001:       SUBROUTINE ZHSEQR( JOB, COMPZ, N, ILO, IHI, H, LDH, W, Z, LDZ,
002:      $                   WORK, LWORK, INFO )
003: *
004: *  -- LAPACK driver routine (version 3.2) --
005: *     Univ. of Tennessee, Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..
006: *     November 2006
007: *
008: *     .. Scalar Arguments ..
009:       INTEGER            IHI, ILO, INFO, LDH, LDZ, LWORK, N
010:       CHARACTER          COMPZ, JOB
011: *     ..
012: *     .. Array Arguments ..
013:       COMPLEX*16         H( LDH, * ), W( * ), WORK( * ), Z( LDZ, * )
014: *     ..
015: *     Purpose
016: *     =======
017: *
018: *     ZHSEQR computes the eigenvalues of a Hessenberg matrix H
019: *     and, optionally, the matrices T and Z from the Schur decomposition
020: *     H = Z T Z**H, where T is an upper triangular matrix (the
021: *     Schur form), and Z is the unitary matrix of Schur vectors.
022: *
023: *     Optionally Z may be postmultiplied into an input unitary
024: *     matrix Q so that this routine can give the Schur factorization
025: *     of a matrix A which has been reduced to the Hessenberg form H
026: *     by the unitary matrix Q:  A = Q*H*Q**H = (QZ)*H*(QZ)**H.
027: *
028: *     Arguments
029: *     =========
030: *
031: *     JOB   (input) CHARACTER*1
032: *           = 'E':  compute eigenvalues only;
033: *           = 'S':  compute eigenvalues and the Schur form T.
034: *
035: *     COMPZ (input) CHARACTER*1
036: *           = 'N':  no Schur vectors are computed;
037: *           = 'I':  Z is initialized to the unit matrix and the matrix Z
038: *                   of Schur vectors of H is returned;
039: *           = 'V':  Z must contain an unitary matrix Q on entry, and
040: *                   the product Q*Z is returned.
041: *
042: *     N     (input) INTEGER
043: *           The order of the matrix H.  N .GE. 0.
044: *
045: *     ILO   (input) INTEGER
046: *     IHI   (input) INTEGER
047: *           It is assumed that H is already upper triangular in rows
048: *           and columns 1:ILO-1 and IHI+1:N. ILO and IHI are normally
049: *           set by a previous call to ZGEBAL, and then passed to ZGEHRD
050: *           when the matrix output by ZGEBAL is reduced to Hessenberg
051: *           form. Otherwise ILO and IHI should be set to 1 and N
052: *           respectively.  If N.GT.0, then 1.LE.ILO.LE.IHI.LE.N.
053: *           If N = 0, then ILO = 1 and IHI = 0.
054: *
055: *     H     (input/output) COMPLEX*16 array, dimension (LDH,N)
056: *           On entry, the upper Hessenberg matrix H.
057: *           On exit, if INFO = 0 and JOB = 'S', H contains the upper
058: *           triangular matrix T from the Schur decomposition (the
059: *           Schur form). If INFO = 0 and JOB = 'E', the contents of
060: *           H are unspecified on exit.  (The output value of H when
061: *           INFO.GT.0 is given under the description of INFO below.)
062: *
063: *           Unlike earlier versions of ZHSEQR, this subroutine may
064: *           explicitly H(i,j) = 0 for i.GT.j and j = 1, 2, ... ILO-1
065: *           or j = IHI+1, IHI+2, ... N.
066: *
067: *     LDH   (input) INTEGER
068: *           The leading dimension of the array H. LDH .GE. max(1,N).
069: *
070: *     W        (output) COMPLEX*16 array, dimension (N)
071: *           The computed eigenvalues. If JOB = 'S', the eigenvalues are
072: *           stored in the same order as on the diagonal of the Schur
073: *           form returned in H, with W(i) = H(i,i).
074: *
075: *     Z     (input/output) COMPLEX*16 array, dimension (LDZ,N)
076: *           If COMPZ = 'N', Z is not referenced.
