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Journal ArticleDOI

Reduction of a band-symmetric generalized eigenvalue problem

01 Jan 1973-Communications of The ACM (ACM)-Vol. 16, Iss: 1, pp 41-44
TL;DR: The algorithm reduces the generalized problem to an ordinary eigenvalue problem for a symmetric band matrix C, whose bandwidth is the same as A and B, and is similar to those of Rutishauser and Schwartz for the reduction of symmetric matrices to band form.
Abstract: An algorithm is described for reducing the generalized eigenvalue problem Ax = lBx to an ordinary problem, in case A and B are symmetric band matrices with B positive definite. If n is the order of the matrix and m the bandwidth, the matrices A and B are partitioned into m-by-m blocks; and the algorithm is described in terms of these blocks. The algorithm reduces the generalized problem to an ordinary eigenvalue problem for a symmetric band matrix C whose bandwidth is the same as A and B. The algorithm is similar to those of Rutishauser and Schwartz for the reduction of symmetric matrices to band form. The calculation of C requires order n2m operation. The round-off error in the calculation of C is of the same order as the sum of the errors at each of the n/m steps of the algorithm, the latter errors being largely determined by the condition of B with respect to inversion.
Citations
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01 Jan 2000

813 citations


Cites background from "Reduction of a band-symmetric gener..."

  • ...= 2A 15 (L+ 1)−3 + A 4 (L+ 1)−4 + 2A 15 (L+ 1)−5 +O ( (L+ 1)−6 ) , (9)...

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  • ...The fixed parameters in (9) - contraction coefficients dγ and exponents ηγ - have typically been optimized in calculations of atoms....

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Journal ArticleDOI
TL;DR: A survey of computational methods in linear algebra can be found in this article, where the authors discuss the means and methods of estimating the quality of numerical solution of computational problems, the generalized inverse of a matrix, the solution of systems with rectangular and poorly conditioned matrices, and more traditional questions such as algebraic eigenvalue problems and systems with a square matrix.
Abstract: The authors' survey paper is devoted to the present state of computational methods in linear algebra. Questions discussed are the means and methods of estimating the quality of numerical solution of computational problems, the generalized inverse of a matrix, the solution of systems with rectangular and poorly conditioned matrices, the inverse eigenvalue problem, and more traditional questions such as algebraic eigenvalue problems and the solution of systems with a square matrix (by direct and iterative methods).

667 citations

Journal ArticleDOI
A. Demmler1, C. Reinsch1
TL;DR: For polynomial splines this matrix is closely related to an oscillation matrix and its eigenvectors show the typical sign distribution as discussed by the authors, which is the basis for a variant of spline smoothing.
Abstract: Spline smoothing can be reduced to the minimization of a certain quadratic form with positive semidefinite matrix. For polynomial splines this matrix is closely related to an oscillation matrix and its eigenvectors show the typical sign distribution. This fact is the basis for a variant of spline smoothing.

105 citations


Cites background from "Reduction of a band-symmetric gener..."

  • ...Both, QrQ and W are positive definite band matrices so that Crawford's [ 5 ] reduction algorithm can be used with advantage for the economical computation of ~k for k = m + 1 ........

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Journal ArticleDOI
TL;DR: In this paper, the authors describe a method of calculating highly excited states of hydrogen in uniform magnetic fields of moderate strength (approximately 5T) using adiabatic approximations and the role of spin-orbit interaction.
Abstract: The authors describe a method of calculating highly excited states of hydrogen in uniform magnetic fields of moderate strength ( approximately 5T). Results for Balmer emission lines and an approximate photoabsorption spectrum for Cs are presented. The authors comment on the use of adiabatic approximations and the role of the spin-orbit interaction.

93 citations

ReportDOI
31 Jan 1992
TL;DR: The LAPACK Users` Guide gives an informal introduction to the design of the algorithms and software, summarizes the contents of the package, describes conventions used in the software and documentation, and includes complete specifications for calling the routines.
Abstract: LAPACK is a transportable library of Fortran 77 subroutines for solving the most common problems in numerical linear algebra: systems of linear equations, linear least squares problems, eigenvalue problems and singular value problems. LAPACK is designed to supersede LINPACK and EISPACK, principally by restructuring the software to achieve much greater efficiency on vector processors, high-performance ``superscalar`` workstations, and shared-memory multi-processors. LAPACK also adds extra functionality, uses some new or improved algorithms, and integrates the two sets of algorithms into a unified package. The LAPACK Users` Guide gives an informal introduction to the design of the algorithms and software, summarizes the contents of the package, describes conventions used in the software and documentation, and includes complete specifications for calling the routines. This edition of the Users` guide describes Release 1.0 of LAPACK.

92 citations


Cites methods from "Reduction of a band-symmetric gener..."

