1: *> \brief \b ZLAQZ0
2: *
3: * =========== DOCUMENTATION ===========
4: *
5: * Online html documentation available at
6: * http://www.netlib.org/lapack/explore-html/
7: *
8: *> \htmlonly
9: *> Download ZLAQZ0 + dependencies
10: *> <a href="http://www.netlib.org/cgi-bin/netlibfiles.tgz?format=tgz&filename=/lapack/lapack_routine/zlaqz0.f">
11: *> [TGZ]</a>
12: *> <a href="http://www.netlib.org/cgi-bin/netlibfiles.zip?format=zip&filename=/lapack/lapack_routine/zlaqz0.f">
13: *> [ZIP]</a>
14: *> <a href="http://www.netlib.org/cgi-bin/netlibfiles.txt?format=txt&filename=/lapack/lapack_routine/zlaqz0.f">
15: *> [TXT]</a>
16: *> \endhtmlonly
17: *
18: * Definition:
19: * ===========
20: *
21: * SUBROUTINE ZLAQZ0( WANTS, WANTQ, WANTZ, N, ILO, IHI, A, LDA, B,
22: * $ LDB, ALPHA, BETA, Q, LDQ, Z, LDZ, WORK, LWORK, RWORK, REC,
23: * $ INFO )
24: * IMPLICIT NONE
25: *
26: * Arguments
27: * CHARACTER, INTENT( IN ) :: WANTS, WANTQ, WANTZ
28: * INTEGER, INTENT( IN ) :: N, ILO, IHI, LDA, LDB, LDQ, LDZ, LWORK,
29: * $ REC
30: * INTEGER, INTENT( OUT ) :: INFO
31: * COMPLEX*16, INTENT( INOUT ) :: A( LDA, * ), B( LDB, * ), Q( LDQ,
32: * $ * ), Z( LDZ, * ), ALPHA( * ), BETA( * ), WORK( * )
33: * DOUBLE PRECISION, INTENT( OUT ) :: RWORK( * )
34: * ..
35: *
36: *
37: *> \par Purpose:
38: * =============
39: *>
40: *> \verbatim
41: *>
42: *> ZLAQZ0 computes the eigenvalues of a real matrix pair (H,T),
43: *> where H is an upper Hessenberg matrix and T is upper triangular,
44: *> using the double-shift QZ method.
45: *> Matrix pairs of this type are produced by the reduction to
46: *> generalized upper Hessenberg form of a real matrix pair (A,B):
47: *>
48: *> A = Q1*H*Z1**H, B = Q1*T*Z1**H,
49: *>
50: *> as computed by ZGGHRD.
51: *>
52: *> If JOB='S', then the Hessenberg-triangular pair (H,T) is
53: *> also reduced to generalized Schur form,
54: *>
55: *> H = Q*S*Z**H, T = Q*P*Z**H,
56: *>
57: *> where Q and Z are unitary matrices, P and S are an upper triangular
58: *> matrices.
59: *>
60: *> Optionally, the unitary matrix Q from the generalized Schur
61: *> factorization may be postmultiplied into an input matrix Q1, and the
62: *> unitary matrix Z may be postmultiplied into an input matrix Z1.
63: *> If Q1 and Z1 are the unitary matrices from ZGGHRD that reduced
64: *> the matrix pair (A,B) to generalized upper Hessenberg form, then the
65: *> output matrices Q1*Q and Z1*Z are the unitary factors from the
66: *> generalized Schur factorization of (A,B):
67: *>
68: *> A = (Q1*Q)*S*(Z1*Z)**H, B = (Q1*Q)*P*(Z1*Z)**H.
69: *>
70: *> To avoid overflow, eigenvalues of the matrix pair (H,T) (equivalently,
71: *> of (A,B)) are computed as a pair of values (alpha,beta), where alpha is
72: *> complex and beta real.
73: *> If beta is nonzero, lambda = alpha / beta is an eigenvalue of the
74: *> generalized nonsymmetric eigenvalue problem (GNEP)
75: *> A*x = lambda*B*x
76: *> and if alpha is nonzero, mu = beta / alpha is an eigenvalue of the
77: *> alternate form of the GNEP
78: *> mu*A*y = B*y.