077: *           If COMPZ = 'I', on entry Z need not be set and on exit,
078: *           if INFO = 0, Z contains the unitary matrix Z of the Schur
079: *           vectors of H.  If COMPZ = 'V', on entry Z must contain an
080: *           N-by-N matrix Q, which is assumed to be equal to the unit
081: *           matrix except for the submatrix Z(ILO:IHI,ILO:IHI). On exit,
082: *           if INFO = 0, Z contains Q*Z.
083: *           Normally Q is the unitary matrix generated by ZUNGHR
084: *           after the call to ZGEHRD which formed the Hessenberg matrix
085: *           H. (The output value of Z when INFO.GT.0 is given under
086: *           the description of INFO below.)
087: *
088: *     LDZ   (input) INTEGER
089: *           The leading dimension of the array Z.  if COMPZ = 'I' or
090: *           COMPZ = 'V', then LDZ.GE.MAX(1,N).  Otherwize, LDZ.GE.1.
091: *
092: *     WORK  (workspace/output) COMPLEX*16 array, dimension (LWORK)
093: *           On exit, if INFO = 0, WORK(1) returns an estimate of
094: *           the optimal value for LWORK.
095: *
096: *     LWORK (input) INTEGER
097: *           The dimension of the array WORK.  LWORK .GE. max(1,N)
098: *           is sufficient and delivers very good and sometimes
099: *           optimal performance.  However, LWORK as large as 11*N
100: *           may be required for optimal performance.  A workspace
101: *           query is recommended to determine the optimal workspace
102: *           size.
103: *
104: *           If LWORK = -1, then ZHSEQR does a workspace query.
105: *           In this case, ZHSEQR checks the input parameters and
106: *           estimates the optimal workspace size for the given
107: *           values of N, ILO and IHI.  The estimate is returned
108: *           in WORK(1).  No error message related to LWORK is
109: *           issued by XERBLA.  Neither H nor Z are accessed.
110: *
111: *
112: *     INFO  (output) INTEGER
113: *             =  0:  successful exit
114: *           .LT. 0:  if INFO = -i, the i-th argument had an illegal
115: *                    value
116: *           .GT. 0:  if INFO = i, ZHSEQR failed to compute all of
117: *                the eigenvalues.  Elements 1:ilo-1 and i+1:n of WR
118: *                and WI contain those eigenvalues which have been
119: *                successfully computed.  (Failures are rare.)
120: *
121: *                If INFO .GT. 0 and JOB = 'E', then on exit, the
122: *                remaining unconverged eigenvalues are the eigen-
123: *                values of the upper Hessenberg matrix rows and
124: *                columns ILO through INFO of the final, output
125: *                value of H.
126: *
127: *                If INFO .GT. 0 and JOB   = 'S', then on exit
128: *
129: *           (*)  (initial value of H)*U  = U*(final value of H)
130: *
131: *                where U is a unitary matrix.  The final
132: *                value of  H is upper Hessenberg and triangular in
133: *                rows and columns INFO+1 through IHI.
134: *
135: *                If INFO .GT. 0 and COMPZ = 'V', then on exit
136: *
137: *                  (final value of Z)  =  (initial value of Z)*U
138: *
139: *                where U is the unitary matrix in (*) (regard-
140: *                less of the value of JOB.)
141: *
142: *                If INFO .GT. 0 and COMPZ = 'I', then on exit
143: *                      (final value of Z)  = U
144: *                where U is the unitary matrix in (*) (regard-
145: *                less of the value of JOB.)
146: *
147: *                If INFO .GT. 0 and COMPZ = 'N', then Z is not
148: *                accessed.
149: *
150: *     ================================================================
151: *             Default values supplied by
152: *             ILAENV(ISPEC,'ZHSEQR',JOB(:1)//COMPZ(:1),N,ILO,IHI,LWORK).
