Annotation of rpl/lapack/lapack/zhgeqz.f, revision 1.20
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: *
269: *> \ingroup complex16GEcomputational
270: *
271: *> \par Further Details:
272: * =====================
273: *>
274: *> \verbatim
275: *>
276: *> We assume that complex ABS works as long as its value is less than
277: *> overflow.
278: *> \endverbatim
279: *>
280: * =====================================================================
1.1 bertrand 281: SUBROUTINE ZHGEQZ( JOB, COMPQ, COMPZ, N, ILO, IHI, H, LDH, T, LDT,
282: $ ALPHA, BETA, Q, LDQ, Z, LDZ, WORK, LWORK,
283: $ RWORK, INFO )
284: *
1.20 ! bertrand 285: * -- LAPACK computational routine --
1.1 bertrand 286: * -- LAPACK is a software package provided by Univ. of Tennessee, --
287: * -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
288: *
289: * .. Scalar Arguments ..
290: CHARACTER COMPQ, COMPZ, JOB
291: INTEGER IHI, ILO, INFO, LDH, LDQ, LDT, LDZ, LWORK, N
292: * ..
293: * .. Array Arguments ..
294: DOUBLE PRECISION RWORK( * )
295: COMPLEX*16 ALPHA( * ), BETA( * ), H( LDH, * ),
296: $ Q( LDQ, * ), T( LDT, * ), WORK( * ),
297: $ Z( LDZ, * )
298: * ..
299: *
300: * =====================================================================
301: *
302: * .. Parameters ..
303: COMPLEX*16 CZERO, CONE
304: PARAMETER ( CZERO = ( 0.0D+0, 0.0D+0 ),
305: $ CONE = ( 1.0D+0, 0.0D+0 ) )
306: DOUBLE PRECISION ZERO, ONE
307: PARAMETER ( ZERO = 0.0D+0, ONE = 1.0D+0 )
308: DOUBLE PRECISION HALF
309: PARAMETER ( HALF = 0.5D+0 )
310: * ..
311: * .. Local Scalars ..
312: LOGICAL ILAZR2, ILAZRO, ILQ, ILSCHR, ILZ, LQUERY
313: INTEGER ICOMPQ, ICOMPZ, IFIRST, IFRSTM, IITER, ILAST,
314: $ ILASTM, IN, ISCHUR, ISTART, J, JC, JCH, JITER,
315: $ JR, MAXIT
316: DOUBLE PRECISION ABSB, ANORM, ASCALE, ATOL, BNORM, BSCALE, BTOL,
317: $ C, SAFMIN, TEMP, TEMP2, TEMPR, ULP
318: COMPLEX*16 ABI22, AD11, AD12, AD21, AD22, CTEMP, CTEMP2,
1.20 ! bertrand 319: $ CTEMP3, ESHIFT, S, SHIFT, SIGNBC,
! 320: $ U12, X, ABI12, Y
1.1 bertrand 321: * ..
322: * .. External Functions ..
1.20 ! bertrand 323: COMPLEX*16 ZLADIV
1.1 bertrand 324: LOGICAL LSAME
325: DOUBLE PRECISION DLAMCH, ZLANHS
1.20 ! bertrand 326: EXTERNAL ZLADIV, LSAME, DLAMCH, ZLANHS
1.1 bertrand 327: * ..
328: * .. External Subroutines ..
329: EXTERNAL XERBLA, ZLARTG, ZLASET, ZROT, ZSCAL
330: * ..
331: * .. Intrinsic Functions ..
332: INTRINSIC ABS, DBLE, DCMPLX, DCONJG, DIMAG, MAX, MIN,
333: $ SQRT
334: * ..
335: * .. Statement Functions ..
336: DOUBLE PRECISION ABS1
337: * ..
338: * .. Statement Function definitions ..
339: ABS1( X ) = ABS( DBLE( X ) ) + ABS( DIMAG( X ) )
340: * ..
341: * .. Executable Statements ..
342: *
343: * Decode JOB, COMPQ, COMPZ
344: *
345: IF( LSAME( JOB, 'E' ) ) THEN
346: ILSCHR = .FALSE.
