1: *> \brief \b DLAQZ0
2: *
3: * =========== DOCUMENTATION ===========
4: *
5: * Online html documentation available at
6: * http://www.netlib.org/lapack/explore-html/
7: *
8: *> \htmlonly
9: *> Download DLAQZ0 + dependencies
10: *> <a href="http://www.netlib.org/cgi-bin/netlibfiles.tgz?format=tgz&filename=/lapack/lapack_routine/dlaqz0.f">
11: *> [TGZ]</a>
12: *> <a href="http://www.netlib.org/cgi-bin/netlibfiles.zip?format=zip&filename=/lapack/lapack_routine/dlaqz0.f">
13: *> [ZIP]</a>
14: *> <a href="http://www.netlib.org/cgi-bin/netlibfiles.txt?format=txt&filename=/lapack/lapack_routine/dlaqz0.f">
15: *> [TXT]</a>
16: *> \endhtmlonly
17: *
18: * Definition:
19: * ===========
20: *
21: * RECURSIVE SUBROUTINE DLAQZ0( WANTS, WANTQ, WANTZ, N, ILO, IHI, A,
22: * $ LDA, B, LDB, ALPHAR, ALPHAI, BETA,
23: * $ Q, LDQ, Z, LDZ, WORK, LWORK, REC,
24: * $ INFO )
25: * IMPLICIT NONE
26: *
27: * Arguments
28: * CHARACTER, INTENT( IN ) :: WANTS, WANTQ, WANTZ
29: * INTEGER, INTENT( IN ) :: N, ILO, IHI, LDA, LDB, LDQ, LDZ, LWORK,
30: * $ REC
31: *
32: * INTEGER, INTENT( OUT ) :: INFO
33: *
34: * DOUBLE PRECISION, INTENT( INOUT ) :: A( LDA, * ), B( LDB, * ),
35: * $ Q( LDQ, * ), Z( LDZ, * ), ALPHAR( * ),
36: * $ ALPHAI( * ), BETA( * ), WORK( * )
37: * ..
38: *
39: *
40: *> \par Purpose:
41: * =============
42: *>
43: *> \verbatim
44: *>
45: *> DLAQZ0 computes the eigenvalues of a real matrix pair (H,T),
46: *> where H is an upper Hessenberg matrix and T is upper triangular,
47: *> using the double-shift QZ method.
48: *> Matrix pairs of this type are produced by the reduction to
49: *> generalized upper Hessenberg form of a real matrix pair (A,B):
50: *>
51: *> A = Q1*H*Z1**T, B = Q1*T*Z1**T,
52: *>
53: *> as computed by DGGHRD.
54: *>
55: *> If JOB='S', then the Hessenberg-triangular pair (H,T) is
56: *> also reduced to generalized Schur form,
57: *>
58: *> H = Q*S*Z**T, T = Q*P*Z**T,
59: *>
60: *> where Q and Z are orthogonal matrices, P is an upper triangular
61: *> matrix, and S is a quasi-triangular matrix with 1-by-1 and 2-by-2
62: *> diagonal blocks.
63: *>
64: *> The 1-by-1 blocks correspond to real eigenvalues of the matrix pair
65: *> (H,T) and the 2-by-2 blocks correspond to complex conjugate pairs of
66: *> eigenvalues.
67: *>
68: *> Additionally, the 2-by-2 upper triangular diagonal blocks of P
69: *> corresponding to 2-by-2 blocks of S are reduced to positive diagonal
70: *> form, i.e., if S(j+1,j) is non-zero, then P(j+1,j) = P(j,j+1) = 0,
71: *> P(j,j) > 0, and P(j+1,j+1) > 0.
72: *>
73: *> Optionally, the orthogonal matrix Q from the generalized Schur
74: *> factorization may be postmultiplied into an input matrix Q1, and the
75: *> orthogonal matrix Z may be postmultiplied into an input matrix Z1.
76: *> If Q1 and Z1 are the orthogonal matrices from DGGHRD that reduced
77: *> the matrix pair (A,B) to generalized upper Hessenberg form, then the
78: *> output matrices Q1*Q and Z1*Z are the orthogonal factors from the
79: *> generalized Schur factorization of (A,B):
80: *>
81: *> A = (Q1*Q)*S*(Z1*Z)**T, B = (Q1*Q)*P*(Z1*Z)**T.
82: *>
83: *> To avoid overflow, eigenvalues of the matrix pair (H,T) (equivalently,
84: *> of (A,B)) are computed as a pair of values (alpha,beta), where alpha is
85: *> complex and beta real.
