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Trefethen L.N., Bau D. — Numerical Linear Algebra
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.


Язык: en

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

Операции: Положить на полку | Скопировать ссылку для форума | Скопировать ID
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Предметный указатель
"Fast matrix inverse"      248
$e_{j}$      7
$L^{2}[-1,1]$      52 285
$Pentium^{TM}$ microprocessor      100
$\chi^{2}$ (Chi-squared) distribution      240
$\infty$-norm      18 20 21
$\pi$, 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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