Page 439 Iiii Ad Hoc Shifts, 165 Aggressive Early Deflation

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Index

ad hoc shifts, 165 for generalized eigenvalue problem, aggressive early deflation, 205–207 244–246 algebraic multiplicity, 35 butterfly form, 211 algebraic Riccati equation, 100 Arnoldi process, 372 cache memory, 132 block, 418 efficient use in QR algorithm, 203 Hamiltonian skew symmetric, 420 Cauchy-Schwarz-Bunyakovski (CSB) in- implicitly restarted, 364, 374 equality, 4 on a product, 412 Cayley transform, 377, 381 skew Hamiltonian, 401 of a Hamiltonian matrix, 400 skew symmetric, 381 characteristic equation, 34 unitary (= isometric), 382 of generalized eigenvalue problem, ARPACK, 364 236 augmented pencil, 268 Chebyshev polynomials, 354–359 Cholesky decomposition, 16 backward error, 92 Cholesky LR algorithm, 167 backward stability, 92 column compression, 57 and residuals, 107–109 column space, 23 and unitary matrices, 117 companion matrix, 39, 142, 147 balancing, 200 complex orthogonal matrix, 121 Bartels-Stewart algorithm, 196, 199 complex symmetric matrix, 121 variant, 200 condition number basis, 20 of eigenvalue, 97 orthonormal, 20 of eigenvector, 105 standard, 20 of invariant subspace, 100, 105 Bauer-Fike theorem, 94–95 of matrix, 6 variant, 110 related to SVD, 30 biorthonormal vectors, 390 conjugate transpose, 3 block Krylov process, 418 continuity of eigenvalues, 92 blocking for efficient cache use, 132 Lipschitz, 96 breakdown of unsymmetric Lanczos pro- contraction mapping theorem cess, 391, 397 application of, 102 bulge pencil, 269–272 proof of, 112 bulge-chasing algorithm, generic, 175– convergence 177 of GR algorithms, 223–228 bi-directional, 287 of QR algorithm, 225 close-up view, 188–191 of Krylov process, 368

439 From "The Matrix Eigenvalue Problem" by David S. Watkins. This book is available for purchase at www.siam.org/catalog. i i

i i i i eigen 2007/6/6 page 440 i i

of subspace iteration, 215 algebraic multiplicity of, 35 CS decomposition, 75 associated with invariant subspace, cyclic matrix, 295 38 perfect shuffle of, 299 condition number of, 97 continuity of, 92 D-orthogonal matrix, 13 dominant, 154 decomposition geometric multiplicity of, 69 Cholesky, 16 infinite, 235, 252–257 CS,75 eigenvectors, 33 GR, 129–133 dominant, 154 HR, 14, 130 left, right, 34 Jordan, see Jordan canonical form linear independence of, 37 LR, 10, 130 elementary matrix, 118 QR, 11, 130, 134 elementary reflector, 119 RG, 149 elimination matrix, 117–124 Schur, 44 equivalence of matrix pairs, strict, 236 singular value, 28, 305–309, 318 exact shift, 183, 193 spectral, 73 in implicit restart, 365, 367 SR, 13, 130 exceptional shifts, 165 symplectic URV, 324, 325 defective matrix, 37, 53 factorization, see decomposition deflating pair of subspaces, 258 flip matrix, 147 deflation, 164 flop, 131 aggressive early, 205–207 full rank, 23 of infinite eigenvalue, 254 degree Galois theory, 153, 240 of a GR iteration, 159 Gauss transform, 118–119 of a GZ iteration, 244 with pivoting, 119 derogatory matrix, 183 generalized eigenvalue problem, 235 diagonal matrix, 9 as a product eigenvalue problem, 319 diagonalizable matrix, 37 geometric multiplicity, 69 dimension, 20 Gerschgorin disk theorem, 93 direct rotator, 89 Givens transformation, 120 direct sum, 21 GR algorithm, 159, 175 direct vs. iterative methods, 153 convergence of, 223–228 distance between subspaces, 81 cubic, 227 divide-and-conquer method, 208 quadratic, 226 dominant eigenvalue/vector, 154 explicit, 159 dominant invariant subspace, 154 for a product, 295–297, 301–302 duality implicit, 175 and Krylov subspaces, 147 GR decomposition, 129 in subspace iteration, 156 condensed, 133 operation count for, 130 eigenspace, 33 Gram-Schmidt process, 134 dominant, 154 and orthoprojectors, 138 eigenvalue, 33 applied twice, 135