86: *> If beta is nonzero, lambda = alpha / beta is an eigenvalue of the
87: *> generalized nonsymmetric eigenvalue problem (GNEP)
88: *> A*x = lambda*B*x
89: *> and if alpha is nonzero, mu = beta / alpha is an eigenvalue of the
90: *> alternate form of the GNEP
91: *> mu*A*y = B*y.
92: *> Real eigenvalues can be read directly from the generalized Schur
93: *> form:
94: *> alpha = S(i,i), beta = P(i,i).
95: *>
96: *> Ref: C.B. Moler & G.W. Stewart, "An Algorithm for Generalized Matrix
97: *> Eigenvalue Problems", SIAM J. Numer. Anal., 10(1973),
98: *> pp. 241--256.
99: *>
100: *> Ref: B. Kagstrom, D. Kressner, "Multishift Variants of the QZ
101: *> Algorithm with Aggressive Early Deflation", SIAM J. Numer.
102: *> Anal., 29(2006), pp. 199--227.
103: *>
104: *> Ref: T. Steel, D. Camps, K. Meerbergen, R. Vandebril "A multishift,
105: *> multipole rational QZ method with agressive early deflation"
106: *> \endverbatim
107: *
108: * Arguments:
109: * ==========
110: *
111: *> \param[in] WANTS
112: *> \verbatim
113: *> WANTS is CHARACTER*1
114: *> = 'E': Compute eigenvalues only;
115: *> = 'S': Compute eigenvalues and the Schur form.
116: *> \endverbatim
117: *>
118: *> \param[in] WANTQ
119: *> \verbatim
120: *> WANTQ is CHARACTER*1
121: *> = 'N': Left Schur vectors (Q) are not computed;
122: *> = 'I': Q is initialized to the unit matrix and the matrix Q
123: *> of left Schur vectors of (A,B) is returned;
124: *> = 'V': Q must contain an orthogonal matrix Q1 on entry and
125: *> the product Q1*Q is returned.
126: *> \endverbatim
127: *>
128: *> \param[in] WANTZ
129: *> \verbatim
130: *> WANTZ is CHARACTER*1
131: *> = 'N': Right Schur vectors (Z) are not computed;
132: *> = 'I': Z is initialized to the unit matrix and the matrix Z
133: *> of right Schur vectors of (A,B) is returned;
134: *> = 'V': Z must contain an orthogonal matrix Z1 on entry and
135: *> the product Z1*Z is returned.
136: *> \endverbatim
137: *>
138: *> \param[in] N
139: *> \verbatim
140: *> N is INTEGER
141: *> The order of the matrices A, B, Q, and Z. N >= 0.
142: *> \endverbatim
143: *>
144: *> \param[in] ILO
145: *> \verbatim
146: *> ILO is INTEGER
147: *> \endverbatim
148: *>
149: *> \param[in] IHI
150: *> \verbatim
151: *> IHI is INTEGER
152: *> ILO and IHI mark the rows and columns of A which are in
153: *> Hessenberg form. It is assumed that A is already upper
154: *> triangular in rows and columns 1:ILO-1 and IHI+1:N.
155: *> If N > 0, 1 <= ILO <= IHI <= N; if N = 0, ILO=1 and IHI=0.
156: *> \endverbatim
157: *>
158: *> \param[in,out] A
159: *> \verbatim
160: *> A is DOUBLE PRECISION array, dimension (LDA, N)
161: *> On entry, the N-by-N upper Hessenberg matrix A.
162: *> On exit, if JOB = 'S', A contains the upper quasi-triangular
163: *> matrix S from the generalized Schur factorization.
164: *> If JOB = 'E', the diagonal blocks of A match those of S, but
165: *> the rest of A is unspecified.
166: *> \endverbatim
167: *>
168: *> \param[in] LDA
169: *> \verbatim
170: *> LDA is INTEGER
171: *> The leading dimension of the array A. LDA >= max( 1, N ).
172: *> \endverbatim
173: *>
174: *> \param[in,out] B
175: *> \verbatim
176: *> B is DOUBLE PRECISION array, dimension (LDB, N)
177: *> On entry, the N-by-N upper triangular matrix B.
178: *> On exit, if JOB = 'S', B contains the upper triangular
179: *> matrix P from the generalized Schur factorization;
180: *> 2-by-2 diagonal blocks of P corresponding to 2-by-2 blocks of S
181: *> are reduced to positive diagonal form, i.e., if A(j+1,j) is
182: *> non-zero, then B(j+1,j) = B(j,j+1) = 0, B(j,j) > 0, and
183: *> B(j+1,j+1) > 0.
