1.2.1 Jordan Canonical Form ...................... 2 1.2.2 Polynomial Interpolation ..................... 4 1.2.3 Cauchy Integral Theorem ..................... 7 1.2.4 Equivalence of Definitions .................... 8 1.2.5 Example: Function of Identity Plus Rank- 1 Matrix ...... 8 1.2.6 Example: Function of Discrete Fourier Transform Matrix ... 10 1.3 Properties ................................. 10 1.4 Nonprimary Matrix Functions ...................... 14 1.5 Existence of (Real) Matrix Square Roots and Logarithms ....... 16 1.6 Classification of Matrix Square Roots and Logarithms ......... 17 1.7 Principal Square Root and Logarithm .................. 20 1.8 f {AB) and f (В А)............................. 21 1.9 Miscellany ................................. 23 1.10 A Brief History of Matrix Functions ................... 26 1.11 Notes and References ........................... 27 Problems .................................. 29 2 Applications 35 2.1 Differential Equations ........................... 35 2.1.1 Exponential Integrators ...................... 36 2.2 Nuclear Magnetic Resonance ....................... 37 2.3 Markov Models .............................. 37 2.4 Control Theory .............................. 39 2.5 The Nonsymmetric Eigenvalue Problem ................. 41 2.6 Orthogonalization and the Orthogonal Procrustes Problem ...... 42 2.7 Theoretical Particle Physics ....................... 43 2.8 Other Matrix Functions .......................... 44 2.9 Nonlinear Matrix Equations ....................... 44 2.10 Geometric Mean .............................. 46 2.11 Pseudospectra ............................... 47 2.12 Algebras .................................. 47 vii Contents 2.13 Sensitivity Analysis ............................ 48 2.14 Other Applications ............................ 48 2.14.1 Boundary Value Problems .................... 48 2.14.2 Semidefinite Programming .................... 48 2.14.3 Matrix Sector Function ...................... 48 2.14.4 Matrix Disk Function ....................... 49 2.14.5 The Average Eye in Optics .................... 50 2.14.6 Computer Graphics ........................ 50 2.14.7 Bregman Divergences ....................... 50 2.14.8 Structured Matrix Interpolation ................. 50 2.14.9 The Lambert W Function and Delay Differential Equations . 51 2.15 Notes and References ........................... 51 Problems .................................. 52 Conditioning 55 3.1 Condition Numbers ............................ 55 3.2 Properties of the Frédiét Derivative ................... 57 3.3 Bounding the Condition Number .................... 63 3.4 Computing or Estimating the Condition Number ........... 64 3.5 Notes and References ........................... 69 Problems .................................. 70 Techniques for General Functions 71 4.1 Matrix Powers ............................... 71 4.2 Polynomial Evaluation .......................... 72 4.3 Taylor Series ................................ 76 4.4 Rational Approximation ......................... 78 4.4.1 Best ¿(χ Approximation ..................... 79 4.4.2 Padé Approximation ....................... 79 4.4.3 Evaluating Rational Functions .................. 80 4.5 Diagoualization .............................. 81 4.6 Schur Decomposition and Triangular Matrices ............. 84 4.7 Block Diagonalization ........................... 89 4.8 Interpolating Polynomial and Characteristic Polynomial ....... 89 4.9 Matrix Iterations ............................. 91 4.9.1 Order of Convergence ....................... 91 4.9.2 Termination Criteria ....................... 92 4.9.3 Convergence ............................ 93 4.9.4 Numerical Stability ........................ 95 4.10 Preprocessing ............................... 99 4.11 Bounds for lì{A) ............................ 