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Index A-optimality, 356 augmented associativity matrix, 314 absolute error, 395, 404, 434 augmented connectivity matrix, 314 ACM Transactions on Mathematical Automatically Tuned Linear Algebra Software, 505 Software (ATLAS), 456 adj(·), 53 axpy, 10, 36, 65 adjacency matrix, 265, 266, 314 axpy elementary operator matrix, 65 adjoint (see also conjugate transpose), 44 backward error analysis, 404, 410 adjoint, classical (see also adjugate), 53 Banachiewicz factorization, 195 adjugate, 53, 488 base, 380 adjugate and inverse, 93 base point, 379 affine group, 90, 141 basis, 14, Exercise 2.3:, 38 affine space, 32 batch algorithm, 421 Beowulf (cluster computing), 471 affine transformation, 176 bias, in exponent of floating-point Aitken’s integral, 167 number, 381 algebraic multiplicity, 113 big endian, 391 algorithm, definition, 417 big O (order), 406, 413 Amdahl’s law, 416 big omega (order), 407 angle between matrices, 132 bilinear form, 69, 105 angle between vectors, 26, 174, 287 bit, 375 ANSI (standards), 387, 447, 448 bitmap, 376 Applied Statistics algorithms, 505 BLACS (software), 460, 470 approximation of a matrix, 137, 271, BLAS (software), 454, 455, 460, 470, 354, 439 472 approximation of a vector, 30 BMvN distribution, 169, 473 arithmetic mean, 24, 26 Bolzano-Weierstrass theorem for Arnoldi method, 252 orthogonal matrices, 105 artificial ill-conditioning, 206 Boolean matrix, 314 ASCII code, 375 Box M statistic, 298 association matrix, 261, 265, 287, 295, byte, 375 296, 299 ATLAS (Automatically Tuned Linear C (programming language), 387, 401, Algebra Software), 456 447–461 520 Index C++ (programming language), 388 complex data type, 388, 401, 402, 447 CALGO (Collected Algorithms of the condition (problem or data), 408 ACM), 505 condition number, 202, 218, 225, 346, cancellation error, 399, 410 409, 411, 431, 440 canonical form, equivalent, 86 condition number for nonfull rank canonical form, similar, 116 matrices, 225 canonical singular value factorization, condition number for nonsquare 127 matrices, 225 Cartesian geometry, 24, 57 condition number with respect to catastrophic cancellation, 397 computing a sample standard Cauchy matrix, 313 deviation, 411 Cauchy-Schwarz inequality, 16, 75 condition number with respect to Cauchy-Schwarz inequality for matrices, inversion, 203, 218 75, 140 conditional inverse, 102 Cayley multiplication, 59 cone, 14, 32 Cayley-Hamilton theorem, 109 configuration matrix, 299 CDF (Common Data Format), 376 conjugate gradient method, 213–217 centered matrix, 223, 293 conjugate norm, 71 centered vector, 35 conjugate transpose, 44, 104 chaining of operations, 396 conjugate vectors, 71, 105 character data, 376 connected vertices, 263, 267 character string, 376 connectivity matrix, 265, 266, 314 characteristic equation, 108 consistency property, 128 characteristic polynomial, 108 consistency test, 435, 475 characteristic value (see also eigenvalue), consistent system of equations, 82, 206, 106 211 characteristic vector (see also eigenvec- constrained least squares, equality tor), 106 constraints, 337, Exercise 9.4d:, chasing, 250 366 Chebyshev norm, 17 continuous function, 147 Cholesky decomposition, 194, 276, 354 contrast, 333 classification, 313 convergence criterion, 417 cluster analysis, 313 convergence of a sequence of matrices, cluster computing, 471 105, 118, 134 Cochran’s theorem, 283, 325 convergence of a sequence of vectors, 20 cofactor, 52, 488 convergence of powers of a matrix, 135, Collected Algorithms of the ACM 305 (CALGO), 505 convergence rate, 417 collinearity, 202, 329, 350 convex cone, 14, 32, 279 column rank, 76 Corr(·, ·), 37 column space, 41, 69, 81, 82 correlation, 37 column-major, 430, 446, 448 correlation matrix, 295, Exercise 8.8:, column-sum norm, 130 318, 342 Common Data Format (CDF), 376 correlation matrix, positive definite companion matrix, 109, 241 approximation, 353 complementary projection matrix, 286 cost matrix, 299 complete graph, 262 Cov(·, ·), 36 complete pivoting, 210 covariance, 36 completing the Gramian, 139 covariance matrix, 295 Index 521 cross product of vectors, 33 dissimilarity matrix, 299 cross products matrix, 196, 288 distance matrix, 299 cross products, computing sum of distributed linear algebra machine, 470 Exercise 10.17c:, 426 distribution vector, 307 Crout method, 187 divide and conquer, 415 curse of dimensionality, 419 dominant eigenvalue, 111 Doolittle method, 187 D-optimality, 356–358, 439 dot product of matrices, 74 daxpy,10 dot product of vectors, 15, 69 decomposable matrix, 303 double precision, 385, 391 defective (deficient) matrix, 116, 117 doubly stochastic matrix, 306 deficient (defective) matrix, 116, 117 Drazin inverse, 286 deflation, 243–244 dual cone, 32 degrees of freedom, 291, 292, 331, 350 derivative with respect to a vector or E(·), 