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69 Adjacency Matrix Index A-optimality, 439 angle between matrices, 168 absolute error, 484, 494, 526 angle between vectors, 37, 229, 357 ACM Transactions on Mathematical ANSI (standards), 475, 562, 563 Software, 552, 617 Applied Statistics algorithms, 617 adj(·), 69 approximation and estimation, 431 adjacency matrix, 331–334, 390 approximation of a matrix, 174, 257, Exercise 8.20:, 396 339, 437, 533 augmented, 390 approximation of a vector, 41–43 adjoint (see also conjugate transpose), arithmetic mean, 35, 37 59 Arnoldi method, 318 adjoint, classical (see also adjugate), 69 artificial ill-conditioning, 269 adjugate, 69, 598 ASCII code, 460 adjugate and inverse, 118 association matrix, 328–338, 357, 365, affine group, 115, 179 366, 369–371 affine space, 43 adjacency matrix, 331–332, 390 affine transformation, 228 connectivity matrix, 331–332, 390 Aitken’s integral, 217 dissimilarity matrix, 369 AL(·) (affine group), 115 distance matrix, 369 algebraic multiplicity, 144 incidence matrix, 331–332, 390 similarity matrix, 369 algorithm, 264, 499–513 ATLAS (Automatically Tuned Linear batch, 512 Algebra Software), 555 definition, 509 augmented adjacency matrix, 390 direct, 264 augmented connectivity matrix, 390 divide and conquer, 506 Automatically Tuned Linear Algebra greedy, 506 Software (ATLAS), 555 iterative, 264, 508–511, 564 autoregressive process, 447–450 reverse communication, 564 axpy, 12, 50, 83, 554, 555 online, 512 axpy elementary operator matrix, 83 out-of-core, 512 real-time, 512 back substitution, 274, 406 algorithmically singular, 121 backward error analysis, 495, 500 Anaconda, 553, 556, 569 Banach space, 33 angle(·, ·), 37 Banachiewicz factorization, 255 632 Index banded matrix, 58 Cartesian geometry, 35, 73 inverse, 121 catastrophic cancellation, 486 Bartlett decomposition, 346 Cauchy matrix, 389 base, 467 Cauchy-Schwarz inequality, 24, 98 base point, 466 Cauchy-Schwarz inequality for matrices, basis, 21–23 98, 178 Exercise 2.6:, 52 Cayley multiplication, 75, 94 orthonormal, 40–41 Cayley-Hamilton theorem, 138 batch algorithm, 512 CDF (Common Data Format), 463 Bauer-Fike theorem, 307 centered matrix, 288, 363 Beowulf (cluster computing), 559 centered vector, 49 bias, in exponent of floating-point chaining of operations, 485 number, 468 character data, 461 big endian, 480 character string, 461 big integer, 466, 492 characteristic equation, 138 big O (order), 497, 503, 591 characteristic polynomial, 138 big omega (order), 497 characteristic value (see also eigenvalue), bilinear form, 91, 134 135 bit, 460 characteristic vector (see also eigenvec- bitmap, 461 tor), 135 BLACS (software), 557, 558 chasing, 317 BLAS (software), 553–556 Chebyshev norm, 28 CUDA, 560 chi-squared distribution, 400 PBLAS, 558 noncentral, 400 PLASMA, 559 PDF, equation (9.3), 400 PSBLAS, 558 Cholesky decomposition, 253–256, 345, block diagonal matrix, 62 436 determinant of, 70 computing, 558, 574 inverse of, 121 root-free, 255 multiplication, 78 circulant matrix, 383 BMvN distribution, 219, 548 classification, 389 Bolzano-Weierstrass theorem for cluster analysis, 389 orthogonal matrices, 133 cluster computing, 559 Boolean matrix, 390 coarray (Fortran construct), 563 Box M statistic, 368 Cochran’s theorem, 353–356, 401 bra·ket notation, 24 cofactor, 68, 598 byte, 460 Collected Algorithms of the ACM (CALGO), 617 C (programming language), 474, 489, collinearity, 265, 406, 430 568–569 column rank, 99 C++ (programming language), 475, column space, 55, 89, 104, 105 568–569 column-major, 522, 539, 545 CALGO (Collected Algorithms of the column-sum norm, 165 ACM), 617 Common Data Format (CDF), 463 cancellation error, 487, 500 companion matrix, 139, 305 canonical form, equivalent, 110 compatible linear systems, 106 canonical form, similar, 149 compensated summation, 486 canonical singular value factorization, complementary projection matrix, 356 162 complementary vector spaces, 19 Index 633 complete graph, 329 convergence of powers of a matrix, 171, complete pivoting, 275 375 complete space, 33 convergence rate, 509 completing the Gramian, 177 convex combination, 12 complex data type, 491 convex cone, 44–46, 346, 349, 371 complex vectors/matrices, 33, 132, 387 convex function, 26, 197 Conda, 553 convex optimization, 346 condition (problem or data), 499 convex set, 12 condition number, 265, 270, 290, 426, convexity, 26 427, 499, 502, 523, 533 coordinate, 4 computing the number, 533, 574 Cor(·, ·), 51 inverse of matrix, 266, 270 correlation, 51, 90 nonfull rank matrices, 290 correlation matrix, 365, 422 nonsquare matrices, 290 Exercise 8.8:, 394 sample standard deviation, 502 positive definite approximation, 436 conditional inverse, 128 pseudo-correlation matrix, 437 cone, 43–46, 327 sample, 422 Exercise 2.18:, 53 cost matrix, 369 ·, · convex cone, 44, 346 Cov( ), 50 