347: ISCHUR = 1
348: ELSE IF( LSAME( JOB, 'S' ) ) THEN
349: ILSCHR = .TRUE.
350: ISCHUR = 2
351: ELSE
1.20 ! bertrand 352: ILSCHR = .TRUE.
1.1 bertrand 353: ISCHUR = 0
354: END IF
355: *
356: IF( LSAME( COMPQ, 'N' ) ) THEN
357: ILQ = .FALSE.
358: ICOMPQ = 1
359: ELSE IF( LSAME( COMPQ, 'V' ) ) THEN
360: ILQ = .TRUE.
361: ICOMPQ = 2
362: ELSE IF( LSAME( COMPQ, 'I' ) ) THEN
363: ILQ = .TRUE.
364: ICOMPQ = 3
365: ELSE
1.20 ! bertrand 366: ILQ = .TRUE.
1.1 bertrand 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
1.20 ! bertrand 380: ILZ = .TRUE.
1.1 bertrand 381: ICOMPZ = 0
382: END IF
383: *
384: * Check Argument Values
385: *
386: INFO = 0
387: WORK( 1 ) = MAX( 1, N )
388: LQUERY = ( LWORK.EQ.-1 )
389: IF( ISCHUR.EQ.0 ) THEN
390: INFO = -1
391: ELSE IF( ICOMPQ.EQ.0 ) THEN
392: INFO = -2
393: ELSE IF( ICOMPZ.EQ.0 ) THEN
394: INFO = -3
395: ELSE IF( N.LT.0 ) THEN
396: INFO = -4
397: ELSE IF( ILO.LT.1 ) THEN
398: INFO = -5
399: ELSE IF( IHI.GT.N .OR. IHI.LT.ILO-1 ) THEN
400: INFO = -6
401: ELSE IF( LDH.LT.N ) THEN
402: INFO = -8
403: ELSE IF( LDT.LT.N ) THEN
404: INFO = -10
405: ELSE IF( LDQ.LT.1 .OR. ( ILQ .AND. LDQ.LT.N ) ) THEN
406: INFO = -14
407: ELSE IF( LDZ.LT.1 .OR. ( ILZ .AND. LDZ.LT.N ) ) THEN
408: INFO = -16
409: ELSE IF( LWORK.LT.MAX( 1, N ) .AND. .NOT.LQUERY ) THEN
410: INFO = -18
411: END IF
412: IF( INFO.NE.0 ) THEN
413: CALL XERBLA( 'ZHGEQZ', -INFO )
414: RETURN
415: ELSE IF( LQUERY ) THEN
416: RETURN
417: END IF
418: *
419: * Quick return if possible
420: *
421: * WORK( 1 ) = CMPLX( 1 )
422: IF( N.LE.0 ) THEN
423: WORK( 1 ) = DCMPLX( 1 )
424: RETURN
425: END IF
426: *
427: * Initialize Q and Z
428: *
429: IF( ICOMPQ.EQ.3 )
430: $ CALL ZLASET( 'Full', N, N, CZERO, CONE, Q, LDQ )
431: IF( ICOMPZ.EQ.3 )
432: $ CALL ZLASET( 'Full', N, N, CZERO, CONE, Z, LDZ )
433: *
434: * Machine Constants
435: *
436: IN = IHI + 1 - ILO
437: SAFMIN = DLAMCH( 'S' )
438: ULP = DLAMCH( 'E' )*DLAMCH( 'B' )
439: ANORM = ZLANHS( 'F', IN, H( ILO, ILO ), LDH, RWORK )
440: BNORM = ZLANHS( 'F', IN, T( ILO, ILO ), LDT, RWORK )
441: ATOL = MAX( SAFMIN, ULP*ANORM )
442: BTOL = MAX( SAFMIN, ULP*BNORM )
443: ASCALE = ONE / MAX( SAFMIN, ANORM )
444: BSCALE = ONE / MAX( SAFMIN, BNORM )
445: *
446: *
447: * Set Eigenvalues IHI+1:N
448: *
449: DO 10 J = IHI + 1, N
450: ABSB = ABS( T( J, J ) )
451: IF( ABSB.GT.SAFMIN ) THEN
452: SIGNBC = DCONJG( T( J, J ) / ABSB )
453: T( J, J ) = ABSB
454: IF( ILSCHR ) THEN
455: CALL ZSCAL( J-1, SIGNBC, T( 1, J ), 1 )
456: CALL ZSCAL( J, SIGNBC, H( 1, J ), 1 )
457: ELSE
1.14 bertrand 458: CALL ZSCAL( 1, SIGNBC, H( J, J ), 1 )
1.1 bertrand 459: END IF
460: IF( ILZ )
461: $ CALL ZSCAL( N, SIGNBC, Z( 1, J ), 1 )
462: ELSE
463: T( J, J ) = CZERO
464: END IF
465: ALPHA( J ) = H( J, J )
466: BETA( J ) = T( J, J )
467: 10 CONTINUE
468: *
469: * If IHI < ILO, skip QZ steps
470: *
471: IF( IHI.LT.ILO )
472: $ GO TO 190
473: *
474: * MAIN QZ ITERATION LOOP
475: *
476: * Initialize dynamic indices
477: *
478: * Eigenvalues ILAST+1:N have been found.
