Annotation of rpl/lapack/lapack/dhseqr.f, revision 1.9
1.9 ! bertrand 1: *> \brief \b DHSEQR
! 2: *
! 3: * =========== DOCUMENTATION ===========
! 4: *
! 5: * Online html documentation available at
! 6: * http://www.netlib.org/lapack/explore-html/
! 7: *
! 8: *> \htmlonly
! 9: *> Download DHSEQR + dependencies
! 10: *> <a href="http://www.netlib.org/cgi-bin/netlibfiles.tgz?format=tgz&filename=/lapack/lapack_routine/dhseqr.f">
! 11: *> [TGZ]</a>
! 12: *> <a href="http://www.netlib.org/cgi-bin/netlibfiles.zip?format=zip&filename=/lapack/lapack_routine/dhseqr.f">
! 13: *> [ZIP]</a>
! 14: *> <a href="http://www.netlib.org/cgi-bin/netlibfiles.txt?format=txt&filename=/lapack/lapack_routine/dhseqr.f">
! 15: *> [TXT]</a>
! 16: *> \endhtmlonly
! 17: *
! 18: * Definition:
! 19: * ===========
! 20: *
! 21: * SUBROUTINE DHSEQR( JOB, COMPZ, N, ILO, IHI, H, LDH, WR, WI, Z,
! 22: * LDZ, WORK, LWORK, INFO )
! 23: *
! 24: * .. Scalar Arguments ..
! 25: * INTEGER IHI, ILO, INFO, LDH, LDZ, LWORK, N
! 26: * CHARACTER COMPZ, JOB
! 27: * ..
! 28: * .. Array Arguments ..
! 29: * DOUBLE PRECISION H( LDH, * ), WI( * ), WORK( * ), WR( * ),
! 30: * $ Z( LDZ, * )
! 31: * ..
! 32: *
! 33: *
! 34: *> \par Purpose:
! 35: * =============
! 36: *>
! 37: *> \verbatim
! 38: *>
! 39: *> DHSEQR computes the eigenvalues of a Hessenberg matrix H
! 40: *> and, optionally, the matrices T and Z from the Schur decomposition
! 41: *> H = Z T Z**T, where T is an upper quasi-triangular matrix (the
! 42: *> Schur form), and Z is the orthogonal matrix of Schur vectors.
! 43: *>
! 44: *> Optionally Z may be postmultiplied into an input orthogonal
! 45: *> matrix Q so that this routine can give the Schur factorization
! 46: *> of a matrix A which has been reduced to the Hessenberg form H
! 47: *> by the orthogonal matrix Q: A = Q*H*Q**T = (QZ)*T*(QZ)**T.
! 48: *> \endverbatim
! 49: *
! 50: * Arguments:
! 51: * ==========
! 52: *
! 53: *> \param[in] JOB
! 54: *> \verbatim
! 55: *> JOB is CHARACTER*1
! 56: *> = 'E': compute eigenvalues only;
! 57: *> = 'S': compute eigenvalues and the Schur form T.
! 58: *> \endverbatim
! 59: *>
! 60: *> \param[in] COMPZ
! 61: *> \verbatim
! 62: *> COMPZ is CHARACTER*1
! 63: *> = 'N': no Schur vectors are computed;
! 64: *> = 'I': Z is initialized to the unit matrix and the matrix Z
! 65: *> of Schur vectors of H is returned;
! 66: *> = 'V': Z must contain an orthogonal matrix Q on entry, and
! 67: *> the product Q*Z is returned.
! 68: *> \endverbatim
! 69: *>
! 70: *> \param[in] N
! 71: *> \verbatim
! 72: *> N is INTEGER
! 73: *> The order of the matrix H. N .GE. 0.
! 74: *> \endverbatim
! 75: *>
! 76: *> \param[in] ILO
! 77: *> \verbatim
! 78: *> ILO is INTEGER
! 79: *> \endverbatim
! 80: *>
! 81: *> \param[in] IHI
! 82: *> \verbatim
! 83: *> IHI is INTEGER
! 84: *>
! 85: *> It is assumed that H is already upper triangular in rows
! 86: *> and columns 1:ILO-1 and IHI+1:N. ILO and IHI are normally
! 87: *> set by a previous call to DGEBAL, and then passed to ZGEHRD
! 88: *> when the matrix output by DGEBAL is reduced to Hessenberg
! 89: *> form. Otherwise ILO and IHI should be set to 1 and N
! 90: *> respectively. If N.GT.0, then 1.LE.ILO.LE.IHI.LE.N.