102 4.12 Note« and References ........................... 104 Problems .................................. 105 Matrix Sign Function 107 5.1 Sensitivity and Conditioning ....................... 109 5.2 Schur Method ............................... 112 5.3 Newton s Method ............................. 113 5.4 The Padé Family of Iterations ...................... 115 5.5 Scaling the Newton Iteration ....................... 119 Contents ix 5.6 Terminating the Iterations ........................ 121 5.7 Numerical Stability of Sign Iterations .................. 123 5.8 Numerical Experiments and Algorithm ................. 125 5.9 Best Loo Approximation ......................... 128 5.10 Notes and References ........................... 129 Problems .................................. 131 6 Matrix Square Root 133 6.1 Sensitivity and Conditioning ....................... 133 6.2 Schur Method ............................... 135 6.3 Newton s Method and Its Variants .................... 139 6.4 Stability and Limiting Accuracy ..................... 144 6.4.1 Newton Iteration ......................... 144 6.4.2 DB Iterations ........................... 145 6.4.3 CR Iteration ............................ 146 6.4.4 IN Iteration ............................ 146 6.4.5 Summary .............................. 147 6.5 Scaling the Newton Iteration ....................... 147 6.6 Numerical Experiments .......................... 148 6.7 Iterations via the Matrix Sign Function ................. 152 6.8 Special Matrices .............................. 154 6.8.1 Binomial Iteration ......................... 154 6.8.2 Modified Newton Iterations ................... 157 6.8.3 M-Matrices and ff-Matrices ................... 159 6.8.4 Hermitian Positive Definite Matrices .............. 161 6.9 Computing Small-Normed Square Roots ................ 162 6.10 Comparison of Methods .......................... 164 6.11 Involutory Matrices ............................ 165 6.12 Notes and References ........................... 166 Problems .................................. 168 7 Matrix pth Root 173 7.1 Theory ................................... 173 7.2 Schur Method ............................... 175 7.3 Newton s Method ............................. 177 7.4 Inverse Newton Method .......................... 181 7.5 Schur-Newton Algorithm ......................... 184 7.6 Matrix Sign Method ............................ 186 7.7 Notes and References ........................... 187 Problems .................................. 189 8 The Polar Decomposition 193 8.1 Approximation Properties ........................ 197 8.2 Sensitivity and Conditioning ....................... 199 8.3 Newton s Method ............................. 202 8.4 Obtaining Iterations via the Matrix Sign Function ........... 202 8.5 The Padé Family of Methods ....................... 203 8.6 Scaling the Newton Iteration ....................... 205 8.7 Terminating the Iterations ........................ 207 8.8 Numerical Stability and Choice of Я .................. 209 Contents 8.9 Algorithm .................................210 8.10 Notes and References ...........................213 Problems ..................................216 9 Schur-Parlett Algorithm 221 9.1 Evaluating Functions of the Atomic Blocks ............... 221 9.2 Evaluating the Upper Triangular Part of ƒ (T) ............. 225 9.3 Reordering and Blocking the Schur Form................ 226 9.4 Schur-Parlett Algorithm for f (A) .................... 228 9.5 Preprocessing ............................... 230 9.6 Notes and References ........................... 231 Problems .................................. 231 10 Matrix Exponential 233 10.1 Basic Properties .............................. 233 10.2 Conditioning ................................ 238 10.3 Scaling and Squaring Method ...................... 241 10.4 Schur Algorithms ............................. 250 10.4.1 Newton Divided Difference Interpolation ............ 250 10.4.2 Schur-Fréchet Algorithm ..................... 251 10.4.3 Schur Parlett Algorithm ..................... 251 10.5 Numerical Experiment .......................... 252 10.6 Evaluating the Frédiét Derivative and Its Norm ............ 253 10.6.1 Quadrature ............................ 254 10.6.2 The Kronecker Formulae ..................... 256 10.6.3 Computing and Estimating the Norm .............. 258 10.7 Miscellany ................................. 259 10.7.1 Hermitian Matrices and Best L^ Approximation ....... 259 10.7.2 Essentially Nonnegative Matrices ................ 260 10.7.3 Preprocessing ........................... 261 10.7.4 The V Functions .......................... 261 10.8 Notes and References ........................... 262 Problems .................................. 265 11 Matrix Logarithm 269 11.1 Basic Properties .............................. 269 11.2 Conditioning ................................ 272 11.3 Series Expansions ............................. 273 11.4 Padé Approximation ........................... 274 11.5 Inverse Scaling and Squaring Method .................. 275 11.5.1 Schur Decomposition: Triangular Matrices ........... 276 11.5.2 Full Matrices ........................... 278 11.6 Sdiur Algorithms ............................. 279 11.6.1 Schur Frédiét Algorithm ..................... 279 11.6.2 Schur-Parlett Algorithm ..................... 279 11.7 Numerical Experiment .......................... 280 11.8 Evaluating the Frechei Derivative .................... 281 11.9 Notes and References ........................... 283 Problems .................................. 284 Contents xi 12 Matrix Cosine and Sine 287 12.1 Basic Properties .............................. 287 12.2 Conditioning ................................ 289 12.3 Padé Approximation of Cosine ...................... 290 12.4 Double Angle Algorithm for Cosine ................... 290 12.5 Numerical Experiment .......................... 295 12.6 Double Angle Algorithm for Sine and Cosine .............. 296 12.6.1 Preprocessing ........................... 299 12.7 Notes and References ........................... 299 Problems .................................. 300 13 Function of Matrix Times Vector: f (A) b 301 13.1 Representation via Polynomial Interpolation .............. 301 13.2 Krylov Subspace Methods ........................ 302 13.2.1 The Arnoldi Process ....................... 302 13.2.2 Arnoldi Approximation of f(A)b ................. 304 13.2.3 Lanczos Biorthogonalization ................... 306 13.3 Quadrature ................................. 306 13.3.1 On the Real Line ......................... 306 13.3.2 Contour Integration ........................ 307 13.4 Differential Equations ........................... 308 13.5 Other Methods .............................. 309 13.6 Notes and References ........................... 309 Problems .................................. 310 14 Miscellany 313 14.1 Structured Matrices ............................ 313 14.1.1 Algebras and Groups ....................... 313 14.1.2 Monotone Functions ....................... 315 14.1.3 Other Structures ......................... 315 14.1.4 Data Sparse Representations ................... 316 14.1.5 Computing Structured /(A) for Structured A ......... 316 14.2 Exponential Decay of Functions of Banded Matrices .......... 317 14.3 Approximating Entries of Matrix Functions ............... 318 A Notation 319 В Background: Definitions and Useful Pacts 321 B.I Basic Notation ............................... 321 B.2 Eigenvalues and Jordan Canonical Form ................ 321 B.3 Invariant Subspaces ............................ 323 B.4 Special Classes of Matrices ........................ 