184: *> If JOB = 'E', the diagonal blocks of B match those of P, but
185: *> the rest of B is unspecified.
186: *> \endverbatim
187: *>
188: *> \param[in] LDB
189: *> \verbatim
190: *> LDB is INTEGER
191: *> The leading dimension of the array B. LDB >= max( 1, N ).
192: *> \endverbatim
193: *>
194: *> \param[out] ALPHAR
195: *> \verbatim
196: *> ALPHAR is DOUBLE PRECISION array, dimension (N)
197: *> The real parts of each scalar alpha defining an eigenvalue
198: *> of GNEP.
199: *> \endverbatim
200: *>
201: *> \param[out] ALPHAI
202: *> \verbatim
203: *> ALPHAI is DOUBLE PRECISION array, dimension (N)
204: *> The imaginary parts of each scalar alpha defining an
205: *> eigenvalue of GNEP.
206: *> If ALPHAI(j) is zero, then the j-th eigenvalue is real; if
207: *> positive, then the j-th and (j+1)-st eigenvalues are a
208: *> complex conjugate pair, with ALPHAI(j+1) = -ALPHAI(j).
209: *> \endverbatim
210: *>
211: *> \param[out] BETA
212: *> \verbatim
213: *> BETA is DOUBLE PRECISION array, dimension (N)
214: *> The scalars beta that define the eigenvalues of GNEP.
215: *> Together, the quantities alpha = (ALPHAR(j),ALPHAI(j)) and
216: *> beta = BETA(j) represent the j-th eigenvalue of the matrix
217: *> pair (A,B), in one of the forms lambda = alpha/beta or
218: *> mu = beta/alpha. Since either lambda or mu may overflow,
219: *> they should not, in general, be computed.
220: *> \endverbatim
221: *>
222: *> \param[in,out] Q
223: *> \verbatim
224: *> Q is DOUBLE PRECISION array, dimension (LDQ, N)
225: *> On entry, if COMPQ = 'V', the orthogonal matrix Q1 used in
226: *> the reduction of (A,B) to generalized Hessenberg form.
227: *> On exit, if COMPQ = 'I', the orthogonal matrix of left Schur
228: *> vectors of (A,B), and if COMPQ = 'V', the orthogonal matrix
229: *> of left Schur vectors of (A,B).
230: *> Not referenced if COMPQ = 'N'.
231: *> \endverbatim
232: *>
233: *> \param[in] LDQ
234: *> \verbatim
235: *> LDQ is INTEGER
236: *> The leading dimension of the array Q. LDQ >= 1.
237: *> If COMPQ='V' or 'I', then LDQ >= N.
238: *> \endverbatim
239: *>
240: *> \param[in,out] Z
241: *> \verbatim
242: *> Z is DOUBLE PRECISION array, dimension (LDZ, N)
243: *> On entry, if COMPZ = 'V', the orthogonal matrix Z1 used in
244: *> the reduction of (A,B) to generalized Hessenberg form.
245: *> On exit, if COMPZ = 'I', the orthogonal matrix of
246: *> right Schur vectors of (H,T), and if COMPZ = 'V', the
247: *> orthogonal matrix of right Schur vectors of (A,B).
248: *> Not referenced if COMPZ = 'N'.
249: *> \endverbatim
250: *>
251: *> \param[in] LDZ
252: *> \verbatim
253: *> LDZ is INTEGER
254: *> The leading dimension of the array Z. LDZ >= 1.
255: *> If COMPZ='V' or 'I', then LDZ >= N.
256: *> \endverbatim
257: *>
258: *> \param[out] WORK
259: *> \verbatim
260: *> WORK is DOUBLE PRECISION array, dimension (MAX(1,LWORK))
261: *> On exit, if INFO >= 0, WORK(1) returns the optimal LWORK.
262: *> \endverbatim
263: *>
264: *> \param[in] LWORK
265: *> \verbatim
266: *> LWORK is INTEGER
267: *> The dimension of the array WORK. LWORK >= max(1,N).
268: *>
269: *> If LWORK = -1, then a workspace query is assumed; the routine
270: *> only calculates the optimal size of the WORK array, returns
271: *> this value as the first entry of the WORK array, and no error
272: *> message related to LWORK is issued by XERBLA.
273: *> \endverbatim
274: *>
275: *> \param[in] REC
276: *> \verbatim
277: *> REC is INTEGER
278: *> REC indicates the current recursion level. Should be set
279: *> to 0 on first call.
280: *> \endverbatim
281: *>
282: *> \param[out] INFO
283: *> \verbatim
284: *> INFO is INTEGER
285: *> = 0: successful exit
286: *> < 0: if INFO = -i, the i-th argument had an illegal value
287: *> = 1,...,N: the QZ iteration did not converge. (A,B) is not
288: *> in Schur form, but ALPHAR(i), ALPHAI(i), and
289: *> BETA(i), i=INFO+1,...,N should be correct.
290: *> \endverbatim
291: *
292: * Authors:
293: * ========
294: *
295: *> \author Thijs Steel, KU Leuven
296: *
297: *> \date May 2020
298: *
299: *> \ingroup doubleGEcomputational
300: *>
301: * =====================================================================
302: RECURSIVE SUBROUTINE DLAQZ0( WANTS, WANTQ, WANTZ, N, ILO, IHI, A,
303: $ LDA, B, LDB, ALPHAR, ALPHAI, BETA,
304: $ Q, LDQ, Z, LDZ, WORK, LWORK, REC,
305: $ INFO )
306: IMPLICIT NONE
307:
308: * Arguments
309: CHARACTER, INTENT( IN ) :: WANTS, WANTQ, WANTZ
310: INTEGER, INTENT( IN ) :: N, ILO, IHI, LDA, LDB, LDQ, LDZ, LWORK,
311: $ REC
312:
313: INTEGER, INTENT( OUT ) :: INFO
314:
315: DOUBLE PRECISION, INTENT( INOUT ) :: A( LDA, * ), B( LDB, * ),
316: $ Q( LDQ, * ), Z( LDZ, * ), ALPHAR( * ),
317: $ ALPHAI( * ), BETA( * ), WORK( * )
318:
319: * Parameters
320: DOUBLE PRECISION :: ZERO, ONE, HALF
321: PARAMETER( ZERO = 0.0D0, ONE = 1.0D0, HALF = 0.5D0 )
322:
323: * Local scalars
324: DOUBLE PRECISION :: SMLNUM, ULP, ESHIFT, SAFMIN, SAFMAX, C1, S1,
325: $ TEMP, SWAP, BNORM, BTOL
326: INTEGER :: ISTART, ISTOP, IITER, MAXIT, ISTART2, K, LD, NSHIFTS,
327: $ NBLOCK, NW, NMIN, NIBBLE, N_UNDEFLATED, N_DEFLATED,
328: $ NS, SWEEP_INFO, SHIFTPOS, LWORKREQ, K2, ISTARTM,
329: $ ISTOPM, IWANTS, IWANTQ, IWANTZ, NORM_INFO, AED_INFO,
330: $ NWR, NBR, NSR, ITEMP1, ITEMP2, RCOST, I
331: LOGICAL :: ILSCHUR, ILQ, ILZ
332: CHARACTER :: JBCMPZ*3
333:
334: * External Functions
335: EXTERNAL :: XERBLA, DHGEQZ, DLASET, DLAQZ3, DLAQZ4, DLABAD,
336: $ DLARTG, DROT
337: DOUBLE PRECISION, EXTERNAL :: DLAMCH, DLANHS
338: LOGICAL, EXTERNAL :: LSAME
339: INTEGER, EXTERNAL :: ILAENV
340:
341: *
342: * Decode wantS,wantQ,wantZ
343: *
344: IF( LSAME( WANTS, 'E' ) ) THEN
345: ILSCHUR = .FALSE.
346: IWANTS = 1
347: ELSE IF( LSAME( WANTS, 'S' ) ) THEN
348: ILSCHUR = .TRUE.
349: IWANTS = 2
350: ELSE
351: IWANTS = 0
352: END IF
353:
354: IF( LSAME( WANTQ, 'N' ) ) THEN
355: ILQ = .FALSE.
356: IWANTQ = 1
357: ELSE IF( LSAME( WANTQ, 'V' ) ) THEN
358: ILQ = .TRUE.
359: IWANTQ = 2
360: ELSE IF( LSAME( WANTQ, 'I' ) ) THEN
361: ILQ = .TRUE.
362: IWANTQ = 3
363: ELSE
364: IWANTQ = 0
365: END IF
366:
367: IF( LSAME( WANTZ, 'N' ) ) THEN
368: ILZ = .FALSE.
369: IWANTZ = 1
370: ELSE IF( LSAME( WANTZ, 'V' ) ) THEN
371: ILZ = .TRUE.
372: IWANTZ = 2
373: ELSE IF( LSAME( WANTZ, 'I' ) ) THEN
374: ILZ = .TRUE.
375: IWANTZ = 3
376: ELSE
377: IWANTZ = 0
378: END IF
379: *
380: * Check Argument Values
381: *
382: INFO = 0
383: IF( IWANTS.EQ.0 ) THEN
384: INFO = -1
385: ELSE IF( IWANTQ.EQ.0 ) THEN
386: INFO = -2
387: ELSE IF( IWANTZ.EQ.0 ) THEN
388: INFO = -3
389: ELSE IF( N.LT.0 ) THEN
390: INFO = -4
391: ELSE IF( ILO.LT.1 ) THEN
392: INFO = -5
393: ELSE IF( IHI.GT.N .OR. IHI.LT.ILO-1 ) THEN
394: INFO = -6
395: ELSE IF( LDA.LT.N ) THEN
396: INFO = -8
397: ELSE IF( LDB.LT.N ) THEN
398: INFO = -10
399: ELSE IF( LDQ.LT.1 .OR. ( ILQ .AND. LDQ.LT.N ) ) THEN
400: INFO = -15
401: ELSE IF( LDZ.LT.1 .OR. ( ILZ .AND. LDZ.LT.N ) ) THEN
402: INFO = -17
403: END IF
404: IF( INFO.NE.0 ) THEN
405: CALL XERBLA( 'DLAQZ0', -INFO )
406: RETURN
407: END IF
408:
409: *
410: * Quick return if possible
411: *
412: IF( N.LE.0 ) THEN
413: WORK( 1 ) = DBLE( 1 )
414: RETURN
415: END IF
416:
417: *
418: * Get the parameters
419: *
420: JBCMPZ( 1:1 ) = WANTS
421: JBCMPZ( 2:2 ) = WANTQ
422: JBCMPZ( 3:3 ) = WANTZ
423:
424: NMIN = ILAENV( 12, 'DLAQZ0', JBCMPZ, N, ILO, IHI, LWORK )
425:
426: NWR = ILAENV( 13, 'DLAQZ0', JBCMPZ, N, ILO, IHI, LWORK )
427: NWR = MAX( 2, NWR )
428: NWR = MIN( IHI-ILO+1, ( N-1 ) / 3, NWR )
429:
430: NIBBLE = ILAENV( 14, 'DLAQZ0', JBCMPZ, N, ILO, IHI, LWORK )
431:
432: NSR = ILAENV( 15, 'DLAQZ0', JBCMPZ, N, ILO, IHI, LWORK )
433: NSR = MIN( NSR, ( N+6 ) / 9, IHI-ILO )
434: NSR = MAX( 2, NSR-MOD( NSR, 2 ) )
435:
436: RCOST = ILAENV( 17, 'DLAQZ0', JBCMPZ, N, ILO, IHI, LWORK )
437: ITEMP1 = INT( NSR/SQRT( 1+2*NSR/( DBLE( RCOST )/100*N ) ) )
438: ITEMP1 = ( ( ITEMP1-1 )/4 )*4+4
439: NBR = NSR+ITEMP1
440:
441: IF( N .LT. NMIN .OR. REC .GE. 2 ) THEN
442: CALL DHGEQZ( WANTS, WANTQ, WANTZ, N, ILO, IHI, A, LDA, B, LDB,
443: $ ALPHAR, ALPHAI, BETA, Q, LDQ, Z, LDZ, WORK,
444: $ LWORK, INFO )
445: RETURN
446: END IF
447:
448: *
449: * Find out required workspace
450: *
451:
452: * Workspace query to dlaqz3
453: NW = MAX( NWR, NMIN )
454: CALL DLAQZ3( ILSCHUR, ILQ, ILZ, N, ILO, IHI, NW, A, LDA, B, LDB,
455: $ Q, LDQ, Z, LDZ, N_UNDEFLATED, N_DEFLATED, ALPHAR,
456: $ ALPHAI, BETA, WORK, NW, WORK, NW, WORK, -1, REC,
457: $ AED_INFO )
458: ITEMP1 = INT( WORK( 1 ) )
459: * Workspace query to dlaqz4
460: CALL DLAQZ4( ILSCHUR, ILQ, ILZ, N, ILO, IHI, NSR, NBR, ALPHAR,
461: $ ALPHAI, BETA, A, LDA, B, LDB, Q, LDQ, Z, LDZ, WORK,
462: $ NBR, WORK, NBR, WORK, -1, SWEEP_INFO )
463: ITEMP2 = INT( WORK( 1 ) )
464:
465: LWORKREQ = MAX( ITEMP1+2*NW**2, ITEMP2+2*NBR**2 )
466: IF ( LWORK .EQ.-1 ) THEN
467: WORK( 1 ) = DBLE( LWORKREQ )
468: RETURN
469: ELSE IF ( LWORK .LT. LWORKREQ ) THEN
470: INFO = -19
471: END IF
472: IF( INFO.NE.0 ) THEN
473: CALL XERBLA( 'DLAQZ0', INFO )
474: RETURN
475: END IF
476: *
477: * Initialize Q and Z
478: *
479: IF( IWANTQ.EQ.3 ) CALL DLASET( 'FULL', N, N, ZERO, ONE, Q, LDQ )
480: IF( IWANTZ.EQ.3 ) CALL DLASET( 'FULL', N, N, ZERO, ONE, Z, LDZ )
481:
482: * Get machine constants
483: SAFMIN = DLAMCH( 'SAFE MINIMUM' )
484: SAFMAX = ONE/SAFMIN
485: CALL DLABAD( SAFMIN, SAFMAX )
486: ULP = DLAMCH( 'PRECISION' )
487: SMLNUM = SAFMIN*( DBLE( N )/ULP )
488:
489: BNORM = DLANHS( 'F', IHI-ILO+1, B( ILO, ILO ), LDB, WORK )
490: BTOL = MAX( SAFMIN, ULP*BNORM )
491:
492: ISTART = ILO
493: ISTOP = IHI
494: MAXIT = 3*( IHI-ILO+1 )
495: LD = 0
496:
497: DO IITER = 1, MAXIT
498: IF( IITER .GE. MAXIT ) THEN
499: INFO = ISTOP+1
500: GOTO 80
501: END IF
502: IF ( ISTART+1 .GE. ISTOP ) THEN
503: ISTOP = ISTART
504: EXIT
505: END IF
506:
507: * Check deflations at the end
508: IF ( ABS( A( ISTOP-1, ISTOP-2 ) ) .LE. MAX( SMLNUM,
509: $ ULP*( ABS( A( ISTOP-1, ISTOP-1 ) )+ABS( A( ISTOP-2,
510: $ ISTOP-2 ) ) ) ) ) THEN
511: A( ISTOP-1, ISTOP-2 ) = ZERO
512: ISTOP = ISTOP-2
513: LD = 0
514: ESHIFT = ZERO
515: ELSE IF ( ABS( A( ISTOP, ISTOP-1 ) ) .LE. MAX( SMLNUM,
516: $ ULP*( ABS( A( ISTOP, ISTOP ) )+ABS( A( ISTOP-1,
517: $ ISTOP-1 ) ) ) ) ) THEN
518: A( ISTOP, ISTOP-1 ) = ZERO
519: ISTOP = ISTOP-1
520: LD = 0
521: ESHIFT = ZERO
522: END IF
523: * Check deflations at the start
524: IF ( ABS( A( ISTART+2, ISTART+1 ) ) .LE. MAX( SMLNUM,
525: $ ULP*( ABS( A( ISTART+1, ISTART+1 ) )+ABS( A( ISTART+2,
526: $ ISTART+2 ) ) ) ) ) THEN
527: A( ISTART+2, ISTART+1 ) = ZERO
528: ISTART = ISTART+2
529: LD = 0
530: ESHIFT = ZERO
531: ELSE IF ( ABS( A( ISTART+1, ISTART ) ) .LE. MAX( SMLNUM,
532: $ ULP*( ABS( A( ISTART, ISTART ) )+ABS( A( ISTART+1,
533: $ ISTART+1 ) ) ) ) ) THEN
534: A( ISTART+1, ISTART ) = ZERO
535: ISTART = ISTART+1
536: LD = 0
537: ESHIFT = ZERO
538: END IF
539:
540: IF ( ISTART+1 .GE. ISTOP ) THEN
541: EXIT
542: END IF
543:
544: * Check interior deflations
545: ISTART2 = ISTART
546: DO K = ISTOP, ISTART+1, -1
547: IF ( ABS( A( K, K-1 ) ) .LE. MAX( SMLNUM, ULP*( ABS( A( K,
548: $ K ) )+ABS( A( K-1, K-1 ) ) ) ) ) THEN
549: A( K, K-1 ) = ZERO
550: ISTART2 = K
551: EXIT
552: END IF
553: END DO
554:
555: * Get range to apply rotations to
556: IF ( ILSCHUR ) THEN
557: ISTARTM = 1
558: ISTOPM = N
559: ELSE
560: ISTARTM = ISTART2
561: ISTOPM = ISTOP
562: END IF
563:
564: * Check infinite eigenvalues, this is done without blocking so might
565: * slow down the method when many infinite eigenvalues are present
566: K = ISTOP
567: DO WHILE ( K.GE.ISTART2 )
568:
569: IF( ABS( B( K, K ) ) .LT. BTOL ) THEN
570: * A diagonal element of B is negligable, move it
571: * to the top and deflate it
572:
573: DO K2 = K, ISTART2+1, -1
574: CALL DLARTG( B( K2-1, K2 ), B( K2-1, K2-1 ), C1, S1,
575: $ TEMP )
576: B( K2-1, K2 ) = TEMP
577: B( K2-1, K2-1 ) = ZERO
578:
579: CALL DROT( K2-2-ISTARTM+1, B( ISTARTM, K2 ), 1,
580: $ B( ISTARTM, K2-1 ), 1, C1, S1 )
581: CALL DROT( MIN( K2+1, ISTOP )-ISTARTM+1, A( ISTARTM,
582: $ K2 ), 1, A( ISTARTM, K2-1 ), 1, C1, S1 )
583: IF ( ILZ ) THEN
584: CALL DROT( N, Z( 1, K2 ), 1, Z( 1, K2-1 ), 1, C1,
585: $ S1 )
586: END IF
587:
588: IF( K2.LT.ISTOP ) THEN
589: CALL DLARTG( A( K2, K2-1 ), A( K2+1, K2-1 ), C1,
590: $ S1, TEMP )
591: A( K2, K2-1 ) = TEMP
592: A( K2+1, K2-1 ) = ZERO
593:
594: CALL DROT( ISTOPM-K2+1, A( K2, K2 ), LDA, A( K2+1,
595: $ K2 ), LDA, C1, S1 )
596: CALL DROT( ISTOPM-K2+1, B( K2, K2 ), LDB, B( K2+1,
597: $ K2 ), LDB, C1, S1 )
598: IF( ILQ ) THEN
599: CALL DROT( N, Q( 1, K2 ), 1, Q( 1, K2+1 ), 1,
600: $ C1, S1 )
601: END IF
602: END IF
603:
604: END DO
605:
606: IF( ISTART2.LT.ISTOP )THEN
607: CALL DLARTG( A( ISTART2, ISTART2 ), A( ISTART2+1,
608: $ ISTART2 ), C1, S1, TEMP )
609: A( ISTART2, ISTART2 ) = TEMP
610: A( ISTART2+1, ISTART2 ) = ZERO
611:
612: CALL DROT( ISTOPM-( ISTART2+1 )+1, A( ISTART2,
613: $ ISTART2+1 ), LDA, A( ISTART2+1,
614: $ ISTART2+1 ), LDA, C1, S1 )
615: CALL DROT( ISTOPM-( ISTART2+1 )+1, B( ISTART2,
616: $ ISTART2+1 ), LDB, B( ISTART2+1,
617: $ ISTART2+1 ), LDB, C1, S1 )
618: IF( ILQ ) THEN
619: CALL DROT( N, Q( 1, ISTART2 ), 1, Q( 1,
620: $ ISTART2+1 ), 1, C1, S1 )
621: END IF
622: END IF
623:
624: ISTART2 = ISTART2+1
625:
626: END IF
627: K = K-1
628: END DO
629:
630: * istart2 now points to the top of the bottom right
631: * unreduced Hessenberg block
632: IF ( ISTART2 .GE. ISTOP ) THEN
633: ISTOP = ISTART2-1
634: LD = 0
635: ESHIFT = ZERO
636: CYCLE
637: END IF
638:
639: NW = NWR
640: NSHIFTS = NSR
641: NBLOCK = NBR
642:
643: IF ( ISTOP-ISTART2+1 .LT. NMIN ) THEN
644: * Setting nw to the size of the subblock will make AED deflate
645: * all the eigenvalues. This is slightly more efficient than just
646: * using DHGEQZ because the off diagonal part gets updated via BLAS.
647: IF ( ISTOP-ISTART+1 .LT. NMIN ) THEN
648: NW = ISTOP-ISTART+1
649: ISTART2 = ISTART
650: ELSE
651: NW = ISTOP-ISTART2+1
652: END IF
653: END IF
654:
655: *
656: * Time for AED
657: *
658: CALL DLAQZ3( ILSCHUR, ILQ, ILZ, N, ISTART2, ISTOP, NW, A, LDA,
659: $ B, LDB, Q, LDQ, Z, LDZ, N_UNDEFLATED, N_DEFLATED,
660: $ ALPHAR, ALPHAI, BETA, WORK, NW, WORK( NW**2+1 ),
661: $ NW, WORK( 2*NW**2+1 ), LWORK-2*NW**2, REC,
662: $ AED_INFO )
663:
664: IF ( N_DEFLATED > 0 ) THEN
665: ISTOP = ISTOP-N_DEFLATED
666: LD = 0
667: ESHIFT = ZERO
668: END IF
669:
670: IF ( 100*N_DEFLATED > NIBBLE*( N_DEFLATED+N_UNDEFLATED ) .OR.
671: $ ISTOP-ISTART2+1 .LT. NMIN ) THEN
672: * AED has uncovered many eigenvalues. Skip a QZ sweep and run
673: * AED again.
674: CYCLE
675: END IF
676:
677: LD = LD+1
678:
679: NS = MIN( NSHIFTS, ISTOP-ISTART2 )
680: NS = MIN( NS, N_UNDEFLATED )
681: SHIFTPOS = ISTOP-N_DEFLATED-N_UNDEFLATED+1
682: *
683: * Shuffle shifts to put double shifts in front
684: * This ensures that we don't split up a double shift
685: *
686: DO I = SHIFTPOS, SHIFTPOS+N_UNDEFLATED-1, 2
687: IF( ALPHAI( I ).NE.-ALPHAI( I+1 ) ) THEN
688: *
689: SWAP = ALPHAR( I )
690: ALPHAR( I ) = ALPHAR( I+1 )
691: ALPHAR( I+1 ) = ALPHAR( I+2 )
692: ALPHAR( I+2 ) = SWAP
693:
694: SWAP = ALPHAI( I )
695: ALPHAI( I ) = ALPHAI( I+1 )
696: ALPHAI( I+1 ) = ALPHAI( I+2 )
697: ALPHAI( I+2 ) = SWAP
698:
699: SWAP = BETA( I )
700: BETA( I ) = BETA( I+1 )
701: BETA( I+1 ) = BETA( I+2 )
702: BETA( I+2 ) = SWAP
703: END IF
704: END DO
705:
706: IF ( MOD( LD, 6 ) .EQ. 0 ) THEN
707: *
708: * Exceptional shift. Chosen for no particularly good reason.
709: *
710: IF( ( DBLE( MAXIT )*SAFMIN )*ABS( A( ISTOP,
711: $ ISTOP-1 ) ).LT.ABS( A( ISTOP-1, ISTOP-1 ) ) ) THEN
712: ESHIFT = A( ISTOP, ISTOP-1 )/B( ISTOP-1, ISTOP-1 )
713: ELSE
714: ESHIFT = ESHIFT+ONE/( SAFMIN*DBLE( MAXIT ) )
715: END IF
716: ALPHAR( SHIFTPOS ) = ONE
717: ALPHAR( SHIFTPOS+1 ) = ZERO
718: ALPHAI( SHIFTPOS ) = ZERO
719: ALPHAI( SHIFTPOS+1 ) = ZERO
720: BETA( SHIFTPOS ) = ESHIFT
721: BETA( SHIFTPOS+1 ) = ESHIFT
722: NS = 2
723: END IF
724:
725: *
726: * Time for a QZ sweep
727: *
728: CALL DLAQZ4( ILSCHUR, ILQ, ILZ, N, ISTART2, ISTOP, NS, NBLOCK,
729: $ ALPHAR( SHIFTPOS ), ALPHAI( SHIFTPOS ),
730: $ BETA( SHIFTPOS ), A, LDA, B, LDB, Q, LDQ, Z, LDZ,
731: $ WORK, NBLOCK, WORK( NBLOCK**2+1 ), NBLOCK,
732: $ WORK( 2*NBLOCK**2+1 ), LWORK-2*NBLOCK**2,
733: $ SWEEP_INFO )
734:
735: END DO
736:
737: *
738: * Call DHGEQZ to normalize the eigenvalue blocks and set the eigenvalues
739: * If all the eigenvalues have been found, DHGEQZ will not do any iterations
740: * and only normalize the blocks. In case of a rare convergence failure,
741: * the single shift might perform better.
742: *
743: 80 CALL DHGEQZ( WANTS, WANTQ, WANTZ, N, ILO, IHI, A, LDA, B, LDB,
744: $ ALPHAR, ALPHAI, BETA, Q, LDQ, Z, LDZ, WORK, LWORK,
745: $ NORM_INFO )
746:
747: INFO = NORM_INFO
748:
749: END SUBROUTINE
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