  • ...Index by Keyword absolute error, 80, 81 absolute gap, 105, 113, 114, 117 accuracy, 77, 90, 115 high, 87, 104, 105, 113, 114 angle between vectors and subspaces, 82–84, 94, 103, 106, 112, 115, 118, 120, 130 arguments ABNRM, 109, 125 arrays, 136 BALANC, 109, 124 BBNRM, 125 BERR, 91 description conventions, 135 DIAG, 142 dimensions, 136 FERR, 89, 91 ILO and IHI, 43, 109, 125 INFO, 137, 152 LDA, 136, 152 LSCALE, 125 LWORK, 137 options, 136 order of, 134 RANK, 94 RCOND, 82 RCONDE, 108, 111, 122, 126 RCONDV, 108, 111, 122, 127 RSCALE, 125 SCALE, 109 UPLO, 136 work space, 137 ARPACK, xvi, xx auxiliary routines, 11 Auxiliary Routines, index of: see Appendix B, 169 avoiding poor performance, 154 backward error, 85, 86, 89–91, 93, 105 backward stability, 85, 108, 113, 118, 122, 131 componentwise, 87, 115 normwise, 86 balancing and conditioning, eigenproblems, 50 balancing of eigenproblems, 109, 124 basis, orthonormal, 19 bidiagonal form, 45, 64, 144 BLAS, 4, 55, 154 Level 1, 56 Level 2, 56, 58 Level 3, 56, 58 Level 3, fast, 77, 132 quick reference guide: see Appendix C, 181 block algorithm, 56 block size, 148 determination of, 138 from ILAENV, 138 tuning ILAENV, 138 block width, 58 blocked algorithms, performance, 59 bug reports, 7 checklist, 150 mailing alias, 150 cache, 54, 55 CAPSS, xvi, xx Cholesky factorization blocked form, 58 split, 47 chordal distance, 118, 119 CLAPACK, 8 cluster eigenvalues, 105 error bound, 108 generalized eigenvalues 391 392 INDEX BY KEYWORD error bound, 122 singular values, 114 commercial use, 5 complete orthogonal factorization, 34 computational routines, 11, 25 Computational Routines, index of: see Ap- pendix A, 156 condensed form reduction to, 64 condition number, 43, 44, 51, 82, 86, 88, 90, 92, 93, 108–110, 116, 117, 119, 122, 124, 125, 131 estimate, 87, 90, 111, 127 Cosine-Sine decomposition, 24 Crawford’s algorithm, 47 crossover point, 147 Cuppen’s divide and conquer algorithm, 40 cycle time, 54 data movement, 55 debugging release notes, 151 debugging hints installation, 151 deflating subspaces, 22, 49, 51 error bound, 122 diagonal pivoting method, 165, 166, 178, 179 distributed memory, xvi, xx divide and conquer, 21 least squares, 16 SVD, 20 documentation installation guide, 150 documentation, structure, 134 driver routine generalized least squares, 16 generalized nonsymmetric eigenvalue prob- lem, 21, 22 generalized SVD, 23 generalized symmetric definite eigenvalue problem, 20 linear equations, 14 linear least squares, 16 nonsymmetric eigenvalue problem, 19 symmetric tridiagonal eigenvalue problem, 12 driver routines, 11, 14 divide and conquer, 16, 20, 21 expert, 14, 21 simple, 14, 21 Driver Routines, index of: see Appendix A, 156 effective rank, 33 efficiency, 56 eigendecomposition blocked form, 63 multishift QR iteration, 67 symmetric, 103 eigenvalue, 17, 18, 39, 42, 103 error bound, 103, 104, 106, 108, 110, 115, 117, 118, 120, 125 generalized, 48 ordering of, 23 GNEP, 21 GSEP, 21 infinite, 48 NEP, 18 nontrivial, 25 ordering of, 19, 43, 51 sensitivity of, 43, 44 generalized, 51 SEP, 39 trivial, 25 eigenvalue problem ill-conditioned, 22 singular, 22, 128 eigenvector, 17, 39, 103 error bound, 103, 105, 106, 108, 110, 115, 117, 118, 120, 125 GNEP, 21 left, 22 right, 22 GSEP, 21 left, 18, 42, 50 generalized, 48 NEP, 18 right, 18, 42, 50 generalized, 48 INDEX BY KEYWORD 393 SEP, 39 EISPACK, 4, 139, 142, 149 converting from: see Appendix D, 186 elementary Householder matrix, see House- holder matrix, 31, 32, 34, 45, 143 elementary reflector, see Householder matrix, 31, 34, 45, 143 equality-constrained least squares, 17 equilibration, 14, 27 errata, 7 error absolute, 80, 81 analysis, 85 backward, 85, 86, 89–91, 93, 105 measurement of, 80 matrix, 80, 81 scalar, 80 subspace, 80, 82 vector, 80 relative, 78, 80, 81, 83, 86, 90 error bounds, 77 clustered eigenvalues, 107, 121 generalized nonsymmetric eigenproblem, 119, 121 generalized singular value decomposition, 129–131 generalized symmetric definite eigenprob- lem, 115, 117 linear equations, 88, 89 linear least squares, 91, 93 nonsymmetric eigenproblem, 106, 107 required for fast Level 3 BLAS, 132 singular value decomposition, 112, 113 symmetric eigenproblem, 103, 104 error handler, XERBLA, 138, 152 failures, 152 common causes, 151 error handling, 137 INFO, 137 FAQ LAPACK, 3 floating-point arithmetic, 78 guard digit, 80 IEEE standard, 79 infinity, 80 machine precision, 78 NaN, 80 Not-a-Number, 80 overflow, 78–80, 82, 90, 94, 105, 113 roundoff error, 78 underflow, 78–80 forward error, 28 forward stability, 87 componentwise relative, 87 gap, 105, 106, 113, 114, 117 general linear model problem, 36 generalized eigenproblem nonsymmetric, 21 symmetric definite, 46 symmetric definite banded, 47 generalized eigenvalue error bound, 122 generalized eigenvector error bound, 122 generalized Hessenberg form reduction to, 49 generalized least squares, 16 generalized orthogonal factorization, 35 generalized Schur vectors, 22 generalized singular value, 23 generalized singular value decomposition, 23, 51 special cases, 24 Givens rotation, 39, 45, 47 GLM, 17, 36, 159 GNEP, 21 GQR, 17, 34–36 GRQ, 17, 37 GSEP, 20 GSVD, 23, 25, 51 guard digit, 80 GUPTRI, 129 Hessenberg form, 42, 64 reduction to, 64 upper, 42 Householder matrix, 63, 143 complex, 143 394 INDEX BY KEYWORD Householder transformation - blocked form, 63 Householder vector, 143 ill-conditioned, 86 ill-posed, 86 infinity, 80 INFO, 137, 152 input error, 77 installation, 6 ILAENV, 146 LAPACK, 