001:       SUBROUTINE ZHGEQZ( JOB, COMPQ, COMPZ, N, ILO, IHI, H, LDH, T, LDT,
002:      $                   ALPHA, BETA, Q, LDQ, Z, LDZ, WORK, LWORK,
003:      $                   RWORK, INFO )
004: *
005: *  -- LAPACK routine (version 3.2) --
006: *  -- LAPACK is a software package provided by Univ. of Tennessee,    --
007: *  -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
008: *     November 2006
009: *
010: *     .. Scalar Arguments ..
011:       CHARACTER          COMPQ, COMPZ, JOB
012:       INTEGER            IHI, ILO, INFO, LDH, LDQ, LDT, LDZ, LWORK, N
013: *     ..
014: *     .. Array Arguments ..
015:       DOUBLE PRECISION   RWORK( * )
016:       COMPLEX*16         ALPHA( * ), BETA( * ), H( LDH, * ),
017:      $                   Q( LDQ, * ), T( LDT, * ), WORK( * ),
018:      $                   Z( LDZ, * )
019: *     ..
020: *
021: *  Purpose
022: *  =======
023: *
024: *  ZHGEQZ computes the eigenvalues of a complex matrix pair (H,T),
025: *  where H is an upper Hessenberg matrix and T is upper triangular,
026: *  using the single-shift QZ method.
027: *  Matrix pairs of this type are produced by the reduction to
028: *  generalized upper Hessenberg form of a complex matrix pair (A,B):
029: *  
030: *     A = Q1*H*Z1**H,  B = Q1*T*Z1**H,
031: *  
032: *  as computed by ZGGHRD.
033: *  
034: *  If JOB='S', then the Hessenberg-triangular pair (H,T) is
035: *  also reduced to generalized Schur form,
036: *  
037: *     H = Q*S*Z**H,  T = Q*P*Z**H,
038: *  
039: *  where Q and Z are unitary matrices and S and P are upper triangular.
040: *  
041: *  Optionally, the unitary matrix Q from the generalized Schur
042: *  factorization may be postmultiplied into an input matrix Q1, and the
043: *  unitary matrix Z may be postmultiplied into an input matrix Z1.
044: *  If Q1 and Z1 are the unitary matrices from ZGGHRD that reduced
045: *  the matrix pair (A,B) to generalized Hessenberg form, then the output
046: *  matrices Q1*Q and Z1*Z are the unitary factors from the generalized
047: *  Schur factorization of (A,B):
048: *  
049: *     A = (Q1*Q)*S*(Z1*Z)**H,  B = (Q1*Q)*P*(Z1*Z)**H.
050: *  
051: *  To avoid overflow, eigenvalues of the matrix pair (H,T)
052: *  (equivalently, of (A,B)) are computed as a pair of complex values
053: *  (alpha,beta).  If beta is nonzero, lambda = alpha / beta is an
054: *  eigenvalue of the generalized nonsymmetric eigenvalue problem (GNEP)
055: *     A*x = lambda*B*x
056: *  and if alpha is nonzero, mu = beta / alpha is an eigenvalue of the
057: *  alternate form of the GNEP
058: *     mu*A*y = B*y.
059: *  The values of alpha and beta for the i-th eigenvalue can be read
060: *  directly from the generalized Schur form:  alpha = S(i,i),
061: *  beta = P(i,i).
062: *
063: *  Ref: C.B. Moler & G.W. Stewart, "An Algorithm for Generalized Matrix
064: *       Eigenvalue Problems", SIAM J. Numer. Anal., 10(1973),
065: *       pp. 241--256.
066: *
067: *  Arguments
068: *  =========
069: *
070: *  JOB     (input) CHARACTER*1
071: *          = 'E': Compute eigenvalues only;
072: *          = 'S': Computer eigenvalues and the Schur form.
073: *
074: *  COMPQ   (input) CHARACTER*1
075: *          = 'N': Left Schur vectors (Q) are not computed;
076: *          = 'I': Q is initialized to the unit matrix and the matrix Q
077: *                 of left Schur vectors of (H,T) is returned;
078: *          = 'V': Q must contain a unitary matrix Q1 on entry and
079: *                 the product Q1*Q is returned.
080: *
081: *  COMPZ   (input) CHARACTER*1
082: *          = 'N': Right Schur vectors (Z) are not computed;
083: *          = 'I': Q is initialized to the unit matrix and the matrix Z
084: *                 of right Schur vectors of (H,T) is returned;
085: *          = 'V': Z must contain a unitary matrix Z1 on entry and
086: *                 the product Z1*Z is returned.
