Annotation of rpl/lapack/lapack/zhgeqz.f, revision 1.19
1.8 bertrand 1: *> \brief \b ZHGEQZ
2: *
3: * =========== DOCUMENTATION ===========
4: *
1.17 bertrand 5: * Online html documentation available at
6: * http://www.netlib.org/lapack/explore-html/
1.8 bertrand 7: *
8: *> \htmlonly
1.17 bertrand 9: *> Download ZHGEQZ + dependencies
10: *> <a href="http://www.netlib.org/cgi-bin/netlibfiles.tgz?format=tgz&filename=/lapack/lapack_routine/zhgeqz.f">
11: *> [TGZ]</a>
12: *> <a href="http://www.netlib.org/cgi-bin/netlibfiles.zip?format=zip&filename=/lapack/lapack_routine/zhgeqz.f">
13: *> [ZIP]</a>
14: *> <a href="http://www.netlib.org/cgi-bin/netlibfiles.txt?format=txt&filename=/lapack/lapack_routine/zhgeqz.f">
1.8 bertrand 15: *> [TXT]</a>
1.17 bertrand 16: *> \endhtmlonly
1.8 bertrand 17: *
18: * Definition:
19: * ===========
20: *
21: * SUBROUTINE ZHGEQZ( JOB, COMPQ, COMPZ, N, ILO, IHI, H, LDH, T, LDT,
22: * ALPHA, BETA, Q, LDQ, Z, LDZ, WORK, LWORK,
23: * RWORK, INFO )
1.17 bertrand 24: *
1.8 bertrand 25: * .. Scalar Arguments ..
26: * CHARACTER COMPQ, COMPZ, JOB
27: * INTEGER IHI, ILO, INFO, LDH, LDQ, LDT, LDZ, LWORK, N
28: * ..
29: * .. Array Arguments ..
30: * DOUBLE PRECISION RWORK( * )
31: * COMPLEX*16 ALPHA( * ), BETA( * ), H( LDH, * ),
32: * $ Q( LDQ, * ), T( LDT, * ), WORK( * ),
33: * $ Z( LDZ, * )
34: * ..
1.17 bertrand 35: *
1.8 bertrand 36: *
37: *> \par Purpose:
38: * =============
39: *>
40: *> \verbatim
41: *>
42: *> ZHGEQZ computes the eigenvalues of a complex matrix pair (H,T),
43: *> where H is an upper Hessenberg matrix and T is upper triangular,
44: *> using the single-shift QZ method.
45: *> Matrix pairs of this type are produced by the reduction to
46: *> generalized upper Hessenberg form of a complex matrix pair (A,B):
1.17 bertrand 47: *>
1.8 bertrand 48: *> A = Q1*H*Z1**H, B = Q1*T*Z1**H,
1.17 bertrand 49: *>
1.8 bertrand 50: *> as computed by ZGGHRD.
1.17 bertrand 51: *>
1.8 bertrand 52: *> If JOB='S', then the Hessenberg-triangular pair (H,T) is
53: *> also reduced to generalized Schur form,
1.17 bertrand 54: *>
1.8 bertrand 55: *> H = Q*S*Z**H, T = Q*P*Z**H,
1.17 bertrand 56: *>
1.8 bertrand 57: *> where Q and Z are unitary matrices and S and P are upper triangular.
1.17 bertrand 58: *>
1.8 bertrand 59: *> Optionally, the unitary matrix Q from the generalized Schur
60: *> factorization may be postmultiplied into an input matrix Q1, and the
61: *> unitary matrix Z may be postmultiplied into an input matrix Z1.
62: *> If Q1 and Z1 are the unitary matrices from ZGGHRD that reduced
63: *> the matrix pair (A,B) to generalized Hessenberg form, then the output
64: *> matrices Q1*Q and Z1*Z are the unitary factors from the generalized
65: *> Schur factorization of (A,B):
1.17 bertrand 66: *>
1.8 bertrand 67: *> A = (Q1*Q)*S*(Z1*Z)**H, B = (Q1*Q)*P*(Z1*Z)**H.
1.17 bertrand 68: *>
1.8 bertrand 69: *> To avoid overflow, eigenvalues of the matrix pair (H,T)
70: *> (equivalently, of (A,B)) are computed as a pair of complex values
71: *> (alpha,beta). If beta is nonzero, lambda = alpha / beta is an
72: *> eigenvalue of the generalized nonsymmetric eigenvalue problem (GNEP)
73: *> A*x = lambda*B*x
74: *> and if alpha is nonzero, mu = beta / alpha is an eigenvalue of the
75: *> alternate form of the GNEP
76: *> mu*A*y = B*y.
77: *> The values of alpha and beta for the i-th eigenvalue can be read
78: *> directly from the generalized Schur form: alpha = S(i,i),
79: *> beta = P(i,i).
80: *>
81: *> Ref: C.B. Moler & G.W. Stewart, "An Algorithm for Generalized Matrix
82: *> Eigenvalue Problems", SIAM J. Numer. Anal., 10(1973),
83: *> pp. 241--256.