! 91: *> If N = 0, then ILO = 1 and IHI = 0.
! 92: *> \endverbatim
! 93: *>
! 94: *> \param[in,out] H
! 95: *> \verbatim
! 96: *> H is DOUBLE PRECISION array, dimension (LDH,N)
! 97: *> On entry, the upper Hessenberg matrix H.
! 98: *> On exit, if INFO = 0 and JOB = 'S', then H contains the
! 99: *> upper quasi-triangular matrix T from the Schur decomposition
! 100: *> (the Schur form); 2-by-2 diagonal blocks (corresponding to
! 101: *> complex conjugate pairs of eigenvalues) are returned in
! 102: *> standard form, with H(i,i) = H(i+1,i+1) and
! 103: *> H(i+1,i)*H(i,i+1).LT.0. If INFO = 0 and JOB = 'E', the
! 104: *> contents of H are unspecified on exit. (The output value of
! 105: *> H when INFO.GT.0 is given under the description of INFO
! 106: *> below.)
! 107: *>
! 108: *> Unlike earlier versions of DHSEQR, this subroutine may
! 109: *> explicitly H(i,j) = 0 for i.GT.j and j = 1, 2, ... ILO-1
! 110: *> or j = IHI+1, IHI+2, ... N.
! 111: *> \endverbatim
! 112: *>
! 113: *> \param[in] LDH
! 114: *> \verbatim
! 115: *> LDH is INTEGER
! 116: *> The leading dimension of the array H. LDH .GE. max(1,N).
! 117: *> \endverbatim
! 118: *>
! 119: *> \param[out] WR
! 120: *> \verbatim
! 121: *> WR is DOUBLE PRECISION array, dimension (N)
! 122: *> \endverbatim
! 123: *>
! 124: *> \param[out] WI
! 125: *> \verbatim
! 126: *> WI is DOUBLE PRECISION array, dimension (N)
! 127: *>
! 128: *> The real and imaginary parts, respectively, of the computed
! 129: *> eigenvalues. If two eigenvalues are computed as a complex
! 130: *> conjugate pair, they are stored in consecutive elements of
! 131: *> WR and WI, say the i-th and (i+1)th, with WI(i) .GT. 0 and
! 132: *> WI(i+1) .LT. 0. If JOB = 'S', the eigenvalues are stored in
! 133: *> the same order as on the diagonal of the Schur form returned
! 134: *> in H, with WR(i) = H(i,i) and, if H(i:i+1,i:i+1) is a 2-by-2
! 135: *> diagonal block, WI(i) = sqrt(-H(i+1,i)*H(i,i+1)) and
! 136: *> WI(i+1) = -WI(i).
! 137: *> \endverbatim
! 138: *>
! 139: *> \param[in,out] Z
! 140: *> \verbatim
! 141: *> Z is DOUBLE PRECISION array, dimension (LDZ,N)
! 142: *> If COMPZ = 'N', Z is not referenced.
! 143: *> If COMPZ = 'I', on entry Z need not be set and on exit,
! 144: *> if INFO = 0, Z contains the orthogonal matrix Z of the Schur
! 145: *> vectors of H. If COMPZ = 'V', on entry Z must contain an
! 146: *> N-by-N matrix Q, which is assumed to be equal to the unit
! 147: *> matrix except for the submatrix Z(ILO:IHI,ILO:IHI). On exit,
! 148: *> if INFO = 0, Z contains Q*Z.
! 149: *> Normally Q is the orthogonal matrix generated by DORGHR
! 150: *> after the call to DGEHRD which formed the Hessenberg matrix
! 151: *> H. (The output value of Z when INFO.GT.0 is given under
! 152: *> the description of INFO below.)
! 153: *> \endverbatim
! 154: *>
! 155: *> \param[in] LDZ
! 156: *> \verbatim
! 157: *> LDZ is INTEGER
! 158: *> The leading dimension of the array Z. if COMPZ = 'I' or
! 159: *> COMPZ = 'V', then LDZ.GE.MAX(1,N). Otherwize, LDZ.GE.1.
! 160: *> \endverbatim
! 161: *>
! 162: *> \param[out] WORK
! 163: *> \verbatim
! 164: *> WORK is DOUBLE PRECISION array, dimension (LWORK)
! 165: *> On exit, if INFO = 0, WORK(1) returns an estimate of
! 166: *> the optimal value for LWORK.
! 167: *> \endverbatim
! 168: *>
! 169: *> \param[in] LWORK
! 170: *> \verbatim
! 171: *> LWORK is INTEGER
! 172: *> The dimension of the array WORK. LWORK .GE. max(1,N)
! 173: *> is sufficient and delivers very good and sometimes
! 174: *> optimal performance. However, LWORK as large as 11*N
! 175: *> may be required for optimal performance. A workspace
! 176: *> query is recommended to determine the optimal workspace
! 177: *> size.
! 178: *>
! 179: *> If LWORK = -1, then DHSEQR does a workspace query.
! 180: *> In this case, DHSEQR checks the input parameters and
! 181: *> estimates the optimal workspace size for the given
! 182: *> values of N, ILO and IHI. The estimate is returned
! 183: *> in WORK(1). No error message related to LWORK is
! 184: *> issued by XERBLA. Neither H nor Z are accessed.
! 185: *> \endverbatim
! 186: *>
! 187: *> \param[out] INFO
! 188: *> \verbatim
! 189: *> INFO is INTEGER
! 190: *> = 0: successful exit
! 191: *> .LT. 0: if INFO = -i, the i-th argument had an illegal
! 192: *> value
! 193: *> .GT. 0: if INFO = i, DHSEQR failed to compute all of
! 194: *> the eigenvalues. Elements 1:ilo-1 and i+1:n of WR
! 195: *> and WI contain those eigenvalues which have been
! 196: *> successfully computed. (Failures are rare.)
! 197: *>
! 198: *> If INFO .GT. 0 and JOB = 'E', then on exit, the
! 199: *> remaining unconverged eigenvalues are the eigen-
! 200: *> values of the upper Hessenberg matrix rows and
! 201: *> columns ILO through INFO of the final, output
! 202: *> value of H.
! 203: *>
! 204: *> If INFO .GT. 0 and JOB = 'S', then on exit
! 205: *>
! 206: *> (*) (initial value of H)*U = U*(final value of H)
! 207: *>
! 208: *> where U is an orthogonal matrix. The final
! 209: *> value of H is upper Hessenberg and quasi-triangular
! 210: *> in rows and columns INFO+1 through IHI.
! 211: *>
! 212: *> If INFO .GT. 0 and COMPZ = 'V', then on exit
! 213: *>
! 214: *> (final value of Z) = (initial value of Z)*U
! 215: *>
! 216: *> where U is the orthogonal matrix in (*) (regard-
! 217: *> less of the value of JOB.)
! 218: *>
! 219: *> If INFO .GT. 0 and COMPZ = 'I', then on exit
! 220: *> (final value of Z) = U
! 221: *> where U is the orthogonal matrix in (*) (regard-
! 222: *> less of the value of JOB.)
! 223: *>
! 224: *> If INFO .GT. 0 and COMPZ = 'N', then Z is not
! 225: *> accessed.
! 226: *> \endverbatim
! 227: *
! 228: * Authors:
! 229: * ========
! 230: *
! 231: *> \author Univ. of Tennessee
! 232: *> \author Univ. of California Berkeley
! 233: *> \author Univ. of Colorado Denver
! 234: *> \author NAG Ltd.
! 235: *
! 236: *> \date November 2011
! 237: *
! 238: *> \ingroup doubleOTHERcomputational
! 239: *
! 240: *> \par Contributors:
! 241: * ==================
! 242: *>
! 243: *> Karen Braman and Ralph Byers, Department of Mathematics,
! 244: *> University of Kansas, USA
! 245: *
! 246: *> \par Further Details:
! 247: * =====================
! 248: *>
! 249: *> \verbatim
! 250: *>
! 251: *> Default values supplied by
! 252: *> ILAENV(ISPEC,'DHSEQR',JOB(:1)//COMPZ(:1),N,ILO,IHI,LWORK).
! 253: *> It is suggested that these defaults be adjusted in order
! 254: *> to attain best performance in each particular
! 255: *> computational environment.
! 256: *>
! 257: *> ISPEC=12: The DLAHQR vs DLAQR0 crossover point.
! 258: *> Default: 75. (Must be at least 11.)
! 259: *>
! 260: *> ISPEC=13: Recommended deflation window size.
! 261: *> This depends on ILO, IHI and NS. NS is the
! 262: *> number of simultaneous shifts returned
! 263: *> by ILAENV(ISPEC=15). (See ISPEC=15 below.)
! 264: *> The default for (IHI-ILO+1).LE.500 is NS.
! 265: *> The default for (IHI-ILO+1).GT.500 is 3*NS/2.
! 266: *>
! 267: *> ISPEC=14: Nibble crossover point. (See IPARMQ for
! 268: *> details.) Default: 14% of deflation window
! 269: *> size.
! 270: *>
! 271: *> ISPEC=15: Number of simultaneous shifts in a multishift
! 272: *> QR iteration.
! 273: *>
! 274: *> If IHI-ILO+1 is ...
! 275: *>
! 276: *> greater than ...but less ... the
! 277: *> or equal to ... than default is
! 278: *>
! 279: *> 1 30 NS = 2(+)
! 280: *> 30 60 NS = 4(+)
! 281: *> 60 150 NS = 10(+)
! 282: *> 150 590 NS = **
! 283: *> 590 3000 NS = 64
! 284: *> 3000 6000 NS = 128
! 285: *> 6000 infinity NS = 256
! 286: *>
! 287: *> (+) By default some or all matrices of this order
! 288: *> are passed to the implicit double shift routine
! 289: *> DLAHQR and this parameter is ignored. See
! 290: *> ISPEC=12 above and comments in IPARMQ for
! 291: *> details.
! 292: *>
! 293: *> (**) The asterisks (**) indicate an ad-hoc
! 294: *> function of N increasing from 10 to 64.
! 295: *>
! 296: *> ISPEC=16: Select structured matrix multiply.
! 297: *> If the number of simultaneous shifts (specified
! 298: *> by ISPEC=15) is less than 14, then the default
! 299: *> for ISPEC=16 is 0. Otherwise the default for
! 300: *> ISPEC=16 is 2.
! 301: *> \endverbatim
! 302: *
! 303: *> \par References:
! 304: * ================
! 305: *>
! 306: *> K. Braman, R. Byers and R. Mathias, The Multi-Shift QR
! 307: *> Algorithm Part I: Maintaining Well Focused Shifts, and Level 3
! 308: *> Performance, SIAM Journal of Matrix Analysis, volume 23, pages
! 309: *> 929--947, 2002.
! 310: *> \n
! 311: *> K. Braman, R. Byers and R. Mathias, The Multi-Shift QR
! 312: *> Algorithm Part II: Aggressive Early Deflation, SIAM Journal
! 313: *> of Matrix Analysis, volume 23, pages 948--973, 2002.
! 314: *
! 315: * =====================================================================
1.1 bertrand 316: SUBROUTINE DHSEQR( JOB, COMPZ, N, ILO, IHI, H, LDH, WR, WI, Z,
317: $ LDZ, WORK, LWORK, INFO )
318: *
1.9 ! bertrand 319: * -- LAPACK computational routine (version 3.4.0) --
! 320: * -- LAPACK is a software package provided by Univ. of Tennessee, --
! 321: * -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
! 322: * November 2011
1.1 bertrand 323: *
324: * .. Scalar Arguments ..
325: INTEGER IHI, ILO, INFO, LDH, LDZ, LWORK, N
326: CHARACTER COMPZ, JOB
327: * ..
328: * .. Array Arguments ..
329: DOUBLE PRECISION H( LDH, * ), WI( * ), WORK( * ), WR( * ),
330: $ Z( LDZ, * )
331: * ..
332: *
1.9 ! bertrand 333: * =====================================================================
1.1 bertrand 334: *
335: * .. Parameters ..
336: *
337: * ==== Matrices of order NTINY or smaller must be processed by
338: * . DLAHQR because of insufficient subdiagonal scratch space.
339: * . (This is a hard limit.) ====
340: INTEGER NTINY
341: PARAMETER ( NTINY = 11 )
342: *
343: * ==== NL allocates some local workspace to help small matrices
344: * . through a rare DLAHQR failure. NL .GT. NTINY = 11 is
345: * . required and NL .LE. NMIN = ILAENV(ISPEC=12,...) is recom-
346: * . mended. (The default value of NMIN is 75.) Using NL = 49
347: * . allows up to six simultaneous shifts and a 16-by-16
348: * . deflation window. ====
349: INTEGER NL
350: PARAMETER ( NL = 49 )
351: DOUBLE PRECISION ZERO, ONE
352: PARAMETER ( ZERO = 0.0d0, ONE = 1.0d0 )
353: * ..
354: * .. Local Arrays ..
355: DOUBLE PRECISION HL( NL, NL ), WORKL( NL )
356: * ..
357: * .. Local Scalars ..
358: INTEGER I, KBOT, NMIN
359: LOGICAL INITZ, LQUERY, WANTT, WANTZ
360: * ..
361: * .. External Functions ..
362: INTEGER ILAENV
363: LOGICAL LSAME
364: EXTERNAL ILAENV, LSAME
365: * ..
366: * .. External Subroutines ..
367: EXTERNAL DLACPY, DLAHQR, DLAQR0, DLASET, XERBLA
368: * ..
369: * .. Intrinsic Functions ..
370: INTRINSIC DBLE, MAX, MIN
371: * ..
372: * .. Executable Statements ..
373: *
374: * ==== Decode and check the input parameters. ====
375: *
376: WANTT = LSAME( JOB, 'S' )
377: INITZ = LSAME( COMPZ, 'I' )
378: WANTZ = INITZ .OR. LSAME( COMPZ, 'V' )
379: WORK( 1 ) = DBLE( MAX( 1, N ) )
380: LQUERY = LWORK.EQ.-1
381: *
382: INFO = 0
383: IF( .NOT.LSAME( JOB, 'E' ) .AND. .NOT.WANTT ) THEN
384: INFO = -1
385: ELSE IF( .NOT.LSAME( COMPZ, 'N' ) .AND. .NOT.WANTZ ) THEN
386: INFO = -2
387: ELSE IF( N.LT.0 ) THEN
388: INFO = -3
389: ELSE IF( ILO.LT.1 .OR. ILO.GT.MAX( 1, N ) ) THEN
390: INFO = -4
391: ELSE IF( IHI.LT.MIN( ILO, N ) .OR. IHI.GT.N ) THEN
392: INFO = -5
393: ELSE IF( LDH.LT.MAX( 1, N ) ) THEN
394: INFO = -7
395: ELSE IF( LDZ.LT.1 .OR. ( WANTZ .AND. LDZ.LT.MAX( 1, N ) ) ) THEN
396: INFO = -11
397: ELSE IF( LWORK.LT.MAX( 1, N ) .AND. .NOT.LQUERY ) THEN
398: INFO = -13
399: END IF
400: *
401: IF( INFO.NE.0 ) THEN
402: *
403: * ==== Quick return in case of invalid argument. ====
404: *
405: CALL XERBLA( 'DHSEQR', -INFO )
406: RETURN
407: *
408: ELSE IF( N.EQ.0 ) THEN
409: *
410: * ==== Quick return in case N = 0; nothing to do. ====
411: *
412: RETURN
413: *
414: ELSE IF( LQUERY ) THEN
415: *
416: * ==== Quick return in case of a workspace query ====
417: *
418: CALL DLAQR0( WANTT, WANTZ, N, ILO, IHI, H, LDH, WR, WI, ILO,
419: $ IHI, Z, LDZ, WORK, LWORK, INFO )