145 xLAMCH, 78, 146 cost of, 155 installation guide, 145 invariant subspaces, 22, 43, 108 error bound, 105, 108 inverse iteration, 40, 42 iterative refinement, 28, 132 JLAPACK, 8 LAPACK commercial use of, 5 prebuilt libraries, availability of, 5 LAPACK++, xx LAPACK90, 8 LDLT factorization, 26 linear equations, 25 linear least squares problem, 15, 31–33 generalized, 16 equality-constrained (LSE), 17, 38 regression model (GLM), 17 overdetermined, 91 overdetermined system, 94 rank-deficient, 32, 34, 46 regularization, 94 underdetermined, 94 weighted, 17 linear systems, solution of, 14, 25 LINPACK, 4, 56, 139 converting from: see Appendix D, 186 LLS, 15, 16 local memory, 55 LQ factorization, 32 LSE, 17, 38, 159 LU factorization, 26 blocked form, 60 matrix types, 26 LWORK, 137 LWORK query, 137 machine parameters, 146 machine precision, 78 matrix inversion, 14 minimum norm least squares solution, 16 minimum norm solution, 15, 32, 34, 46 multishift QR algorithm, tuning, 149 naming scheme, 12 auxiliary, 13 driver and computational, 12 NaN, 80 NEP, 18 netlib, 5 mirror repositories, 5 nonsymmetric eigenproblem, 42 generalized, 21, 48 norm ∞-norm, 81 1-norm, 81 2-norm, 81, 84 Frobenius norm, 81, 84 infinity-norm, 81 matrix, 81 one-norm, 81 two-norm, 81 vector, 80, 81 normalization, 48 Not-a-Number, 80 numerical error, sources of, 77 input error, 77 roundoff error, 77, 78 orthogonal (unitary) factorizations, 31 orthogonal (unitary) transformation, 34, 64 orthogonal factorization generalized, 35 overdetermined system, 15, 16, 93, 94 overflow, 78–80, 82, 90, 94, 105, 113 parallelism INDEX BY KEYWORD 395 compiler directives, 55 loop-based, 55 ParPre, xvi pencil, 22 performance, 4, 54 block size, 148 crossover point, 148 LWORK, 154 recommendations, 154 sensitivity, 154 permutation, 50 portability, 54, 56, 78 prebuilt libraries lapack, 5 QL factorization, 34 implicit, 40 QR decomposition with pivoting, 52 QR factorization, 31 blocked form, 63 generalized (GQR), 17, 34, 35 implicit, 40 with pivoting, 32 query, 137 quotient singular value decomposition, 23, 51 QZ method, 49 rank numerical determination of, 33, 52, 92 reduction bidiagonal, 45 tridiagonal, 39 upper Hessenberg, 42 regression, generalized linear, 17 regularization, 94 relative error, 78, 80, 81, 83, 86, 90 relative gap, 106, 114 reliability, see test suites, 6, 150 reporting bugs, see bug reports, 150 roundoff error, 77, 78 RQ factorization, 34 generalized (GRQ), 17, 37 s and dif, 125 s and sep, 110 ScaLAPACK, 8 scaling, 27, 109, 110, 124 Schur decomposition generalized, 48 Schur factorization, 18, 42 generalized, 22 Schur form, 42, 43 generalized, 49, 51 Schur vectors, 19, 42 generalized, 22, 49 SEP, 17, 39 separation of matrices, 110 separation of two matrix pairs, 125 shared memory, 54 similarity transformation, 43 singular value, 45 error bound, 112–114, 130, 132 generalized, 23 singular value decomposition generalized, 23, 25, 51 special cases, 24 singular value decomposition (SVD), 19, 44 singular vector error bound, 112–114 singular vectors left, 19, 45 right, 19, 45 source code, 5 spectral factorization, 18 spectral projector, 110 split Cholesky factorization, 47 stability, 31, 77 backward, 85–87, 108, 113, 115, 118, 122, 131 forward, 87 storage scheme, 139 band, 141 band LU, 141 bidiagonal, 142 conventional, 139 diagonal of Hermitian matrix, 143 Hermitian, 140 orthogonal or unitary matrices, 143 packed, 140 symmetric, 140 396 INDEX BY KEYWORD symmetric tridiagonal, 142 triangular, 140 unsymmetric tridiagonal, 142 Strassen’s method, 132, 133 subspaces, 84 angle between, 82–84, 94, 103, 106, 112, 115, 118, 120, 130 deflating, 22, 49, 51 invariant, 22, 43, 158 support, 7 SVD, 19 Sylvester equation, 43, 86, 111 generalized, 51, 126, 127 symmetric eigenproblem, 39 symmetric indefinite factorization, 26 blocked form, 60 test suites, 6, 150 tridiagonal form, 39, 64, 105, 144 troubleshooting, 150 tuning block multishift QR: NS, MAXB, 146 block size: NB, NBMIN, and NX, 146 divide and conquer, 147 SVD: NXSVD, 147 underdetermined system, 15, 16, 32, 94 underflow, 78–80 upper Hessenberg form, 42 vector registers, 55 vectorization, 55 workspace query, 137 workstation, super-scalar, 54 wrong results, 153 Index by Routine Name BAKVEC (EISPACK), 191 BALANC (EISPACK), 191 BALBAK (EISPACK), 191 BANDR (EISPACK), 187, 191 BANDV (EISPACK), 187, 191 BISECT (EISPACK), 191 BQR (EISPACK), 187, 191 CBABK2 (EISPACK), 191 CBAL (EISPACK), 191 CBDSDC, 147 CBDSQR, 45, 46, 114, 157, 210 CG (EISPACK), 191 CGBBRD, 45, 46, 157, 211 CGBCON, 27, 29, 157, 212 CGBEQU, 27, 29, 157, 175, 213 CGBRFS, 27, 29, 157, 213 CGBSV, 15, 157, 214 CGBSVX, 15, 85, 157, 215 CGBTRF, 27, 29, 147, 157, 217 CGBTRS, 27, 29, 157, 218 CGEBAK, 43, 44, 157, 191, 219 CGEBAL, 42–44, 109, 157, 191, 219 CGEBRD, 45, 46, 64, 66, 147, 157, 160, 161, 220 CGECON, 27, 29, 157, 221 CGEEQU, 27, 29, 157, 175, 221 CGEES, 19, 20, 154, 157, 222 CGEESX, 19, 20, 83, 107, 154, 158, 223 CGEEV, 19, 20, 154, 158, 191, 225 CGEEVX, 19, 20, 106, 107, 154, 158, 226 CGEHRD, 42–44, 64, 147, 158, 160, 161, 191, 228 CGELQF, 32, 35, 46, 147, 158, 160, 161, 229 CGELS, 16, 91, 93, 158, 229 CGELSD, 16, 66, 91, 93, 147, 158, 230 CGELSS, 16, 91, 93, 147, 158, 232 CGELSX, 16, 91, 93 CGELSY, 16, 91, 93, 158, 233 CGEQLF, 34, 35, 147, 158, 160, 161, 234 CGEQP3, 33, 35, 147, 158, 235 CGEQPF, 33 CGEQRF, 31, 33, 35, 45, 147, 158, 160, 161, 235 CGERFS, 27, 29, 158, 236 CGERQF, 34, 35, 147, 158, 160, 161, 236 CGESDD, 20, 46, 66, 94, 112, 147, 158, 237 CGESV, 15, 