From "The Matrix Eigenvalue Problem" by David S. Watkins. This book is available for purchase at www.siam.org/catalog. i i

i i i i eigen 2007/6/6 page 441 i i

GZ algorithm, 243, 320 existence of, 43, 44 implicit, 244 isometric Arnoldi process, 382 reverse, 279 isotropic subspace, 326, 400 iterative vs. direct methods, 153 Hamiltonian Lanczos process, 404 as a product Krylov process, 416 Jacobi’s method, 208 stability test, 408 Jordan block, 61 Hamiltonian matrix, 17, 227 Jordan canonical form, 53–65 eigenvalue symmetry of, 41 complex, 63 Schur form, 326 practical determination, 63–64 SR algorithm for, 209 real, 64 symplectic URV decomposition of, 324, 325 Kronecker canonical form, 236 HAPACK, 210 Kronecker product, 65 Hermitian matrix, 16, 50, 207–208, 317, Krylov matrix, 177 374 left, 281 Hessenberg matrix, upper, 140 Krylov process, 359–360 and GR algorithms, 162 as a partial similarity transformation, and Krylov process, 360 361 and Krylov subspaces, 141 block, 418 m-Hessenberg matrix, 175 convergence of, 368 proper (= unreduced), 141, 175 implicit restart of, 364–370 skew Hamiltonian, 211 on a product, 412 Hessenberg-triangular form, 239 residual theorem, 362 Householder transformation, 119 Krylov sequence, 35, 352 HR algorithm, 209 Krylov subspaces as HZ algorithm, 249–251 and Hessenberg form, 141 cubic convergence of, 227 approximating eigenvectors from, 352 on a product, 339 defined, 141 HR decomposition, 14, 130 duality and, 147 hyperbolic matrix, 13 of a skew-Hamiltonian matrix, 401 elimination matrices, 123–124 Krylov-Schur algorithm, 367, 374 HZ algorithm, 249–251 Lanczos process, 374, 390 implicit restart, 352, 364–370 breakdown of, 391, 397 explained as subspace iteration, 366– Hamiltonian, 404 367 Hamiltonian skew symmetric, 420 first method, 364 skew Hamiltonian, 402 second method (thick restart), 367 skew symmetric, 381 unitary, 387 symmetric, 374 infinite eigenvalue, 235 symplectic, 408 removal of, 254 unsymmetric, 390 inner product, 4 LAPACK invariant subspace, 38 efficient cache use by, 133 dominant, 154 QR routines, 200–207 eigenvalues associated with, 38 QZ routines, 248

From "The Matrix Eigenvalue Problem" by David S. Watkins. This book is available for purchase at www.siam.org/catalog. i i

i i i i eigen 2007/6/6 page 442 i i

sep estimator, 104 sparse, 351 linear independence, 20 symmetric, 16, 50, 207–208, 317, Lipschitz continuity, 96 374 locking, 368 symplectic, 12, 41, 211, 408–411 LR algorithm, 160, 179 totally nonnegative, 348 and quotient-difference algorithm, 169 triangular, 9, 36 Cholesky variant, 167 tridiagonal, 141 LR decomposition, 10, 130 unit triangular, 9 with partial pivoting, 10, 130 unitary, 10, 343, 377, 382 memory hierarchy, 132 matrix minor, leading principal, 14 companion, 39, 142, 147 MRRR method, 208 complex orthogonal, 121 multiplicity of eigenvalue complex symmetric, 121 algebraic, 35 cyclic, 295 geometric, 69 defective, 37, 53 derogatory, 183 nilpotent matrix, 56 diagonal, 9 nonderogatory matrix, 183 diagonalizable, 37 norm elementary, 118 matrix, 5 flip, 147 spectral, 5 Hamiltonian, 17, 41, 209, 227, 321– and SVD, 30 342, 404–408 vector, 3 Hamiltonian skew symmetric, 408, normal matrix, 16, 227 419 null space, 22 Hermitian, 16, 50, 207–208, 317, 374 nullity, 22 Hessenberg, 140, 162, 360 hyperbolic, 13 oblique projector, 27 Krylov, 177 orthogonal complement, 21 left, 281 orthogonal matrix, 11 m-Hessenberg, 175 complex, 121 nilpotent, 56 orthonormal set, 20 nonderogatory, 183 orthoprojector, 27 normal, 16, 227 orthogonal, 11 parallel QR algorithm, 201 Pascal, 348 Pascal matrix, 348 permutation, 10 pencil (= matrix pair), 236 positive definite, 16 augmented, 268 pseudosymmetric, 145, 208, 227, 250, regular vs. singular, 236 338 perfect shift, 183, 193 quasitriangular, 45 perfect shuffle, 12, 299 semisimple, 37 permutation matrix, 10 signature, 13 pipelined QR iterations, 201 skew Hamiltonian, 17, 142, 144, 211 plane rotator, 120 skew Hermitian, 16 positive definite matrix, 16 skew symmetric, 16, 317 power method, 154

From "The Matrix Eigenvalue Problem" by David S. Watkins. This book is available for purchase at www.siam.org/catalog. i i

i i i i eigen 2007/6/6 page 443 i i

principal angles, 76–79 resolution of the identity, 73 relationship to SVD, 79 RG algorithm, 278 principal submatrix, leading, 10 RG decomposition, 149 product eigenvalue problem, 293 Riccati equation, 100 Krylov subspace methods for, 412 Ritz value, 374, 380 removal of zero eigenvalues, 304– Ritz vector, 380 305 rotator, 120 role of cyclic matrix, 295 projection theorem, 20 Schur decomposition, 44 projector, 26 Schur parameters, 343 oblique vs. orthogonal, 27 Schur theorem, 43 spectral, 72 for a Hamiltonian matrix, 326 proper upper Hessenberg matrix, 141 for a skew-Hamiltonian matrix, 144 pseudosymmetric matrix, 145, 227, 250 generalized, 236 and product eigenvalue problem, 338 real, 45 HR algorithm for, 208 semisimple matrix, 37 Pythagorean theorem, 5 diagonalizability of, 53 separation (sep), 102 QR algorithm, 160, 179 shift blurring, 201, 267, 273 convergence of, 225 shift-and-invert strategy, 354 cubic convergence of, 227 shifts of origin, 155, 162–166 efficient cache use in, 203 ad hoc, 165 explicit, 160 convergence acceleration, 162–164 for Hamiltonian matrices, 210 exact, 183, 193 for the SVD, 308 exceptional, 165 for unitary matrices, 343 generalized Rayleigh quotient, 165 implicit, 179 in a GZ iteration, 243 parallel, 201 transmission of, 267 QR decomposition, 11, 130 Wilkinson, 166, 201, 208 and Gram-Schmidt process, 134 SHIRA, 402 quasitriangular matrix, 45 shuffle, perfect, 12, 299 quotient-difference (qd) algorithm, 167– signature matrix, 13 171 similarity of matrices, 42 as a product LR algorithm, 311 simple eigenvalue, 96 differential form (dqds), 170, 311 simultaneous iteration, 156 QZ algorithm, 244, 248 singular value decomposition (SVD), 28 killed by underflow, 276 as a product eigenvalue problem, 305– 309 range of matrix, 22 of a product, 318 rank, 22 QR algorithm for, 308 full, 23 related to eigenvalue problem, 52 Rayleigh quotient shift, generalized, 165 singular vector, 29 real Schur theorem, 45 skew-Hamiltonian matrix, 17, 211 reorthogonalization, 373 Arnoldi process for, 401 residuals, and backward stability, 107– Hessenberg form of, 142 109 Jordan form of, 144

From "The Matrix Eigenvalue Problem" by David S. Watkins. This book is available for purchase at www.siam.org/catalog. i i

i i i i eigen 2007/6/6 page 444 i i

Lanczos process for, 402 butterﬂy form, 211 real Schur form of, 144 eigenvalue symmetry of, 41 skew-Hermitian matrix, 16 elimination matrices, 121–123 skew-symmetric matrix, 16, 317 SR algorithm for, 211 span, 19 symplectic shear, 122 spanning set, 20 symplectic URV decomposition, 324, 325 sparse matrix, 351 spectral decomposition, 73 tensor product, 65 spectral projector, 72 thick restart, 367 spectral theorem totally nonnegative matrix, 348 for normal matrices, 48 trace, 46, 50 for real symmetric matrices, 50 transpose, 3 spectrum, 33 triangle inequality, 3 SR algorithm, 209 triangular matrix, 9 cubic convergence of, 227 eigenvalues of, 36 for symplectic matrix, 211 unit, 9 reduced to HR algorithm, 341–342 tridiagonal matrix, 141 SR decomposition, 13, 130 stability, see backward stability ultimate shift strategy, 193 standard basis, 20 unitary matrix, 10 strict equivalence of matrix pairs, 236 Arnoldi process for, 382 structure preservation principle, 160 Cayley transform of, 377, 381 subspace, 19 QR algorithm for, 343 invariant, 38 unsymmetric Lanczos process, 390 eigenvalues associated with, 38 breakdown of, 391, 397 isotropic, 326 real case, 393 Krylov, see Krylov subspaces stability test, 397 matrix representation of, 23 Van Loan’s VZalgorithm, 320 subspace iteration, 153 convergence of, 215 Weyr characteristic, 61 duality in, 156 Wilkinson shift, 166, 201, 208 multiple steps of, 155 Wintner-Murnaghan theorem, 45 superlinear convergence of, 217 SVD, see singular value decomposition Sylvester equation, 54 generalized, 262, 264 use in block swapping, 195 symmetric Lanczos process, 374 block, 419 computing singular values by, 414 symmetric matrix, 16, 50, 207–208, 317, 374 complex, 121 symplectic Lanczos process, 408 stability test, 411 symplectic matrix, 12

From "The Matrix Eigenvalue Problem" by David S. Watkins. This book is available for purchase at www.siam.org/catalog. i i

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