479: * Column operations modify rows IFRSTM:whatever
480: * Row operations modify columns whatever:ILASTM
481: *
482: * If only eigenvalues are being computed, then
483: * IFRSTM is the row of the last splitting row above row ILAST;
484: * this is always at least ILO.
485: * IITER counts iterations since the last eigenvalue was found,
486: * to tell when to use an extraordinary shift.
487: * MAXIT is the maximum number of QZ sweeps allowed.
488: *
489: ILAST = IHI
490: IF( ILSCHR ) THEN
491: IFRSTM = 1
492: ILASTM = N
493: ELSE
494: IFRSTM = ILO
495: ILASTM = IHI
496: END IF
497: IITER = 0
498: ESHIFT = CZERO
499: MAXIT = 30*( IHI-ILO+1 )
500: *
501: DO 170 JITER = 1, MAXIT
502: *
503: * Check for too many iterations.
504: *
505: IF( JITER.GT.MAXIT )
506: $ GO TO 180
507: *
508: * Split the matrix if possible.
509: *
510: * Two tests:
511: * 1: H(j,j-1)=0 or j=ILO
512: * 2: T(j,j)=0
513: *
514: * Special case: j=ILAST
515: *
516: IF( ILAST.EQ.ILO ) THEN
517: GO TO 60
518: ELSE
1.20 ! bertrand 519: IF( ABS1( H( ILAST, ILAST-1 ) ).LE.MAX( SAFMIN, ULP*(
! 520: $ ABS1( H( ILAST, ILAST ) ) + ABS1( H( ILAST-1, ILAST-1 )
! 521: $ ) ) ) ) THEN
1.1 bertrand 522: H( ILAST, ILAST-1 ) = CZERO
523: GO TO 60
524: END IF
525: END IF
526: *
527: IF( ABS( T( ILAST, ILAST ) ).LE.BTOL ) THEN
528: T( ILAST, ILAST ) = CZERO
529: GO TO 50
530: END IF
531: *
532: * General case: j<ILAST
533: *
534: DO 40 J = ILAST - 1, ILO, -1
535: *
536: * Test 1: for H(j,j-1)=0 or j=ILO
537: *
538: IF( J.EQ.ILO ) THEN
539: ILAZRO = .TRUE.
540: ELSE
1.20 ! bertrand 541: IF( ABS1( H( J, J-1 ) ).LE.MAX( SAFMIN, ULP*(
! 542: $ ABS1( H( J, J ) ) + ABS1( H( J-1, J-1 ) )
! 543: $ ) ) ) THEN
1.1 bertrand 544: H( J, J-1 ) = CZERO
545: ILAZRO = .TRUE.
546: ELSE
547: ILAZRO = .FALSE.
548: END IF
549: END IF
550: *
551: * Test 2: for T(j,j)=0
552: *
553: IF( ABS( T( J, J ) ).LT.BTOL ) THEN
554: T( J, J ) = CZERO
555: *
556: * Test 1a: Check for 2 consecutive small subdiagonals in A
557: *
558: ILAZR2 = .FALSE.