168 matrix, 145 E-optimality, 356 det(·), 50 echelon form, 86 determinant, 50–58, 276, 278, 356, 439 edge of a graph, 262 determinant as a volume, 57 effective degrees of freedom, 292, 350 determinant of a partitioned matrix, 96 determinant of the inverse, 92 eigenpair, 106 determinant of the transpose, 54 eigenspace, 113 diag(·), 45 eigenvalue, 105–128, 131, 241–256 diag(·) (matrix arguments), 47 eigenvalues of a graph, 314 diagonal element, 42 eigenvalues of a polynomial Exer- diagonal expansion, 57 cise 3.17:, 141 diagonal factorization, 116, 119 eigenvector, 105–128, 241–256 diagonal matrix, 42 eigenvector, left, 106, 123 diagonalizable matrix, 116–119 eigenvectors, linear independence of, diagonalization, 116 112 diagonally dominant matrix, 42, 46, 78, EISPACK, 457 277 elementary operation, 61 differential, 149 elementary operator matrix, 62, 78, 186, differentiation of vectors and matrices, 207 145 elliptic metric, 71 digraph, 266 elliptic norm, 71 digraph of a matrix, 266 endian, 391 dim(·), 12 equivalence of norms, 19, 133 dimension of vector space, 11 equivalence relation, 361 dimension reduction, 287, 345 equivalent canonical factorization, 87 direct method for solving linear systems, equivalent canonical form, 86, 87 201 equivalent matrices, 86 direct product, 73 error bound, 406 direct sum, 13, 48 error of approximation, 407 direct sum of matrices, 47 error, cancellation, 399, 410 directed dissimilarity matrix, 299 error, discretization, 408 direction cosines, 27, 178 error, measures of, 219, 395, 404–406, discrete Legendre polynomials, 309 434 discretization error, 408, 418 error, rounding, 399, 404, 405 522 Index error, rounding, models of, 405, g4 inverse (see also Moore-Penrose Exercise 10.9:, 424 inverse), 102 error, truncation, 408 gamma function, 169, 484 error-free computations, 399 GAMS (Guide to Available Mathemati- errors-in-variables, 329 cal Software), 445 essentially disjoint vector spaces, 12, 48 Gauss (software), 461 estimable combinations of parameters, Gauss-Seidel method, 212 332 Gaussian elimination, 66, 186, 207, 251 Euclidean distance, 22, 299 general linear group, 90, 105 Euclidean distance matrix, 299 generalized eigenvalue, 126, 252 Euclidean matrix norm (see also generalized inverse, 97, 100–103, 189, Frobenius norm), 131 289 Euclidean vector norm, 17 generalized least squares, 337 Euler’s constant Exercise 10.2:, 423 generalized least squares with equality Euler’s integral, 484 constraints Exercise 9.4d:, 366 Euler’s rotation theorem, 177 generalized variance, 296 exact computations, 399 generating set, 14 exception, in computer operations, 394, generating set of a cone, 15 398 generation of random numbers, 358 exponent, 380 geometric multiplicity, 113 exponential order, 413 geometry, 24, 57, 175, 178 extended precision, 385 Givens transformation (rotation), extrapolation, 418 182–185, 192, 251 GMRES, 216 factorization of a matrix, 85, 87, 114, GNU Scientific Library (GSL), 457 116, 173–174, 185–198, 206, 209 graceful underflow, 383 fan-in algorithm, 397, 416 gradient of a function, 151, 152 fast Givens rotation, 185, 433 gradual underflow, 383, 398 fill-in, 197, 434 Gram-Schmidt transformation, 27, 29, Fisher information, 163 192, 432 fixed-point representation, 379 Gramian matrix, 90, 92, 196, 224, flat, 32 288–290 floating-point representation, 379 graph of a matrix, 265 FLOP, or flop, 415 graph theory, 8, 262, 313 FLOPS, or flops, 415 greedy algorithm, 416 Fortran, 388, 389, 415, 447–461 group, 90, 105 Fourier coefficient, 29, 30, 76, 122, 128, GSL (GNU Scientific Library), 457 133 guard digit, 396 Fourier expansion, 25, 29, 75, 122, 128, 133 H¨oldernorm, 17 Frobenius norm, 131–134, 138, 248, 271, H¨older’sinequality, 38 299 Haar distribution, 169, Exercise 4.7:, full precision, 390 171, Exercise 8.8:, 318, 473 full rank, 77, 78, 80, 87, 88 Haar invariant measure, 169 full rank factorization, 85 Hadamard matrix, 310 full rank partitioning, 80, 95 Hadamard multiplication, 72 half precision, 390 g1 inverse, 102 Hankel matrix, 312 g2 inverse, 102 Hankel norm, 312 Index 523 hat matrix, 290, 331 intersection graph, 267 HDF (Hierarchical Data Format), 376 interval arithmetic, 402, 403 Helmert matrix, 308, 333 invariance property, 175 Hemes formula, 221, 339 invariant vector (eigenvector), 106 Hermite form, 87 inverse of a matrix, 83 Hermitian matrix, 42, 45 inverse of a partitioned matrix, 95 Hessenberg matrix, 44, 250 inverse of products or sums of matrices, Hessian of a function, 153 93 hidden bit, 381 inverse of the transpose, 83 Hierarchical Data Format (HDF), 376 inverse, determinant of, 92 high-performance computing, 412 IRLS (iteratively reweighted least Hilbert matrix, 472 squares), 232 Hilbert-Schmidt norm (see also irreducible Markov chain, 361 Frobenius norm), 132 irreducible
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