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chgeqz.f
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1 *> \brief \b CHGEQZ
2 *
3 * =========== DOCUMENTATION ===========
4 *
5 * Online html documentation available at
6 * http://www.netlib.org/lapack/explore-html/
7 *
8 *> \htmlonly
9 *> Download CHGEQZ + dependencies
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11 *> [TGZ]</a>
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13 *> [ZIP]</a>
14 *> <a href="http://www.netlib.org/cgi-bin/netlibfiles.txt?format=txt&filename=/lapack/lapack_routine/chgeqz.f">
15 *> [TXT]</a>
16 *> \endhtmlonly
17 *
18 * Definition:
19 * ===========
20 *
21 * SUBROUTINE CHGEQZ( JOB, COMPQ, COMPZ, N, ILO, IHI, H, LDH, T, LDT,
22 * ALPHA, BETA, Q, LDQ, Z, LDZ, WORK, LWORK,
23 * RWORK, INFO )
24 *
25 * .. Scalar Arguments ..
26 * CHARACTER COMPQ, COMPZ, JOB
27 * INTEGER IHI, ILO, INFO, LDH, LDQ, LDT, LDZ, LWORK, N
28 * ..
29 * .. Array Arguments ..
30 * REAL RWORK( * )
31 * COMPLEX ALPHA( * ), BETA( * ), H( LDH, * ),
32 * $ Q( LDQ, * ), T( LDT, * ), WORK( * ),
33 * $ Z( LDZ, * )
34 * ..
35 *
36 *
37 *> \par Purpose:
38 * =============
39 *>
40 *> \verbatim
41 *>
42 *> CHGEQZ 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):
47 *>
48 *> A = Q1*H*Z1**H, B = Q1*T*Z1**H,
49 *>
50 *> as computed by CGGHRD.
51 *>
52 *> If JOB='S', then the Hessenberg-triangular pair (H,T) is
53 *> also reduced to generalized Schur form,
54 *>
55 *> H = Q*S*Z**H, T = Q*P*Z**H,
56 *>
57 *> where Q and Z are unitary matrices and S and P are upper triangular.
58 *>
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 CGGHRD 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):
66 *>
67 *> A = (Q1*Q)*S*(Z1*Z)**H, B = (Q1*Q)*P*(Z1*Z)**H.
68 *>
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 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 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 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 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 array, dimension (LDQ, N)
193 *> On entry, if COMPZ = 'V', the unitary matrix Q1 used in the
194 *> reduction of (A,B) to generalized Hessenberg form.
195 *> On exit, if COMPZ = 'I', the unitary matrix of left Schur
196 *> vectors of (H,T), and if COMPZ = 'V', the unitary matrix of
197 *> left Schur vectors of (A,B).
198 *> Not referenced if COMPZ = 'N'.
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 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 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 REAL 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 *
264 *> \author Univ. of Tennessee
265 *> \author Univ. of California Berkeley
266 *> \author Univ. of Colorado Denver
267 *> \author NAG Ltd.
268 *
269 *> \date April 2012
270 *
271 *> \ingroup complexGEcomputational
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 * =====================================================================
283  SUBROUTINE chgeqz( JOB, COMPQ, COMPZ, N, ILO, IHI, H, LDH, T, LDT,
284  $ alpha, beta, q, ldq, z, ldz, work, lwork,
285  $ rwork, info )
286 *
287 * -- LAPACK computational routine (version 3.4.1) --
288 * -- LAPACK is a software package provided by Univ. of Tennessee, --
289 * -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
290 * April 2012
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  REAL rwork( * )
298  COMPLEX alpha( * ), beta( * ), h( ldh, * ),
299  $ q( ldq, * ), t( ldt, * ), work( * ),
300  $ z( ldz, * )
301 * ..
302 *
303 * =====================================================================
304 *
305 * .. Parameters ..