84: *> \endverbatim
85: *
86: * Arguments:
87: * ==========
88: *
89: *> \param[in] JOB
90: *> \verbatim
91: *> JOB is CHARACTER*1
92: *> = 'E': Compute eigenvalues only;
93: *> = 'S': Computer eigenvalues and the Schur form.
94: *> \endverbatim
95: *>
96: *> \param[in] COMPQ
97: *> \verbatim
98: *> COMPQ is CHARACTER*1
99: *> = 'N': Left Schur vectors (Q) are not computed;
100: *> = 'I': Q is initialized to the unit matrix and the matrix Q
101: *> of left Schur vectors of (H,T) is returned;
102: *> = 'V': Q must contain a unitary matrix Q1 on entry and
103: *> the product Q1*Q is returned.
104: *> \endverbatim
105: *>
106: *> \param[in] COMPZ
107: *> \verbatim
108: *> COMPZ is CHARACTER*1
109: *> = 'N': Right Schur vectors (Z) are not computed;
110: *> = 'I': Q is initialized to the unit matrix and the matrix Z
111: *> of right Schur vectors of (H,T) is returned;
112: *> = 'V': Z must contain a unitary matrix Z1 on entry and
113: *> the product Z1*Z is returned.
114: *> \endverbatim
115: *>
116: *> \param[in] N
117: *> \verbatim
118: *> N is INTEGER
119: *> The order of the matrices H, T, Q, and Z. N >= 0.
120: *> \endverbatim
121: *>
122: *> \param[in] ILO
123: *> \verbatim
124: *> ILO is INTEGER
125: *> \endverbatim
126: *>
127: *> \param[in] IHI
128: *> \verbatim
129: *> IHI is INTEGER
130: *> ILO and IHI mark the rows and columns of H which are in
131: *> Hessenberg form. It is assumed that A is already upper
132: *> triangular in rows and columns 1:ILO-1 and IHI+1:N.
133: *> If N > 0, 1 <= ILO <= IHI <= N; if N = 0, ILO=1 and IHI=0.
134: *> \endverbatim
135: *>
136: *> \param[in,out] H
137: *> \verbatim
138: *> H is COMPLEX*16 array, dimension (LDH, N)
139: *> On entry, the N-by-N upper Hessenberg matrix H.
140: *> On exit, if JOB = 'S', H contains the upper triangular
141: *> matrix S from the generalized Schur factorization.
142: *> If JOB = 'E', the diagonal of H matches that of S, but
143: *> the rest of H is unspecified.
144: *> \endverbatim
145: *>
146: *> \param[in] LDH
147: *> \verbatim
148: *> LDH is INTEGER
149: *> The leading dimension of the array H. LDH >= max( 1, N ).
150: *> \endverbatim
151: *>
152: *> \param[in,out] T
153: *> \verbatim
154: *> T is COMPLEX*16 array, dimension (LDT, N)
155: *> On entry, the N-by-N upper triangular matrix T.
156: *> On exit, if JOB = 'S', T contains the upper triangular
157: *> matrix P from the generalized Schur factorization.
158: *> If JOB = 'E', the diagonal of T matches that of P, but
159: *> the rest of T is unspecified.
160: *> \endverbatim
161: *>
162: *> \param[in] LDT
163: *> \verbatim
164: *> LDT is INTEGER
165: *> The leading dimension of the array T. LDT >= max( 1, N ).
166: *> \endverbatim
167: *>
168: *> \param[out] ALPHA
169: *> \verbatim
170: *> ALPHA is COMPLEX*16 array, dimension (N)
171: *> The complex scalars alpha that define the eigenvalues of
172: *> GNEP. ALPHA(i) = S(i,i) in the generalized Schur
173: *> factorization.
174: *> \endverbatim
175: *>
176: *> \param[out] BETA
177: *> \verbatim
178: *> BETA is COMPLEX*16 array, dimension (N)
179: *> The real non-negative scalars beta that define the
180: *> eigenvalues of GNEP. BETA(i) = P(i,i) in the generalized
181: *> Schur factorization.
182: *>
183: *> Together, the quantities alpha = ALPHA(j) and beta = BETA(j)
184: *> represent the j-th eigenvalue of the matrix pair (A,B), in
185: *> one of the forms lambda = alpha/beta or mu = beta/alpha.
186: *> Since either lambda or mu may overflow, they should not,
187: *> in general, be computed.
188: *> \endverbatim
189: *>
190: *> \param[in,out] Q
191: *> \verbatim
192: *> Q is COMPLEX*16 array, dimension (LDQ, N)
1.15 bertrand 193: *> On entry, if COMPQ = 'V', the unitary matrix Q1 used in the
1.8 bertrand 194: *> reduction of (A,B) to generalized Hessenberg form.
1.15 bertrand 195: *> On exit, if COMPQ = 'I', the unitary matrix of left Schur
196: *> vectors of (H,T), and if COMPQ = 'V', the unitary matrix of
1.8 bertrand 197: *> left Schur vectors of (A,B).
1.15 bertrand 198: *> Not referenced if COMPQ = 'N'.
1.8 bertrand 199: *> \endverbatim
200: *>
201: *> \param[in] LDQ
202: *> \verbatim
203: *> LDQ is INTEGER
204: *> The leading dimension of the array Q. LDQ >= 1.
205: *> If COMPQ='V' or 'I', then LDQ >= N.
206: *> \endverbatim
207: *>
208: *> \param[in,out] Z
209: *> \verbatim
210: *> Z is COMPLEX*16 array, dimension (LDZ, N)
211: *> On entry, if COMPZ = 'V', the unitary matrix Z1 used in the
212: *> reduction of (A,B) to generalized Hessenberg form.
213: *> On exit, if COMPZ = 'I', the unitary matrix of right Schur
214: *> vectors of (H,T), and if COMPZ = 'V', the unitary matrix of
215: *> right Schur vectors of (A,B).
216: *> Not referenced if COMPZ = 'N'.
217: *> \endverbatim
218: *>
219: *> \param[in] LDZ
220: *> \verbatim
221: *> LDZ is INTEGER
222: *> The leading dimension of the array Z. LDZ >= 1.
223: *> If COMPZ='V' or 'I', then LDZ >= N.
224: *> \endverbatim
225: *>
226: *> \param[out] WORK
227: *> \verbatim
228: *> WORK is COMPLEX*16 array, dimension (MAX(1,LWORK))
229: *> On exit, if INFO >= 0, WORK(1) returns the optimal LWORK.
230: *> \endverbatim
231: *>
232: *> \param[in] LWORK
233: *> \verbatim
234: *> LWORK is INTEGER
235: *> The dimension of the array WORK. LWORK >= max(1,N).
236: *>
237: *> If LWORK = -1, then a workspace query is assumed; the routine
238: *> only calculates the optimal size of the WORK array, returns
239: *> this value as the first entry of the WORK array, and no error
240: *> message related to LWORK is issued by XERBLA.
241: *> \endverbatim
242: *>
243: *> \param[out] RWORK
244: *> \verbatim
245: *> RWORK is DOUBLE PRECISION array, dimension (N)
246: *> \endverbatim
247: *>
248: *> \param[out] INFO
249: *> \verbatim
250: *> INFO is INTEGER
251: *> = 0: successful exit
252: *> < 0: if INFO = -i, the i-th argument had an illegal value
253: *> = 1,...,N: the QZ iteration did not converge. (H,T) is not
254: *> in Schur form, but ALPHA(i) and BETA(i),
255: *> i=INFO+1,...,N should be correct.
256: *> = N+1,...,2*N: the shift calculation failed. (H,T) is not
257: *> in Schur form, but ALPHA(i) and BETA(i),
258: *> i=INFO-N+1,...,N should be correct.
259: *> \endverbatim
260: *
261: * Authors:
262: * ========
263: *
1.17 bertrand 264: *> \author Univ. of Tennessee
265: *> \author Univ. of California Berkeley
266: *> \author Univ. of Colorado Denver
267: *> \author NAG Ltd.
1.8 bertrand 268: *
1.10 bertrand 269: *> \date April 2012
1.8 bertrand 270: *
271: *> \ingroup complex16GEcomputational
272: *
273: *> \par Further Details:
274: * =====================
275: *>
276: *> \verbatim
277: *>
278: *> We assume that complex ABS works as long as its value is less than
279: *> overflow.
280: *> \endverbatim
281: *>
282: * =====================================================================
1.1 bertrand 283: SUBROUTINE ZHGEQZ( JOB, COMPQ, COMPZ, N, ILO, IHI, H, LDH, T, LDT,
284: $ ALPHA, BETA, Q, LDQ, Z, LDZ, WORK, LWORK,
285: $ RWORK, INFO )
286: *
1.17 bertrand 287: * -- LAPACK computational routine (version 3.7.0) --
1.1 bertrand 288: * -- LAPACK is a software package provided by Univ. of Tennessee, --
289: * -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
1.10 bertrand 290: * April 2012
1.1 bertrand 291: *
292: * .. Scalar Arguments ..
293: CHARACTER COMPQ, COMPZ, JOB
294: INTEGER IHI, ILO, INFO, LDH, LDQ, LDT, LDZ, LWORK, N
295: * ..
296: * .. Array Arguments ..
297: DOUBLE PRECISION RWORK( * )
298: COMPLEX*16 ALPHA( * ), BETA( * ), H( LDH, * ),
299: $ Q( LDQ, * ), T( LDT, * ), WORK( * ),
300: $ Z( LDZ, * )
301: * ..
302: *
303: * =====================================================================
304: *
305: * .. Parameters ..
306: COMPLEX*16 CZERO, CONE
307: PARAMETER ( CZERO = ( 0.0D+0, 0.0D+0 ),
308: $ CONE = ( 1.0D+0, 0.0D+0 ) )
309: DOUBLE PRECISION ZERO, ONE
310: PARAMETER ( ZERO = 0.0D+0, ONE = 1.0D+0 )
311: DOUBLE PRECISION HALF
312: PARAMETER ( HALF = 0.5D+0 )
313: * ..
314: * .. Local Scalars ..