79: *> Eigenvalues can be read directly from the generalized Schur
80: *> form:
81: *> alpha = S(i,i), beta = P(i,i).
82: *>
83: *> Ref: C.B. Moler & G.W. Stewart, "An Algorithm for Generalized Matrix
84: *> Eigenvalue Problems", SIAM J. Numer. Anal., 10(1973),
85: *> pp. 241--256.
86: *>
87: *> Ref: B. Kagstrom, D. Kressner, "Multishift Variants of the QZ
88: *> Algorithm with Aggressive Early Deflation", SIAM J. Numer.
89: *> Anal., 29(2006), pp. 199--227.
90: *>
91: *> Ref: T. Steel, D. Camps, K. Meerbergen, R. Vandebril "A multishift,
92: *> multipole rational QZ method with agressive early deflation"
93: *> \endverbatim
94: *
95: * Arguments:
96: * ==========
97: *
98: *> \param[in] WANTS
99: *> \verbatim
100: *> WANTS is CHARACTER*1
101: *> = 'E': Compute eigenvalues only;
102: *> = 'S': Compute eigenvalues and the Schur form.
103: *> \endverbatim
104: *>
105: *> \param[in] WANTQ
106: *> \verbatim
107: *> WANTQ is CHARACTER*1
108: *> = 'N': Left Schur vectors (Q) are not computed;
109: *> = 'I': Q is initialized to the unit matrix and the matrix Q
110: *> of left Schur vectors of (A,B) is returned;
111: *> = 'V': Q must contain an unitary matrix Q1 on entry and
112: *> the product Q1*Q is returned.
113: *> \endverbatim
114: *>
115: *> \param[in] WANTZ
116: *> \verbatim
117: *> WANTZ is CHARACTER*1
118: *> = 'N': Right Schur vectors (Z) are not computed;
119: *> = 'I': Z is initialized to the unit matrix and the matrix Z
120: *> of right Schur vectors of (A,B) is returned;
121: *> = 'V': Z must contain an unitary matrix Z1 on entry and
122: *> the product Z1*Z is returned.
123: *> \endverbatim
124: *>
125: *> \param[in] N
126: *> \verbatim
127: *> N is INTEGER
128: *> The order of the matrices A, B, Q, and Z. N >= 0.
129: *> \endverbatim
130: *>
131: *> \param[in] ILO
132: *> \verbatim
133: *> ILO is INTEGER
134: *> \endverbatim
135: *>
136: *> \param[in] IHI
137: *> \verbatim
138: *> IHI is INTEGER
139: *> ILO and IHI mark the rows and columns of A which are in
140: *> Hessenberg form. It is assumed that A is already upper
141: *> triangular in rows and columns 1:ILO-1 and IHI+1:N.
142: *> If N > 0, 1 <= ILO <= IHI <= N; if N = 0, ILO=1 and IHI=0.
143: *> \endverbatim
144: *>
145: *> \param[in,out] A
146: *> \verbatim
147: *> A is COMPLEX*16 array, dimension (LDA, N)
148: *> On entry, the N-by-N upper Hessenberg matrix A.
149: *> On exit, if JOB = 'S', A contains the upper triangular
150: *> matrix S from the generalized Schur factorization.
151: *> If JOB = 'E', the diagonal blocks of A match those of S, but
152: *> the rest of A is unspecified.
153: *> \endverbatim
154: *>
155: *> \param[in] LDA
156: *> \verbatim
157: *> LDA is INTEGER
158: *> The leading dimension of the array A. LDA >= max( 1, N ).
159: *> \endverbatim
160: *>
161: *> \param[in,out] B
162: *> \verbatim
163: *> B is COMPLEX*16 array, dimension (LDB, N)
164: *> On entry, the N-by-N upper triangular matrix B.
165: *> On exit, if JOB = 'S', B contains the upper triangular
166: *> matrix P from the generalized Schur factorization;
167: *> If JOB = 'E', the diagonal blocks of B match those of P, but
168: *> the rest of B is unspecified.