087: *
088: *  N       (input) INTEGER
089: *          The order of the matrices H, T, Q, and Z.  N >= 0.
090: *
091: *  ILO     (input) INTEGER
092: *  IHI     (input) INTEGER
093: *          ILO and IHI mark the rows and columns of H which are in
094: *          Hessenberg form.  It is assumed that A is already upper
095: *          triangular in rows and columns 1:ILO-1 and IHI+1:N.
096: *          If N > 0, 1 <= ILO <= IHI <= N; if N = 0, ILO=1 and IHI=0.
097: *
098: *  H       (input/output) COMPLEX*16 array, dimension (LDH, N)
099: *          On entry, the N-by-N upper Hessenberg matrix H.
100: *          On exit, if JOB = 'S', H contains the upper triangular
101: *          matrix S from the generalized Schur factorization.
102: *          If JOB = 'E', the diagonal of H matches that of S, but
103: *          the rest of H is unspecified.
104: *
105: *  LDH     (input) INTEGER
106: *          The leading dimension of the array H.  LDH >= max( 1, N ).
107: *
108: *  T       (input/output) COMPLEX*16 array, dimension (LDT, N)
109: *          On entry, the N-by-N upper triangular matrix T.
110: *          On exit, if JOB = 'S', T contains the upper triangular
111: *          matrix P from the generalized Schur factorization.
112: *          If JOB = 'E', the diagonal of T matches that of P, but
113: *          the rest of T is unspecified.
114: *
115: *  LDT     (input) INTEGER
116: *          The leading dimension of the array T.  LDT >= max( 1, N ).
117: *
118: *  ALPHA   (output) COMPLEX*16 array, dimension (N)
119: *          The complex scalars alpha that define the eigenvalues of
120: *          GNEP.  ALPHA(i) = S(i,i) in the generalized Schur
121: *          factorization.
122: *
123: *  BETA    (output) COMPLEX*16 array, dimension (N)
124: *          The real non-negative scalars beta that define the
125: *          eigenvalues of GNEP.  BETA(i) = P(i,i) in the generalized
126: *          Schur factorization.
127: *
128: *          Together, the quantities alpha = ALPHA(j) and beta = BETA(j)
129: *          represent the j-th eigenvalue of the matrix pair (A,B), in
130: *          one of the forms lambda = alpha/beta or mu = beta/alpha.
131: *          Since either lambda or mu may overflow, they should not,
132: *          in general, be computed.
133: *
134: *  Q       (input/output) COMPLEX*16 array, dimension (LDQ, N)
135: *          On entry, if COMPZ = 'V', the unitary matrix Q1 used in the
136: *          reduction of (A,B) to generalized Hessenberg form.
137: *          On exit, if COMPZ = 'I', the unitary matrix of left Schur
138: *          vectors of (H,T), and if COMPZ = 'V', the unitary matrix of
139: *          left Schur vectors of (A,B).
140: *          Not referenced if COMPZ = 'N'.
141: *
142: *  LDQ     (input) INTEGER
143: *          The leading dimension of the array Q.  LDQ >= 1.
144: *          If COMPQ='V' or 'I', then LDQ >= N.
145: *
146: *  Z       (input/output) COMPLEX*16 array, dimension (LDZ, N)
147: *          On entry, if COMPZ = 'V', the unitary matrix Z1 used in the
148: *          reduction of (A,B) to generalized Hessenberg form.
149: *          On exit, if COMPZ = 'I', the unitary matrix of right Schur
150: *          vectors of (H,T), and if COMPZ = 'V', the unitary matrix of
151: *          right Schur vectors of (A,B).
152: *          Not referenced if COMPZ = 'N'.
153: *
154: *  LDZ     (input) INTEGER
155: *          The leading dimension of the array Z.  LDZ >= 1.
156: *          If COMPZ='V' or 'I', then LDZ >= N.
157: *
158: *  WORK    (workspace/output) COMPLEX*16 array, dimension (MAX(1,LWORK))
159: *          On exit, if INFO >= 0, WORK(1) returns the optimal LWORK.
160: *
161: *  LWORK   (input) INTEGER
162: *          The dimension of the array WORK.  LWORK >= max(1,N).
163: *
164: *          If LWORK = -1, then a workspace query is assumed; the routine
165: *          only calculates the optimal size of the WORK array, returns
166: *          this value as the first entry of the WORK array, and no error
167: *          message related to LWORK is issued by XERBLA.
168: *
169: *  RWORK   (workspace) DOUBLE PRECISION array, dimension (N)
170: *
171: *  INFO    (output) INTEGER
172: *          = 0: successful exit
173: *          < 0: if INFO = -i, the i-th argument had an illegal value
174: *          = 1,...,N: the QZ iteration did not converge.  (H,T) is not
175: *                     in Schur form, but ALPHA(i) and BETA(i),
176: *                     i=INFO+1,...,N should be correct.