559: IF( .NOT.ILAZRO ) THEN
560: IF( ABS1( H( J, J-1 ) )*( ASCALE*ABS1( H( J+1,
561: $ J ) ) ).LE.ABS1( H( J, J ) )*( ASCALE*ATOL ) )
562: $ ILAZR2 = .TRUE.
563: END IF
564: *
565: * If both tests pass (1 & 2), i.e., the leading diagonal
566: * element of B in the block is zero, split a 1x1 block off
567: * at the top. (I.e., at the J-th row/column) The leading
568: * diagonal element of the remainder can also be zero, so
569: * this may have to be done repeatedly.
570: *
571: IF( ILAZRO .OR. ILAZR2 ) THEN
572: DO 20 JCH = J, ILAST - 1
573: CTEMP = H( JCH, JCH )
574: CALL ZLARTG( CTEMP, H( JCH+1, JCH ), C, S,
575: $ H( JCH, JCH ) )
576: H( JCH+1, JCH ) = CZERO
577: CALL ZROT( ILASTM-JCH, H( JCH, JCH+1 ), LDH,
578: $ H( JCH+1, JCH+1 ), LDH, C, S )
579: CALL ZROT( ILASTM-JCH, T( JCH, JCH+1 ), LDT,
580: $ T( JCH+1, JCH+1 ), LDT, C, S )
581: IF( ILQ )
582: $ CALL ZROT( N, Q( 1, JCH ), 1, Q( 1, JCH+1 ), 1,
583: $ C, DCONJG( S ) )
584: IF( ILAZR2 )
585: $ H( JCH, JCH-1 ) = H( JCH, JCH-1 )*C
586: ILAZR2 = .FALSE.
587: IF( ABS1( T( JCH+1, JCH+1 ) ).GE.BTOL ) THEN
588: IF( JCH+1.GE.ILAST ) THEN
589: GO TO 60
590: ELSE
591: IFIRST = JCH + 1
592: GO TO 70
593: END IF
594: END IF
595: T( JCH+1, JCH+1 ) = CZERO
596: 20 CONTINUE
597: GO TO 50
598: ELSE
599: *
600: * Only test 2 passed -- chase the zero to T(ILAST,ILAST)
601: * Then process as in the case T(ILAST,ILAST)=0
602: *
603: DO 30 JCH = J, ILAST - 1
604: CTEMP = T( JCH, JCH+1 )
605: CALL ZLARTG( CTEMP, T( JCH+1, JCH+1 ), C, S,
606: $ T( JCH, JCH+1 ) )
607: T( JCH+1, JCH+1 ) = CZERO
608: IF( JCH.LT.ILASTM-1 )
609: $ CALL ZROT( ILASTM-JCH-1, T( JCH, JCH+2 ), LDT,
610: $ T( JCH+1, JCH+2 ), LDT, C, S )
611: CALL ZROT( ILASTM-JCH+2, H( JCH, JCH-1 ), LDH,
612: $ H( JCH+1, JCH-1 ), LDH, C, S )
613: IF( ILQ )
614: $ CALL ZROT( N, Q( 1, JCH ), 1, Q( 1, JCH+1 ), 1,
615: $ C, DCONJG( S ) )
616: CTEMP = H( JCH+1, JCH )
617: CALL ZLARTG( CTEMP, H( JCH+1, JCH-1 ), C, S,
618: $ H( JCH+1, JCH ) )
619: H( JCH+1, JCH-1 ) = CZERO
620: CALL ZROT( JCH+1-IFRSTM, H( IFRSTM, JCH ), 1,
621: $ H( IFRSTM, JCH-1 ), 1, C, S )
622: CALL ZROT( JCH-IFRSTM, T( IFRSTM, JCH ), 1,
623: $ T( IFRSTM, JCH-1 ), 1, C, S )
624: IF( ILZ )
625: $ CALL ZROT( N, Z( 1, JCH ), 1, Z( 1, JCH-1 ), 1,
626: $ C, S )
627: 30 CONTINUE
628: GO TO 50
629: END IF
630: ELSE IF( ILAZRO ) THEN
631: *
632: * Only test 1 passed -- work on J:ILAST
633: *
634: IFIRST = J
635: GO TO 70
636: END IF
637: *
638: * Neither test passed -- try next J
639: *
640: 40 CONTINUE
641: *
642: * (Drop-through is "impossible")
643: *
644: INFO = 2*N + 1
645: GO TO 210
646: *
647: * T(ILAST,ILAST)=0 -- clear H(ILAST,ILAST-1) to split off a
648: * 1x1 block.