169: *> \endverbatim
170: *>
171: *> \param[in] LDB
172: *> \verbatim
173: *> LDB is INTEGER
174: *> The leading dimension of the array B. LDB >= max( 1, N ).
175: *> \endverbatim
176: *>
177: *> \param[out] ALPHA
178: *> \verbatim
179: *> ALPHA is COMPLEX*16 array, dimension (N)
180: *> Each scalar alpha defining an eigenvalue
181: *> of GNEP.
182: *> \endverbatim
183: *>
184: *> \param[out] BETA
185: *> \verbatim
186: *> BETA is COMPLEX*16 array, dimension (N)
187: *> The scalars beta that define the eigenvalues of GNEP.
188: *> Together, the quantities alpha = ALPHA(j) and
189: *> beta = BETA(j) represent the j-th eigenvalue of the matrix
190: *> pair (A,B), in one of the forms lambda = alpha/beta or
191: *> mu = beta/alpha. Since either lambda or mu may overflow,
192: *> they should not, in general, be computed.
193: *> \endverbatim
194: *>
195: *> \param[in,out] Q
196: *> \verbatim
197: *> Q is COMPLEX*16 array, dimension (LDQ, N)
198: *> On entry, if COMPQ = 'V', the unitary matrix Q1 used in
199: *> the reduction of (A,B) to generalized Hessenberg form.
200: *> On exit, if COMPQ = 'I', the unitary matrix of left Schur
201: *> vectors of (A,B), and if COMPQ = 'V', the unitary matrix
202: *> of left Schur vectors of (A,B).
203: *> Not referenced if COMPQ = 'N'.
204: *> \endverbatim
205: *>
206: *> \param[in] LDQ
207: *> \verbatim
208: *> LDQ is INTEGER
209: *> The leading dimension of the array Q. LDQ >= 1.
210: *> If COMPQ='V' or 'I', then LDQ >= N.
211: *> \endverbatim
212: *>
213: *> \param[in,out] Z
214: *> \verbatim
215: *> Z is COMPLEX*16 array, dimension (LDZ, N)
216: *> On entry, if COMPZ = 'V', the unitary matrix Z1 used in
217: *> the reduction of (A,B) to generalized Hessenberg form.
218: *> On exit, if COMPZ = 'I', the unitary matrix of
219: *> right Schur vectors of (H,T), and if COMPZ = 'V', the
220: *> unitary matrix of right Schur vectors of (A,B).
221: *> Not referenced if COMPZ = 'N'.
222: *> \endverbatim
223: *>
224: *> \param[in] LDZ
225: *> \verbatim
226: *> LDZ is INTEGER
227: *> The leading dimension of the array Z. LDZ >= 1.
228: *> If COMPZ='V' or 'I', then LDZ >= N.
229: *> \endverbatim
230: *>
231: *> \param[out] WORK
232: *> \verbatim
233: *> WORK is COMPLEX*16 array, dimension (MAX(1,LWORK))
234: *> On exit, if INFO >= 0, WORK(1) returns the optimal LWORK.
235: *> \endverbatim
236: *>
237: *> \param[out] RWORK
238: *> \verbatim
239: *> RWORK is DOUBLE PRECISION array, dimension (N)
240: *> \endverbatim
241: *>
242: *> \param[in] LWORK
243: *> \verbatim
244: *> LWORK is INTEGER
245: *> The dimension of the array WORK. LWORK >= max(1,N).
246: *>
247: *> If LWORK = -1, then a workspace query is assumed; the routine
248: *> only calculates the optimal size of the WORK array, returns
249: *> this value as the first entry of the WORK array, and no error
250: *> message related to LWORK is issued by XERBLA.
251: *> \endverbatim
252: *>
253: *> \param[in] REC
254: *> \verbatim
255: *> REC is INTEGER
256: *> REC indicates the current recursion level. Should be set
257: *> to 0 on first call.
258: *> \endverbatim
259: *>
260: *> \param[out] INFO
261: *> \verbatim
262: *> INFO is INTEGER
263: *> = 0: successful exit
264: *> < 0: if INFO = -i, the i-th argument had an illegal value
265: *> = 1,...,N: the QZ iteration did not converge. (A,B) is not
266: *> in Schur form, but ALPHA(i) and
267: *> BETA(i), i=INFO+1,...,N should be correct.
268: *> \endverbatim
269: *
270: * Authors:
271: * ========
272: *
273: *> \author Thijs Steel, KU Leuven
274: *
275: *> \date May 2020
276: *
277: *> \ingroup complex16GEcomputational
278: *>
279: * =====================================================================
280: RECURSIVE SUBROUTINE ZLAQZ0( WANTS, WANTQ, WANTZ, N, ILO, IHI, A,
281: $ LDA, B, LDB, ALPHA, BETA, Q, LDQ, Z,
282: $ LDZ, WORK, LWORK, RWORK, REC,
283: $ INFO )
284: IMPLICIT NONE
285:
286: * Arguments
287: CHARACTER, INTENT( IN ) :: WANTS, WANTQ, WANTZ
288: INTEGER, INTENT( IN ) :: N, ILO, IHI, LDA, LDB, LDQ, LDZ, LWORK,
289: $ REC
290: INTEGER, INTENT( OUT ) :: INFO
291: COMPLEX*16, INTENT( INOUT ) :: A( LDA, * ), B( LDB, * ), Q( LDQ,
292: $ * ), Z( LDZ, * ), ALPHA( * ), BETA( * ), WORK( * )
293: DOUBLE PRECISION, INTENT( OUT ) :: RWORK( * )
294:
295: * Parameters
296: COMPLEX*16 CZERO, CONE
297: PARAMETER ( CZERO = ( 0.0D+0, 0.0D+0 ), CONE = ( 1.0D+0,
298: $ 0.0D+0 ) )
299: DOUBLE PRECISION :: ZERO, ONE, HALF
300: PARAMETER( ZERO = 0.0D0, ONE = 1.0D0, HALF = 0.5D0 )
301:
302: * Local scalars
303: DOUBLE PRECISION :: SMLNUM, ULP, SAFMIN, SAFMAX, C1, TEMPR,
304: $ BNORM, BTOL
305: COMPLEX*16 :: ESHIFT, S1, TEMP
306: INTEGER :: ISTART, ISTOP, IITER, MAXIT, ISTART2, K, LD, NSHIFTS,
307: $ NBLOCK, NW, NMIN, NIBBLE, N_UNDEFLATED, N_DEFLATED,
308: $ NS, SWEEP_INFO, SHIFTPOS, LWORKREQ, K2, ISTARTM,
309: $ ISTOPM, IWANTS, IWANTQ, IWANTZ, NORM_INFO, AED_INFO,
310: $ NWR, NBR, NSR, ITEMP1, ITEMP2, RCOST
311: LOGICAL :: ILSCHUR, ILQ, ILZ
312: CHARACTER :: JBCMPZ*3
313:
314: * External Functions
315: EXTERNAL :: XERBLA, ZHGEQZ, ZLAQZ2, ZLAQZ3, ZLASET, DLABAD,
316: $ ZLARTG, ZROT
317: DOUBLE PRECISION, EXTERNAL :: DLAMCH, ZLANHS
318: LOGICAL, EXTERNAL :: LSAME
319: INTEGER, EXTERNAL :: ILAENV
320:
321: *
322: * Decode wantS,wantQ,wantZ
323: *
324: IF( LSAME( WANTS, 'E' ) ) THEN
325: ILSCHUR = .FALSE.
326: IWANTS = 1
327: ELSE IF( LSAME( WANTS, 'S' ) ) THEN
328: ILSCHUR = .TRUE.
329: IWANTS = 2
330: ELSE
331: IWANTS = 0
332: END IF
333:
334: IF( LSAME( WANTQ, 'N' ) ) THEN
335: ILQ = .FALSE.
336: IWANTQ = 1
337: ELSE IF( LSAME( WANTQ, 'V' ) ) THEN
338: ILQ = .TRUE.
339: IWANTQ = 2
340: ELSE IF( LSAME( WANTQ, 'I' ) ) THEN
341: ILQ = .TRUE.