153: *             It is suggested that these defaults be adjusted in order
154: *             to attain best performance in each particular
155: *             computational environment.
156: *
157: *            ISPEC=12: The ZLAHQR vs ZLAQR0 crossover point.
158: *                      Default: 75. (Must be at least 11.)
159: *
160: *            ISPEC=13: Recommended deflation window size.
161: *                      This depends on ILO, IHI and NS.  NS is the
162: *                      number of simultaneous shifts returned
163: *                      by ILAENV(ISPEC=15).  (See ISPEC=15 below.)
164: *                      The default for (IHI-ILO+1).LE.500 is NS.
165: *                      The default for (IHI-ILO+1).GT.500 is 3*NS/2.
166: *
167: *            ISPEC=14: Nibble crossover point. (See IPARMQ for
168: *                      details.)  Default: 14% of deflation window
169: *                      size.
170: *
171: *            ISPEC=15: Number of simultaneous shifts in a multishift
172: *                      QR iteration.
173: *
174: *                      If IHI-ILO+1 is ...
175: *
176: *                      greater than      ...but less    ... the
177: *                      or equal to ...      than        default is
178: *
179: *                           1               30          NS =   2(+)
180: *                          30               60          NS =   4(+)
181: *                          60              150          NS =  10(+)
182: *                         150              590          NS =  **
183: *                         590             3000          NS =  64
184: *                        3000             6000          NS = 128
185: *                        6000             infinity      NS = 256
186: *
187: *                  (+)  By default some or all matrices of this order
188: *                       are passed to the implicit double shift routine
189: *                       ZLAHQR and this parameter is ignored.  See
190: *                       ISPEC=12 above and comments in IPARMQ for
191: *                       details.
192: *
193: *                 (**)  The asterisks (**) indicate an ad-hoc
194: *                       function of N increasing from 10 to 64.
195: *
196: *            ISPEC=16: Select structured matrix multiply.
197: *                      If the number of simultaneous shifts (specified
198: *                      by ISPEC=15) is less than 14, then the default
199: *                      for ISPEC=16 is 0.  Otherwise the default for
200: *                      ISPEC=16 is 2.
201: *
202: *     ================================================================
203: *     Based on contributions by
204: *        Karen Braman and Ralph Byers, Department of Mathematics,
205: *        University of Kansas, USA
206: *
207: *     ================================================================
208: *     References:
209: *       K. Braman, R. Byers and R. Mathias, The Multi-Shift QR
210: *       Algorithm Part I: Maintaining Well Focused Shifts, and Level 3
211: *       Performance, SIAM Journal of Matrix Analysis, volume 23, pages
212: *       929--947, 2002.
213: *
214: *       K. Braman, R. Byers and R. Mathias, The Multi-Shift QR
215: *       Algorithm Part II: Aggressive Early Deflation, SIAM Journal
216: *       of Matrix Analysis, volume 23, pages 948--973, 2002.
217: *
218: *     ================================================================
219: *     .. Parameters ..
220: *
221: *     ==== Matrices of order NTINY or smaller must be processed by
222: *     .    ZLAHQR because of insufficient subdiagonal scratch space.
223: *     .    (This is a hard limit.) ====
224:       INTEGER            NTINY
225:       PARAMETER          ( NTINY = 11 )
226: *
227: *     ==== NL allocates some local workspace to help small matrices
228: *     .    through a rare ZLAHQR failure.  NL .GT. NTINY = 11 is
229: *     .    required and NL .LE. NMIN = ILAENV(ISPEC=12,...) is recom-
230: *     .    mended.  (The default value of NMIN is 75.)  Using NL = 49
231: *     .    allows up to six simultaneous shifts and a 16-by-16
232: *     .    deflation window.  ====
233:       INTEGER            NL
234:       PARAMETER          ( NL = 49 )
235:       COMPLEX*16         ZERO, ONE
236:       PARAMETER          ( ZERO = ( 0.0d0, 0.0d0 ),
237:      $                   ONE = ( 1.0d0, 0.0d0 ) )
238:       DOUBLE PRECISION   RZERO
239:       PARAMETER          ( RZERO = 0.0d0 )
240: *     ..
241: *     .. Local Arrays ..
242:       COMPLEX*16         HL( NL, NL ), WORKL( NL )
243: *     ..
244: *     .. Local Scalars ..
245:       INTEGER            KBOT, NMIN
246:       LOGICAL            INITZ, LQUERY, WANTT, WANTZ
247: *     ..
248: *     .. External Functions ..
249:       INTEGER            ILAENV
250:       LOGICAL            LSAME
251:       EXTERNAL           ILAENV, LSAME
252: *     ..
253: *     .. External Subroutines ..
254:       EXTERNAL           XERBLA, ZCOPY, ZLACPY, ZLAHQR, ZLAQR0, ZLASET
255: *     ..
256: *     .. Intrinsic Functions ..
257:       INTRINSIC          DBLE, DCMPLX, MAX, MIN
258: *     ..
259: *     .. Executable Statements ..
260: *
261: *     ==== Decode and check the input parameters. ====
262: *
263:       WANTT = LSAME( JOB, 'S' )
264:       INITZ = LSAME( COMPZ, 'I' )
265:       WANTZ = INITZ .OR. LSAME( COMPZ, 'V' )
266:       WORK( 1 ) = DCMPLX( DBLE( MAX( 1, N ) ), RZERO )
267:       LQUERY = LWORK.EQ.-1
268: *
269:       INFO = 0
270:       IF( .NOT.LSAME( JOB, 'E' ) .AND. .NOT.WANTT ) THEN
271:          INFO = -1
272:       ELSE IF( .NOT.LSAME( COMPZ, 'N' ) .AND. .NOT.WANTZ ) THEN
273:          INFO = -2
274:       ELSE IF( N.LT.0 ) THEN
275:          INFO = -3
276:       ELSE IF( ILO.LT.1 .OR. ILO.GT.MAX( 1, N ) ) THEN
277:          INFO = -4
278:       ELSE IF( IHI.LT.MIN( ILO, N ) .OR. IHI.GT.N ) THEN
279:          INFO = -5
280:       ELSE IF( LDH.LT.MAX( 1, N ) ) THEN
281:          INFO = -7
282:       ELSE IF( LDZ.LT.1 .OR. ( WANTZ .AND. LDZ.LT.MAX( 1, N ) ) ) THEN
283:          INFO = -10
284:       ELSE IF( LWORK.LT.MAX( 1, N ) .AND. .NOT.LQUERY ) THEN
285:          INFO = -12
286:       END IF
287: *
288:       IF( INFO.NE.0 ) THEN
289: *
290: *        ==== Quick return in case of invalid argument. ====
291: *
292:          CALL XERBLA( 'ZHSEQR', -INFO )
293:          RETURN
294: *
295:       ELSE IF( N.EQ.0 ) THEN
296: *
297: *        ==== Quick return in case N = 0; nothing to do. ====
298: *
299:          RETURN
300: *
301:       ELSE IF( LQUERY ) THEN
302: *
303: *        ==== Quick return in case of a workspace query ====
304: *
305:          CALL ZLAQR0( WANTT, WANTZ, N, ILO, IHI, H, LDH, W, ILO, IHI, Z,
306:      $                LDZ, WORK, LWORK, INFO )
307: *        ==== Ensure reported workspace size is backward-compatible with
308: *        .    previous LAPACK versions. ====
309:          WORK( 1 ) = DCMPLX( MAX( DBLE( WORK( 1 ) ), DBLE( MAX( 1,
310:      $               N ) ) ), RZERO )
311:          RETURN
312: *
313:       ELSE
314: *
315: *        ==== copy eigenvalues isolated by ZGEBAL ====
316: *
317:          IF( ILO.GT.1 )
318:      $      CALL ZCOPY( ILO-1, H, LDH+1, W, 1 )
319:          IF( IHI.LT.N )
320:      $      CALL ZCOPY( N-IHI, H( IHI+1, IHI+1 ), LDH+1, W( IHI+1 ), 1 )
321: *
322: *        ==== Initialize Z, if requested ====
323: *
324:          IF( INITZ )
325:      $      CALL ZLASET( 'A', N, N, ZERO, ONE, Z, LDZ )
326: *
327: *        ==== Quick return if possible ====
328: *
329:          IF( ILO.EQ.IHI ) THEN
330:             W( ILO ) = H( ILO, ILO )
331:             RETURN
332:          END IF
333: *
334: *        ==== ZLAHQR/ZLAQR0 crossover point ====
335: *
336:          NMIN = ILAENV( 12, 'ZHSEQR', JOB( : 1 ) // COMPZ( : 1 ), N,
337:      $          ILO, IHI, LWORK )
338:          NMIN = MAX( NTINY, NMIN )
339: *
340: *        ==== ZLAQR0 for big matrices; ZLAHQR for small ones ====
341: *
342:          IF( N.GT.NMIN ) THEN
343:             CALL ZLAQR0( WANTT, WANTZ, N, ILO, IHI, H, LDH, W, ILO, IHI,
344:      $                   Z, LDZ, WORK, LWORK, INFO )
345:          ELSE
346: *
347: *           ==== Small matrix ====
348: *
349:             CALL ZLAHQR( WANTT, WANTZ, N, ILO, IHI, H, LDH, W, ILO, IHI,
350:      $                   Z, LDZ, INFO )
351: *
352:             IF( INFO.GT.0 ) THEN
353: *
354: *              ==== A rare ZLAHQR failure!  ZLAQR0 sometimes succeeds
355: *              .    when ZLAHQR fails. ====
356: *
357:                KBOT = INFO
358: *
359:                IF( N.GE.NL ) THEN
360: *
361: *                 ==== Larger matrices have enough subdiagonal scratch
362: *                 .    space to call ZLAQR0 directly. ====
363: *
364:                   CALL ZLAQR0( WANTT, WANTZ, N, ILO, KBOT, H, LDH, W,
365:      $                         ILO, IHI, Z, LDZ, WORK, LWORK, INFO )
366: *
367:                ELSE
368: *
369: *                 ==== Tiny matrices don't have enough subdiagonal
370: *                 .    scratch space to benefit from ZLAQR0.  Hence,
371: *                 .    tiny matrices must be copied into a larger
372: *                 .    array before calling ZLAQR0. ====
373: *
374:                   CALL ZLACPY( 'A', N, N, H, LDH, HL, NL )
375:                   HL( N+1, N ) = ZERO
376:                   CALL ZLASET( 'A', NL, NL-N, ZERO, ZERO, HL( 1, N+1 ),
377:      $                         NL )
378:                   CALL ZLAQR0( WANTT, WANTZ, NL, ILO, KBOT, HL, NL, W,
379:      $                         ILO, IHI, Z, LDZ, WORKL, NL, INFO )
380:                   IF( WANTT .OR. INFO.NE.0 )
381:      $               CALL ZLACPY( 'A', N, N, HL, NL, H, LDH )
382:                END IF
383:             END IF
384:          END IF
385: *
386: *        ==== Clear out the trash, if necessary. ====
387: *
388:          IF( ( WANTT .OR. INFO.NE.0 ) .AND. N.GT.2 )
389:      $      CALL ZLASET( 'L', N-2, N-2, ZERO, ZERO, H( 3, 1 ), LDH )
390: *
391: *        ==== Ensure reported workspace size is backward-compatible with
392: *        .    previous LAPACK versions. ====
393: *
394:          WORK( 1 ) = DCMPLX( MAX( DBLE( MAX( 1, N ) ),
395:      $               DBLE( WORK( 1 ) ) ), RZERO )
396:       END IF
397: *
398: *     ==== End of ZHSEQR ====
399: *
400:       END
401: