Annotation of rpl/lapack/lapack/dhseqr.f, revision 1.19
1.9 bertrand 1: *> \brief \b DHSEQR
2: *
3: * =========== DOCUMENTATION ===========
4: *
1.15 bertrand 5: * Online html documentation available at
6: * http://www.netlib.org/lapack/explore-html/
1.9 bertrand 7: *
8: *> \htmlonly
1.15 bertrand 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">
1.9 bertrand 15: *> [TXT]</a>
1.15 bertrand 16: *> \endhtmlonly
1.9 bertrand 17: *
18: * Definition:
19: * ===========
20: *
21: * SUBROUTINE DHSEQR( JOB, COMPZ, N, ILO, IHI, H, LDH, WR, WI, Z,
22: * LDZ, WORK, LWORK, INFO )
1.15 bertrand 23: *
1.9 bertrand 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: * ..
1.15 bertrand 32: *
1.9 bertrand 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
1.18 bertrand 73: *> The order of the matrix H. N >= 0.
1.9 bertrand 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
1.18 bertrand 90: *> respectively. If N > 0, then 1 <= ILO <= IHI <= N.
1.9 bertrand 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
1.18 bertrand 103: *> H(i+1,i)*H(i,i+1) < 0. If INFO = 0 and JOB = 'E', the
1.9 bertrand 104: *> contents of H are unspecified on exit. (The output value of
1.18 bertrand 105: *> H when INFO > 0 is given under the description of INFO
1.9 bertrand 106: *> below.)
107: *>
108: *> Unlike earlier versions of DHSEQR, this subroutine may
1.18 bertrand 109: *> explicitly H(i,j) = 0 for i > j and j = 1, 2, ... ILO-1
1.9 bertrand 110: *> or j = IHI+1, IHI+2, ... N.
111: *> \endverbatim
112: *>
113: *> \param[in] LDH
114: *> \verbatim
115: *> LDH is INTEGER
1.18 bertrand 116: *> The leading dimension of the array H. LDH >= max(1,N).
1.9 bertrand 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
1.18 bertrand 131: *> WR and WI, say the i-th and (i+1)th, with WI(i) > 0 and
132: *> WI(i+1) < 0. If JOB = 'S', the eigenvalues are stored in
1.9 bertrand 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
1.18 bertrand 151: *> H. (The output value of Z when INFO > 0 is given under
1.9 bertrand 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
1.18 bertrand 159: *> COMPZ = 'V', then LDZ >= MAX(1,N). Otherwise, LDZ >= 1.
1.9 bertrand 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
1.18 bertrand 172: *> The dimension of the array WORK. LWORK >= max(1,N)
1.9 bertrand 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
1.18 bertrand 190: *> = 0: successful exit
191: *> < 0: if INFO = -i, the i-th argument had an illegal
1.9 bertrand 192: *> value
1.18 bertrand 193: *> > 0: if INFO = i, DHSEQR failed to compute all of
1.9 bertrand 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: *>
1.18 bertrand 198: *> If INFO > 0 and JOB = 'E', then on exit, the
1.9 bertrand 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: *>
1.18 bertrand 204: *> If INFO > 0 and JOB = 'S', then on exit
1.9 bertrand 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: *>
1.18 bertrand 212: *> If INFO > 0 and COMPZ = 'V', then on exit
1.9 bertrand 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: *>
1.18 bertrand 219: *> If INFO > 0 and COMPZ = 'I', then on exit
1.9 bertrand 220: *> (final value of Z) = U
221: *> where U is the orthogonal matrix in (*) (regard-
222: *> less of the value of JOB.)
223: *>
1.18 bertrand 224: *> If INFO > 0 and COMPZ = 'N', then Z is not
1.9 bertrand 225: *> accessed.
226: *> \endverbatim
227: *
228: * Authors:
229: * ========
230: *
1.15 bertrand 231: *> \author Univ. of Tennessee
232: *> \author Univ. of California Berkeley
233: *> \author Univ. of Colorado Denver
234: *> \author NAG Ltd.
1.9 bertrand 235: *
236: *> \ingroup doubleOTHERcomputational
237: *
238: *> \par Contributors:
239: * ==================
240: *>
241: *> Karen Braman and Ralph Byers, Department of Mathematics,
242: *> University of Kansas, USA
243: *
244: *> \par Further Details:
245: * =====================
246: *>
247: *> \verbatim
248: *>
249: *> Default values supplied by
250: *> ILAENV(ISPEC,'DHSEQR',JOB(:1)//COMPZ(:1),N,ILO,IHI,LWORK).
251: *> It is suggested that these defaults be adjusted in order
252: *> to attain best performance in each particular
253: *> computational environment.
254: *>
255: *> ISPEC=12: The DLAHQR vs DLAQR0 crossover point.
256: *> Default: 75. (Must be at least 11.)
257: *>
258: *> ISPEC=13: Recommended deflation window size.
259: *> This depends on ILO, IHI and NS. NS is the
260: *> number of simultaneous shifts returned
261: *> by ILAENV(ISPEC=15). (See ISPEC=15 below.)
1.18 bertrand 262: *> The default for (IHI-ILO+1) <= 500 is NS.
263: *> The default for (IHI-ILO+1) > 500 is 3*NS/2.
1.9 bertrand 264: *>
265: *> ISPEC=14: Nibble crossover point. (See IPARMQ for
266: *> details.) Default: 14% of deflation window
267: *> size.
268: *>
269: *> ISPEC=15: Number of simultaneous shifts in a multishift
270: *> QR iteration.
271: *>
272: *> If IHI-ILO+1 is ...
273: *>
274: *> greater than ...but less ... the
275: *> or equal to ... than default is
276: *>
277: *> 1 30 NS = 2(+)
278: *> 30 60 NS = 4(+)
279: *> 60 150 NS = 10(+)
280: *> 150 590 NS = **
281: *> 590 3000 NS = 64
282: *> 3000 6000 NS = 128
283: *> 6000 infinity NS = 256
284: *>
285: *> (+) By default some or all matrices of this order
286: *> are passed to the implicit double shift routine
287: *> DLAHQR and this parameter is ignored. See
288: *> ISPEC=12 above and comments in IPARMQ for
289: *> details.
290: *>
291: *> (**) The asterisks (**) indicate an ad-hoc
292: *> function of N increasing from 10 to 64.
293: *>
294: *> ISPEC=16: Select structured matrix multiply.
295: *> If the number of simultaneous shifts (specified
296: *> by ISPEC=15) is less than 14, then the default
297: *> for ISPEC=16 is 0. Otherwise the default for
298: *> ISPEC=16 is 2.
299: *> \endverbatim
300: *
301: *> \par References:
302: * ================
303: *>
304: *> K. Braman, R. Byers and R. Mathias, The Multi-Shift QR
305: *> Algorithm Part I: Maintaining Well Focused Shifts, and Level 3
306: *> Performance, SIAM Journal of Matrix Analysis, volume 23, pages
307: *> 929--947, 2002.
308: *> \n
309: *> K. Braman, R. Byers and R. Mathias, The Multi-Shift QR
310: *> Algorithm Part II: Aggressive Early Deflation, SIAM Journal
311: *> of Matrix Analysis, volume 23, pages 948--973, 2002.
312: *
313: * =====================================================================
1.1 bertrand 314: SUBROUTINE DHSEQR( JOB, COMPZ, N, ILO, IHI, H, LDH, WR, WI, Z,
315: $ LDZ, WORK, LWORK, INFO )
316: *
1.19 ! bertrand 317: * -- LAPACK computational routine --
1.9 bertrand 318: * -- LAPACK is a software package provided by Univ. of Tennessee, --
319: * -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
1.1 bertrand 320: *
321: * .. Scalar Arguments ..
322: INTEGER IHI, ILO, INFO, LDH, LDZ, LWORK, N
323: CHARACTER COMPZ, JOB
324: * ..
325: * .. Array Arguments ..
326: DOUBLE PRECISION H( LDH, * ), WI( * ), WORK( * ), WR( * ),
327: $ Z( LDZ, * )
328: * ..
329: *
1.9 bertrand 330: * =====================================================================
1.1 bertrand 331: *
332: * .. Parameters ..
333: *
334: * ==== Matrices of order NTINY or smaller must be processed by
335: * . DLAHQR because of insufficient subdiagonal scratch space.
336: * . (This is a hard limit.) ====
337: INTEGER NTINY
1.19 ! bertrand 338: PARAMETER ( NTINY = 15 )
1.1 bertrand 339: *
340: * ==== NL allocates some local workspace to help small matrices
1.19 ! bertrand 341: * . through a rare DLAHQR failure. NL > NTINY = 15 is
1.18 bertrand 342: * . required and NL <= NMIN = ILAENV(ISPEC=12,...) is recom-
1.1 bertrand 343: * . mended. (The default value of NMIN is 75.) Using NL = 49
344: * . allows up to six simultaneous shifts and a 16-by-16
345: * . deflation window. ====
346: INTEGER NL
347: PARAMETER ( NL = 49 )
348: DOUBLE PRECISION ZERO, ONE
349: PARAMETER ( ZERO = 0.0d0, ONE = 1.0d0 )
350: * ..
351: * .. Local Arrays ..
352: DOUBLE PRECISION HL( NL, NL ), WORKL( NL )
353: * ..
354: * .. Local Scalars ..
355: INTEGER I, KBOT, NMIN
356: LOGICAL INITZ, LQUERY, WANTT, WANTZ
357: * ..
358: * .. External Functions ..
359: INTEGER ILAENV
360: LOGICAL LSAME
361: EXTERNAL ILAENV, LSAME
362: * ..
363: * .. External Subroutines ..
364: EXTERNAL DLACPY, DLAHQR, DLAQR0, DLASET, XERBLA
365: * ..
366: * .. Intrinsic Functions ..
367: INTRINSIC DBLE, MAX, MIN
368: * ..
369: * .. Executable Statements ..
370: *
371: * ==== Decode and check the input parameters. ====
372: *
373: WANTT = LSAME( JOB, 'S' )
374: INITZ = LSAME( COMPZ, 'I' )
375: WANTZ = INITZ .OR. LSAME( COMPZ, 'V' )
376: WORK( 1 ) = DBLE( MAX( 1, N ) )
377: LQUERY = LWORK.EQ.-1
378: *
379: INFO = 0
380: IF( .NOT.LSAME( JOB, 'E' ) .AND. .NOT.WANTT ) THEN
381: INFO = -1
382: ELSE IF( .NOT.LSAME( COMPZ, 'N' ) .AND. .NOT.WANTZ ) THEN
383: INFO = -2
384: ELSE IF( N.LT.0 ) THEN
385: INFO = -3
386: ELSE IF( ILO.LT.1 .OR. ILO.GT.MAX( 1, N ) ) THEN
387: INFO = -4
388: ELSE IF( IHI.LT.MIN( ILO, N ) .OR. IHI.GT.N ) THEN
389: INFO = -5
390: ELSE IF( LDH.LT.MAX( 1, N ) ) THEN
391: INFO = -7
392: ELSE IF( LDZ.LT.1 .OR. ( WANTZ .AND. LDZ.LT.MAX( 1, N ) ) ) THEN
393: INFO = -11
394: ELSE IF( LWORK.LT.MAX( 1, N ) .AND. .NOT.LQUERY ) THEN
395: INFO = -13
396: END IF
397: *
398: IF( INFO.NE.0 ) THEN
399: *
400: * ==== Quick return in case of invalid argument. ====
401: *
402: CALL XERBLA( 'DHSEQR', -INFO )
403: RETURN
404: *
405: ELSE IF( N.EQ.0 ) THEN
406: *
407: * ==== Quick return in case N = 0; nothing to do. ====
408: *
409: RETURN
410: *
411: ELSE IF( LQUERY ) THEN
412: *
413: * ==== Quick return in case of a workspace query ====
414: *
415: CALL DLAQR0( WANTT, WANTZ, N, ILO, IHI, H, LDH, WR, WI, ILO,
416: $ IHI, Z, LDZ, WORK, LWORK, INFO )
417: * ==== Ensure reported workspace size is backward-compatible with