342: IWANTQ = 3
343: ELSE
344: IWANTQ = 0
345: END IF
346:
347: IF( LSAME( WANTZ, 'N' ) ) THEN
348: ILZ = .FALSE.
349: IWANTZ = 1
350: ELSE IF( LSAME( WANTZ, 'V' ) ) THEN
351: ILZ = .TRUE.
352: IWANTZ = 2
353: ELSE IF( LSAME( WANTZ, 'I' ) ) THEN
354: ILZ = .TRUE.
355: IWANTZ = 3
356: ELSE
357: IWANTZ = 0
358: END IF
359: *
360: * Check Argument Values
361: *
362: INFO = 0
363: IF( IWANTS.EQ.0 ) THEN
364: INFO = -1
365: ELSE IF( IWANTQ.EQ.0 ) THEN
366: INFO = -2
367: ELSE IF( IWANTZ.EQ.0 ) THEN
368: INFO = -3
369: ELSE IF( N.LT.0 ) THEN
370: INFO = -4
371: ELSE IF( ILO.LT.1 ) THEN
372: INFO = -5
373: ELSE IF( IHI.GT.N .OR. IHI.LT.ILO-1 ) THEN
374: INFO = -6
375: ELSE IF( LDA.LT.N ) THEN
376: INFO = -8
377: ELSE IF( LDB.LT.N ) THEN
378: INFO = -10
379: ELSE IF( LDQ.LT.1 .OR. ( ILQ .AND. LDQ.LT.N ) ) THEN
380: INFO = -15
381: ELSE IF( LDZ.LT.1 .OR. ( ILZ .AND. LDZ.LT.N ) ) THEN
382: INFO = -17
383: END IF
384: IF( INFO.NE.0 ) THEN
385: CALL XERBLA( 'ZLAQZ0', -INFO )
386: RETURN
387: END IF
388:
389: *
390: * Quick return if possible
391: *
392: IF( N.LE.0 ) THEN
393: WORK( 1 ) = DBLE( 1 )
394: RETURN
395: END IF
396:
397: *
398: * Get the parameters
399: *
400: JBCMPZ( 1:1 ) = WANTS
401: JBCMPZ( 2:2 ) = WANTQ
402: JBCMPZ( 3:3 ) = WANTZ
403:
404: NMIN = ILAENV( 12, 'ZLAQZ0', JBCMPZ, N, ILO, IHI, LWORK )
405:
406: NWR = ILAENV( 13, 'ZLAQZ0', JBCMPZ, N, ILO, IHI, LWORK )
407: NWR = MAX( 2, NWR )
408: NWR = MIN( IHI-ILO+1, ( N-1 ) / 3, NWR )
409:
410: NIBBLE = ILAENV( 14, 'ZLAQZ0', JBCMPZ, N, ILO, IHI, LWORK )
411:
412: NSR = ILAENV( 15, 'ZLAQZ0', JBCMPZ, N, ILO, IHI, LWORK )
413: NSR = MIN( NSR, ( N+6 ) / 9, IHI-ILO )
414: NSR = MAX( 2, NSR-MOD( NSR, 2 ) )
415:
416: RCOST = ILAENV( 17, 'ZLAQZ0', JBCMPZ, N, ILO, IHI, LWORK )
417: ITEMP1 = INT( NSR/SQRT( 1+2*NSR/( DBLE( RCOST )/100*N ) ) )
418: ITEMP1 = ( ( ITEMP1-1 )/4 )*4+4
419: NBR = NSR+ITEMP1
420:
421: IF( N .LT. NMIN .OR. REC .GE. 2 ) THEN
422: CALL ZHGEQZ( WANTS, WANTQ, WANTZ, N, ILO, IHI, A, LDA, B, LDB,
423: $ ALPHA, BETA, Q, LDQ, Z, LDZ, WORK, LWORK, RWORK,
424: $ INFO )
425: RETURN
426: END IF
427:
428: *
429: * Find out required workspace
430: *
431:
432: * Workspace query to ZLAQZ2
433: NW = MAX( NWR, NMIN )
434: CALL ZLAQZ2( ILSCHUR, ILQ, ILZ, N, ILO, IHI, NW, A, LDA, B, LDB,
435: $ Q, LDQ, Z, LDZ, N_UNDEFLATED, N_DEFLATED, ALPHA,
436: $ BETA, WORK, NW, WORK, NW, WORK, -1, RWORK, REC,
437: $ AED_INFO )