306  COMPLEX czero, cone
307  parameter( czero = ( 0.0e+0, 0.0e+0 ),
308  $ cone = ( 1.0e+0, 0.0e+0 ) )
309  REAL zero, one
310  parameter( zero = 0.0e+0, one = 1.0e+0 )
311  REAL half
312  parameter( half = 0.5e+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  REAL absb, anorm, ascale, atol, bnorm, bscale, btol,
320  $ c, safmin, temp, temp2, tempr, ulp
321  COMPLEX 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  REAL clanhs, slamch
328  EXTERNAL lsame, clanhs, slamch
329 * ..
330 * .. External Subroutines ..
331  EXTERNAL clartg, claset, crot, cscal, xerbla
332 * ..
333 * .. Intrinsic Functions ..
334  INTRINSIC abs, aimag, cmplx, conjg, max, min, REAL, sqrt
335 * ..
336 * .. Statement Functions ..
337  REAL abs1
338 * ..
339 * .. Statement Function definitions ..
340  abs1( x ) = abs( REAL( X ) ) + abs( aimag( x ) )
341 * ..
342 * .. Executable Statements ..
343 *
344 * Decode JOB, COMPQ, COMPZ
345 *
346  IF( lsame( job, 'E' ) ) THEN
347  ilschr = .false.
348  ischur = 1
349  ELSE IF( lsame( job, 'S' ) ) THEN
350  ilschr = .true.
351  ischur = 2
352  ELSE
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
366  icompq = 0
367  END IF
368 *
369  IF( lsame( compz, 'N' ) ) THEN
370  ilz = .false.
371  icompz = 1
372  ELSE IF( lsame( compz, 'V' ) ) THEN
373  ilz = .true.
374  icompz = 2
375  ELSE IF( lsame( compz, 'I' ) ) THEN
376  ilz = .true.
377  icompz = 3
378  ELSE
379  icompz = 0
380  END IF
381 *
382 * Check Argument Values
383 *
384  info = 0
385  work( 1 ) = max( 1, n )
386  lquery = ( lwork.EQ.-1 )
387  IF( ischur.EQ.0 ) THEN
388  info = -1
389  ELSE IF( icompq.EQ.0 ) THEN
390  info = -2
391  ELSE IF( icompz.EQ.0 ) THEN
392  info = -3
393  ELSE IF( n.LT.0 ) THEN
394  info = -4
395  ELSE IF( ilo.LT.1 ) THEN
396  info = -5
397  ELSE IF( ihi.GT.n .OR. ihi.LT.ilo-1 ) THEN
398  info = -6
399  ELSE IF( ldh.LT.n ) THEN
400  info = -8
401  ELSE IF( ldt.LT.n ) THEN
402  info = -10
403  ELSE IF( ldq.LT.1 .OR. ( ilq .AND. ldq.LT.n ) ) THEN
404  info = -14
405  ELSE IF( ldz.LT.1 .OR. ( ilz .AND. ldz.LT.n ) ) THEN
406  info = -16
407  ELSE IF( lwork.LT.max( 1, n ) .AND. .NOT.lquery ) THEN
408  info = -18
409  END IF
410  IF( info.NE.0 ) THEN
411  CALL xerbla( 'CHGEQZ', -info )
412  RETURN
413  ELSE IF( lquery ) THEN
414  RETURN
415  END IF
416 *
417 * Quick return if possible
418 *
419 * WORK( 1 ) = CMPLX( 1 )
420  IF( n.LE.0 ) THEN
421  work( 1 ) = cmplx( 1 )
422  RETURN
423  END IF
424 *
425 * Initialize Q and Z
426 *
427  IF( icompq.EQ.3 )
428  $ CALL claset( 'Full', n, n, czero, cone, q, ldq )
429  IF( icompz.EQ.3 )
430  $ CALL claset( 'Full', n, n, czero, cone, z, ldz )
431 *
432 * Machine Constants
433 *
434  in = ihi + 1 - ilo
435  safmin = slamch( 'S' )
436  ulp = slamch( 'E' )*slamch( 'B' )
437  anorm = clanhs( 'F', in, h( ilo, ilo ), ldh, rwork )