covariance, 50 of nonnegative definite matrices, 346 covariance matrix, see variance- of nonnegative matrices, 371 covariance matrix of positive definite matrices, 349 CRAN (Comprehensive R Archive of positive matrices, 371 Network), 552, 576 conference matrix, 381 cross product of vectors, 47 configuration matrix, 369 Exercise 2.19:, 54 conjugate gradient method, 279–283 cross products matrix, 256, 358 preconditioning, 282 cross products, computing sum of conjugate norm, 93 Exercise 10.18c:, 519 conjugate transpose, 59, 132 Crout method, 245 conjugate vectors, 93, 134 cuBLAS, 560 connected vertices, 329, 334 CUDA, 559 connectivity matrix, 331–334, 390 curl, 192 Exercise 8.20:, 396 curse of dimensionality, 511 augmented, 390 cuSPARSE, 560 consistency property of matrix norms, 164 D-optimality, 439–440, 533 consistency test, 527, 551 daxpy, 12 consistent system of equations, 105, decomposable matrix, 373 272, 277 decomposition, see also factorization of constrained least squares, equality a matrix constraints, 413 additive, 353, 354 Exercise 9.4d:, 451 Bartlett decomposition, 346 continuous function, 186 multiplicative, 108 contrast, 410 nonnegative matrix factorization, convergence criterion, 508 257, 337 convergence of a sequence of matrices, singular value decomposition, 134, 152, 171 161–164, 320, 336, 425, 532 convergence of a sequence of vectors, 32 spectral decomposition, 155 634 Index defective (deficient) matrix, 149, 150 dimension of vector space, 14 deficient (defective) matrix, 149, 150 dimension reduction, 20, 356, 425 deflation, 308–310 direct method for solving linear systems, degrees of freedom, 360, 362, 407, 430 272–277 del, 192 direct product (of matrices), 95 derivative with respect to a matrix, 195 direct product (of sets), 5 derivative with respect to a vector, direct product (of vector spaces), 20 189–195 basis for, 23 det(·), 66 direct sum decomposition of a vector determinant, 66–74 space, 19 as criterion for optimal design, 439 direct sum of matrices, 62 computing, 533 direct sum of vector spaces, 18–20, 64 derivative of, 195 basis for, 22 Jacobian, 217 direct sum decomposition, 19 of block diagonal matrix, 70 directed dissimilarity matrix, 369 of Cayley product, 88 direction cosines, 38, 231 of diagonal matrix, 70 discrete Fourier transform, 384 of elementary operator matrix, 86 discrete Legendre polynomials, 380 of inverse, 117 discretization error, 498, 509 of Kronecker product, 96 dissimilarity matrix, 369 of nonnegative definite matrix, 345 distance, 32 of partitioned matrix, 70, 122 between matrices, 175 of permutation matrix, 87 between vectors, 32 of positive definite matrix, 347 distance matrix, 369 of transpose, 70 distributed computing, 463, 508, 544 of triangular matrix, 70 distributed linear algebra machine, 558 relation to eigenvalues, 141 distribution vector, 377 relation to geometric volume, 73, 217 div, 192 diag(·), 56, 60, 596 divergence, 192 with matrix arguments, 62 divide and conquer, 506 diagonal element, 56 document-term matrix, 336 diagonal expansion, 73 dominant eigenvalue, 142 diagonal factorization, 148, 152 Doolittle method, 245 diagonal matrix, 57 dot product of matrices, 97 determinant of, 70 dot product of vectors, 23, 91 inverse of, 120 double cone, 43 multiplication, 78 double precision, 472, 480 diagonalizable matrix, 148–152, 306, doubly stochastic matrix, 377 343 Drazin inverse, 129–130 orthogonally, 154 dual cone, 44 unitarily, 147, 343, 387 diagonally dominant matrix, 57, 62, Epq, E(π), Ep(a), Epq(a) (elementary 100, 348 operator matrices), 84 differential, 188 E(·) (expectation operator), 216 differentiation of vectors and matrices, E-optimality, 439 183–220 echelon form, 111 digraph, 332 edge of a graph, 329 of a matrix, 333 EDP (exact dot product), 493 dim(·), 14 effective degrees of freedom, 362, 430 Index 635 efficiency, computational, 503–508 Euclidean distance matrix, 369 eigenpair, 134 Euclidean matrix norm (see also eigenspace, 144 Frobenius norm), 167 eigenvalue, 134–164, 166, 305–322 Euclidean vector norm, 27 computing, 306–318, 558, 574 Euler’s constant Exercise 10.2:, 516 Jacobi method, 313–316 Euler’s integral, 593 Krylov methods, 318 Euler’s rotation theorem, 231 power method, 311–313 exact computations, 493 QR method, 316–318 exact dot product (EDP), 493 of a graph, 391 exception, in computer operations, 483, of a polynomial Exercise 3.26:, 181 487 relation to singular value, 163 expectation, 212–220 upper bound on, 142, 145, 306 exponent, 467 eigenvector, 134–164, 305–322 exponential order, 503 left eigenvector, 135, 158 exponential, matrix, 153, 184 eigenvectors, linear independence of, extended precision, 472 143 extrapolation, 509 EISPACK, 556 elementary operation, 79 factorization of a matrix, 108, 112, 147, elementary operator matrix, 79–87, 101, 148,
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