649: *
650: 50 CONTINUE
651: CTEMP = H( ILAST, ILAST )
652: CALL ZLARTG( CTEMP, H( ILAST, ILAST-1 ), C, S,
653: $ H( ILAST, ILAST ) )
654: H( ILAST, ILAST-1 ) = CZERO
655: CALL ZROT( ILAST-IFRSTM, H( IFRSTM, ILAST ), 1,
656: $ H( IFRSTM, ILAST-1 ), 1, C, S )
657: CALL ZROT( ILAST-IFRSTM, T( IFRSTM, ILAST ), 1,
658: $ T( IFRSTM, ILAST-1 ), 1, C, S )
659: IF( ILZ )
660: $ CALL ZROT( N, Z( 1, ILAST ), 1, Z( 1, ILAST-1 ), 1, C, S )
661: *
662: * H(ILAST,ILAST-1)=0 -- Standardize B, set ALPHA and BETA
663: *
664: 60 CONTINUE
665: ABSB = ABS( T( ILAST, ILAST ) )
666: IF( ABSB.GT.SAFMIN ) THEN
667: SIGNBC = DCONJG( T( ILAST, ILAST ) / ABSB )
668: T( ILAST, ILAST ) = ABSB
669: IF( ILSCHR ) THEN
670: CALL ZSCAL( ILAST-IFRSTM, SIGNBC, T( IFRSTM, ILAST ), 1 )
671: CALL ZSCAL( ILAST+1-IFRSTM, SIGNBC, H( IFRSTM, ILAST ),
672: $ 1 )
673: ELSE
1.14 bertrand 674: CALL ZSCAL( 1, SIGNBC, H( ILAST, ILAST ), 1 )
1.1 bertrand 675: END IF
676: IF( ILZ )
677: $ CALL ZSCAL( N, SIGNBC, Z( 1, ILAST ), 1 )
678: ELSE
679: T( ILAST, ILAST ) = CZERO
680: END IF
681: ALPHA( ILAST ) = H( ILAST, ILAST )
682: BETA( ILAST ) = T( ILAST, ILAST )
683: *
684: * Go to next block -- exit if finished.
685: *
686: ILAST = ILAST - 1
687: IF( ILAST.LT.ILO )
688: $ GO TO 190
689: *
690: * Reset counters
691: *
692: IITER = 0
693: ESHIFT = CZERO
694: IF( .NOT.ILSCHR ) THEN
695: ILASTM = ILAST
696: IF( IFRSTM.GT.ILAST )
697: $ IFRSTM = ILO
698: END IF
699: GO TO 160
700: *
701: * QZ step
702: *
703: * This iteration only involves rows/columns IFIRST:ILAST. We
704: * assume IFIRST < ILAST, and that the diagonal of B is non-zero.
705: *
706: 70 CONTINUE
707: IITER = IITER + 1
708: IF( .NOT.ILSCHR ) THEN
709: IFRSTM = IFIRST
710: END IF
711: *
712: * Compute the Shift.
713: *
714: * At this point, IFIRST < ILAST, and the diagonal elements of
715: * T(IFIRST:ILAST,IFIRST,ILAST) are larger than BTOL (in
716: * magnitude)
717: *
718: IF( ( IITER / 10 )*10.NE.IITER ) THEN
719: *
720: * The Wilkinson shift (AEP p.512), i.e., the eigenvalue of
721: * the bottom-right 2x2 block of A inv(B) which is nearest to
722: * the bottom-right element.
723: *
724: * We factor B as U*D, where U has unit diagonals, and
725: * compute (A*inv(D))*inv(U).
726: *
727: U12 = ( BSCALE*T( ILAST-1, ILAST ) ) /
728: $ ( BSCALE*T( ILAST, ILAST ) )
729: AD11 = ( ASCALE*H( ILAST-1, ILAST-1 ) ) /
730: $ ( BSCALE*T( ILAST-1, ILAST-1 ) )
731: AD21 = ( ASCALE*H( ILAST, ILAST-1 ) ) /
732: $ ( BSCALE*T( ILAST-1, ILAST-1 ) )
733: AD12 = ( ASCALE*H( ILAST-1, ILAST ) ) /
734: $ ( BSCALE*T( ILAST, ILAST ) )
735: AD22 = ( ASCALE*H( ILAST, ILAST ) ) /
736: $ ( BSCALE*T( ILAST, ILAST ) )
737: ABI22 = AD22 - U12*AD21
1.20 ! bertrand 738: ABI12 = AD12 - U12*AD11
1.1 bertrand 739: *
1.20 ! bertrand 740: SHIFT = ABI22
! 741: CTEMP = SQRT( ABI12 )*SQRT( AD21 )
! 742: TEMP = ABS1( CTEMP )
! 743: IF( CTEMP.NE.ZERO ) THEN
! 744: X = HALF*( AD11-SHIFT )
! 745: TEMP2 = ABS1( X )
! 746: TEMP = MAX( TEMP, ABS1( X ) )
! 747: Y = TEMP*SQRT( ( X / TEMP )**2+( CTEMP / TEMP )**2 )
! 748: IF( TEMP2.GT.ZERO ) THEN
! 749: IF( DBLE( X / TEMP2 )*DBLE( Y )+
! 750: $ DIMAG( X / TEMP2 )*DIMAG( Y ).LT.ZERO )Y = -Y
! 751: END IF
! 752: SHIFT = SHIFT - CTEMP*ZLADIV( CTEMP, ( X+Y ) )
1.1 bertrand 753: END IF
754: ELSE
755: *
756: * Exceptional shift. Chosen for no particularly good reason.
757: *
1.20 ! bertrand 758: IF( ( IITER / 20 )*20.EQ.IITER .AND.
! 759: $ BSCALE*ABS1(T( ILAST, ILAST )).GT.SAFMIN ) THEN
! 760: ESHIFT = ESHIFT + ( ASCALE*H( ILAST,
! 761: $ ILAST ) )/( BSCALE*T( ILAST, ILAST ) )
! 762: ELSE
! 763: ESHIFT = ESHIFT + ( ASCALE*H( ILAST,
! 764: $ ILAST-1 ) )/( BSCALE*T( ILAST-1, ILAST-1 ) )
! 765: END IF
1.1 bertrand 766: SHIFT = ESHIFT
767: END IF
768: *
769: * Now check for two consecutive small subdiagonals.
770: *
771: DO 80 J = ILAST - 1, IFIRST + 1, -1
772: ISTART = J
773: CTEMP = ASCALE*H( J, J ) - SHIFT*( BSCALE*T( J, J ) )
774: TEMP = ABS1( CTEMP )
775: TEMP2 = ASCALE*ABS1( H( J+1, J ) )
776: TEMPR = MAX( TEMP, TEMP2 )
777: IF( TEMPR.LT.ONE .AND. TEMPR.NE.ZERO ) THEN
778: TEMP = TEMP / TEMPR
779: TEMP2 = TEMP2 / TEMPR
780: END IF
781: IF( ABS1( H( J, J-1 ) )*TEMP2.LE.TEMP*ATOL )
782: $ GO TO 90
783: 80 CONTINUE
784: *
785: ISTART = IFIRST
786: CTEMP = ASCALE*H( IFIRST, IFIRST ) -
787: $ SHIFT*( BSCALE*T( IFIRST, IFIRST ) )
788: 90 CONTINUE
789: *
790: * Do an implicit-shift QZ sweep.