420: * ==== Ensure reported workspace size is backward-compatible with
421: * . previous LAPACK versions. ====
422: WORK( 1 ) = MAX( DBLE( MAX( 1, N ) ), WORK( 1 ) )
423: RETURN
424: *
425: ELSE
426: *
427: * ==== copy eigenvalues isolated by DGEBAL ====
428: *
429: DO 10 I = 1, ILO - 1
430: WR( I ) = H( I, I )
431: WI( I ) = ZERO
432: 10 CONTINUE
433: DO 20 I = IHI + 1, N
434: WR( I ) = H( I, I )
435: WI( I ) = ZERO
436: 20 CONTINUE
437: *
438: * ==== Initialize Z, if requested ====
439: *
440: IF( INITZ )
441: $ CALL DLASET( 'A', N, N, ZERO, ONE, Z, LDZ )
442: *
443: * ==== Quick return if possible ====
444: *
445: IF( ILO.EQ.IHI ) THEN
446: WR( ILO ) = H( ILO, ILO )
447: WI( ILO ) = ZERO
448: RETURN
449: END IF
450: *
451: * ==== DLAHQR/DLAQR0 crossover point ====
452: *
453: NMIN = ILAENV( 12, 'DHSEQR', JOB( : 1 ) // COMPZ( : 1 ), N,
454: $ ILO, IHI, LWORK )
455: NMIN = MAX( NTINY, NMIN )
456: *
457: * ==== DLAQR0 for big matrices; DLAHQR for small ones ====
458: *
459: IF( N.GT.NMIN ) THEN
460: CALL DLAQR0( WANTT, WANTZ, N, ILO, IHI, H, LDH, WR, WI, ILO,
461: $ IHI, Z, LDZ, WORK, LWORK, INFO )
462: ELSE
463: *
464: * ==== Small matrix ====
465: *
466: CALL DLAHQR( WANTT, WANTZ, N, ILO, IHI, H, LDH, WR, WI, ILO,
467: $ IHI, Z, LDZ, INFO )
468: *
469: IF( INFO.GT.0 ) THEN
470: *
471: * ==== A rare DLAHQR failure! DLAQR0 sometimes succeeds
472: * . when DLAHQR fails. ====
473: *
474: KBOT = INFO
475: *
476: IF( N.GE.NL ) THEN
477: *
478: * ==== Larger matrices have enough subdiagonal scratch
479: * . space to call DLAQR0 directly. ====
480: *
481: CALL DLAQR0( WANTT, WANTZ, N, ILO, KBOT, H, LDH, WR,
482: $ WI, ILO, IHI, Z, LDZ, WORK, LWORK, INFO )
483: *
484: ELSE
485: *
486: * ==== Tiny matrices don't have enough subdiagonal
487: * . scratch space to benefit from DLAQR0. Hence,
488: * . tiny matrices must be copied into a larger
489: * . array before calling DLAQR0. ====
490: *
491: CALL DLACPY( 'A', N, N, H, LDH, HL, NL )
492: HL( N+1, N ) = ZERO
493: CALL DLASET( 'A', NL, NL-N, ZERO, ZERO, HL( 1, N+1 ),
494: $ NL )
495: CALL DLAQR0( WANTT, WANTZ, NL, ILO, KBOT, HL, NL, WR,
496: $ WI, ILO, IHI, Z, LDZ, WORKL, NL, INFO )
497: IF( WANTT .OR. INFO.NE.0 )
498: $ CALL DLACPY( 'A', N, N, HL, NL, H, LDH )
499: END IF
500: END IF
501: END IF
502: *
503: * ==== Clear out the trash, if necessary. ====
504: *
505: IF( ( WANTT .OR. INFO.NE.0 ) .AND. N.GT.2 )
506: $ CALL DLASET( 'L', N-2, N-2, ZERO, ZERO, H( 3, 1 ), LDH )
507: *
508: * ==== Ensure reported workspace size is backward-compatible with
509: * . previous LAPACK versions. ====
510: *
511: WORK( 1 ) = MAX( DBLE( MAX( 1, N ) ), WORK( 1 ) )
512: END IF
513: *
514: * ==== End of DHSEQR ====
515: *
516: END
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