numeric-linalg
Educational material on the SciPy implementation of numerical linear algebra algorithms
| Name | Size | Mode | |
| .. | |||
| lapack/SRC/dhseqr.f | 18741B | -rw-r--r-- |
001 002 003 004 005 006 007 008 009 010 011 012 013 014 015 016 017 018 019 020 021 022 023 024 025 026 027 028 029 030 031 032 033 034 035 036 037 038 039 040 041 042 043 044 045 046 047 048 049 050 051 052 053 054 055 056 057 058 059 060 061 062 063 064 065 066 067 068 069 070 071 072 073 074 075 076 077 078 079 080 081 082 083 084 085 086 087 088 089 090 091 092 093 094 095 096 097 098 099 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120 121 122 123 124 125 126 127 128 129 130 131 132 133 134 135 136 137 138 139 140 141 142 143 144 145 146 147 148 149 150 151 152 153 154 155 156 157 158 159 160 161 162 163 164 165 166 167 168 169 170 171 172 173 174 175 176 177 178 179 180 181 182 183 184 185 186 187 188 189 190 191 192 193 194 195 196 197 198 199 200 201 202 203 204 205 206 207 208 209 210 211 212 213 214 215 216 217 218 219 220 221 222 223 224 225 226 227 228 229 230 231 232 233 234 235 236 237 238 239 240 241 242 243 244 245 246 247 248 249 250 251 252 253 254 255 256 257 258 259 260 261 262 263 264 265 266 267 268 269 270 271 272 273 274 275 276 277 278 279 280 281 282 283 284 285 286 287 288 289 290 291 292 293 294 295 296 297 298 299 300 301 302 303 304 305 306 307 308 309 310 311 312 313 314 315 316 317 318 319 320 321 322 323 324 325 326 327 328 329 330 331 332 333 334 335 336 337 338 339 340 341 342 343 344 345 346 347 348 349 350 351 352 353 354 355 356 357 358 359 360 361 362 363 364 365 366 367 368 369 370 371 372 373 374 375 376 377 378 379 380 381 382 383 384 385 386 387 388 389 390 391 392 393 394 395 396 397 398 399 400 401 402 403 404 405 406 407 408 409 410 411 412 413 414 415 416 417 418 419 420 421 422 423 424 425 426 427 428 429 430 431 432 433 434 435 436 437 438 439 440 441 442 443 444 445 446 447 448 449 450 451 452 453 454 455 456 457 458 459 460 461 462 463 464 465 466 467 468 469 470 471 472 473 474 475 476 477 478 479 480 481 482 483 484 485 486 487 488 489 490 491 492 493 494 495 496 497 498 499 500 501 502 503 504 505 506 507 508 509 510 511 512 513 514 515 516 517 518 519
*> \brief \b DHSEQR * * =========== DOCUMENTATION =========== * * Online html documentation available at * http://www.netlib.org/lapack/explore-html/ * *> \htmlonly *> Download DHSEQR + dependencies *> <a href="http://www.netlib.org/cgi-bin/netlibfiles.tgz?format=tgz&filename=/lapack/lapack_routine/dhseqr.f"> *> [TGZ]</a> *> <a href="http://www.netlib.org/cgi-bin/netlibfiles.zip?format=zip&filename=/lapack/lapack_routine/dhseqr.f"> *> [ZIP]</a> *> <a href="http://www.netlib.org/cgi-bin/netlibfiles.txt?format=txt&filename=/lapack/lapack_routine/dhseqr.f"> *> [TXT]</a> *> \endhtmlonly * * Definition: * =========== * * SUBROUTINE DHSEQR( JOB, COMPZ, N, ILO, IHI, H, LDH, WR, WI, Z, * LDZ, WORK, LWORK, INFO ) * * .. Scalar Arguments .. * INTEGER IHI, ILO, INFO, LDH, LDZ, LWORK, N * CHARACTER COMPZ, JOB * .. * .. Array Arguments .. * DOUBLE PRECISION H( LDH, * ), WI( * ), WORK( * ), WR( * ), * $ Z( LDZ, * ) * .. * * *> \par Purpose: * ============= *> *> \verbatim *> *> DHSEQR computes the eigenvalues of a Hessenberg matrix H *> and, optionally, the matrices T and Z from the Schur decomposition *> H = Z T Z**T, where T is an upper quasi-triangular matrix (the *> Schur form), and Z is the orthogonal matrix of Schur vectors. *> *> Optionally Z may be postmultiplied into an input orthogonal *> matrix Q so that this routine can give the Schur factorization *> of a matrix A which has been reduced to the Hessenberg form H *> by the orthogonal matrix Q: A = Q*H*Q**T = (QZ)*T*(QZ)**T. *> \endverbatim * * Arguments: * ========== * *> \param[in] JOB *> \verbatim *> JOB is CHARACTER*1 *> = 'E': compute eigenvalues only; *> = 'S': compute eigenvalues and the Schur form T. *> \endverbatim *> *> \param[in] COMPZ *> \verbatim *> COMPZ is CHARACTER*1 *> = 'N': no Schur vectors are computed; *> = 'I': Z is initialized to the unit matrix and the matrix Z *> of Schur vectors of H is returned; *> = 'V': Z must contain an orthogonal matrix Q on entry, and *> the product Q*Z is returned. *> \endverbatim *> *> \param[in] N *> \verbatim *> N is INTEGER *> The order of the matrix H. N >= 0. *> \endverbatim *> *> \param[in] ILO *> \verbatim *> ILO is INTEGER *> \endverbatim *> *> \param[in] IHI *> \verbatim *> IHI is INTEGER *> *> It is assumed that H is already upper triangular in rows *> and columns 1:ILO-1 and IHI+1:N. ILO and IHI are normally *> set by a previous call to DGEBAL, and then passed to ZGEHRD *> when the matrix output by DGEBAL is reduced to Hessenberg *> form. Otherwise ILO and IHI should be set to 1 and N *> respectively. If N > 0, then 1 <= ILO <= IHI <= N. *> If N = 0, then ILO = 1 and IHI = 0. *> \endverbatim *> *> \param[in,out] H *> \verbatim *> H is DOUBLE PRECISION array, dimension (LDH,N) *> On entry, the upper Hessenberg matrix H. *> On exit, if INFO = 0 and JOB = 'S', then H contains the *> upper quasi-triangular matrix T from the Schur decomposition *> (the Schur form); 2-by-2 diagonal blocks (corresponding to *> complex conjugate pairs of eigenvalues) are returned in *> standard form, with H(i,i) = H(i+1,i+1) and *> H(i+1,i)*H(i,i+1) < 0. If INFO = 0 and JOB = 'E', the *> contents of H are unspecified on exit. (The output value of *> H when INFO > 0 is given under the description of INFO *> below.) *> *> Unlike earlier versions of DHSEQR, this subroutine may *> explicitly H(i,j) = 0 for i > j and j = 1, 2, ... ILO-1 *> or j = IHI+1, IHI+2, ... N. *> \endverbatim *> *> \param[in] LDH *> \verbatim *> LDH is INTEGER *> The leading dimension of the array H. LDH >= max(1,N). *> \endverbatim *> *> \param[out] WR *> \verbatim *> WR is DOUBLE PRECISION array, dimension (N) *> \endverbatim *> *> \param[out] WI *> \verbatim *> WI is DOUBLE PRECISION array, dimension (N) *> *> The real and imaginary parts, respectively, of the computed *> eigenvalues. If two eigenvalues are computed as a complex *> conjugate pair, they are stored in consecutive elements of *> WR and WI, say the i-th and (i+1)th, with WI(i) > 0 and *> WI(i+1) < 0. If JOB = 'S', the eigenvalues are stored in *> the same order as on the diagonal of the Schur form returned *> in H, with WR(i) = H(i,i) and, if H(i:i+1,i:i+1) is a 2-by-2 *> diagonal block, WI(i) = sqrt(-H(i+1,i)*H(i,i+1)) and *> WI(i+1) = -WI(i). *> \endverbatim *> *> \param[in,out] Z *> \verbatim *> Z is DOUBLE PRECISION array, dimension (LDZ,N) *> If COMPZ = 'N', Z is not referenced. *> If COMPZ = 'I', on entry Z need not be set and on exit, *> if INFO = 0, Z contains the orthogonal matrix Z of the Schur *> vectors of H. If COMPZ = 'V', on entry Z must contain an *> N-by-N matrix Q, which is assumed to be equal to the unit *> matrix except for the submatrix Z(ILO:IHI,ILO:IHI). On exit, *> if INFO = 0, Z contains Q*Z. *> Normally Q is the orthogonal matrix generated by DORGHR *> after the call to DGEHRD which formed the Hessenberg matrix *> H. (The output value of Z when INFO > 0 is given under *> the description of INFO below.) *> \endverbatim *> *> \param[in] LDZ *> \verbatim *> LDZ is INTEGER *> The leading dimension of the array Z. if COMPZ = 'I' or *> COMPZ = 'V', then LDZ >= MAX(1,N). Otherwise, LDZ >= 1. *> \endverbatim *> *> \param[out] WORK *> \verbatim *> WORK is DOUBLE PRECISION array, dimension (LWORK) *> On exit, if INFO = 0, WORK(1) returns an estimate of *> the optimal value for LWORK. *> \endverbatim *> *> \param[in] LWORK *> \verbatim *> LWORK is INTEGER *> The dimension of the array WORK. LWORK >= max(1,N) *> is sufficient and delivers very good and sometimes *> optimal performance. However, LWORK as large as 11*N *> may be required for optimal performance. A workspace *> query is recommended to determine the optimal workspace *> size. *> *> If LWORK = -1, then DHSEQR does a workspace query. *> In this case, DHSEQR checks the input parameters and *> estimates the optimal workspace size for the given *> values of N, ILO and IHI. The estimate is returned *> in WORK(1). No error message related to LWORK is *> issued by XERBLA. Neither H nor Z are accessed. *> \endverbatim *> *> \param[out] INFO *> \verbatim *> INFO is INTEGER *> = 0: successful exit *> < 0: if INFO = -i, the i-th argument had an illegal *> value *> > 0: if INFO = i, DHSEQR failed to compute all of *> the eigenvalues. Elements 1:ilo-1 and i+1:n of WR *> and WI contain those eigenvalues which have been *> successfully computed. (Failures are rare.) *> *> If INFO > 0 and JOB = 'E', then on exit, the *> remaining unconverged eigenvalues are the eigen- *> values of the upper Hessenberg matrix rows and *> columns ILO through INFO of the final, output *> value of H. *> *> If INFO > 0 and JOB = 'S', then on exit *> *> (*) (initial value of H)*U = U*(final value of H) *> *> where U is an orthogonal matrix. The final *> value of H is upper Hessenberg and quasi-triangular *> in rows and columns INFO+1 through IHI. *> *> If INFO > 0 and COMPZ = 'V', then on exit *> *> (final value of Z) = (initial value of Z)*U *> *> where U is the orthogonal matrix in (*) (regard- *> less of the value of JOB.) *> *> If INFO > 0 and COMPZ = 'I', then on exit *> (final value of Z) = U *> where U is the orthogonal matrix in (*) (regard- *> less of the value of JOB.) *> *> If INFO > 0 and COMPZ = 'N', then Z is not *> accessed. *> \endverbatim * * Authors: * ======== * *> \author Univ. of Tennessee *> \author Univ. of California Berkeley *> \author Univ. of Colorado Denver *> \author NAG Ltd. * *> \ingroup hseqr * *> \par Contributors: * ================== *> *> Karen Braman and Ralph Byers, Department of Mathematics, *> University of Kansas, USA * *> \par Further Details: * ===================== *> *> \verbatim *> *> Default values supplied by *> ILAENV(ISPEC,'DHSEQR',JOB(:1)//COMPZ(:1),N,ILO,IHI,LWORK). *> It is suggested that these defaults be adjusted in order *> to attain best performance in each particular *> computational environment. *> *> ISPEC=12: The DLAHQR vs DLAQR0 crossover point. *> Default: 75. (Must be at least 11.) *> *> ISPEC=13: Recommended deflation window size. *> This depends on ILO, IHI and NS. NS is the *> number of simultaneous shifts returned *> by ILAENV(ISPEC=15). (See ISPEC=15 below.) *> The default for (IHI-ILO+1) <= 500 is NS. *> The default for (IHI-ILO+1) > 500 is 3*NS/2. *> *> ISPEC=14: Nibble crossover point. (See IPARMQ for *> details.) Default: 14% of deflation window *> size. *> *> ISPEC=15: Number of simultaneous shifts in a multishift *> QR iteration. *> *> If IHI-ILO+1 is ... *> *> greater than ...but less ... the *> or equal to ... than default is *> *> 1 30 NS = 2(+) *> 30 60 NS = 4(+) *> 60 150 NS = 10(+) *> 150 590 NS = ** *> 590 3000 NS = 64 *> 3000 6000 NS = 128 *> 6000 infinity NS = 256 *> *> (+) By default some or all matrices of this order *> are passed to the implicit double shift routine *> DLAHQR and this parameter is ignored. See *> ISPEC=12 above and comments in IPARMQ for *> details. *> *> (**) The asterisks (**) indicate an ad-hoc *> function of N increasing from 10 to 64. *> *> ISPEC=16: Select structured matrix multiply. *> If the number of simultaneous shifts (specified *> by ISPEC=15) is less than 14, then the default *> for ISPEC=16 is 0. Otherwise the default for *> ISPEC=16 is 2. *> \endverbatim * *> \par References: * ================ *> *> K. Braman, R. Byers and R. Mathias, The Multi-Shift QR *> Algorithm Part I: Maintaining Well Focused Shifts, and Level 3 *> Performance, SIAM Journal of Matrix Analysis, volume 23, pages *> 929--947, 2002. *> \n *> K. Braman, R. Byers and R. Mathias, The Multi-Shift QR *> Algorithm Part II: Aggressive Early Deflation, SIAM Journal *> of Matrix Analysis, volume 23, pages 948--973, 2002. * * ===================================================================== SUBROUTINE DHSEQR( JOB, COMPZ, N, ILO, IHI, H, LDH, WR, WI, Z, $ LDZ, WORK, LWORK, INFO ) * * -- LAPACK computational routine -- * -- LAPACK is a software package provided by Univ. of Tennessee, -- * -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..-- * * .. Scalar Arguments .. INTEGER IHI, ILO, INFO, LDH, LDZ, LWORK, N CHARACTER COMPZ, JOB * .. * .. Array Arguments .. DOUBLE PRECISION H( LDH, * ), WI( * ), WORK( * ), WR( * ), $ Z( LDZ, * ) * .. * * ===================================================================== * * .. Parameters .. * * ==== Matrices of order NTINY or smaller must be processed by * . DLAHQR because of insufficient subdiagonal scratch space. * . (This is a hard limit.) ==== INTEGER NTINY PARAMETER ( NTINY = 15 ) * * ==== NL allocates some local workspace to help small matrices * . through a rare DLAHQR failure. NL > NTINY = 15 is * . required and NL <= NMIN = ILAENV(ISPEC=12,...) is recom- * . mended. (The default value of NMIN is 75.) Using NL = 49 * . allows up to six simultaneous shifts and a 16-by-16 * . deflation window. ==== INTEGER NL PARAMETER ( NL = 49 ) DOUBLE PRECISION ZERO, ONE PARAMETER ( ZERO = 0.0d0, ONE = 1.0d0 ) * .. * .. Local Arrays .. DOUBLE PRECISION HL( NL, NL ), WORKL( NL ) * .. * .. Local Scalars .. INTEGER I, KBOT, NMIN LOGICAL INITZ, LQUERY, WANTT, WANTZ * .. * .. External Functions .. INTEGER ILAENV LOGICAL LSAME EXTERNAL ILAENV, LSAME * .. * .. External Subroutines .. EXTERNAL DLACPY, DLAHQR, DLAQR0, DLASET, $ XERBLA * .. * .. Intrinsic Functions .. INTRINSIC DBLE, MAX, MIN * .. * .. Executable Statements .. * * ==== Decode and check the input parameters. ==== * WANTT = LSAME( JOB, 'S' ) INITZ = LSAME( COMPZ, 'I' ) WANTZ = INITZ .OR. LSAME( COMPZ, 'V' ) WORK( 1 ) = DBLE( MAX( 1, N ) ) LQUERY = LWORK.EQ.-1 * INFO = 0 IF( .NOT.LSAME( JOB, 'E' ) .AND. .NOT.WANTT ) THEN INFO = -1 ELSE IF( .NOT.LSAME( COMPZ, 'N' ) .AND. .NOT.WANTZ ) THEN INFO = -2 ELSE IF( N.LT.0 ) THEN INFO = -3 ELSE IF( ILO.LT.1 .OR. ILO.GT.MAX( 1, N ) ) THEN INFO = -4 ELSE IF( IHI.LT.MIN( ILO, N ) .OR. IHI.GT.N ) THEN INFO = -5 ELSE IF( LDH.LT.MAX( 1, N ) ) THEN INFO = -7 ELSE IF( LDZ.LT.1 .OR. ( WANTZ .AND. LDZ.LT.MAX( 1, N ) ) ) THEN INFO = -11 ELSE IF( LWORK.LT.MAX( 1, N ) .AND. .NOT.LQUERY ) THEN INFO = -13 END IF * IF( INFO.NE.0 ) THEN * * ==== Quick return in case of invalid argument. ==== * CALL XERBLA( 'DHSEQR', -INFO ) RETURN * ELSE IF( N.EQ.0 ) THEN * * ==== Quick return in case N = 0; nothing to do. ==== * RETURN * ELSE IF( LQUERY ) THEN * * ==== Quick return in case of a workspace query ==== * CALL DLAQR0( WANTT, WANTZ, N, ILO, IHI, H, LDH, WR, WI, ILO, $ IHI, Z, LDZ, WORK, LWORK, INFO ) * ==== Ensure reported workspace size is backward-compatible with * . previous LAPACK versions. ==== WORK( 1 ) = MAX( DBLE( MAX( 1, N ) ), WORK( 1 ) ) RETURN * ELSE * * ==== copy eigenvalues isolated by DGEBAL ==== * DO 10 I = 1, ILO - 1 WR( I ) = H( I, I ) WI( I ) = ZERO 10 CONTINUE DO 20 I = IHI + 1, N WR( I ) = H( I, I ) WI( I ) = ZERO 20 CONTINUE * * ==== Initialize Z, if requested ==== * IF( INITZ ) $ CALL DLASET( 'A', N, N, ZERO, ONE, Z, LDZ ) * * ==== Quick return if possible ==== * IF( ILO.EQ.IHI ) THEN WR( ILO ) = H( ILO, ILO ) WI( ILO ) = ZERO RETURN END IF * * ==== DLAHQR/DLAQR0 crossover point ==== * NMIN = ILAENV( 12, 'DHSEQR', JOB( : 1 ) // COMPZ( : 1 ), N, $ ILO, IHI, LWORK ) NMIN = MAX( NTINY, NMIN ) * * ==== DLAQR0 for big matrices; DLAHQR for small ones ==== * IF( N.GT.NMIN ) THEN CALL DLAQR0( WANTT, WANTZ, N, ILO, IHI, H, LDH, WR, WI, $ ILO, $ IHI, Z, LDZ, WORK, LWORK, INFO ) ELSE * * ==== Small matrix ==== * CALL DLAHQR( WANTT, WANTZ, N, ILO, IHI, H, LDH, WR, WI, $ ILO, $ IHI, Z, LDZ, INFO ) * IF( INFO.GT.0 ) THEN * * ==== A rare DLAHQR failure! DLAQR0 sometimes succeeds * . when DLAHQR fails. ==== * KBOT = INFO * IF( N.GE.NL ) THEN * * ==== Larger matrices have enough subdiagonal scratch * . space to call DLAQR0 directly. ==== * CALL DLAQR0( WANTT, WANTZ, N, ILO, KBOT, H, LDH, $ WR, $ WI, ILO, IHI, Z, LDZ, WORK, LWORK, INFO ) * ELSE * * ==== Tiny matrices don't have enough subdiagonal * . scratch space to benefit from DLAQR0. Hence, * . tiny matrices must be copied into a larger * . array before calling DLAQR0. ==== * CALL DLACPY( 'A', N, N, H, LDH, HL, NL ) HL( N+1, N ) = ZERO CALL DLASET( 'A', NL, NL-N, ZERO, ZERO, HL( 1, $ N+1 ), $ NL ) CALL DLAQR0( WANTT, WANTZ, NL, ILO, KBOT, HL, NL, $ WR, $ WI, ILO, IHI, Z, LDZ, WORKL, NL, INFO ) IF( WANTT .OR. INFO.NE.0 ) $ CALL DLACPY( 'A', N, N, HL, NL, H, LDH ) END IF END IF END IF * * ==== Clear out the trash, if necessary. ==== * IF( ( WANTT .OR. INFO.NE.0 ) .AND. N.GT.2 ) $ CALL DLASET( 'L', N-2, N-2, ZERO, ZERO, H( 3, 1 ), LDH ) * * ==== Ensure reported workspace size is backward-compatible with * . previous LAPACK versions. ==== * WORK( 1 ) = MAX( DBLE( MAX( 1, N ) ), WORK( 1 ) ) END IF * * ==== End of DHSEQR ==== * END