438  bnorm = clanhs( 'F', in, t( ilo, ilo ), ldt, rwork )
439  atol = max( safmin, ulp*anorm )
440  btol = max( safmin, ulp*bnorm )
441  ascale = one / max( safmin, anorm )
442  bscale = one / max( safmin, bnorm )
443 *
444 *
445 * Set Eigenvalues IHI+1:N
446 *
447  DO 10 j = ihi + 1, n
448  absb = abs( t( j, j ) )
449  IF( absb.GT.safmin ) THEN
450  signbc = conjg( t( j, j ) / absb )
451  t( j, j ) = absb
452  IF( ilschr ) THEN
453  CALL cscal( j-1, signbc, t( 1, j ), 1 )
454  CALL cscal( j, signbc, h( 1, j ), 1 )
455  ELSE
456  h( j, j ) = h( j, j )*signbc
457  END IF
458  IF( ilz )
459  $ CALL cscal( n, signbc, z( 1, j ), 1 )
460  ELSE
461  t( j, j ) = czero
462  END IF
463  alpha( j ) = h( j, j )
464  beta( j ) = t( j, j )
465  10 CONTINUE
466 *
467 * If IHI < ILO, skip QZ steps
468 *
469  IF( ihi.LT.ilo )
470  $ go to 190
471 *
472 * MAIN QZ ITERATION LOOP
473 *
474 * Initialize dynamic indices
475 *
476 * Eigenvalues ILAST+1:N have been found.
477 * Column operations modify rows IFRSTM:whatever
478 * Row operations modify columns whatever:ILASTM
479 *
480 * If only eigenvalues are being computed, then
481 * IFRSTM is the row of the last splitting row above row ILAST;
482 * this is always at least ILO.
483 * IITER counts iterations since the last eigenvalue was found,
484 * to tell when to use an extraordinary shift.
485 * MAXIT is the maximum number of QZ sweeps allowed.
486 *
487  ilast = ihi
488  IF( ilschr ) THEN
489  ifrstm = 1
490  ilastm = n
491  ELSE
492  ifrstm = ilo
493  ilastm = ihi
494  END IF
495  iiter = 0
496  eshift = czero
497  maxit = 30*( ihi-ilo+1 )
498 *
499  DO 170 jiter = 1, maxit
500 *
501 * Check for too many iterations.
502 *
503  IF( jiter.GT.maxit )
504  $ go to 180
505 *
506 * Split the matrix if possible.
507 *
508 * Two tests:
509 * 1: H(j,j-1)=0 or j=ILO
510 * 2: T(j,j)=0
511 *
512 * Special case: j=ILAST
513 *
514  IF( ilast.EQ.ilo ) THEN
515  go to 60
516  ELSE
517  IF( abs1( h( ilast, ilast-1 ) ).LE.atol ) THEN
518  h( ilast, ilast-1 ) = czero
519  go to 60
520  END IF
521  END IF
522 *
523  IF( abs( t( ilast, ilast ) ).LE.btol ) THEN
524  t( ilast, ilast ) = czero
525  go to 50
526  END IF
527 *
528 * General case: j<ILAST
529 *
530  DO 40 j = ilast - 1, ilo, -1
531 *
532 * Test 1: for H(j,j-1)=0 or j=ILO
533 *
534  IF( j.EQ.ilo ) THEN
535  ilazro = .true.
536  ELSE
537  IF( abs1( h( j, j-1 ) ).LE.atol ) THEN
538  h( j, j-1 ) = czero
539  ilazro = .true.
540  ELSE
541  ilazro = .false.
542  END IF
543  END IF
544 *
545 * Test 2: for T(j,j)=0
546 *
547  IF( abs( t( j, j ) ).LT.btol ) THEN
548  t( j, j ) = czero
549 *
550 * Test 1a: Check for 2 consecutive small subdiagonals in A
551 *
552  ilazr2 = .false.
553  IF( .NOT.ilazro ) THEN
554  IF( abs1( h( j, j-1 ) )*( ascale*abs1( h( j+1,
555  $ j ) ) ).LE.abs1( h( j, j ) )*( ascale*atol ) )
556  $ ilazr2 = .true.