418: * . previous LAPACK versions. ====
419: WORK( 1 ) = MAX( DBLE( MAX( 1, N ) ), WORK( 1 ) )
420: RETURN
421: *
422: ELSE
423: *
424: * ==== copy eigenvalues isolated by DGEBAL ====
425: *
426: DO 10 I = 1, ILO - 1
427: WR( I ) = H( I, I )
428: WI( I ) = ZERO
429: 10 CONTINUE
430: DO 20 I = IHI + 1, N
431: WR( I ) = H( I, I )
432: WI( I ) = ZERO
433: 20 CONTINUE
434: *
435: * ==== Initialize Z, if requested ====
436: *
437: IF( INITZ )
438: $ CALL DLASET( 'A', N, N, ZERO, ONE, Z, LDZ )
439: *
440: * ==== Quick return if possible ====
441: *
442: IF( ILO.EQ.IHI ) THEN
443: WR( ILO ) = H( ILO, ILO )
444: WI( ILO ) = ZERO
445: RETURN
446: END IF
447: *
448: * ==== DLAHQR/DLAQR0 crossover point ====
449: *
450: NMIN = ILAENV( 12, 'DHSEQR', JOB( : 1 ) // COMPZ( : 1 ), N,
451: $ ILO, IHI, LWORK )
452: NMIN = MAX( NTINY, NMIN )
453: *
454: * ==== DLAQR0 for big matrices; DLAHQR for small ones ====
455: *
456: IF( N.GT.NMIN ) THEN
457: CALL DLAQR0( WANTT, WANTZ, N, ILO, IHI, H, LDH, WR, WI, ILO,
458: $ IHI, Z, LDZ, WORK, LWORK, INFO )
459: ELSE
460: *
461: * ==== Small matrix ====
462: *
463: CALL DLAHQR( WANTT, WANTZ, N, ILO, IHI, H, LDH, WR, WI, ILO,
464: $ IHI, Z, LDZ, INFO )
465: *
466: IF( INFO.GT.0 ) THEN
467: *
468: * ==== A rare DLAHQR failure! DLAQR0 sometimes succeeds
469: * . when DLAHQR fails. ====
470: *
471: KBOT = INFO
472: *
473: IF( N.GE.NL ) THEN
474: *
475: * ==== Larger matrices have enough subdiagonal scratch
476: * . space to call DLAQR0 directly. ====
477: *
478: CALL DLAQR0( WANTT, WANTZ, N, ILO, KBOT, H, LDH, WR,
479: $ WI, ILO, IHI, Z, LDZ, WORK, LWORK, INFO )
480: *
481: ELSE
482: *
483: * ==== Tiny matrices don't have enough subdiagonal
484: * . scratch space to benefit from DLAQR0. Hence,
485: * . tiny matrices must be copied into a larger
486: * . array before calling DLAQR0. ====
487: *
488: CALL DLACPY( 'A', N, N, H, LDH, HL, NL )
489: HL( N+1, N ) = ZERO
490: CALL DLASET( 'A', NL, NL-N, ZERO, ZERO, HL( 1, N+1 ),
491: $ NL )
492: CALL DLAQR0( WANTT, WANTZ, NL, ILO, KBOT, HL, NL, WR,
493: $ WI, ILO, IHI, Z, LDZ, WORKL, NL, INFO )
494: IF( WANTT .OR. INFO.NE.0 )
495: $ CALL DLACPY( 'A', N, N, HL, NL, H, LDH )
496: END IF
497: END IF
498: END IF
499: *
500: * ==== Clear out the trash, if necessary. ====
501: *
502: IF( ( WANTT .OR. INFO.NE.0 ) .AND. N.GT.2 )
503: $ CALL DLASET( 'L', N-2, N-2, ZERO, ZERO, H( 3, 1 ), LDH )
504: *
505: * ==== Ensure reported workspace size is backward-compatible with
506: * . previous LAPACK versions. ====
507: *
508: WORK( 1 ) = MAX( DBLE( MAX( 1, N ) ), WORK( 1 ) )
509: END IF
510: *
511: * ==== End of DHSEQR ====
512: *
513: END
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