177: *          = N+1,...,2*N: the shift calculation failed.  (H,T) is not
178: *                     in Schur form, but ALPHA(i) and BETA(i),
179: *                     i=INFO-N+1,...,N should be correct.
180: *
181: *  Further Details
182: *  ===============
183: *
184: *  We assume that complex ABS works as long as its value is less than
185: *  overflow.
186: *
187: *  =====================================================================
188: *
189: *     .. Parameters ..
190:       COMPLEX*16         CZERO, CONE
191:       PARAMETER          ( CZERO = ( 0.0D+0, 0.0D+0 ),
192:      $                   CONE = ( 1.0D+0, 0.0D+0 ) )
193:       DOUBLE PRECISION   ZERO, ONE
194:       PARAMETER          ( ZERO = 0.0D+0, ONE = 1.0D+0 )
195:       DOUBLE PRECISION   HALF
196:       PARAMETER          ( HALF = 0.5D+0 )
197: *     ..
198: *     .. Local Scalars ..
199:       LOGICAL            ILAZR2, ILAZRO, ILQ, ILSCHR, ILZ, LQUERY
200:       INTEGER            ICOMPQ, ICOMPZ, IFIRST, IFRSTM, IITER, ILAST,
201:      $                   ILASTM, IN, ISCHUR, ISTART, J, JC, JCH, JITER,
202:      $                   JR, MAXIT
203:       DOUBLE PRECISION   ABSB, ANORM, ASCALE, ATOL, BNORM, BSCALE, BTOL,
204:      $                   C, SAFMIN, TEMP, TEMP2, TEMPR, ULP
205:       COMPLEX*16         ABI22, AD11, AD12, AD21, AD22, CTEMP, CTEMP2,
206:      $                   CTEMP3, ESHIFT, RTDISC, S, SHIFT, SIGNBC, T1,
207:      $                   U12, X
208: *     ..
209: *     .. External Functions ..
210:       LOGICAL            LSAME
211:       DOUBLE PRECISION   DLAMCH, ZLANHS
212:       EXTERNAL           LSAME, DLAMCH, ZLANHS
213: *     ..
214: *     .. External Subroutines ..
215:       EXTERNAL           XERBLA, ZLARTG, ZLASET, ZROT, ZSCAL
216: *     ..
217: *     .. Intrinsic Functions ..
218:       INTRINSIC          ABS, DBLE, DCMPLX, DCONJG, DIMAG, MAX, MIN,
219:      $                   SQRT
220: *     ..
221: *     .. Statement Functions ..
222:       DOUBLE PRECISION   ABS1
223: *     ..
224: *     .. Statement Function definitions ..
225:       ABS1( X ) = ABS( DBLE( X ) ) + ABS( DIMAG( X ) )
226: *     ..
227: *     .. Executable Statements ..
228: *
229: *     Decode JOB, COMPQ, COMPZ
230: *
231:       IF( LSAME( JOB, 'E' ) ) THEN
232:          ILSCHR = .FALSE.
233:          ISCHUR = 1
234:       ELSE IF( LSAME( JOB, 'S' ) ) THEN
235:          ILSCHR = .TRUE.
236:          ISCHUR = 2
237:       ELSE
238:          ISCHUR = 0
239:       END IF
240: *
241:       IF( LSAME( COMPQ, 'N' ) ) THEN
242:          ILQ = .FALSE.
243:          ICOMPQ = 1
244:       ELSE IF( LSAME( COMPQ, 'V' ) ) THEN
245:          ILQ = .TRUE.
246:          ICOMPQ = 2
247:       ELSE IF( LSAME( COMPQ, 'I' ) ) THEN
248:          ILQ = .TRUE.
249:          ICOMPQ = 3
250:       ELSE
251:          ICOMPQ = 0
252:       END IF
253: *
254:       IF( LSAME( COMPZ, 'N' ) ) THEN
255:          ILZ = .FALSE.
256:          ICOMPZ = 1
257:       ELSE IF( LSAME( COMPZ, 'V' ) ) THEN
258:          ILZ = .TRUE.
259:          ICOMPZ = 2
260:       ELSE IF( LSAME( COMPZ, 'I' ) ) THEN
261:          ILZ = .TRUE.