158, 238 CGESVD, 20, 46, 94, 112, 114, 147, 159, 239 CGESVX, 15, 85, 159, 240 CGETRF, 27, 29, 147, 157, 159, 242 CGETRI, 27, 29, 147, 159, 243 CGETRS, 27, 29, 159, 243 CGGBAK, 50, 52, 159, 244 CGGBAL, 50, 52, 124, 159, 245 CGGES, 23, 25, 159, 246 CGGESX, 23, 25, 121, 159, 248 CGGEV, 23, 25, 159, 250 CGGEVX, 23, 25, 120, 121, 159, 251 CGGGLM, 17, 159, 254 CGGHRD, 49, 52, 159, 255 CGGLSE, 17, 159, 256 CGGQRF, 36, 159, 256 CGGRQF, 37, 159, 257 CGGSVD, 25, 130, 159, 259 CGGSVP, 52, 53, 159, 167, 261 CGTCON, 27, 29, 160, 262 CGTRFS, 27, 29, 160, 263 CGTSV, 15, 160, 264 CGTSVX, 15, 160, 264 CGTTRF, 27, 29, 160, 266 CGTTRS, 27, 29, 160, 266 CH (EISPACK), 191 CHBEV, 20, 103, 163, 308 397 398 INDEX BY ROUTINE NAME CHBEVD, 20, 80, 103, 163, 309 CHBEVX, 20, 103, 163, 310 CHBGST, 47, 48, 163, 311 CHBGV, 25, 115, 163, 312 CHBGVD, 25, 80, 115, 163, 313 CHBGVX, 25, 115, 163, 314 CHBTRD, 39, 41, 163, 316 CHECON, 27, 30, 165, 190, 340 CHEEV, 20, 103, 165, 191, 341 CHEEVD, 20, 80, 103, 165, 342 CHEEVR, 20, 80, 103, 147, 166, 343 CHEEVX, 20, 103, 166, 345 CHEGST, 48, 147, 166, 346 CHEGV, 25, 86, 115, 166, 347 CHEGVD, 25, 80, 115, 166, 348 CHEGVX, 25, 115, 166, 349 CHERFS, 27, 30, 166, 351 CHESV, 15, 166, 352 CHESVX, 15, 166, 353 CHETRD, 39, 41, 147, 160, 161, 166, 192, 355 CHETRF, 27, 30, 147, 165, 166, 190, 355 CHETRI, 27, 30, 166, 190, 356 CHETRS, 27, 30, 166, 190, 357 CHGEQZ, 49, 52, 160, 267 CHICO (LINPACK), 190 CHIDI (LINPACK), 190 CHIFA (LINPACK), 190 CHISL (LINPACK), 190 CHPCO (LINPACK), 190 CHPCON, 27, 30, 164, 190, 317 CHPDI (LINPACK), 190 CHPEV, 20, 103, 164, 317 CHPEVD, 20, 80, 103, 164, 318 CHPEVX, 20, 103, 164, 319 CHPFA (LINPACK), 190 CHPGST, 48, 164, 320 CHPGV, 25, 115, 164, 321 CHPGVD, 25, 80, 115, 164, 322 CHPGVX, 25, 115, 164, 323 CHPRFS, 27, 30, 164, 325 CHPSL (LINPACK), 190 CHPSV, 15, 164, 326 CHPSVX, 15, 164, 327 CHPTRD, 39, 41, 160, 164, 192, 328 CHPTRF, 27, 30, 164–166, 190, 329 CHPTRI, 27, 30, 165, 190, 329 CHPTRS, 27, 30, 165, 190, 330 CHSEIN, 42, 44, 160, 191, 269 CHSEQR, 42–44, 146, 149, 154, 160, 191, 271 CINVIT (EISPACK), 191 COMBAK (EISPACK), 191 COMHES (EISPACK), 191 COMLR (EISPACK), 191 COMLR2 (EISPACK), 191 COMQR (EISPACK), 67, 191 COMQR2 (EISPACK), 191 CORTB (EISPACK), 191 CORTH (EISPACK), 191 CPBCON, 27, 29, 161, 285 CPBEQU, 27, 29, 161, 175, 286 CPBRFS, 27, 29, 161, 286 CPBSTF, 47, 48, 161, 287 CPBSV, 15, 161, 288 CPBSVX, 15, 161, 289 CPBTRF, 27, 29, 147, 161, 290 CPBTRS, 27, 29, 161, 291 CPOCON, 27, 29, 161, 292 CPOEQU, 27, 29, 162, 175, 292 CPORFS, 27, 29, 162, 293 CPOSV, 15, 162, 293 CPOSVX, 15, 162, 294 CPOTRF, 27, 29, 147, 161, 162, 166, 296 CPOTRI, 27, 29, 147, 162, 296 CPOTRS, 27, 29, 162, 297 CPPCON, 27, 29, 162, 297 CPPEQU, 27, 29, 162, 175, 298 CPPRFS, 27, 29, 162, 298 CPPSV, 15, 162, 299 CPPSVX, 15, 162, 300 CPPTRF, 27, 29, 162, 164, 302 CPPTRI, 27, 29, 162, 302 CPPTRS, 27, 29, 162, 302 CPTCON, 27, 29, 106, 163, 303 CPTEQR, 40, 41, 105, 163, 303 CPTRFS, 27, 29, 163, 304 CPTSV, 15, 163, 305 CPTSVX, 15, 163, 306 CPTTRF, 27, 29, 163, 307 CPTTRS, 27, 29, 163, 307 CSPCON, 27, 30, 164, 317 INDEX BY ROUTINE NAME 399 CSPRFS, 27, 30, 164, 325 CSPSV, 15, 164, 326 CSPSVX, 15, 164, 327 CSPTRF, 27, 30, 164–166, 329 CSPTRI, 27, 30, 165, 329 CSPTRS, 27, 30, 165, 330 CSTEDC, 40, 41, 65, 66, 80, 147, 165, 332 CSTEGR, 40, 41, 64, 66, 68, 80, 105, 147, 165, 333 CSTEIN, 39–41, 66, 104, 165, 335 CSTEQR, 39–41, 65, 165, 336 CSYCON, 27, 30, 165, 340 CSYEVR, 80 CSYRFS, 27, 30, 166, 351 CSYSV, 15, 166, 352 CSYSVX, 15, 166, 353 CSYTRD, 64 CSYTRF, 27, 30, 147, 165, 166, 355 CSYTRI, 27, 30, 166, 356 CSYTRS, 27, 30, 166, 357 CTBCON, 27, 30, 166, 357 CTBRFS, 27, 30, 167, 358 CTBTRS, 27, 30, 167, 359 CTGEVC, 50, 52, 167, 360 CTGEXC, 51, 52, 126, 167, 361 CTGSEN, 51, 52, 121, 167, 363 CTGSJA, 52, 53, 119, 167, 365 CTGSNA, 51, 52, 121, 167, 367 CTGSYL, 51, 52, 126, 147, 167, 369 CTPCON, 27, 30, 167, 370 CTPRFS, 27, 30, 167, 371 CTPTRI, 27, 30, 167, 372 CTPTRS, 27, 30, 167, 372 CTRCON, 27, 30, 94, 167, 373 CTREVC, 42, 44, 167, 191, 374 CTREXC, 43, 44, 110, 167, 375 CTRRFS, 27, 30, 167, 376 CTRSEN, 44, 107, 168, 377 CTRSNA, 43, 44, 107, 168, 378 CTRSYL, 43, 44, 111, 168, 380 CTRTRI, 27, 30, 147, 168, 381 CTRTRS, 27, 30, 32, 168, 381 CTZRQF, 34, 147 CTZRZF, 34, 35, 161, 168, 382 CUNGBR, 45, 46, 160, 273 CUNGHR, 42, 44, 160, 191, 274 CUNGLQ, 32, 35, 147, 160, 274 CUNGQL, 35, 147, 160, 275 CUNGQR, 31, 33, 35, 147, 160, 276 CUNGRQ, 35, 147, 160, 276 CUNGTR, 41, 160, 277 CUNMBR, 45, 46, 161, 277 CUNMHR, 42, 44, 161, 191, 279 CUNMLQ, 32, 35, 147, 161, 280 CUNMQL, 35, 147, 161, 280 CUNMQR, 31–33, 35, 36, 38, 147, 161, 281 CUNMRQ, 35, 37, 38, 147, 161, 282 CUNMRZ, 34, 35, 161, 283 CUNMTR, 39, 41, 161, 192, 284 CUPGTR, 39, 41, 160, 272 CUPMTR, 41, 160, 192, 272 DBDSDC, 45, 46, 80, 114, 147, 209 DBDSQR, 45, 46, 114, 210 DCHDC (LINPACK), 188 DCHDD (LINPACK), 188 DCHEX (LINPACK), 188 DCHUD (LINPACK), 188 DDISNA, 105, 113, 210 DGBBRD, 45, 46, 211 DGBCO (LINPACK), 188 DGBCON, 27, 29, 188, 212 DGBDI (LINPACK), 188 DGBEQU, 27, 29, 213 DGBFA (LINPACK), 188 DGBRFS, 27, 29, 213 DGBSL (LINPACK), 188 DGBSV, 15, 214 DGBSVX, 15, 85, 215 DGBTRF, 27, 29, 188, 217 DGBTRS, 27, 29, 188, 218 DGEBAK, 43, 44, 219 DGEBAL, 42–44, 109, 219 DGEBRD, 45, 46, 64–66, 220 DGECO (LINPACK), 188 DGECON, 27, 29, 188, 221 DGEDI (LINPACK), 188 DGEEQU, 27, 29, 221 DGEES, 19, 20, 154, 222 DGEESX, 19, 20, 83, 107, 154, 223 400 INDEX BY ROUTINE NAME DGEEV, 19, 20, 67, 154, 225 DGEEVX, 19, 20, 106, 107, 154, 226 DGEFA (LINPACK), 188 DGEHRD, 42–44, 64, 65, 228 DGELQF, 32, 35, 46, 229 DGELS, 16, 69, 91, 93, 229 DGELSD, 16, 66, 69, 80, 91, 93, 147, 230 DGELSS, 16, 69, 91, 93, 147, 232 DGELSX, 16, 69, 91, 93 DGELSY, 16, 69, 91, 93, 233 DGEQLF, 34, 35, 234 DGEQP3, 33, 35, 235 DGEQPF, 33, 189 DGEQRF, 31, 33, 35, 45, 63, 189, 235 DGERFS, 27, 29, 236 DGERQF, 34, 35, 236 DGESDD, 20, 46, 66, 67, 80, 94, 112, 147, 237 DGESL (LINPACK), 188 DGESV, 15, 67, 238 DGESVD, 20, 46, 66, 67, 94, 112, 114, 147, 189, 239 DGESVX, 15, 85, 240 DGETRF, 27, 29, 60, 188, 242 DGETRI, 27, 29, 188, 243 DGETRS, 27, 29, 188, 243 DGGBAK, 50, 52, 244 DGGBAL, 50, 52, 124, 245 DGGES, 23, 25, 246 DGGESX, 23, 25, 121, 248 DGGEV, 23, 25, 250 DGGEVX, 23, 25, 120, 121, 251 DGGGLM, 17, 254 DGGHRD, 49, 52, 255 DGGLSE, 17, 256 DGGQRF, 36, 256 DGGRQF, 37, 257 DGGSVD, 25, 130, 259 DGGSVP, 52, 53, 261 DGTCON, 27, 29, 262 DGTRFS, 27, 29, 263 DGTSL (LINPACK), 188 DGTSV, 15, 188, 264 DGTSVX, 15, 264 DGTTRF, 27, 29, 266 DGTTRS, 27, 29, 266 DHEEVR, 80 DHGEQZ, 49, 52, 267 DHSEIN, 42, 44, 269 DHSEQR, 42–44, 146, 149, 154, 271 DLALSD, 80 DLAMCH, 78, 146, 174 DLASV2, 79 DOPGTR, 39, 41, 272 DOPMTR, 39, 41, 272 DORGBR, 45, 46, 273 DORGHR, 42, 44, 274 DORGLQ, 32, 35, 274 DORGQL, 35, 275 DORGQR, 31, 33, 35, 276 DORGRQ, 35, 276 DORGTR, 39, 41, 277 DORMBR, 45, 46, 277 DORMHR, 42, 44, 279 DORMLQ, 32, 35, 280 DORMQL, 35, 280 DORMQR, 31–33, 35, 36, 38, 189, 281 DORMRQ, 35, 37, 38, 282 DORMRZ, 34, 35, 283 DORMTR, 39, 41, 284 DPBCO (LINPACK), 188 DPBCON, 27, 29, 188, 285 DPBDI (LINPACK), 188 DPBEQU, 27, 29, 286 DPBFA (LINPACK), 188 DPBRFS, 27, 29, 286 DPBSL (LINPACK), 188 DPBSTF, 47, 48, 287 DPBSV, 15, 288 DPBSVX, 15, 289 DPBTRF, 27, 29, 188, 290 DPBTRS, 27, 29, 188, 291 DPOCO (LINPACK), 188 DPOCON, 27, 29, 188, 292 DPODI (LINPACK), 188 DPOEQU, 27, 29, 292 DPOFA (LINPACK), 188 DPORFS, 27, 29, 293 DPOSL (LINPACK), 188 DPOSV, 15, 293 DPOSVX, 15, 294 INDEX BY ROUTINE NAME 401 DPOTRF, 27, 29, 60, 188, 296 DPOTRI, 27, 29, 188, 296 DPOTRS, 27, 29, 188, 297 DPPCO (LINPACK), 188, 189 DPPCON, 27, 29, 188, 297 DPPDI (LINPACK), 189 DPPEQU, 27, 29, 298 DPPFA (LINPACK), 189 DPPRFS, 27, 29, 298 DPPSL (LINPACK), 189 DPPSV, 15, 299 DPPSVX, 15, 300 DPPTRF, 27, 29, 188, 189, 302 DPPTRI, 27, 29, 189, 302 DPPTRS, 27, 29, 189, 302 DPTCON, 27, 29, 106, 303 DPTEQR, 40, 41, 105, 303 DPTRFS, 27, 29, 304 DPTSL (LINPACK), 189 DPTSV, 15, 189, 305 DPTSVX, 15, 306 DPTTRF, 27, 29, 307 DPTTRS, 27, 29, 307 DQRDC (LINPACK), 189 DQRSL (LINPACK), 189 DSBEV, 20, 103, 308 DSBEVD, 20, 80, 103, 309 DSBEVX, 20, 103, 310 DSBGST, 47, 48, 311 DSBGV, 25, 115, 312 DSBGVD, 25, 80, 115, 313 DSBGVX, 25, 115, 314 DSBTRD, 39, 41, 316 DSICO (LINPACK), 189 DSIDI (LINPACK), 189 DSIFA (LINPACK), 189 DSISL (LINPACK), 189 DSPCO (LINPACK), 189 DSPCON, 27, 30, 189, 317 DSPDI (LINPACK), 189 DSPEV, 20, 103, 317 DSPEVD, 20, 80, 103, 318 DSPEVX, 20, 103, 319 DSPFA (LINPACK), 189 DSPGST, 48, 320 DSPGV, 25, 115, 321 DSPGVD, 25, 80, 115, 322 DSPGVX, 25, 115, 323 DSPRFS, 27, 30, 325 DSPSL (LINPACK), 189 DSPSV, 15, 326 DSPSVX, 15, 327 DSPTRD, 39, 41, 328 DSPTRF, 27, 30, 189, 329 DSPTRI, 27, 30, 189, 329 DSPTRS, 27, 30, 189, 330 DSTEBZ, 40, 41, 105, 331 DSTEDC, 40, 41, 65, 66, 69, 80, 147, 332 DSTEGR, 40, 41, 64, 66, 68, 69, 80, 105, 147, 333 DSTEIN, 39–41, 66, 104, 335 DSTEQR, 39–41, 65, 69, 336 DSTERF, 40, 41, 64, 336 DSTEV, 20, 69, 103, 337 DSTEVD, 20, 69, 80, 103, 337 DSTEVR, 20, 69, 80, 103, 147, 338 DSTEVX, 20, 103, 105, 339 DSVDC (LINPACK), 189 DSYCON, 27, 30, 189, 340 DSYEV, 20, 69, 103, 341 DSYEVD, 20, 69, 80, 103, 342 DSYEVR, 20, 69, 80, 103, 147, 343 DSYEVX, 20, 69, 103, 345 DSYGST, 48, 346 DSYGV, 25, 86, 115, 347 DSYGVD, 25, 80, 115, 348 DSYGVX, 25, 115, 349 DSYRFS, 27, 30, 351 DSYSV, 15, 352 DSYSVX, 15, 353 DSYTRD, 39, 41, 64, 65, 355 DSYTRF, 27, 30, 60, 61, 189, 355 DSYTRI, 27, 30, 189, 356 DSYTRS, 27, 30, 189, 357 DTBCON, 27, 30, 357 DTBRFS, 27, 30, 358 DTBTRS, 27, 30, 359 DTGEVC, 50, 52, 360 DTGEXC, 51, 52, 126, 361 DTGSEN, 51, 52, 121, 363 402 INDEX BY ROUTINE NAME DTGSJA, 52, 53, 119, 365 DTGSNA, 51, 52, 121, 367 DTGSYL, 51, 52, 126, 369 DTPCON, 27, 30, 370 DTPRFS, 27, 30, 371 DTPTRI, 27, 30, 372 DTPTRS, 27, 30, 372 DTRCO (LINPACK), 189 DTRCON, 27, 30, 94, 189, 373 DTRDI (LINPACK), 189 DTREVC, 42, 44, 374 DTREXC, 43, 44, 110, 375 DTRRFS, 27, 30, 376 DTRSEN, 44, 107, 377 DTRSL (LINPACK), 189 DTRSNA, 43, 44, 107, 378 DTRSYL, 43, 44, 111, 380 DTRTRI, 27, 30, 189, 381 DTRTRS, 27, 30, 32, 189, 381 DTZRQF, 34 DTZRZF, 34, 35, 382 ELMBAK (EISPACK), 191 ELMHES (EISPACK), 191 ELTRAN (EISPACK), 191 FIGI (EISPACK), 191, 192 FIGI2 (EISPACK), 192 HQR (EISPACK), 67, 192 HQR2 (EISPACK), 192 HTRIB3 (EISPACK), 192 HTRIBK (EISPACK), 192 HTRID3 (EISPACK), 187, 192 HTRIDI (EISPACK), 187, 192 ILAENV, 138, 146–148, 154, 170 IMTQL1 (EISPACK), 187, 192 IMTQL2 (EISPACK), 187, 192 IMTQLV (EISPACK), 192 INVIT (EISPACK), 192 LSAME, 170 LSAMEN, 170 MINFIT (EISPACK), 192 ORTBAK (EISPACK), 192 ORTHES (EISPACK), 192 ORTRAN (EISPACK), 192 QZHES (EISPACK), 192 QZIT (EISPACK), 192 QZVAL (EISPACK), 192 QZVEC (EISPACK), 192 RATQR (EISPACK), 187, 192 REBAK (EISPACK), 192 REBAKB (EISPACK), 192 REDUC (EISPACK), 192, 193 REDUC2 (EISPACK), 192, 193 RG (EISPACK), 193 RGG (EISPACK), 193 RS (EISPACK), 193 RSB (EISPACK), 187, 193 RSG (EISPACK), 193 RSGAB (EISPACK), 193 RSGBA (EISPACK), 193 RSM (EISPACK), 193 RSP (EISPACK), 193 RST (EISPACK), 187, 193 RT (EISPACK), 193 SBDSDC, 45, 46, 80, 114, 147, 157, 209 SBDSQR, 45, 46, 114, 157, 210 SCHDC (LINPACK), 188 SCHDD (LINPACK), 188 SCHEX (LINPACK), 188 SCHUD (LINPACK), 188 SDISNA, 105, 