791: *
792: * Initial Q
793: *
794: CTEMP2 = ASCALE*H( ISTART+1, ISTART )
795: CALL ZLARTG( CTEMP, CTEMP2, C, S, CTEMP3 )
796: *
797: * Sweep
798: *
799: DO 150 J = ISTART, ILAST - 1
800: IF( J.GT.ISTART ) THEN
801: CTEMP = H( J, J-1 )
802: CALL ZLARTG( CTEMP, H( J+1, J-1 ), C, S, H( J, J-1 ) )
803: H( J+1, J-1 ) = CZERO
804: END IF
805: *
806: DO 100 JC = J, ILASTM
807: CTEMP = C*H( J, JC ) + S*H( J+1, JC )
808: H( J+1, JC ) = -DCONJG( S )*H( J, JC ) + C*H( J+1, JC )
809: H( J, JC ) = CTEMP
810: CTEMP2 = C*T( J, JC ) + S*T( J+1, JC )
811: T( J+1, JC ) = -DCONJG( S )*T( J, JC ) + C*T( J+1, JC )
812: T( J, JC ) = CTEMP2
813: 100 CONTINUE
814: IF( ILQ ) THEN
815: DO 110 JR = 1, N
816: CTEMP = C*Q( JR, J ) + DCONJG( S )*Q( JR, J+1 )
817: Q( JR, J+1 ) = -S*Q( JR, J ) + C*Q( JR, J+1 )
818: Q( JR, J ) = CTEMP
819: 110 CONTINUE
820: END IF
821: *
822: CTEMP = T( J+1, J+1 )
823: CALL ZLARTG( CTEMP, T( J+1, J ), C, S, T( J+1, J+1 ) )
824: T( J+1, J ) = CZERO
825: *
826: DO 120 JR = IFRSTM, MIN( J+2, ILAST )
827: CTEMP = C*H( JR, J+1 ) + S*H( JR, J )
828: H( JR, J ) = -DCONJG( S )*H( JR, J+1 ) + C*H( JR, J )
829: H( JR, J+1 ) = CTEMP
830: 120 CONTINUE
831: DO 130 JR = IFRSTM, J
832: CTEMP = C*T( JR, J+1 ) + S*T( JR, J )
833: T( JR, J ) = -DCONJG( S )*T( JR, J+1 ) + C*T( JR, J )
834: T( JR, J+1 ) = CTEMP
835: 130 CONTINUE
836: IF( ILZ ) THEN
837: DO 140 JR = 1, N
838: CTEMP = C*Z( JR, J+1 ) + S*Z( JR, J )
839: Z( JR, J ) = -DCONJG( S )*Z( JR, J+1 ) + C*Z( JR, J )
840: Z( JR, J+1 ) = CTEMP
841: 140 CONTINUE
842: END IF
843: 150 CONTINUE
844: *
845: 160 CONTINUE
846: *
847: 170 CONTINUE
848: *
849: * Drop-through = non-convergence
850: *
851: 180 CONTINUE
852: INFO = ILAST
853: GO TO 210
854: *
855: * Successful completion of all QZ steps
856: *
857: 190 CONTINUE
858: *
859: * Set Eigenvalues 1:ILO-1
860: *
861: DO 200 J = 1, ILO - 1
862: ABSB = ABS( T( J, J ) )
863: IF( ABSB.GT.SAFMIN ) THEN
864: SIGNBC = DCONJG( T( J, J ) / ABSB )
865: T( J, J ) = ABSB
866: IF( ILSCHR ) THEN
867: CALL ZSCAL( J-1, SIGNBC, T( 1, J ), 1 )
868: CALL ZSCAL( J, SIGNBC, H( 1, J ), 1 )
869: ELSE
1.14 bertrand 870: CALL ZSCAL( 1, SIGNBC, H( J, J ), 1 )
1.1 bertrand 871: END IF
872: IF( ILZ )
873: $ CALL ZSCAL( N, SIGNBC, Z( 1, J ), 1 )
874: ELSE
875: T( J, J ) = CZERO
876: END IF
877: ALPHA( J ) = H( J, J )
878: BETA( J ) = T( J, J )
879: 200 CONTINUE
880: *
881: * Normal Termination
882: *
883: INFO = 0
884: *
885: * Exit (other than argument error) -- return optimal workspace size
886: *
887: 210 CONTINUE
888: WORK( 1 ) = DCMPLX( N )
889: RETURN
890: *
891: * End of ZHGEQZ
892: *
893: END
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