438: ITEMP1 = INT( WORK( 1 ) )
439: * Workspace query to ZLAQZ3
440: CALL ZLAQZ3( ILSCHUR, ILQ, ILZ, N, ILO, IHI, NSR, NBR, ALPHA,
441: $ BETA, A, LDA, B, LDB, Q, LDQ, Z, LDZ, WORK, NBR,
442: $ WORK, NBR, WORK, -1, SWEEP_INFO )
443: ITEMP2 = INT( WORK( 1 ) )
444:
445: LWORKREQ = MAX( ITEMP1+2*NW**2, ITEMP2+2*NBR**2 )
446: IF ( LWORK .EQ.-1 ) THEN
447: WORK( 1 ) = DBLE( LWORKREQ )
448: RETURN
449: ELSE IF ( LWORK .LT. LWORKREQ ) THEN
450: INFO = -19
451: END IF
452: IF( INFO.NE.0 ) THEN
453: CALL XERBLA( 'ZLAQZ0', INFO )
454: RETURN
455: END IF
456: *
457: * Initialize Q and Z
458: *
459: IF( IWANTQ.EQ.3 ) CALL ZLASET( 'FULL', N, N, CZERO, CONE, Q,
460: $ LDQ )
461: IF( IWANTZ.EQ.3 ) CALL ZLASET( 'FULL', N, N, CZERO, CONE, Z,
462: $ LDZ )
463:
464: * Get machine constants
465: SAFMIN = DLAMCH( 'SAFE MINIMUM' )
466: SAFMAX = ONE/SAFMIN
467: CALL DLABAD( SAFMIN, SAFMAX )
468: ULP = DLAMCH( 'PRECISION' )
469: SMLNUM = SAFMIN*( DBLE( N )/ULP )
470:
471: BNORM = ZLANHS( 'F', IHI-ILO+1, B( ILO, ILO ), LDB, RWORK )
472: BTOL = MAX( SAFMIN, ULP*BNORM )
473:
474: ISTART = ILO
475: ISTOP = IHI
476: MAXIT = 30*( IHI-ILO+1 )
477: LD = 0
478:
479: DO IITER = 1, MAXIT
480: IF( IITER .GE. MAXIT ) THEN
481: INFO = ISTOP+1
482: GOTO 80
483: END IF
484: IF ( ISTART+1 .GE. ISTOP ) THEN
485: ISTOP = ISTART
486: EXIT
487: END IF
488:
489: * Check deflations at the end
490: IF ( ABS( A( ISTOP, ISTOP-1 ) ) .LE. MAX( SMLNUM,
491: $ ULP*( ABS( A( ISTOP, ISTOP ) )+ABS( A( ISTOP-1,
492: $ ISTOP-1 ) ) ) ) ) THEN
493: A( ISTOP, ISTOP-1 ) = CZERO
494: ISTOP = ISTOP-1
495: LD = 0
496: ESHIFT = CZERO
497: END IF
498: * Check deflations at the start
499: IF ( ABS( A( ISTART+1, ISTART ) ) .LE. MAX( SMLNUM,
500: $ ULP*( ABS( A( ISTART, ISTART ) )+ABS( A( ISTART+1,
501: $ ISTART+1 ) ) ) ) ) THEN
502: A( ISTART+1, ISTART ) = CZERO
503: ISTART = ISTART+1
504: LD = 0
505: ESHIFT = CZERO
506: END IF
507:
508: IF ( ISTART+1 .GE. ISTOP ) THEN
509: EXIT
510: END IF
511:
512: * Check interior deflations
513: ISTART2 = ISTART
514: DO K = ISTOP, ISTART+1, -1
515: IF ( ABS( A( K, K-1 ) ) .LE. MAX( SMLNUM, ULP*( ABS( A( K,
516: $ K ) )+ABS( A( K-1, K-1 ) ) ) ) ) THEN
517: A( K, K-1 ) = CZERO
518: ISTART2 = K
519: EXIT
520: END IF
521: END DO
522:
523: * Get range to apply rotations to
524: IF ( ILSCHUR ) THEN
525: ISTARTM = 1
526: ISTOPM = N
527: ELSE
528: ISTARTM = ISTART2
529: ISTOPM = ISTOP
530: END IF
531:
532: * Check infinite eigenvalues, this is done without blocking so might
533: * slow down the method when many infinite eigenvalues are present
534: K = ISTOP
535: DO WHILE ( K.GE.ISTART2 )
536:
537: IF( ABS( B( K, K ) ) .LT. BTOL ) THEN
538: * A diagonal element of B is negligable, move it
539: * to the top and deflate it
540:
541: DO K2 = K, ISTART2+1, -1
542: CALL ZLARTG( B( K2-1, K2 ), B( K2-1, K2-1 ), C1, S1,
543: $ TEMP )
544: B( K2-1, K2 ) = TEMP
545: B( K2-1, K2-1 ) = CZERO
546:
547: CALL ZROT( K2-2-ISTARTM+1, B( ISTARTM, K2 ), 1,
548: $ B( ISTARTM, K2-1 ), 1, C1, S1 )
549: CALL ZROT( MIN( K2+1, ISTOP )-ISTARTM+1, A( ISTARTM,
550: $ K2 ), 1, A( ISTARTM, K2-1 ), 1, C1, S1 )
551: IF ( ILZ ) THEN
552: CALL ZROT( N, Z( 1, K2 ), 1, Z( 1, K2-1 ), 1, C1,
553: $ S1 )
554: END IF
555:
556: IF( K2.LT.ISTOP ) THEN
557: CALL ZLARTG( A( K2, K2-1 ), A( K2+1, K2-1 ), C1,
558: $ S1, TEMP )
559: A( K2, K2-1 ) = TEMP
560: A( K2+1, K2-1 ) = CZERO
561:
562: CALL ZROT( ISTOPM-K2+1, A( K2, K2 ), LDA, A( K2+1,
563: $ K2 ), LDA, C1, S1 )
564: CALL ZROT( ISTOPM-K2+1, B( K2, K2 ), LDB, B( K2+1,
565: $ K2 ), LDB, C1, S1 )
566: IF( ILQ ) THEN
567: CALL ZROT( N, Q( 1, K2 ), 1, Q( 1, K2+1 ), 1,
568: $ C1, DCONJG( S1 ) )
569: END IF
570: END IF
571:
572: END DO
573:
574: IF( ISTART2.LT.ISTOP )THEN
575: CALL ZLARTG( A( ISTART2, ISTART2 ), A( ISTART2+1,
576: $ ISTART2 ), C1, S1, TEMP )
577: A( ISTART2, ISTART2 ) = TEMP
578: A( ISTART2+1, ISTART2 ) = CZERO
579:
580: CALL ZROT( ISTOPM-( ISTART2+1 )+1, A( ISTART2,
581: $ ISTART2+1 ), LDA, A( ISTART2+1,
582: $ ISTART2+1 ), LDA, C1, S1 )
583: CALL ZROT( ISTOPM-( ISTART2+1 )+1, B( ISTART2,
584: $ ISTART2+1 ), LDB, B( ISTART2+1,
585: $ ISTART2+1 ), LDB, C1, S1 )
586: IF( ILQ ) THEN
587: CALL ZROT( N, Q( 1, ISTART2 ), 1, Q( 1,
588: $ ISTART2+1 ), 1, C1, DCONJG( S1 ) )
589: END IF
590: END IF
591:
592: ISTART2 = ISTART2+1
593:
594: END IF
595: K = K-1
596: END DO
597:
598: * istart2 now points to the top of the bottom right
599: * unreduced Hessenberg block
600: IF ( ISTART2 .GE. ISTOP ) THEN
601: ISTOP = ISTART2-1
602: LD = 0
603: ESHIFT = CZERO
604: CYCLE
605: END IF
606:
607: NW = NWR
608: NSHIFTS = NSR
609: NBLOCK = NBR
610:
611: IF ( ISTOP-ISTART2+1 .LT. NMIN ) THEN
612: * Setting nw to the size of the subblock will make AED deflate
613: * all the eigenvalues. This is slightly more efficient than just
614: * using qz_small because the off diagonal part gets updated via BLAS.
615: IF ( ISTOP-ISTART+1 .LT. NMIN ) THEN
616: NW = ISTOP-ISTART+1
617: ISTART2 = ISTART
618: ELSE
619: NW = ISTOP-ISTART2+1
620: END IF
621: END IF
622:
623: *
624: * Time for AED
625: *
626: CALL ZLAQZ2( ILSCHUR, ILQ, ILZ, N, ISTART2, ISTOP, NW, A, LDA,
627: $ B, LDB, Q, LDQ, Z, LDZ, N_UNDEFLATED, N_DEFLATED,
628: $ ALPHA, BETA, WORK, NW, WORK( NW**2+1 ), NW,
629: $ WORK( 2*NW**2+1 ), LWORK-2*NW**2, RWORK, REC,
630: $ AED_INFO )
631:
632: IF ( N_DEFLATED > 0 ) THEN
633: ISTOP = ISTOP-N_DEFLATED
634: LD = 0
635: ESHIFT = CZERO
636: END IF
637:
638: IF ( 100*N_DEFLATED > NIBBLE*( N_DEFLATED+N_UNDEFLATED ) .OR.
639: $ ISTOP-ISTART2+1 .LT. NMIN ) THEN
640: * AED has uncovered many eigenvalues. Skip a QZ sweep and run
641: * AED again.
642: CYCLE
643: END IF
644:
645: LD = LD+1
646:
647: NS = MIN( NSHIFTS, ISTOP-ISTART2 )
648: NS = MIN( NS, N_UNDEFLATED )
649: SHIFTPOS = ISTOP-N_DEFLATED-N_UNDEFLATED+1
650:
651: IF ( MOD( LD, 6 ) .EQ. 0 ) THEN
652: *
653: * Exceptional shift. Chosen for no particularly good reason.
654: *
655: IF( ( DBLE( MAXIT )*SAFMIN )*ABS( A( ISTOP,
656: $ ISTOP-1 ) ).LT.ABS( A( ISTOP-1, ISTOP-1 ) ) ) THEN
657: ESHIFT = A( ISTOP, ISTOP-1 )/B( ISTOP-1, ISTOP-1 )
658: ELSE
659: ESHIFT = ESHIFT+CONE/( SAFMIN*DBLE( MAXIT ) )
660: END IF
661: ALPHA( SHIFTPOS ) = CONE
662: BETA( SHIFTPOS ) = ESHIFT
663: NS = 1
664: END IF
665:
666: *
667: * Time for a QZ sweep
668: *
669: CALL ZLAQZ3( ILSCHUR, ILQ, ILZ, N, ISTART2, ISTOP, NS, NBLOCK,
670: $ ALPHA( SHIFTPOS ), BETA( SHIFTPOS ), A, LDA, B,
671: $ LDB, Q, LDQ, Z, LDZ, WORK, NBLOCK, WORK( NBLOCK**
672: $ 2+1 ), NBLOCK, WORK( 2*NBLOCK**2+1 ),
673: $ LWORK-2*NBLOCK**2, SWEEP_INFO )
674:
675: END DO
676:
677: *
678: * Call ZHGEQZ to normalize the eigenvalue blocks and set the eigenvalues
679: * If all the eigenvalues have been found, ZHGEQZ will not do any iterations
680: * and only normalize the blocks. In case of a rare convergence failure,
681: * the single shift might perform better.
682: *
683: 80 CALL ZHGEQZ( WANTS, WANTQ, WANTZ, N, ILO, IHI, A, LDA, B, LDB,
684: $ ALPHA, BETA, Q, LDQ, Z, LDZ, WORK, LWORK, RWORK,
685: $ NORM_INFO )
686:
687: INFO = NORM_INFO
688:
689: END SUBROUTINE
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