315: LOGICAL ILAZR2, ILAZRO, ILQ, ILSCHR, ILZ, LQUERY
316: INTEGER ICOMPQ, ICOMPZ, IFIRST, IFRSTM, IITER, ILAST,
317: $ ILASTM, IN, ISCHUR, ISTART, J, JC, JCH, JITER,
318: $ JR, MAXIT
319: DOUBLE PRECISION ABSB, ANORM, ASCALE, ATOL, BNORM, BSCALE, BTOL,
320: $ C, SAFMIN, TEMP, TEMP2, TEMPR, ULP
321: COMPLEX*16 ABI22, AD11, AD12, AD21, AD22, CTEMP, CTEMP2,
322: $ CTEMP3, ESHIFT, RTDISC, S, SHIFT, SIGNBC, T1,
323: $ U12, X
324: * ..
325: * .. External Functions ..
326: LOGICAL LSAME
327: DOUBLE PRECISION DLAMCH, ZLANHS
328: EXTERNAL LSAME, DLAMCH, ZLANHS
329: * ..
330: * .. External Subroutines ..
331: EXTERNAL XERBLA, ZLARTG, ZLASET, ZROT, ZSCAL
332: * ..
333: * .. Intrinsic Functions ..
334: INTRINSIC ABS, DBLE, DCMPLX, DCONJG, DIMAG, MAX, MIN,
335: $ SQRT
336: * ..
337: * .. Statement Functions ..
338: DOUBLE PRECISION ABS1
339: * ..
340: * .. Statement Function definitions ..
341: ABS1( X ) = ABS( DBLE( X ) ) + ABS( DIMAG( X ) )
342: * ..
343: * .. Executable Statements ..
344: *
345: * Decode JOB, COMPQ, COMPZ
346: *
347: IF( LSAME( JOB, 'E' ) ) THEN
348: ILSCHR = .FALSE.
349: ISCHUR = 1
350: ELSE IF( LSAME( JOB, 'S' ) ) THEN
351: ILSCHR = .TRUE.
352: ISCHUR = 2
353: ELSE
354: ISCHUR = 0
355: END IF
356: *
357: IF( LSAME( COMPQ, 'N' ) ) THEN
358: ILQ = .FALSE.
359: ICOMPQ = 1
360: ELSE IF( LSAME( COMPQ, 'V' ) ) THEN
361: ILQ = .TRUE.
362: ICOMPQ = 2
363: ELSE IF( LSAME( COMPQ, 'I' ) ) THEN
364: ILQ = .TRUE.
365: ICOMPQ = 3
366: ELSE
367: ICOMPQ = 0
368: END IF
369: *
370: IF( LSAME( COMPZ, 'N' ) ) THEN
371: ILZ = .FALSE.
372: ICOMPZ = 1
373: ELSE IF( LSAME( COMPZ, 'V' ) ) THEN
374: ILZ = .TRUE.
375: ICOMPZ = 2
376: ELSE IF( LSAME( COMPZ, 'I' ) ) THEN
377: ILZ = .TRUE.
378: ICOMPZ = 3
379: ELSE
380: ICOMPZ = 0
381: END IF
382: *
383: * Check Argument Values
384: *
385: INFO = 0
386: WORK( 1 ) = MAX( 1, N )
387: LQUERY = ( LWORK.EQ.-1 )
388: IF( ISCHUR.EQ.0 ) THEN
389: INFO = -1
390: ELSE IF( ICOMPQ.EQ.0 ) THEN
391: INFO = -2
392: ELSE IF( ICOMPZ.EQ.0 ) THEN
393: INFO = -3
394: ELSE IF( N.LT.0 ) THEN
395: INFO = -4
396: ELSE IF( ILO.LT.1 ) THEN
397: INFO = -5
398: ELSE IF( IHI.GT.N .OR. IHI.LT.ILO-1 ) THEN
399: INFO = -6
400: ELSE IF( LDH.LT.N ) THEN
401: INFO = -8
402: ELSE IF( LDT.LT.N ) THEN
403: INFO = -10
404: ELSE IF( LDQ.LT.1 .OR. ( ILQ .AND. LDQ.LT.N ) ) THEN
405: INFO = -14
406: ELSE IF( LDZ.LT.1 .OR. ( ILZ .AND. LDZ.LT.N ) ) THEN
407: INFO = -16
408: ELSE IF( LWORK.LT.MAX( 1, N ) .AND. .NOT.LQUERY ) THEN
409: INFO = -18
410: END IF
411: IF( INFO.NE.0 ) THEN
412: CALL XERBLA( 'ZHGEQZ', -INFO )
413: RETURN
414: ELSE IF( LQUERY ) THEN
415: RETURN
416: END IF
417: *
418: * Quick return if possible
419: *
420: * WORK( 1 ) = CMPLX( 1 )
421: IF( N.LE.0 ) THEN
422: WORK( 1 ) = DCMPLX( 1 )
423: RETURN
424: END IF
425: *
426: * Initialize Q and Z
427: *
428: IF( ICOMPQ.EQ.3 )
429: $ CALL ZLASET( 'Full', N, N, CZERO, CONE, Q, LDQ )
430: IF( ICOMPZ.EQ.3 )
431: $ CALL ZLASET( 'Full', N, N, CZERO, CONE, Z, LDZ )
432: *
433: * Machine Constants
434: *
435: IN = IHI + 1 - ILO
436: SAFMIN = DLAMCH( 'S' )
437: ULP = DLAMCH( 'E' )*DLAMCH( 'B' )
438: ANORM = ZLANHS( 'F', IN, H( ILO, ILO ), LDH, RWORK )
439: BNORM = ZLANHS( 'F', IN, T( ILO, ILO ), LDT, RWORK )
440: ATOL = MAX( SAFMIN, ULP*ANORM )
441: BTOL = MAX( SAFMIN, ULP*BNORM )
442: ASCALE = ONE / MAX( SAFMIN, ANORM )
443: BSCALE = ONE / MAX( SAFMIN, BNORM )
444: *
445: *
446: * Set Eigenvalues IHI+1:N
447: *
448: DO 10 J = IHI + 1, N
449: ABSB = ABS( T( J, J ) )
450: IF( ABSB.GT.SAFMIN ) THEN
451: SIGNBC = DCONJG( T( J, J ) / ABSB )
452: T( J, J ) = ABSB
453: IF( ILSCHR ) THEN
454: CALL ZSCAL( J-1, SIGNBC, T( 1, J ), 1 )
455: CALL ZSCAL( J, SIGNBC, H( 1, J ), 1 )
456: ELSE
1.14 bertrand 457: CALL ZSCAL( 1, SIGNBC, H( J, J ), 1 )
1.1 bertrand 458: END IF
459: IF( ILZ )
460: $ CALL ZSCAL( N, SIGNBC, Z( 1, J ), 1 )
461: ELSE
462: T( J, J ) = CZERO
463: END IF
464: ALPHA( J ) = H( J, J )
465: BETA( J ) = T( J, J )
466: 10 CONTINUE
467: *
468: * If IHI < ILO, skip QZ steps
469: *
470: IF( IHI.LT.ILO )
471: $ GO TO 190
472: *
473: * MAIN QZ ITERATION LOOP
474: *
475: * Initialize dynamic indices
476: *
477: * Eigenvalues ILAST+1:N have been found.
478: * Column operations modify rows IFRSTM:whatever
479: * Row operations modify columns whatever:ILASTM
480: *
481: * If only eigenvalues are being computed, then
482: * IFRSTM is the row of the last splitting row above row ILAST;
483: * this is always at least ILO.
484: * IITER counts iterations since the last eigenvalue was found,
485: * to tell when to use an extraordinary shift.
486: * MAXIT is the maximum number of QZ sweeps allowed.
487: *
488: ILAST = IHI
489: IF( ILSCHR ) THEN
490: IFRSTM = 1
491: ILASTM = N
492: ELSE
493: IFRSTM = ILO
494: ILASTM = IHI
495: END IF
496: IITER = 0
497: ESHIFT = CZERO
498: MAXIT = 30*( IHI-ILO+1 )
499: *
500: DO 170 JITER = 1, MAXIT
501: *
502: * Check for too many iterations.
503: *
504: IF( JITER.GT.MAXIT )
505: $ GO TO 180
506: *
507: * Split the matrix if possible.
508: *
509: * Two tests:
510: * 1: H(j,j-1)=0 or j=ILO
511: * 2: T(j,j)=0
512: *
513: * Special case: j=ILAST
514: *
515: IF( ILAST.EQ.ILO ) THEN
516: GO TO 60
517: ELSE
518: IF( ABS1( H( ILAST, ILAST-1 ) ).LE.ATOL ) THEN
519: H( ILAST, ILAST-1 ) = CZERO
520: GO TO 60
521: END IF
522: END IF
523: *
524: IF( ABS( T( ILAST, ILAST ) ).LE.BTOL ) THEN
525: T( ILAST, ILAST ) = CZERO
526: GO TO 50
527: END IF
528: *
529: * General case: j<ILAST
530: *
531: DO 40 J = ILAST - 1, ILO, -1
532: *
533: * Test 1: for H(j,j-1)=0 or j=ILO
534: *
535: IF( J.EQ.ILO ) THEN
536: ILAZRO = .TRUE.
537: ELSE
538: IF( ABS1( H( J, J-1 ) ).LE.ATOL ) THEN
539: H( J, J-1 ) = CZERO
540: ILAZRO = .TRUE.
541: ELSE
542: ILAZRO = .FALSE.
543: END IF
544: END IF
545: *
546: * Test 2: for T(j,j)=0
547: *
548: IF( ABS( T( J, J ) ).LT.BTOL ) THEN
549: T( J, J ) = CZERO
550: *
551: * Test 1a: Check for 2 consecutive small subdiagonals in A
552: *
553: ILAZR2 = .FALSE.
554: IF( .NOT.ILAZRO ) THEN
555: IF( ABS1( H( J, J-1 ) )*( ASCALE*ABS1( H( J+1,
556: $ J ) ) ).LE.ABS1( H( J, J ) )*( ASCALE*ATOL ) )
557: $ ILAZR2 = .TRUE.
558: END IF
559: *
560: * If both tests pass (1 & 2), i.e., the leading diagonal
561: * element of B in the block is zero, split a 1x1 block off
562: * at the top. (I.e., at the J-th row/column) The leading
563: * diagonal element of the remainder can also be zero, so
564: * this may have to be done repeatedly.
565: *
566: IF( ILAZRO .OR. ILAZR2 ) THEN
567: DO 20 JCH = J, ILAST - 1
568: CTEMP = H( JCH, JCH )
569: CALL ZLARTG( CTEMP, H( JCH+1, JCH ), C, S,
570: $ H( JCH, JCH ) )
571: H( JCH+1, JCH ) = CZERO
572: CALL ZROT( ILASTM-JCH, H( JCH, JCH+1 ), LDH,
573: $ H( JCH+1, JCH+1 ), LDH, C, S )
574: CALL ZROT( ILASTM-JCH, T( JCH, JCH+1 ), LDT,
575: $ T( JCH+1, JCH+1 ), LDT, C, S )
576: IF( ILQ )
577: $ CALL ZROT( N, Q( 1, JCH ), 1, Q( 1, JCH+1 ), 1,
578: $ C, DCONJG( S ) )
579: IF( ILAZR2 )
580: $ H( JCH, JCH-1 ) = H( JCH, JCH-1 )*C
581: ILAZR2 = .FALSE.
582: IF( ABS1( T( JCH+1, JCH+1 ) ).GE.BTOL ) THEN
583: IF( JCH+1.GE.ILAST ) THEN
584: GO TO 60
585: ELSE
586: IFIRST = JCH + 1
587: GO TO 70
588: END IF
589: END IF
590: T( JCH+1, JCH+1 ) = CZERO
591: 20 CONTINUE
592: GO TO 50
593: ELSE
594: *
595: * Only test 2 passed -- chase the zero to T(ILAST,ILAST)
596: * Then process as in the case T(ILAST,ILAST)=0
597: *
598: DO 30 JCH = J, ILAST - 1
599: CTEMP = T( JCH, JCH+1 )
600: CALL ZLARTG( CTEMP, T( JCH+1, JCH+1 ), C, S,
601: $ T( JCH, JCH+1 ) )
602: T( JCH+1, JCH+1 ) = CZERO
603: IF( JCH.LT.ILASTM-1 )
604: $ CALL ZROT( ILASTM-JCH-1, T( JCH, JCH+2 ), LDT,
605: $ T( JCH+1, JCH+2 ), LDT, C, S )
606: CALL ZROT( ILASTM-JCH+2, H( JCH, JCH-1 ), LDH,
607: $ H( JCH+1, JCH-1 ), LDH, C, S )
608: IF( ILQ )
609: $ CALL ZROT( N, Q( 1, JCH ), 1, Q( 1, JCH+1 ), 1,
610: $ C, DCONJG( S ) )
611: CTEMP = H( JCH+1, JCH )
612: CALL ZLARTG( CTEMP, H( JCH+1, JCH-1 ), C, S,
613: $ H( JCH+1, JCH ) )
614: H( JCH+1, JCH-1 ) = CZERO
615: CALL ZROT( JCH+1-IFRSTM, H( IFRSTM, JCH ), 1,
616: $ H( IFRSTM, JCH-1 ), 1, C, S )
617: CALL ZROT( JCH-IFRSTM, T( IFRSTM, JCH ), 1,
618: $ T( IFRSTM, JCH-1 ), 1, C, S )
619: IF( ILZ )
620: $ CALL ZROT( N, Z( 1, JCH ), 1, Z( 1, JCH-1 ), 1,
621: $ C, S )
622: 30 CONTINUE
623: GO TO 50
624: END IF
625: ELSE IF( ILAZRO ) THEN
626: *
627: * Only test 1 passed -- work on J:ILAST
628: *
629: IFIRST = J
630: GO TO 70
631: END IF
632: *
633: * Neither test passed -- try next J
634: *
635: 40 CONTINUE
636: *
637: * (Drop-through is "impossible")
638: *
639: INFO = 2*N + 1
640: GO TO 210
641: *
642: * T(ILAST,ILAST)=0 -- clear H(ILAST,ILAST-1) to split off a
643: * 1x1 block.
644: *
645: 50 CONTINUE
646: CTEMP = H( ILAST, ILAST )
647: CALL ZLARTG( CTEMP, H( ILAST, ILAST-1 ), C, S,
648: $ H( ILAST, ILAST ) )
649: H( ILAST, ILAST-1 ) = CZERO
650: CALL ZROT( ILAST-IFRSTM, H( IFRSTM, ILAST ), 1,
651: $ H( IFRSTM, ILAST-1 ), 1, C, S )
652: CALL ZROT( ILAST-IFRSTM, T( IFRSTM, ILAST ), 1,
653: $ T( IFRSTM, ILAST-1 ), 1, C, S )
654: IF( ILZ )
655: $ CALL ZROT( N, Z( 1, ILAST ), 1, Z( 1, ILAST-1 ), 1, C, S )
656: *
657: * H(ILAST,ILAST-1)=0 -- Standardize B, set ALPHA and BETA
658: *
659: 60 CONTINUE
660: ABSB = ABS( T( ILAST, ILAST ) )
661: IF( ABSB.GT.SAFMIN ) THEN
662: SIGNBC = DCONJG( T( ILAST, ILAST ) / ABSB )
663: T( ILAST, ILAST ) = ABSB
664: IF( ILSCHR ) THEN
665: CALL ZSCAL( ILAST-IFRSTM, SIGNBC, T( IFRSTM, ILAST ), 1 )
666: CALL ZSCAL( ILAST+1-IFRSTM, SIGNBC, H( IFRSTM, ILAST ),
667: $ 1 )
668: ELSE
1.14 bertrand 669: CALL ZSCAL( 1, SIGNBC, H( ILAST, ILAST ), 1 )
1.1 bertrand 670: END IF
671: IF( ILZ )
672: $ CALL ZSCAL( N, SIGNBC, Z( 1, ILAST ), 1 )
673: ELSE
674: T( ILAST, ILAST ) = CZERO
675: END IF
676: ALPHA( ILAST ) = H( ILAST, ILAST )
677: BETA( ILAST ) = T( ILAST, ILAST )
678: *
679: * Go to next block -- exit if finished.
680: *
681: ILAST = ILAST - 1
682: IF( ILAST.LT.ILO )
683: $ GO TO 190
684: *
685: * Reset counters
686: *
687: IITER = 0
688: ESHIFT = CZERO
689: IF( .NOT.ILSCHR ) THEN
690: ILASTM = ILAST
691: IF( IFRSTM.GT.ILAST )
692: $ IFRSTM = ILO
693: END IF
694: GO TO 160
695: *
696: * QZ step
697: *
698: * This iteration only involves rows/columns IFIRST:ILAST. We
699: * assume IFIRST < ILAST, and that the diagonal of B is non-zero.
700: *
701: 70 CONTINUE
702: IITER = IITER + 1
703: IF( .NOT.ILSCHR ) THEN
704: IFRSTM = IFIRST
705: END IF
706: *
707: * Compute the Shift.
708: *
709: * At this point, IFIRST < ILAST, and the diagonal elements of
710: * T(IFIRST:ILAST,IFIRST,ILAST) are larger than BTOL (in
711: * magnitude)
712: *
713: IF( ( IITER / 10 )*10.NE.IITER ) THEN
714: *
715: * The Wilkinson shift (AEP p.512), i.e., the eigenvalue of
716: * the bottom-right 2x2 block of A inv(B) which is nearest to
717: * the bottom-right element.
718: *
719: * We factor B as U*D, where U has unit diagonals, and
720: * compute (A*inv(D))*inv(U).
721: *
722: U12 = ( BSCALE*T( ILAST-1, ILAST ) ) /
723: $ ( BSCALE*T( ILAST, ILAST ) )
724: AD11 = ( ASCALE*H( ILAST-1, ILAST-1 ) ) /
725: $ ( BSCALE*T( ILAST-1, ILAST-1 ) )
726: AD21 = ( ASCALE*H( ILAST, ILAST-1 ) ) /
727: $ ( BSCALE*T( ILAST-1, ILAST-1 ) )
728: AD12 = ( ASCALE*H( ILAST-1, ILAST ) ) /
729: $ ( BSCALE*T( ILAST, ILAST ) )
730: AD22 = ( ASCALE*H( ILAST, ILAST ) ) /
731: $ ( BSCALE*T( ILAST, ILAST ) )
732: ABI22 = AD22 - U12*AD21
733: *
734: T1 = HALF*( AD11+ABI22 )
735: RTDISC = SQRT( T1**2+AD12*AD21-AD11*AD22 )
736: TEMP = DBLE( T1-ABI22 )*DBLE( RTDISC ) +
737: $ DIMAG( T1-ABI22 )*DIMAG( RTDISC )
738: IF( TEMP.LE.ZERO ) THEN
739: SHIFT = T1 + RTDISC
740: ELSE
741: SHIFT = T1 - RTDISC
742: END IF
743: ELSE
744: *
745: * Exceptional shift. Chosen for no particularly good reason.
746: *
1.10 bertrand 747: ESHIFT = ESHIFT + (ASCALE*H(ILAST,ILAST-1))/
748: $ (BSCALE*T(ILAST-1,ILAST-1))
1.1 bertrand 749: SHIFT = ESHIFT
750: END IF
751: *
752: * Now check for two consecutive small subdiagonals.
753: *
754: DO 80 J = ILAST - 1, IFIRST + 1, -1
755: ISTART = J
756: CTEMP = ASCALE*H( J, J ) - SHIFT*( BSCALE*T( J, J ) )
757: TEMP = ABS1( CTEMP )
758: TEMP2 = ASCALE*ABS1( H( J+1, J ) )
759: TEMPR = MAX( TEMP, TEMP2 )
760: IF( TEMPR.LT.ONE .AND. TEMPR.NE.ZERO ) THEN
761: TEMP = TEMP / TEMPR
762: TEMP2 = TEMP2 / TEMPR
763: END IF
764: IF( ABS1( H( J, J-1 ) )*TEMP2.LE.TEMP*ATOL )
765: $ GO TO 90
766: 80 CONTINUE
767: *
768: ISTART = IFIRST
769: CTEMP = ASCALE*H( IFIRST, IFIRST ) -
770: $ SHIFT*( BSCALE*T( IFIRST, IFIRST ) )
771: 90 CONTINUE
772: *
773: * Do an implicit-shift QZ sweep.
774: *
775: * Initial Q
776: *
777: CTEMP2 = ASCALE*H( ISTART+1, ISTART )
778: CALL ZLARTG( CTEMP, CTEMP2, C, S, CTEMP3 )
779: *
780: * Sweep
781: *
782: DO 150 J = ISTART, ILAST - 1
783: IF( J.GT.ISTART ) THEN
784: CTEMP = H( J, J-1 )
785: CALL ZLARTG( CTEMP, H( J+1, J-1 ), C, S, H( J, J-1 ) )
786: H( J+1, J-1 ) = CZERO
787: END IF
788: *
789: DO 100 JC = J, ILASTM
790: CTEMP = C*H( J, JC ) + S*H( J+1, JC )
791: H( J+1, JC ) = -DCONJG( S )*H( J, JC ) + C*H( J+1, JC )
792: H( J, JC ) = CTEMP
793: CTEMP2 = C*T( J, JC ) + S*T( J+1, JC )
794: T( J+1, JC ) = -DCONJG( S )*T( J, JC ) + C*T( J+1, JC )
795: T( J, JC ) = CTEMP2
796: 100 CONTINUE
797: IF( ILQ ) THEN
798: DO 110 JR = 1, N
799: CTEMP = C*Q( JR, J ) + DCONJG( S )*Q( JR, J+1 )
800: Q( JR, J+1 ) = -S*Q( JR, J ) + C*Q( JR, J+1 )
801: Q( JR, J ) = CTEMP
802: 110 CONTINUE
803: END IF
804: *
805: CTEMP = T( J+1, J+1 )
806: CALL ZLARTG( CTEMP, T( J+1, J ), C, S, T( J+1, J+1 ) )
807: T( J+1, J ) = CZERO
808: *
809: DO 120 JR = IFRSTM, MIN( J+2, ILAST )
810: CTEMP = C*H( JR, J+1 ) + S*H( JR, J )
811: H( JR, J ) = -DCONJG( S )*H( JR, J+1 ) + C*H( JR, J )
812: H( JR, J+1 ) = CTEMP
813: 120 CONTINUE
814: DO 130 JR = IFRSTM, J
815: CTEMP = C*T( JR, J+1 ) + S*T( JR, J )
816: T( JR, J ) = -DCONJG( S )*T( JR, J+1 ) + C*T( JR, J )
817: T( JR, J+1 ) = CTEMP
818: 130 CONTINUE
819: IF( ILZ ) THEN
820: DO 140 JR = 1, N
821: CTEMP = C*Z( JR, J+1 ) + S*Z( JR, J )
822: Z( JR, J ) = -DCONJG( S )*Z( JR, J+1 ) + C*Z( JR, J )
823: Z( JR, J+1 ) = CTEMP
824: 140 CONTINUE
825: END IF
826: 150 CONTINUE
827: *
828: 160 CONTINUE
829: *
830: 170 CONTINUE
831: *
832: * Drop-through = non-convergence
833: *
834: 180 CONTINUE
835: INFO = ILAST
836: GO TO 210
837: *
838: * Successful completion of all QZ steps
839: *
840: 190 CONTINUE
841: *
842: * Set Eigenvalues 1:ILO-1
843: *
844: DO 200 J = 1, ILO - 1
845: ABSB = ABS( T( J, J ) )
846: IF( ABSB.GT.SAFMIN ) THEN
847: SIGNBC = DCONJG( T( J, J ) / ABSB )
848: T( J, J ) = ABSB
849: IF( ILSCHR ) THEN
850: CALL ZSCAL( J-1, SIGNBC, T( 1, J ), 1 )
851: CALL ZSCAL( J, SIGNBC, H( 1, J ), 1 )
852: ELSE
1.14 bertrand 853: CALL ZSCAL( 1, SIGNBC, H( J, J ), 1 )
1.1 bertrand 854: END IF
855: IF( ILZ )
856: $ CALL ZSCAL( N, SIGNBC, Z( 1, J ), 1 )
857: ELSE
858: T( J, J ) = CZERO
859: END IF
860: ALPHA( J ) = H( J, J )
861: BETA( J ) = T( J, J )
862: 200 CONTINUE
863: *
864: * Normal Termination
865: *
866: INFO = 0
867: *
868: * Exit (other than argument error) -- return optimal workspace size
869: *
870: 210 CONTINUE
871: WORK( 1 ) = DCMPLX( N )
872: RETURN
873: *
874: * End of ZHGEQZ
875: *
876: END
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