262:          ICOMPZ = 3
263:       ELSE
264:          ICOMPZ = 0
265:       END IF
266: *
267: *     Check Argument Values
268: *
269:       INFO = 0
270:       WORK( 1 ) = MAX( 1, N )
271:       LQUERY = ( LWORK.EQ.-1 )
272:       IF( ISCHUR.EQ.0 ) THEN
273:          INFO = -1
274:       ELSE IF( ICOMPQ.EQ.0 ) THEN
275:          INFO = -2
276:       ELSE IF( ICOMPZ.EQ.0 ) THEN
277:          INFO = -3
278:       ELSE IF( N.LT.0 ) THEN
279:          INFO = -4
280:       ELSE IF( ILO.LT.1 ) THEN
281:          INFO = -5
282:       ELSE IF( IHI.GT.N .OR. IHI.LT.ILO-1 ) THEN
283:          INFO = -6
284:       ELSE IF( LDH.LT.N ) THEN
285:          INFO = -8
286:       ELSE IF( LDT.LT.N ) THEN
287:          INFO = -10
288:       ELSE IF( LDQ.LT.1 .OR. ( ILQ .AND. LDQ.LT.N ) ) THEN
289:          INFO = -14
290:       ELSE IF( LDZ.LT.1 .OR. ( ILZ .AND. LDZ.LT.N ) ) THEN
291:          INFO = -16
292:       ELSE IF( LWORK.LT.MAX( 1, N ) .AND. .NOT.LQUERY ) THEN
293:          INFO = -18
294:       END IF
295:       IF( INFO.NE.0 ) THEN
296:          CALL XERBLA( 'ZHGEQZ', -INFO )
297:          RETURN
298:       ELSE IF( LQUERY ) THEN
299:          RETURN
300:       END IF
301: *
302: *     Quick return if possible
303: *
304: *     WORK( 1 ) = CMPLX( 1 )
305:       IF( N.LE.0 ) THEN
306:          WORK( 1 ) = DCMPLX( 1 )
307:          RETURN
308:       END IF
309: *
310: *     Initialize Q and Z
311: *
312:       IF( ICOMPQ.EQ.3 )
313:      $   CALL ZLASET( 'Full', N, N, CZERO, CONE, Q, LDQ )
314:       IF( ICOMPZ.EQ.3 )
315:      $   CALL ZLASET( 'Full', N, N, CZERO, CONE, Z, LDZ )
316: *
317: *     Machine Constants
318: *
319:       IN = IHI + 1 - ILO
320:       SAFMIN = DLAMCH( 'S' )
321:       ULP = DLAMCH( 'E' )*DLAMCH( 'B' )
322:       ANORM = ZLANHS( 'F', IN, H( ILO, ILO ), LDH, RWORK )
323:       BNORM = ZLANHS( 'F', IN, T( ILO, ILO ), LDT, RWORK )
324:       ATOL = MAX( SAFMIN, ULP*ANORM )
325:       BTOL = MAX( SAFMIN, ULP*BNORM )
326:       ASCALE = ONE / MAX( SAFMIN, ANORM )
327:       BSCALE = ONE / MAX( SAFMIN, BNORM )
328: *
329: *
330: *     Set Eigenvalues IHI+1:N
331: *
332:       DO 10 J = IHI + 1, N
333:          ABSB = ABS( T( J, J ) )
334:          IF( ABSB.GT.SAFMIN ) THEN
335:             SIGNBC = DCONJG( T( J, J ) / ABSB )
336:             T( J, J ) = ABSB
337:             IF( ILSCHR ) THEN
338:                CALL ZSCAL( J-1, SIGNBC, T( 1, J ), 1 )
339:                CALL ZSCAL( J, SIGNBC, H( 1, J ), 1 )
340:             ELSE
341:                H( J, J ) = H( J, J )*SIGNBC
342:             END IF
343:             IF( ILZ )
344:      $         CALL ZSCAL( N, SIGNBC, Z( 1, J ), 1 )
345:          ELSE
346:             T( J, J ) = CZERO
347:          END IF
348:          ALPHA( J ) = H( J, J )
349:          BETA( J ) = T( J, J )
350:    10 CONTINUE
351: *
352: *     If IHI < ILO, skip QZ steps
353: *
354:       IF( IHI.LT.ILO )
355:      $   GO TO 190
356: *
357: *     MAIN QZ ITERATION LOOP
358: *
359: *     Initialize dynamic indices
360: *
361: *     Eigenvalues ILAST+1:N have been found.
362: *        Column operations modify rows IFRSTM:whatever
363: *        Row operations modify columns whatever:ILASTM
364: *
365: *     If only eigenvalues are being computed, then
366: *        IFRSTM is the row of the last splitting row above row ILAST;
367: *        this is always at least ILO.
368: *     IITER counts iterations since the last eigenvalue was found,
369: *        to tell when to use an extraordinary shift.