113, 210 SGBBRD, 45, 46, 157, 211 SGBCO (LINPACK), 188 SGBCON, 27, 29, 157, 188, 212 SGBDI (LINPACK), 188 SGBEQU, 27, 29, 157, 175, 213 SGBFA (LINPACK), 188 SGBRFS, 27, 29, 157, 213 SGBSL (LINPACK), 188 SGBSV, 15, 157, 191, 214 SGBSVX, 15, 85, 157, 215 SGBTRF, 27, 29, 147, 157, 188, 217 SGBTRS, 27, 29, 157, 188, 218 SGEBAK, 43, 44, 157, 191, 219 INDEX BY ROUTINE NAME 403 SGEBAL, 42–44, 109, 157, 191, 219 SGEBRD, 45, 46, 64, 66, 147, 148, 157, 160, 161, 220 SGECO (LINPACK), 188 SGECON, 27, 29, 157, 188, 221 SGEDI (LINPACK), 188 SGEEQU, 27, 29, 157, 175, 221 SGEES, 19, 20, 154, 157, 222 SGEESX, 19, 20, 83, 107, 154, 158, 223 SGEEV, 19, 20, 154, 158, 193, 225 SGEEVX, 19, 20, 106, 107, 154, 158, 226 SGEFA (LINPACK), 188 SGEGV, 193 SGEHRD, 42–44, 64, 147, 158, 160, 161, 191, 192, 228 SGELQF, 32, 35, 46, 147, 158, 160, 161, 178, 179, 229 SGELS, 16, 91, 93, 158, 229 SGELSD, 16, 66, 80, 91, 93, 147, 158, 230 SGELSS, 16, 91, 93, 147, 158, 192, 232 SGELSX, 16, 91–93 SGELSY, 16, 91–93, 158, 233 SGEQLF, 34, 35, 147, 158, 160, 161, 178, 179, 234 SGEQP3, 33, 35, 147, 158, 235 SGEQPF, 33, 189 SGEQRF, 31, 33, 35, 45, 147, 148, 158, 160, 161, 178, 179, 189, 235 SGERFS, 27, 29, 158, 236 SGERQF, 34, 35, 147, 158, 160, 161, 179, 236 SGESDD, 20, 46, 66, 80, 94, 112, 147, 158, 237 SGESL (LINPACK), 188 SGESV, 15, 88, 152, 158, 238 SGESVD, 20, 46, 94, 112, 114, 147, 159, 189, 193, 239 SGESVX, 15, 85, 89, 153, 159, 240 SGETRF, 27, 29, 147, 157, 159, 188, 242 SGETRI, 27, 29, 147, 159, 188, 243 SGETRS, 27, 29, 159, 188, 243 SGGBAK, 50, 52, 159, 244 SGGBAL, 50, 52, 124, 159, 245 SGGES, 23, 25, 159, 246 SGGESX, 23, 25, 121, 159, 248 SGGEV, 23, 25, 159, 250 SGGEVX, 23, 25, 120, 121, 159, 251 SGGGLM, 17, 159, 254 SGGHRD, 49, 52, 159, 192, 255 SGGLSE, 17, 159, 256 SGGQRF, 36, 159, 256 SGGRQF, 37, 159, 257 SGGSVD, 25, 130, 159, 259 SGGSVP, 52, 53, 159, 167, 261 SGTCON, 27, 29, 160, 262 SGTRFS, 27, 29, 160, 263 SGTSL (LINPACK), 188 SGTSV, 15, 160, 188, 264 SGTSVX, 15, 160, 264 SGTTRF, 27, 29, 160, 266 SGTTRS, 27, 29, 160, 266 SHEEVR, 80 SHGEQZ, 49, 52, 160, 192, 267 SHSEIN, 42, 44, 160, 192, 269 SHSEQR, 42–44, 146, 149, 154, 160, 192, 271 SLALSD, 80 SLAMCH, 78, 146, 155, 174 SLASV2, 79 SOPGTR, 39, 41, 160, 272 SOPMTR, 39, 41, 160, 194, 272 SORGBR, 45, 46, 160, 273 SORGHR, 42, 44, 160, 191, 192, 274 SORGLQ, 32, 35, 147, 160, 274 SORGQL, 35, 147, 160, 275 SORGQR, 31, 33, 35, 147, 160, 276 SORGRQ, 35, 147, 160, 276 SORGTR, 39, 41, 160, 194, 277 SORMBR, 45, 46, 161, 277 SORMHR, 42, 44, 161, 191, 192, 279 SORMLQ, 32, 35, 147, 161, 280 SORMQL, 35, 147, 161, 280 SORMQR, 31–33, 35, 36, 38, 147, 161, 189, 281 SORMRQ, 35, 37, 38, 147, 161, 282 SORMRZ, 34, 35, 161, 283 SORMTR, 39, 41, 161, 194, 284 SPBCO (LINPACK), 188 SPBCON, 27, 29, 161, 188, 285 SPBDI (LINPACK), 188 SPBEQU, 27, 29, 161, 175, 286 SPBFA (LINPACK), 188 404 INDEX BY ROUTINE NAME SPBRFS, 27, 29, 161, 286 SPBSL (LINPACK), 188 SPBSTF, 47, 48, 161, 287 SPBSV, 15, 161, 288 SPBSVX, 15, 161, 289 SPBTRF, 27, 29, 147, 161, 188, 290 SPBTRS, 27, 29, 161, 188, 291 SPOCO (LINPACK), 188 SPOCON, 27, 29, 161, 188, 292 SPODI (LINPACK), 188 SPOEQU, 27, 29, 162, 175, 292 SPOFA (LINPACK), 56, 57, 188 SPORFS, 27, 29, 162, 293 SPOSL (LINPACK), 188 SPOSV, 15, 162, 293 SPOSVX, 15, 162, 294 SPOTRF, 27, 29, 59, 147, 161, 162, 166, 179, 188, 296 SPOTRI, 27, 29, 147, 162, 188, 296 SPOTRS, 27, 29, 162, 188, 297 SPPCO (LINPACK), 188, 189 SPPCON, 27, 29, 162, 188, 297 SPPDI (LINPACK), 189 SPPEQU, 27, 29, 162, 175, 298 SPPFA (LINPACK), 189 SPPRFS, 27, 29, 162, 298 SPPSL (LINPACK), 189 SPPSV, 15, 162, 299 SPPSVX, 15, 162, 300 SPPTRF, 27, 29, 162, 164, 188, 189, 302 SPPTRI, 27, 29, 162, 189, 302 SPPTRS, 27, 29, 162, 189, 302 SPTCON, 27, 29, 106, 163, 303 SPTEQR, 40, 41, 105, 163, 303 SPTRFS, 27, 29, 163, 304 SPTSL (LINPACK), 189 SPTSV, 15, 163, 189, 305 SPTSVX, 15, 163, 306 SPTTRF, 27, 29, 163, 307 SPTTRS, 27, 29, 163, 307 SQRDC (LINPACK), 189 SQRSL (LINPACK), 189 SSBEV, 20, 103, 163, 193, 308 SSBEVD, 20, 80, 103, 163, 193, 309 SSBEVX, 20, 103, 163, 191, 310 SSBGST, 47, 48, 163, 311 SSBGV, 25, 115, 163, 312 SSBGVD, 25, 80, 115, 163, 313 SSBGVX, 25, 115, 163, 314 SSBTRD, 39, 41, 163, 191, 316 SSICO (LINPACK), 189 SSIDI (LINPACK), 189 SSIFA (LINPACK), 189 SSISL (LINPACK), 189 SSPCO (LINPACK), 189 SSPCON, 27, 30, 164, 189, 317 SSPDI (LINPACK), 189 SSPEV, 20, 103, 164, 193, 317 SSPEVD, 20, 80, 103, 164, 193, 318 SSPEVX, 20, 103, 164, 319 SSPFA (LINPACK), 189 SSPGST, 48, 164, 320 SSPGV, 25, 115, 164, 321 SSPGVD, 25, 80, 115, 164, 322 SSPGVX, 25, 115, 164, 323 SSPRFS, 27, 30, 164, 325 SSPSL (LINPACK), 189 SSPSV, 15, 164, 326 SSPSVX, 15, 164, 327 SSPTRD, 39, 41, 160, 164, 194, 328 SSPTRF, 27, 30, 164–166, 189, 329 SSPTRI, 27, 30, 165, 189, 329 SSPTRS, 27, 30, 165, 189, 330 SSTEBZ, 40, 41, 105, 147, 165, 172, 191, 192, 194, 331 SSTEDC, 40, 41, 65, 66, 80, 147, 165, 192, 193, 332 SSTEGR, 40, 41, 64, 66, 68, 80, 105, 147, 165, 333 SSTEIN, 39–41, 66, 104, 165, 193, 194, 335 SSTEQR, 39–41, 65, 165, 192, 193, 336 SSTERF, 40, 41, 64, 165, 192, 193, 336 SSTEV, 20, 103, 165, 193, 337 SSTEVD, 20, 80, 103, 165, 193, 337 SSTEVR, 20, 80, 103, 147, 165, 338 SSTEVX, 20, 103, 105, 165, 339 SSVDC (LINPACK), 189 SSYCON, 27, 30, 165, 189, 340 SSYEV, 20, 103, 165, 193, 341 SSYEVD, 20, 80, 103, 165, 193, 342 INDEX BY ROUTINE NAME 405 SSYEVR, 20, 80, 103, 147, 166, 343 SSYEVX, 20, 103, 166, 193, 345 SSYGST, 48, 147, 166, 193, 346 SSYGV, 25, 86, 115, 166, 193, 347 SSYGVD, 25, 80, 115, 166, 348 SSYGVX, 25, 115, 166, 349 SSYRFS, 27, 30, 166, 351 SSYSV, 15, 166, 352 SSYSVX, 15, 166, 353 SSYTRD, 39, 41, 64, 147, 160, 161, 166, 194, 355 SSYTRF, 27, 30, 60, 147, 165, 166, 189, 355 SSYTRI, 27, 30, 166, 189, 356 SSYTRS, 27, 30, 166, 189, 357 STBCON, 27, 30, 166, 357 STBRFS, 27, 30, 167, 358 STBTRS, 27, 30, 167, 359 STGEVC, 50, 52, 167, 192, 360 STGEXC, 51, 52, 126, 167, 361 STGSEN, 51, 52, 121, 167, 363 STGSJA, 52, 53, 119, 167, 365 STGSNA, 51, 52, 121, 167, 367 STGSYL, 51, 52, 126, 147, 167, 369 STPCON, 27, 30, 167, 370 STPRFS, 27, 30, 167, 371 STPTRI, 27, 30, 167, 372 STPTRS, 27, 30, 167, 372 STRCO (LINPACK), 189 STRCON, 27, 30, 94, 167, 189, 373 STRDI (LINPACK), 189 STREVC, 42, 44, 167, 192, 374 STREXC, 43, 44, 110, 167, 375 STRRFS, 27, 30, 167, 376 STRSEN, 44, 107, 168, 377 STRSL (LINPACK), 189 STRSNA, 43, 44, 107, 168, 378 STRSYL, 43, 44, 111, 168, 380 STRTRI, 27, 30, 147, 168, 189, 381 STRTRS, 27, 30, 32, 168, 189, 381 STZRQF, 34, 147 STZRZF, 34, 35, 161, 168, 179, 382 SVD (EISPACK), 193 TINVIT (EISPACK), 193 TQL1 (EISPACK), 187, 193 TQL2 (EISPACK), 187, 193 TQLRAT (EISPACK), 187, 193 TRBAK1 (EISPACK), 194 TRBAK3 (EISPACK), 194 TRED1 (EISPACK), 187, 194 TRED2 (EISPACK), 187, 194 TRED3 (EISPACK), 187, 194 TRIDIB (EISPACK), 194 TSTURM (EISPACK), 194 XERBLA, 138, 152, 179 ZBDSDC, 147 ZBDSQR, 45, 46, 114, 210 ZGBBRD, 45, 46, 211 ZGBCON, 27, 29, 212 ZGBEQU, 27, 29, 213 ZGBRFS, 27, 29, 213 ZGBSV, 15, 214 ZGBSVX, 15, 85, 215 ZGBTRF, 27, 29, 217 ZGBTRS, 27, 29, 218 ZGEBAK, 43, 44, 219 ZGEBAL, 42–44, 109, 219 ZGEBRD, 45, 46, 64, 66, 220 ZGECON, 27, 29, 221 ZGEEQU, 27, 29, 221 ZGEES, 19, 20, 154, 222 ZGEESX, 19, 20, 83, 107, 154, 223 ZGEEV, 19, 20, 154, 225 ZGEEVX, 19, 20, 106, 107, 154, 226 ZGEHRD, 42–44, 64, 228 ZGELQF, 32, 35, 46, 229 ZGELS, 16, 91, 93, 229 ZGELSD, 16, 66, 91, 93, 147, 230 ZGELSS, 16, 91, 93, 147, 232 ZGELSX, 16, 91, 93 ZGELSY, 16, 91, 93, 233 ZGEQLF, 34, 35, 234 ZGEQP3, 33, 35, 235 ZGEQPF, 33 ZGEQRF, 31, 33, 35, 45, 235 ZGERFS, 27, 29, 236 ZGERQF, 34, 35, 236 ZGESDD, 20, 46, 66, 94, 112, 147, 237 ZGESV, 15, 238 406 INDEX BY ROUTINE NAME ZGESVD, 20, 46, 94, 112, 114, 147, 239 ZGESVX, 15, 85, 240 ZGETRF, 27, 29, 242 ZGETRI, 27, 29, 243 ZGETRS, 27, 29, 243 ZGGBAK, 50, 52, 244 ZGGBAL, 50, 52, 124, 245 ZGGES, 23, 25, 246 ZGGESX, 23, 121, 248 ZGGEV, 23, 25, 250 ZGGEVX, 23, 25, 120, 121, 251 ZGGGLM, 17, 254 ZGGHRD, 49, 52, 255 ZGGLSE, 17, 256 ZGGQRF, 36, 256 ZGGRQF, 37, 257 ZGGSVD, 25, 130, 259 ZGGSVP, 52, 53, 261 ZGTCON, 27, 29, 262 ZGTRFS, 27, 29, 263 ZGTSV, 15, 264 ZGTSVX, 15, 264 ZGTTRF, 27, 29, 266 ZGTTRS, 27, 29, 266 ZHBEV, 20, 103, 308 ZHBEVD, 20, 80, 103, 309 ZHBEVX, 20, 103, 310 ZHBGST, 47, 48, 311 ZHBGV, 25, 115, 312 ZHBGVD, 25, 80, 115, 313 ZHBGVX, 25, 115, 314 ZHBTRD, 39, 41, 316 ZHECON, 27, 30, 190, 340 ZHEEV, 20, 103, 341 ZHEEVD, 20, 80, 103, 342 ZHEEVR, 20, 80, 103, 147, 343 ZHEEVX, 20, 103, 345 ZHEGST, 48, 346 ZHEGV, 25, 86, 115, 347 ZHEGVD, 25, 80, 115, 348 ZHEGVX, 25, 115, 349 ZHERFS, 27, 30, 351 ZHESV, 15, 352 ZHESVX, 15, 353 ZHETRD, 39, 41, 355 ZHETRF, 27, 30, 190, 355 ZHETRI, 27, 30, 190, 356 ZHETRS, 27, 30, 190, 357 ZHGEQZ, 49, 52, 267 ZHICO (LINPACK), 190 ZHIDI (LINPACK), 190 ZHIFA (LINPACK), 190 ZHISL (LINPACK), 190 ZHPCO (LINPACK), 190 ZHPCON, 27, 30, 190, 317 ZHPDI (LINPACK), 190 ZHPEV, 20, 103, 317 ZHPEVD, 20, 80, 103, 318 ZHPEVX, 20, 103, 319 ZHPFA (LINPACK), 190 ZHPGST, 48, 320 ZHPGV, 25, 115, 321 ZHPGVD, 25, 80, 115, 322 ZHPGVX, 25, 115, 323 ZHPRFS, 27, 30, 325 ZHPSL (LINPACK), 190 ZHPSV, 15, 326 ZHPSVX, 15, 327 ZHPTRD, 39, 41, 328 ZHPTRF, 27, 30, 190, 329 ZHPTRI, 27, 30, 190, 329 ZHPTRS, 27, 30, 190, 330 ZHSEIN, 42, 44, 269 ZHSEQR, 42–44, 146, 149, 154, 271 ZPBCON, 27, 29, 285 ZPBEQU, 27, 29, 286 ZPBRFS, 27, 29, 286 ZPBSTF, 47, 48, 287 ZPBSV, 15, 288 ZPBSVX, 15, 289 ZPBTRF, 27, 29, 290 ZPBTRS, 27, 29, 291 ZPOCON, 27, 29, 292 ZPOEQU, 27, 29, 292 ZPORFS, 27, 29, 293 ZPOSV, 15, 293 ZPOSVX, 15, 294 ZPOTRF, 27, 29, 296 ZPOTRI, 27, 29, 296 ZPOTRS, 27, 29, 297 INDEX BY ROUTINE NAME 407 ZPPCON, 27, 29, 297 ZPPEQU, 27, 29, 298 ZPPRFS, 27, 29, 298 ZPPSV, 15, 299 ZPPSVX, 15, 300 ZPPTRF, 27, 29, 302 ZPPTRI, 27, 29, 302 ZPPTRS, 27, 29, 302 ZPTCON, 27, 29, 106, 303 ZPTEQR, 40, 41, 105, 303 ZPTRFS, 27, 29, 304 ZPTSV, 15, 305 ZPTSVX, 15, 306 ZPTTRF, 27, 29, 307 ZPTTRS, 27, 29, 307 ZSPCON, 27, 30, 317 ZSPRFS, 27, 30, 325 ZSPSV, 15, 326 ZSPSVX, 15, 327 ZSPTRF, 27, 30, 329 ZSPTRI, 27, 30, 329 ZSPTRS, 27, 30, 330 ZSTEDC, 40, 41, 65, 66, 80, 147, 332 ZSTEGR, 40, 41, 64, 66, 68, 80, 105, 147, 333 ZSTEIN, 39–41, 66, 104, 335 ZSTEQR, 39–41, 65, 336 ZSYCON, 27, 30, 340 ZSYEVR, 80 ZSYRFS, 27, 30, 351 ZSYSV, 15, 352 ZSYSVX, 15, 353 ZSYTRD, 64 ZSYTRF, 27, 30, 355 ZSYTRI, 27, 30, 356 ZSYTRS, 27, 30, 357 ZTBCON, 27, 30, 357 ZTBRFS, 27, 30, 358 ZTBTRS, 27, 30, 359 ZTGEVC, 50, 52, 360 ZTGEXC, 51, 52, 126, 361 ZTGSEN, 51, 52, 121, 363 ZTGSJA, 52, 53, 119, 365 ZTGSNA, 51, 52, 121, 367 ZTGSYL, 51, 52, 126, 369 ZTPCON, 27, 30, 370 ZTPRFS, 27, 30, 371 ZTPTRI, 27, 30, 372 ZTPTRS, 27, 30, 372 ZTRCON, 27, 30, 94, 373 ZTREVC, 42, 44, 374 ZTREXC, 43, 44, 110, 375 ZTRRFS, 27, 30, 376 ZTRSEN, 44, 107, 377 ZTRSNA, 43, 44, 107, 378 ZTRSYL, 43, 44, 111, 380 ZTRTRI, 27, 30, 381 ZTRTRS, 27, 30, 32, 381 ZTZRQF, 34 ZTZRZF, 34, 35, 382 ZUNGBR, 45, 46, 273 ZUNGHR, 42, 44, 274 ZUNGLQ, 32, 35, 274 ZUNGQL, 35, 275 ZUNGQR, 31, 33, 35, 276 ZUNGRQ, 35, 276 ZUNGTR, 41, 277 ZUNMBR, 45, 46, 277 ZUNMHR, 42, 44, 279 ZUNMLQ, 32, 35, 280 ZUNMQL, 35, 280 ZUNMQR, 31–33, 35, 36, 38, 281 ZUNMRQ, 35, 37, 38, 282 ZUNMRZ, 34, 35, 283 ZUNMTR, 39, 41, 284 ZUPGTR, 39, 41, 272 ZUPMTR, 41, 272...

    [...]

  • ...[20] C. R. Crawford, Reduction of a band-symmetric generalized eigenvalue problem, Comm. ACM, 16 (1973), pp. 41–44....

    [...]

  • ...This is known as Crawford’s algorithm (see Crawford [20])....

    [...]

References
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01 Jan 1964

1,573 citations

Journal ArticleDOI
TL;DR: In this paper, Givens proposed the Jacobi rotation method to reduce a full symmetric matrix A = (a ik ) of order n by a sequence of appropriately chosen elementary orthogonal transformations (in the following called Jacobi rotations) to triput diagonal form.
Abstract: The well known method proposed by Givens [1] reduces a full symmetric matrix A = (a ik ) of order n by a sequence of appropriately chosen elementary orthogonal transformations (in the following called Jacobi rotations) to triput diagonal form. This is achieved by (n - 1)(n - 2)/2 Jacobi rotations, each of which annihilates one of the elements a ik with |i - k|>1. If this process is applied in one of its usual ways to a symmetric band matrix A = (a ik ) of order n and with the band width m>1, i.e. with $${a_{ik}} = 0{\rm{ for all }}i{\rm{ and }}k{\rm{ with |}}i - k{\rm{| >}}m,$$ (1) it would of course produce a tridiagonal matrix, too. But the rotations generate immediately nonvanishing elements outside the original band that show the tendency to fill out the matrix. Thus it seems that little profit with respect to computational and storage requirements may be taken from the property of the given matrix A to be of band type.

93 citations


"Reduction of a band-symmetric gener..." refers methods in this paper

  • ...The algorithm of Schwarz to reduce C to tridiagonal form....

    [...]

  • ...Rutishauser [2], 1963, and Schwarz [3], 1968, de-scribed methods for the ordinary eigenvalue problem which in the case of band matrices take full advantage of sparsity....

    [...]

  • ...Rutishauser [2], 1963, and Schwarz [3], 1968, described methods for the ordinary eigenvalue problem which in the case o f band matrices take full advantage o f sparsity....

    [...]

  • ...Schwarz, H. R. Tridiagonalization of a symmetric band matrix....

    [...]

Journal ArticleDOI
TL;DR: The spectrum of $Ax - \lambda Bx = 0$ consists of stable and unstable eigenvalues, which undergo, respectively, small and large changes in response to small changes in A and B.
Abstract: The spectrum of $Ax - \lambda Bx = 0$ consists of stable and unstable eigenvalues, which undergo, respectively, small and large changes in response to small changes in A and B. The algorithm isolates and accurately computes the eigenspace associated with the stable eigenvalues.

52 citations

01 Aug 1966
TL;DR: Two tested programs are supplied to find the eigenvalues of a symmetric tridiagonal matrix using a square-root-free version of the QR algorithm and a compact kind of Sturm sequence algorithm.
Abstract: Two tested programs are supplied to find the eigenvalues of a symmetric tridiagonal matrix. One program uses a square-root-free version of the QR algorithm. The other uses a compact kind of Sturm sequence algorithm. These programs are faster and more accurate than the other comparable programs published previously with which they have been compared.

11 citations


"Reduction of a band-symmetric gener..." refers methods in this paper

  • ...The Sturm sequence algorithm for symmetric tridiagonal matrices of Kahan and Varah [7], 1966, could be used to replace step 3, or the Q-R algorithm for band symmetric matrices could be used to replace steps 2 and 3....

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  • ...Kahan, W., and Varah, J....

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  • ...The Sturm sequence al- gorithm for symmetric tridiagonal matrices of Kahan and Varah [7], 1966, could be used to replace step 3, or the Q-R algorithm for band symmetric matrices could be used to replace steps 2 and 3....

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Book
01 Jan 1974

4 citations


"Reduction of a band-symmetric gener..." refers background in this paper

  • ...That analysis appears in my thesis, Crawford [5], 1970....

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  • ...Reduction of a Band-Symmetric Generalized Eigen- value Problem C.R. Crawford The University of Toronto Consider the generalized eigenvalue problem: Let A and B be n-by-n real symmetric matrices with B positive definite; find real numbers k such that Ax = )~Bx for some nonzero x C V,,(R)....

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  • ...That analysis appears in my thesis, Crawford [5], 1970....

    [...]

  • ...Crawford, C. R....

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