323 B.5 Matrix Factorizations and Decompositions ............... 324 B.6 Pseudoinverse and Orthogonality .................... 325 B.6.1 Pseudoinverse ........................... 325 В. 6.2 Projector and Orthogonal Projector ............... 326 B.6.3 Partial Isometry .......................... 326 B.7 Norms ................................... 326 B.8 Matrix Sequences and Series ....................... 328 B.9 Perturbation Expansions for Matrix Inverse .............. 328 xii Contents 8. 10 Sherman-Morrison-Woodbury Formula ................. 329 8. 11 Nonnegative Matrices ........................... 329 В. 12 Positive (Senii)defmite Ordering ..................... 330 В. 13 Kroiiecker Product and Sum ....................... 331 B.14 Sylvester Equation ............................ 331 B.15 Floating Point Arithmetic ........................ 331 B.16 Divided Differences ............................ 332 Problems ..................................334 С Operation Counts 335 D Matrix Function Toolbox 339 Б Solutions to Problems 343 Bibliography 379 Index 415
adam_txt	Contents List of Figures xiii List of Tables xv Preface xvii 1 Theory of Matrix Functions 1 1.1 Introduction . 1 1.2 Definitions of f (A) . 2 1.2.1 Jordan Canonical Form . 2 1.2.2 Polynomial Interpolation . 4 1.2.3 Cauchy Integral Theorem . 7 1.2.4 Equivalence of Definitions . 8 1.2.5 Example: Function of Identity Plus Rank- 1 Matrix . 8 1.2.6 Example: Function of Discrete Fourier Transform Matrix . 10 1.3 Properties . 10 1.4 Nonprimary Matrix Functions . 14 1.5 Existence of (Real) Matrix Square Roots and Logarithms . 16 1.6 Classification of Matrix Square Roots and Logarithms . 17 1.7 Principal Square Root and Logarithm . 20 1.8 f {AB) and f (В А). 21 1.9 Miscellany . 23 1.10 A Brief History of Matrix Functions . 26 1.11 Notes and References . 27 Problems . 29 2 Applications 35 2.1 Differential Equations . 35 2.1.1 Exponential Integrators . 36 2.2 Nuclear Magnetic Resonance . 37 2.3 Markov Models . 37 2.4 Control Theory . 39 2.5 The Nonsymmetric Eigenvalue Problem . 41 2.6 Orthogonalization and the Orthogonal Procrustes Problem . 42 2.7 Theoretical Particle Physics . 43 2.8 Other Matrix Functions . 44 2.9 Nonlinear Matrix Equations . 44 2.10 Geometric Mean . 46 2.11 Pseudospectra . 47 2.12 Algebras . 47 vii Contents 2.13 Sensitivity Analysis . 48 2.14 Other Applications . 48 2.14.1 Boundary Value Problems . 48 2.14.2 Semidefinite Programming . 48 2.14.3 Matrix Sector Function . 48 2.14.4 Matrix Disk Function . 49 2.14.5 The Average Eye in Optics . 50 2.14.6 Computer Graphics . 50 2.14.7 Bregman Divergences . 50 2.14.8 Structured Matrix Interpolation . 50 2.14.9 The Lambert W Function and Delay Differential Equations . 51 2.15 Notes and References . 51 Problems . 52 Conditioning 55 3.1 Condition Numbers . 55 3.2 Properties of the Frédiét Derivative . 57 3.3 Bounding the Condition Number . 63 3.4 Computing or Estimating the Condition Number . 64 3.5 Notes and References . 69 Problems . 70 Techniques for General Functions 71 4.1 Matrix Powers . 71 4.2 Polynomial Evaluation . 72 4.3 Taylor Series . 76 4.4 Rational Approximation . 78 4.4.1 Best ¿(χ Approximation . 79 4.4.2 Padé Approximation . 79 4.4.3 Evaluating Rational Functions . 80 4.5 Diagoualization . 81 4.6 Schur Decomposition and Triangular Matrices . 84 4.7 Block Diagonalization . 89 4.8 Interpolating Polynomial and Characteristic Polynomial . 89 4.9 Matrix Iterations . 91 4.9.1 Order of Convergence . 91 4.9.2 Termination Criteria . 92 4.9.3 Convergence . 93 4.9.4 Numerical Stability . 95 4.10 Preprocessing . 99 4.11 Bounds for lì{A)\\ . 102 4.12 Note« and References . 104 Problems . 105 Matrix Sign Function 107 5.1 Sensitivity and Conditioning . 109 5.2 Schur Method . 112 5.3 Newton's Method . 113 5.4 The Padé Family of Iterations . 115 5.5 Scaling the Newton Iteration . 119 Contents ix 5.6 Terminating the Iterations . 121 5.7 Numerical Stability of Sign Iterations . 123 5.8 Numerical Experiments and Algorithm . 125 5.9 Best Loo Approximation . 128 5.10 Notes and References . 129 Problems . 131 6 Matrix Square Root 133 6.1 Sensitivity and Conditioning . 133 6.2 Schur Method . 135 6.3 Newton's Method and Its Variants . 139 6.4 Stability and Limiting Accuracy . 144 6.4.1 Newton Iteration . 144 6.4.2 DB Iterations . 145 6.4.3 CR Iteration . 146 6.4.4 IN Iteration . 146 6.4.5 Summary . 147 6.5 Scaling the Newton Iteration . 147 6.6 Numerical Experiments . 148 6.7 Iterations via the Matrix Sign Function . 152 6.8 Special Matrices . 154 6.8.1 Binomial Iteration . 154 6.8.2 Modified Newton Iterations . 157 6.8.3 M-Matrices and ff-Matrices . 159 6.8.4 Hermitian Positive Definite Matrices . 161 6.9 Computing Small-Normed Square Roots . 162 6.10 Comparison of Methods . 164 6.11 Involutory Matrices . 165 6.12 Notes and References . 166 Problems . 168 7 Matrix pth Root 173 7.1 Theory . 173 7.2 Schur Method . 175 7.3 Newton's Method . 177 7.4 Inverse Newton Method . 181 7.5 Schur-Newton Algorithm . 184 7.6 Matrix Sign Method . 186 7.7 Notes and References . 187 Problems . 189 8 The Polar Decomposition 193 8.1 Approximation Properties . 197 8.2 Sensitivity and Conditioning . 199 8.3 Newton's Method . 202 8.4 Obtaining Iterations via the Matrix Sign Function . 202 8.5 The Padé Family of Methods . 203 8.6 Scaling the Newton Iteration . 205 8.7 Terminating the Iterations . 207 8.8 Numerical Stability and Choice of Я . 209 Contents 8.9 Algorithm .210 8.10 Notes and References .213 Problems .216 9 Schur-Parlett Algorithm 221 9.1 Evaluating Functions of the Atomic Blocks . 221 9.2 Evaluating the Upper Triangular Part of ƒ (T) . 225 9.3 Reordering and Blocking the Schur Form. 226 9.4 Schur-Parlett Algorithm for f (A) . 228 9.5 Preprocessing . 230 9.6 Notes and References . 231 Problems . 231 10 Matrix Exponential 233 10.1 Basic Properties . 233 10.2 Conditioning . 238 10.3 Scaling and Squaring Method . 241 10.4 Schur Algorithms . 250 10.4.1 Newton Divided Difference Interpolation . 250 10.4.2 Schur-Fréchet Algorithm . 251 10.4.3 Schur Parlett Algorithm . 251 10.5 Numerical Experiment . 252 10.6 Evaluating the Frédiét Derivative and Its Norm . 253 10.6.1 Quadrature . 254 10.6.2 The Kronecker Formulae . 256 10.6.3 Computing and Estimating the Norm . 258 10.7 Miscellany . 259 10.7.1 Hermitian Matrices and Best L^ Approximation . 259 10.7.2 Essentially Nonnegative Matrices . 260 10.7.3 Preprocessing . 261 10.7.4 The V' Functions . 261 10.8 Notes and References . 262 Problems . 265 11 Matrix Logarithm 269 11.1 Basic Properties . 269 11.2 Conditioning . 272 11.3 Series Expansions . 273 11.4 Padé Approximation . 274 11.5 Inverse Scaling and Squaring Method . 275 11.5.1 Schur Decomposition: Triangular Matrices . 276 11.5.2 Full Matrices . 278 11.6 Sdiur Algorithms . 279 11.6.1 Schur Frédiét Algorithm . 279 11.6.2 Schur-Parlett Algorithm . 279 11.7 Numerical Experiment . 280 11.8 Evaluating the Frechei Derivative . 281 11.9 Notes and References . 283 Problems . 284 Contents xi 12 Matrix Cosine and Sine 287 12.1 Basic Properties . 287 12.2 Conditioning . 289 12.3 Padé Approximation of Cosine . 290 12.4 Double Angle Algorithm for Cosine . 290 12.5 Numerical Experiment . 295 12.6 Double Angle Algorithm for Sine and Cosine . 296 12.6.1 Preprocessing . 299 12.7 Notes and References . 299 Problems . 300 13 Function of Matrix Times Vector: f (A) b 301 13.1 Representation via Polynomial Interpolation . 301 13.2 Krylov Subspace Methods . 302 13.2.1 The Arnoldi Process . 302 13.2.2 Arnoldi Approximation of f(A)b . 304 13.2.3 Lanczos Biorthogonalization . 306 13.3 Quadrature . 306 13.3.1 On the Real Line . 306 13.3.2 Contour Integration . 307 13.4 Differential Equations . 308 13.5 Other Methods . 309 13.6 Notes and References . 309 Problems . 310 14 Miscellany 313 14.1 Structured Matrices . 313 14.1.1 Algebras and Groups . 313 14.1.2 Monotone Functions . 315 14.1.3 Other Structures . 315 14.1.4 Data Sparse Representations . 316 14.1.5 Computing Structured /(A) for Structured A . 316 14.2 Exponential Decay of Functions of Banded Matrices . 317 14.3 Approximating Entries of Matrix Functions . 318 A Notation 319 В Background: Definitions and Useful Pacts 321 B.I Basic Notation . 321 B.2 Eigenvalues and Jordan Canonical Form . 321 B.3 Invariant Subspaces . 323 B.4 Special Classes of Matrices . 323 B.5 Matrix Factorizations and Decompositions . 324 B.6 Pseudoinverse and Orthogonality . 325 B.6.1 Pseudoinverse . 325 В. 6.2 Projector and Orthogonal Projector . 326 B.6.3 Partial Isometry . 326 B.7 Norms . 326 B.8 Matrix Sequences and Series . 328 B.9 Perturbation Expansions for Matrix Inverse . 328 xii Contents 8. 10 Sherman-Morrison-Woodbury Formula . 329 8. 11 Nonnegative Matrices . 329 В. 12 Positive (Senii)defmite Ordering . 330 В. 13 Kroiiecker Product and Sum . 331 B.14 Sylvester Equation . 331 B.15 Floating Point Arithmetic . 331 B.16 Divided Differences . 332 Problems .334 С Operation Counts 335 D Matrix Function Toolbox 339 Б Solutions to Problems 343 Bibliography 379 Index 415
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id	DE-604.BV023244812
illustrated	Illustrated
index_date	2024-07-02T20:25:19Z
indexdate	2024-08-01T11:30:25Z
institution	BVB
isbn	9780898716467
language	English
lccn	2007061811
oai_aleph_id	oai:aleph.bib-bvb.de:BVB01-016430292
oclc_num	268627557
open_access_boolean
owner	DE-91G DE-BY-TUM DE-20 DE-29T DE-706 DE-703 DE-384 DE-824 DE-83 DE-11 DE-M347 DE-188 DE-355 DE-BY-UBR DE-19 DE-BY-UBM DE-739 DE-862 DE-BY-FWS
owner_facet	DE-91G DE-BY-TUM DE-20 DE-29T DE-706 DE-703 DE-384 DE-824 DE-83 DE-11 DE-M347 DE-188 DE-355 DE-BY-UBR DE-19 DE-BY-UBM DE-739 DE-862 DE-BY-FWS
physical	XX, 425 S. Ill., graph. Darst.
publishDate	2008
publishDateSearch	2008
publishDateSort	2008
publisher	SIAM, Soc. for Industrial and Applied Math.
record_format	marc
spellingShingle	Higham, Nicholas J. 1961-2024 Functions of matrices theory and computation Matrices gtt Matrixfuncties gtt Numerieke wiskunde gtt Matrices Functions Factorization (Mathematics) Funktion Mathematik (DE-588)4071510-3 gnd Matrizenzerlegung (DE-588)4376303-0 gnd Matrixfunktion (DE-588)4169117-9 gnd Matrix Mathematik (DE-588)4037968-1 gnd
subject_GND	(DE-588)4071510-3 (DE-588)4376303-0 (DE-588)4169117-9 (DE-588)4037968-1
title	Functions of matrices theory and computation
title_auth	Functions of matrices theory and computation
title_exact_search	Functions of matrices theory and computation
title_exact_search_txtP	Functions of matrices theory and computation
title_full	Functions of matrices theory and computation Nicholas J. Higham
title_fullStr	Functions of matrices theory and computation Nicholas J. Higham
title_full_unstemmed	Functions of matrices theory and computation Nicholas J. Higham
title_short	Functions of matrices
title_sort	functions of matrices theory and computation
title_sub	theory and computation
topic	Matrices gtt Matrixfuncties gtt Numerieke wiskunde gtt Matrices Functions Factorization (Mathematics) Funktion Mathematik (DE-588)4071510-3 gnd Matrizenzerlegung (DE-588)4376303-0 gnd Matrixfunktion (DE-588)4169117-9 gnd Matrix Mathematik (DE-588)4037968-1 gnd
topic_facet	Matrices Matrixfuncties Numerieke wiskunde Functions Factorization (Mathematics) Funktion Mathematik Matrizenzerlegung Matrixfunktion Matrix Mathematik
url	http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=016430292&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA
work_keys_str_mv	AT highamnicholasj functionsofmatricestheoryandcomputation

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