370: *     MAXIT is the maximum number of QZ sweeps allowed.
371: *
372:       ILAST = IHI
373:       IF( ILSCHR ) THEN
374:          IFRSTM = 1
375:          ILASTM = N
376:       ELSE
377:          IFRSTM = ILO
378:          ILASTM = IHI
379:       END IF
380:       IITER = 0
381:       ESHIFT = CZERO
382:       MAXIT = 30*( IHI-ILO+1 )
383: *
384:       DO 170 JITER = 1, MAXIT
385: *
386: *        Check for too many iterations.
387: *
388:          IF( JITER.GT.MAXIT )
389:      $      GO TO 180
390: *
391: *        Split the matrix if possible.
392: *
393: *        Two tests:
394: *           1: H(j,j-1)=0  or  j=ILO
395: *           2: T(j,j)=0
396: *
397: *        Special case: j=ILAST
398: *
399:          IF( ILAST.EQ.ILO ) THEN
400:             GO TO 60
401:          ELSE
402:             IF( ABS1( H( ILAST, ILAST-1 ) ).LE.ATOL ) THEN
403:                H( ILAST, ILAST-1 ) = CZERO
404:                GO TO 60
405:             END IF
406:          END IF
407: *
408:          IF( ABS( T( ILAST, ILAST ) ).LE.BTOL ) THEN
409:             T( ILAST, ILAST ) = CZERO
410:             GO TO 50
411:          END IF
412: *
413: *        General case: j<ILAST
414: *
415:          DO 40 J = ILAST - 1, ILO, -1
416: *
417: *           Test 1: for H(j,j-1)=0 or j=ILO
418: *
419:             IF( J.EQ.ILO ) THEN
420:                ILAZRO = .TRUE.
421:             ELSE
422:                IF( ABS1( H( J, J-1 ) ).LE.ATOL ) THEN
423:                   H( J, J-1 ) = CZERO
424:                   ILAZRO = .TRUE.
425:                ELSE
426:                   ILAZRO = .FALSE.
427:                END IF
428:             END IF
429: *
430: *           Test 2: for T(j,j)=0
431: *
432:             IF( ABS( T( J, J ) ).LT.BTOL ) THEN
433:                T( J, J ) = CZERO
434: *
435: *              Test 1a: Check for 2 consecutive small subdiagonals in A
436: *
437:                ILAZR2 = .FALSE.
438:                IF( .NOT.ILAZRO ) THEN
439:                   IF( ABS1( H( J, J-1 ) )*( ASCALE*ABS1( H( J+1,
440:      $                J ) ) ).LE.ABS1( H( J, J ) )*( ASCALE*ATOL ) )
441:      $                ILAZR2 = .TRUE.
442:                END IF
443: *
444: *              If both tests pass (1 & 2), i.e., the leading diagonal
445: *              element of B in the block is zero, split a 1x1 block off
446: *              at the top. (I.e., at the J-th row/column) The leading
447: *              diagonal element of the remainder can also be zero, so
448: *              this may have to be done repeatedly.
449: *
450:                IF( ILAZRO .OR. ILAZR2 ) THEN
451:                   DO 20 JCH = J, ILAST - 1
452:                      CTEMP = H( JCH, JCH )
453:                      CALL ZLARTG( CTEMP, H( JCH+1, JCH ), C, S,
454:      $                            H( JCH, JCH ) )
455:                      H( JCH+1, JCH ) = CZERO
456:                      CALL ZROT( ILASTM-JCH, H( JCH, JCH+1 ), LDH,
457:      $                          H( JCH+1, JCH+1 ), LDH, C, S )
458:                      CALL ZROT( ILASTM-JCH, T( JCH, JCH+1 ), LDT,
459:      $                          T( JCH+1, JCH+1 ), LDT, C, S )
460:                      IF( ILQ )
461:      $                  CALL ZROT( N, Q( 1, JCH ), 1, Q( 1, JCH+1 ), 1,
462:      $                             C, DCONJG( S ) )
463:                      IF( ILAZR2 )
464:      $                  H( JCH, JCH-1 ) = H( JCH, JCH-1 )*C
465:                      ILAZR2 = .FALSE.
466:                      IF( ABS1( T( JCH+1, JCH+1 ) ).GE.BTOL ) THEN
467:                         IF( JCH+1.GE.ILAST ) THEN
468:                            GO TO 60
469:                         ELSE
470:                            IFIRST = JCH + 1
471:                            GO TO 70
472:                         END IF
473:                      END IF
474:                      T( JCH+1, JCH+1 ) = CZERO
475:    20             CONTINUE
476:                   GO TO 50
477:                ELSE
478: *
479: *                 Only test 2 passed -- chase the zero to T(ILAST,ILAST)
480: *                 Then process as in the case T(ILAST,ILAST)=0
481: *
482:                   DO 30 JCH = J, ILAST - 1
483:                      CTEMP = T( JCH, JCH+1 )
484:                      CALL ZLARTG( CTEMP, T( JCH+1, JCH+1 ), C, S,
485:      $                            T( JCH, JCH+1 ) )
486:                      T( JCH+1, JCH+1 ) = CZERO
487:                      IF( JCH.LT.ILASTM-1 )
488:      $                  CALL ZROT( ILASTM-JCH-1, T( JCH, JCH+2 ), LDT,
489:      $                             T( JCH+1, JCH+2 ), LDT, C, S )
490:                      CALL ZROT( ILASTM-JCH+2, H( JCH, JCH-1 ), LDH,
491:      $                          H( JCH+1, JCH-1 ), LDH, C, S )
492:                      IF( ILQ )
493:      $                  CALL ZROT( N, Q( 1, JCH ), 1, Q( 1, JCH+1 ), 1,
494:      $                             C, DCONJG( S ) )
495:                      CTEMP = H( JCH+1, JCH )
496:                      CALL ZLARTG( CTEMP, H( JCH+1, JCH-1 ), C, S,
497:      $                            H( JCH+1, JCH ) )
498:                      H( JCH+1, JCH-1 ) = CZERO
499:                      CALL ZROT( JCH+1-IFRSTM, H( IFRSTM, JCH ), 1,
500:      $                          H( IFRSTM, JCH-1 ), 1, C, S )
501:                      CALL ZROT( JCH-IFRSTM, T( IFRSTM, JCH ), 1,
502:      $                          T( IFRSTM, JCH-1 ), 1, C, S )
503:                      IF( ILZ )
504:      $                  CALL ZROT( N, Z( 1, JCH ), 1, Z( 1, JCH-1 ), 1,
505:      $                             C, S )
506:    30             CONTINUE
507:                   GO TO 50
508:                END IF
509:             ELSE IF( ILAZRO ) THEN
510: *
511: *              Only test 1 passed -- work on J:ILAST
512: *
513:                IFIRST = J
514:                GO TO 70
515:             END IF
516: *
517: *           Neither test passed -- try next J
518: *
519:    40    CONTINUE
520: *
521: *        (Drop-through is "impossible")
522: *
523:          INFO = 2*N + 1
524:          GO TO 210
525: *
526: *        T(ILAST,ILAST)=0 -- clear H(ILAST,ILAST-1) to split off a
527: *        1x1 block.
528: *
529:    50    CONTINUE
530:          CTEMP = H( ILAST, ILAST )
531:          CALL ZLARTG( CTEMP, H( ILAST, ILAST-1 ), C, S,
532:      $                H( ILAST, ILAST ) )
533:          H( ILAST, ILAST-1 ) = CZERO
534:          CALL ZROT( ILAST-IFRSTM, H( IFRSTM, ILAST ), 1,
535:      $              H( IFRSTM, ILAST-1 ), 1, C, S )
536:          CALL ZROT( ILAST-IFRSTM, T( IFRSTM, ILAST ), 1,
537:      $              T( IFRSTM, ILAST-1 ), 1, C, S )
538:          IF( ILZ )
539:      $      CALL ZROT( N, Z( 1, ILAST ), 1, Z( 1, ILAST-1 ), 1, C, S )
540: *
541: *        H(ILAST,ILAST-1)=0 -- Standardize B, set ALPHA and BETA
542: *
543:    60    CONTINUE
544:          ABSB = ABS( T( ILAST, ILAST ) )
545:          IF( ABSB.GT.SAFMIN ) THEN
546:             SIGNBC = DCONJG( T( ILAST, ILAST ) / ABSB )
547:             T( ILAST, ILAST ) = ABSB
548:             IF( ILSCHR ) THEN
549:                CALL ZSCAL( ILAST-IFRSTM, SIGNBC, T( IFRSTM, ILAST ), 1 )
550:                CALL ZSCAL( ILAST+1-IFRSTM, SIGNBC, H( IFRSTM, ILAST ),
551:      $                     1 )
552:             ELSE
553:                H( ILAST, ILAST ) = H( ILAST, ILAST )*SIGNBC
554:             END IF
555:             IF( ILZ )
556:      $         CALL ZSCAL( N, SIGNBC, Z( 1, ILAST ), 1 )
557:          ELSE
558:             T( ILAST, ILAST ) = CZERO
559:          END IF
560:          ALPHA( ILAST ) = H( ILAST, ILAST )
561:          BETA( ILAST ) = T( ILAST, ILAST )
562: *
563: *        Go to next block -- exit if finished.
564: *
565:          ILAST = ILAST - 1
566:          IF( ILAST.LT.ILO )
567:      $      GO TO 190
568: *
569: *        Reset counters
570: *
571:          IITER = 0
572:          ESHIFT = CZERO
573:          IF( .NOT.ILSCHR ) THEN
574:             ILASTM = ILAST
575:             IF( IFRSTM.GT.ILAST )
576:      $         IFRSTM = ILO
577:          END IF
578:          GO TO 160
579: *
580: *        QZ step
581: *
582: *        This iteration only involves rows/columns IFIRST:ILAST.  We
583: *        assume IFIRST < ILAST, and that the diagonal of B is non-zero.
584: *
585:    70    CONTINUE
586:          IITER = IITER + 1
587:          IF( .NOT.ILSCHR ) THEN
588:             IFRSTM = IFIRST
589:          END IF
590: *
591: *        Compute the Shift.
592: *
593: *        At this point, IFIRST < ILAST, and the diagonal elements of
594: *        T(IFIRST:ILAST,IFIRST,ILAST) are larger than BTOL (in
595: *        magnitude)
596: *
597:          IF( ( IITER / 10 )*10.NE.IITER ) THEN
598: *
599: *           The Wilkinson shift (AEP p.512), i.e., the eigenvalue of
600: *           the bottom-right 2x2 block of A inv(B) which is nearest to
601: *           the bottom-right element.
602: *
603: *           We factor B as U*D, where U has unit diagonals, and
604: *           compute (A*inv(D))*inv(U).
605: *
606:             U12 = ( BSCALE*T( ILAST-1, ILAST ) ) /
607:      $            ( BSCALE*T( ILAST, ILAST ) )
608:             AD11 = ( ASCALE*H( ILAST-1, ILAST-1 ) ) /
609:      $             ( BSCALE*T( ILAST-1, ILAST-1 ) )
610:             AD21 = ( ASCALE*H( ILAST, ILAST-1 ) ) /
611:      $             ( BSCALE*T( ILAST-1, ILAST-1 ) )
612:             AD12 = ( ASCALE*H( ILAST-1, ILAST ) ) /
613:      $             ( BSCALE*T( ILAST, ILAST ) )
614:             AD22 = ( ASCALE*H( ILAST, ILAST ) ) /
615:      $             ( BSCALE*T( ILAST, ILAST ) )
616:             ABI22 = AD22 - U12*AD21
617: *
618:             T1 = HALF*( AD11+ABI22 )
619:             RTDISC = SQRT( T1**2+AD12*AD21-AD11*AD22 )
620:             TEMP = DBLE( T1-ABI22 )*DBLE( RTDISC ) +
621:      $             DIMAG( T1-ABI22 )*DIMAG( RTDISC )
622:             IF( TEMP.LE.ZERO ) THEN
623:                SHIFT = T1 + RTDISC
624:             ELSE
625:                SHIFT = T1 - RTDISC
626:             END IF
627:          ELSE
628: *
629: *           Exceptional shift.  Chosen for no particularly good reason.
630: *
631:             ESHIFT = ESHIFT + DCONJG( ( ASCALE*H( ILAST-1, ILAST ) ) /
632:      $               ( BSCALE*T( ILAST-1, ILAST-1 ) ) )
633:             SHIFT = ESHIFT
634:          END IF
635: *
636: *        Now check for two consecutive small subdiagonals.
637: *
638:          DO 80 J = ILAST - 1, IFIRST + 1, -1
639:             ISTART = J
640:             CTEMP = ASCALE*H( J, J ) - SHIFT*( BSCALE*T( J, J ) )
641:             TEMP = ABS1( CTEMP )
642:             TEMP2 = ASCALE*ABS1( H( J+1, J ) )
643:             TEMPR = MAX( TEMP, TEMP2 )
644:             IF( TEMPR.LT.ONE .AND. TEMPR.NE.ZERO ) THEN
645:                TEMP = TEMP / TEMPR
646:                TEMP2 = TEMP2 / TEMPR
647:             END IF
648:             IF( ABS1( H( J, J-1 ) )*TEMP2.LE.TEMP*ATOL )
649:      $         GO TO 90
650:    80    CONTINUE
651: *
652:          ISTART = IFIRST
653:          CTEMP = ASCALE*H( IFIRST, IFIRST ) -
654:      $           SHIFT*( BSCALE*T( IFIRST, IFIRST ) )
655:    90    CONTINUE
656: *
657: *        Do an implicit-shift QZ sweep.
658: *
659: *        Initial Q
660: *
661:          CTEMP2 = ASCALE*H( ISTART+1, ISTART )
662:          CALL ZLARTG( CTEMP, CTEMP2, C, S, CTEMP3 )
663: *
664: *        Sweep
665: *
666:          DO 150 J = ISTART, ILAST - 1
667:             IF( J.GT.ISTART ) THEN
668:                CTEMP = H( J, J-1 )
669:                CALL ZLARTG( CTEMP, H( J+1, J-1 ), C, S, H( J, J-1 ) )
670:                H( J+1, J-1 ) = CZERO
671:             END IF
672: *
673:             DO 100 JC = J, ILASTM
674:                CTEMP = C*H( J, JC ) + S*H( J+1, JC )
675:                H( J+1, JC ) = -DCONJG( S )*H( J, JC ) + C*H( J+1, JC )
676:                H( J, JC ) = CTEMP
677:                CTEMP2 = C*T( J, JC ) + S*T( J+1, JC )
678:                T( J+1, JC ) = -DCONJG( S )*T( J, JC ) + C*T( J+1, JC )
679:                T( J, JC ) = CTEMP2
680:   100       CONTINUE
681:             IF( ILQ ) THEN
682:                DO 110 JR = 1, N
683:                   CTEMP = C*Q( JR, J ) + DCONJG( S )*Q( JR, J+1 )
684:                   Q( JR, J+1 ) = -S*Q( JR, J ) + C*Q( JR, J+1 )
685:                   Q( JR, J ) = CTEMP
686:   110          CONTINUE
687:             END IF
688: *
689:             CTEMP = T( J+1, J+1 )
690:             CALL ZLARTG( CTEMP, T( J+1, J ), C, S, T( J+1, J+1 ) )
691:             T( J+1, J ) = CZERO
692: *
693:             DO 120 JR = IFRSTM, MIN( J+2, ILAST )
694:                CTEMP = C*H( JR, J+1 ) + S*H( JR, J )
695:                H( JR, J ) = -DCONJG( S )*H( JR, J+1 ) + C*H( JR, J )
696:                H( JR, J+1 ) = CTEMP
697:   120       CONTINUE
698:             DO 130 JR = IFRSTM, J
699:                CTEMP = C*T( JR, J+1 ) + S*T( JR, J )
700:                T( JR, J ) = -DCONJG( S )*T( JR, J+1 ) + C*T( JR, J )
701:                T( JR, J+1 ) = CTEMP
702:   130       CONTINUE
703:             IF( ILZ ) THEN
704:                DO 140 JR = 1, N
705:                   CTEMP = C*Z( JR, J+1 ) + S*Z( JR, J )
706:                   Z( JR, J ) = -DCONJG( S )*Z( JR, J+1 ) + C*Z( JR, J )
707:                   Z( JR, J+1 ) = CTEMP
708:   140          CONTINUE
709:             END IF
710:   150    CONTINUE
711: *
712:   160    CONTINUE
713: *
714:   170 CONTINUE
715: *
716: *     Drop-through = non-convergence
717: *
718:   180 CONTINUE
719:       INFO = ILAST
720:       GO TO 210
721: *
722: *     Successful completion of all QZ steps
723: *
724:   190 CONTINUE
725: *
726: *     Set Eigenvalues 1:ILO-1
727: *
728:       DO 200 J = 1, ILO - 1
729:          ABSB = ABS( T( J, J ) )
730:          IF( ABSB.GT.SAFMIN ) THEN
731:             SIGNBC = DCONJG( T( J, J ) / ABSB )
732:             T( J, J ) = ABSB
733:             IF( ILSCHR ) THEN
734:                CALL ZSCAL( J-1, SIGNBC, T( 1, J ), 1 )
735:                CALL ZSCAL( J, SIGNBC, H( 1, J ), 1 )
736:             ELSE
737:                H( J, J ) = H( J, J )*SIGNBC
738:             END IF
739:             IF( ILZ )
740:      $         CALL ZSCAL( N, SIGNBC, Z( 1, J ), 1 )
741:          ELSE
742:             T( J, J ) = CZERO
743:          END IF
744:          ALPHA( J ) = H( J, J )
745:          BETA( J ) = T( J, J )
746:   200 CONTINUE
747: *
748: *     Normal Termination
749: *
750:       INFO = 0
751: *
752: *     Exit (other than argument error) -- return optimal workspace size
753: *
754:   210 CONTINUE
755:       WORK( 1 ) = DCMPLX( N )
756:       RETURN
757: *
758: *     End of ZHGEQZ
759: *
760:       END
761: