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Trefethen L.N., Bau D. — Numerical Linear Algebra
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Название: Numerical Linear Algebra
Авторы: Trefethen L.N., Bau D.
Аннотация: Numerical Linear Algebra is a concise, insightful, and elegant introduction to the field of numerical linear algebra. Designed for use as a stand-alone textbook in a one-semester, graduate-level course in the topic, it has already been class-tested by MIT and Cornell graduate students from all fields of mathematics, engineering, and the physical sciences. The authors' clear, inviting style and evident love of the field, along with their eloquent presentation of the most fundamental ideas in numerical linear algebra, make it popular with teachers and students alike.
Numerical Linear Algebra aims to expand the reader's view of the field and to present the core, standard material in a novel way. It is a perfect companion volume to the encyclopedic treatment of the topic that already exists in Golub and Van Loan's now-classic Matrix Computations. All of the most important topics in the field, including iterative methods for systems of equations and eigenvalue problems and the underlying principles of conditioning and stability, are covered. Trefethen and Bau offer a fresh perspective on these and other topics, such as an emphasis on connections with polynomial approximation in the complex plane.
Numerical Linear Algebra is presented in the form of 40 lectures, each of which focuses on one or two central ideas. Throughout, the authors emphasize the unity between topics, never allowing the reader to get lost in details and technicalities. The book breaks with tradition by beginning not with Gaussian elimination, but with the QR factorization—a more important and fresher idea for students, and the thread that connects most of the algorithms ofnumerical linear algebra, including methods for least squares, eigenvalue, and singular value problems, as well as iterative methods for all of these and for systems of equations.
Students will benefit from the many exercises that follow each lecture. Well-chosen references and extensive notes enrich the presentation and provide historical context.
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Рубрика: Математика /
Статус предметного указателя: Готов указатель с номерами страниц
ed2k: ed2k stats
Год издания: 2002
Количество страниц: 361
Добавлена в каталог: 07.04.2008
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Предметный указатель
"Fast matrix inverse" 248
7
52 285
microprocessor 100
(Chi-squared) distribution 240
-norm 18 20 21
, calculation of 327
1-norm 18 20
2-norm 18 20 34
2-norm, computation of 36
4-norm 18
A-conjugate vectors 295
A-norm 294
Abel, Niels 192 324 326
Accuracy 103 111
ADI (alternating direction implicit) splitting 318
Algorithm, formal definition 102
Angle between vectors or subspaces 12 214 332
Arnoldi, approximation problem 259
Arnoldi, eigenvalue estimates see "Ritz values"
Arnoldi, iteration 245 250—265 340
Arnoldi, lemniscate 262—263 340
Arnoldi, polynomial 262
Arnoldi, shift-and-invert 319 342
Augmented matrix 139 141
Back substitution 121—128
Backward, error 116
Backward, error analysis 108 111—112 334—335
Backward, stability 104 334
Banded matrix 154 161 337
Base 98
Basis, change of 8 15 32—33 182
Bauer — Fike theorem 201
BCG (biconjugate gradients) 245 303—312 341
Bi-CGSTAB 311 341
Biconjugate gradients see "BCG"
Bidiagonal matrix 265
Bidiagonal reduction 236—240
Bilinear function 12
Biorthogonal vectors 305—306
Biorthogonalization methods 303—312
Bisection 227—229 233
BLAS (Basic Linear Algebra Subroutines) 330
Block, matrix 143 154 230 235 249 317 330
Block, power iteration see "Simultaneous iteration"
Boundary elements 245 248 317
Breakdown of Arnoldi iteration 256
C 63
Cancellation error 73 91 138
Cauchy — Schwarz inequality 21
Cayley transform 16
Cayley — Hamilton theorem 260
Cayuga, Lake 136
CG see "Conjugate gradients"
CGN or CGNR 245 303—305
CGS (conjugate gradients squared) 311
Chaos 335
Characteristic polynomial 110 183 184 190
Chebyshev points 79 279 292
Chebyshev polynomial of a matrix 265 340
Chebyshev polynomials 287 292 300
Cholesky factorization 82 141 172—178 301 337
Circulant matrix 187 305 318 342
Column, pivoting 139—140 143
Column, rank 7
Column, space 7
Column, spaces, sequence of 48 169 245
Communication 59 66
Compact operator 265 331
Companion matrix 192 338
Complementary subspaces 43 332
Complete pivoting 161 336
Complex, arithmetic 59 100
Complex, conjugate 11
Complex, sign 29 72
Complex, symmetric matrix 312
Componentwise analysis 127 227 334 339
Computers, speed of 243—244 339
Condition number of a matrix 94 333
Condition number of an eigenvalue 258
Condition number, absolute 90
Condition number, computation of 94
Condition number, relative 90
Condition number, squaring of 142 235 305
Conditioning 89—96 333
Conjugate, complex 11
Conjugate, gradients 245 293—302 303 341
Conjugate, hermitian 11
Conjugate, residuals iteration 293
Convergence, cubic 195 208 212 221—222
Convergence, linear or geometric 195 262—264
Convergence, quadratic 195 226
Convergence, superlinear 195 337
Coppersmith and Winograd, algorithm of 247 340
Covariance matrix 234
CS decomposition 332
Cuppen, J.J.M. 229
Data-fitting see "Least squares problem"
Davidson method 319
Defective eigenvalue 185
Defective matrix 185
Deflation 212 223 232
Deletion matrix 9 24
Demmel, James W., book by 329
Dense, matrix 244
Dense, subset 37
Determinant 8 10 34 97 161 330
Determinant, computation of 161
Diagonal matrix 15 18 20 32
Diagonalizable matrix see "Nondefective matrix"
Diagonalization 188
Diagonally dominant matrix 162
Dimensions, physical 10 107
Direct algorithm 190 243 247
Divide-and-conquer algorithm 212 229—233 239
Domain decomposition 317 342
Dual norm 24 95 331
Eigenspace 181 183
Eigenvalue decomposition 33 182
Eigenvalue-revealing factorization 188 191
Eigenvalues 8 15 24 181—189
Eigenvalues, algebraic multiplicity of 183—184
Eigenvalues, computation of 110 190—233 257—265
Eigenvalues, defective 185
Eigenvalues, geometric multiplicity of 183—184
Eigenvalues, perturbation of 188 201 258 333
Eigenvalues, simple 184
Eigenvectors 15 43 181
Eigenvectors, computation of 202 218 227
Eigenvectors, localization of 232 233
EISPACK 257 330 337 338
Electric charge 279 283—284
Error, absolute 103
Error, relative 99 103
Euclidean length 12 17 78
ev and ew (abbreviations for eigenvector and eigenvalue) 188 337
EXPONENT 98
Exponential of a matrix 33 182 189 201
Fast Fourier Transform 63
Fast Poisson solver 317
Feynman, Richard 91 334
Field of values see "Numerical range"
Finite differences 244 317
Finite elements 254 317
Finite sections 333
Fixed point arithmetic 98
fl 99
Floating point, arithmetic 66 97—101 334
Floating point, axioms 99
Floating point, numbers 98
Flop (floating point operation) 58
Forsythe and Moler, book by 243 331
FORTRAN 63 324
Forward error analysis 108 112 177
FRACTION 98
Frobenius norm 22 34
Full rank, matrix of 7
Fundamental law of computer science 246 325 340
Galois, Evariste 192 324 326
Gamma function 85
Gauss quadrature 285—292 341
Gauss — Seidel iteration 318 339
Gaussian elimination x 35 54 61 106 147—171 325
Gaussian elimination, stability 152—154 163—171 325 336
Generalized minimal residuals see "GMRES"
Geometric interpretations 12 25 36 55 59 133 201 233 332 335
Gerschgorin's theorem 189 337
Ghost eigenvalues 282—283
Givens rotation 76 195 218 226 268 275
gmres 245 266—275 293 303 340
GMRES, approximation problem 269
GMRES, restarted 275
Golub and Van Loan, book by ix 329
Golub — Kahan bidiagonalization 236—237
Golub, Gene H. 236 330 331 339
Gradient 203 302
Gram — Schmidt orthogonalization 50—51 56—62 70 148 250—253 332
Gram — Schmidt orthogonalization, classical vs. modified 51 57 65—66 140 332
Graphics 63
Green's function 284
Growth factor 163—171 312 336
Guard digit 100
Hadamard inequality 55
Hadamard matrix 16
Hahn — Banach theorem 331
Hein, Piet 18
Henrici, Peter 327
Hermitian conjugate 11
Hermitian matrix 11 15 34 44 162 172 187
Hermitian positive definite matrix 172 294
Hessenberg matrix 193 198 252
Hessenberg orthogonalization 305—306
Hessenberg reduction 193 196—201 250—251 337—338
Hestenes, Magnus 293 341
Higham, Nicholas J. xii 335
Higham, Nicholas J., book by ix 329
Hilbert space 330 331
Hilbert — Schmidt norm see "Frobenius norm"
Hoelder inequality 21
Horn and Johnson, books by 330
Horner's rule 265
Householder, Alston 70 330 332
Householder, reflector 70—73
Householder, Symposia 333
Householder, triangularization 64 69—76 114—120 147 251 332
Householder, tridiagonalization 196—201 251
Hydrodynamic stability 258
Hyperellipse 20 25 36 95
Hyperplane 71
ICCG (incomplete Cholesky factorization) 316
Ideal Arnoldi polynomial see "Chebyshev polynomial of a matrix"
Idempotent matrix 41
Identity 8
IEEE arithmetic 97 334
Ill-conditioned matrix 94
Ill-conditioned problem 89 91
Ill-posed problem 334
ILU (incomplete LU factorization) 316
Image processing 36 68
Incomplete factorization 316 342
Infinitesimal perturbation 90 133 135
Inner product 12 52 109 285
Integral, equation 245 331
Integral, operator 6 53 286
Interlacing eigenvalues 227—228
Interpolation 10 see
Intersection of subspaces 36 55
Invariant subspace 183
Inverse 8
Inverse, computation of 161
Invertible matrix see "Nonsingular matrix"
Irreducible matrix 227
Iteration 206—207 210 219 338
Iterative methods x 69 192 243—249 326 339—340
Jacobi — Davidson methods 319 342
Jacobi, algorithm 225—227 233 338—339
Jacobi, Carl Gustav Jacob 225
Jacobi, iteration 318
Jacobi, matrix 287—292
Jacobi, polynomial 287
Jacobi, preconditioner 316
Jacobi, rotation 226
Jacobian 90 132—133 258
Jordan form 337
Kahan, William M. 236 334 339
Karmarkar algorithm 326
Kronecker delta function 14
Krylov matrix 253
Krylov sequence 245
Krylov subspace iteration 241—327
Krylov subspaces 245 253
Lanczos iteration 245 250 276—284 298 303 340
Lanczos lemniscate 284
Lanczos polynomial 280
LAPACK 166 205 232 243 257 338
Least squares problem 36 77—85 129—144 305 333
Least squares problem, rank-deficient 143 335
Lebesgue constants 96 334 341
Legendre points 292
Legendre polynomial 53 54 64 68 285—
Lemniscate 262—263
LHC (Lawson — Hanson — Chan) bidiagonaiization 237—239
LINPACK 166 243
Look-ahead Lanczos 311 341
Low-rank approximation 35—36 331
Low-rank approximation, computation of 36
LU factorization 147 154 160
Machine epsilon 66 98 100
Mantissa 98
Mass-spring system 9
MathWorks, Inc., The 63 330 332
MATLAB 31 62 63—68 166 205 257 324 332
Matrix, augmented 139 141
Matrix, banded 154 161 337
Matrix, bidiagonal 265
Matrix, block 143 154 230 235 249 317 330
Matrix, circulant 187 305 318 342
Matrix, companion 192 338
Matrix, complex symmetric 312
Matrix, covariance 234
Matrix, defective 185
Matrix, deletion 9 24
Matrix, dense 244
Matrix, diagonal 15 18 20 32
Matrix, diagonalizable see "Nondefective matrix"
Matrix, diagonally dominant 162
Matrix, Hadamard 16
Matrix, hermitian 11 15 34 44 162 172 187
Matrix, hermitian positive definite 172 294
Matrix, Hessenberg 193 198 252
Matrix, idempotent 41
Matrix, identity 8
Matrix, ill-conditioned 94
Matrix, irreducible 227
Matrix, nondefective 185—186
Matrix, nonnormal 186 258
Matrix, nonsingular 7
Matrix, normal 92 173 187 201
Matrix, orthogonal 14 218
Matrix, permutation 34 157 220
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