557  END IF
558 *
559 * If both tests pass (1 & 2), i.e., the leading diagonal
560 * element of B in the block is zero, split a 1x1 block off
561 * at the top. (I.e., at the J-th row/column) The leading
562 * diagonal element of the remainder can also be zero, so
563 * this may have to be done repeatedly.
564 *
565  IF( ilazro .OR. ilazr2 ) THEN
566  DO 20 jch = j, ilast - 1
567  ctemp = h( jch, jch )
568  CALL clartg( ctemp, h( jch+1, jch ), c, s,
569  $ h( jch, jch ) )
570  h( jch+1, jch ) = czero
571  CALL crot( ilastm-jch, h( jch, jch+1 ), ldh,
572  $ h( jch+1, jch+1 ), ldh, c, s )
573  CALL crot( ilastm-jch, t( jch, jch+1 ), ldt,
574  $ t( jch+1, jch+1 ), ldt, c, s )
575  IF( ilq )
576  $ CALL crot( n, q( 1, jch ), 1, q( 1, jch+1 ), 1,
577  $ c, conjg( s ) )
578  IF( ilazr2 )
579  $ h( jch, jch-1 ) = h( jch, jch-1 )*c
580  ilazr2 = .false.
581  IF( abs1( t( jch+1, jch+1 ) ).GE.btol ) THEN
582  IF( jch+1.GE.ilast ) THEN
583  go to 60
584  ELSE
585  ifirst = jch + 1
586  go to 70
587  END IF
588  END IF
589  t( jch+1, jch+1 ) = czero
590  20 CONTINUE
591  go to 50
592  ELSE
593 *
594 * Only test 2 passed -- chase the zero to T(ILAST,ILAST)
595 * Then process as in the case T(ILAST,ILAST)=0
596 *
597  DO 30 jch = j, ilast - 1
598  ctemp = t( jch, jch+1 )
599  CALL clartg( ctemp, t( jch+1, jch+1 ), c, s,
600  $ t( jch, jch+1 ) )
601  t( jch+1, jch+1 ) = czero
602  IF( jch.LT.ilastm-1 )
603  $ CALL crot( ilastm-jch-1, t( jch, jch+2 ), ldt,
604  $ t( jch+1, jch+2 ), ldt, c, s )
605  CALL crot( ilastm-jch+2, h( jch, jch-1 ), ldh,
606  $ h( jch+1, jch-1 ), ldh, c, s )
607  IF( ilq )
608  $ CALL crot( n, q( 1, jch ), 1, q( 1, jch+1 ), 1,
609  $ c, conjg( s ) )
610  ctemp = h( jch+1, jch )
611  CALL clartg( ctemp, h( jch+1, jch-1 ), c, s,
612  $ h( jch+1, jch ) )
613  h( jch+1, jch-1 ) = czero
614  CALL crot( jch+1-ifrstm, h( ifrstm, jch ), 1,
615  $ h( ifrstm, jch-1 ), 1, c, s )
616  CALL crot( jch-ifrstm, t( ifrstm, jch ), 1,
617  $ t( ifrstm, jch-1 ), 1, c, s )
618  IF( ilz )
619  $ CALL crot( n, z( 1, jch ), 1, z( 1, jch-1 ), 1,
620  $ c, s )
621  30 CONTINUE
622  go to 50
623  END IF
624  ELSE IF( ilazro ) THEN
625 *
626 * Only test 1 passed -- work on J:ILAST
627 *
628  ifirst = j
629  go to 70
630  END IF
631 *
632 * Neither test passed -- try next J
633 *
634  40 CONTINUE
635 *
636 * (Drop-through is "impossible")
637 *
638  info = 2*n + 1
639  go to 210
640 *
641 * T(ILAST,ILAST)=0 -- clear H(ILAST,ILAST-1) to split off a
642 * 1x1 block.
643 *
644  50 CONTINUE
645  ctemp = h( ilast, ilast )
646  CALL clartg( ctemp, h( ilast, ilast-1 ), c, s,
647  $ h( ilast, ilast ) )
648  h( ilast, ilast-1 ) = czero
649  CALL crot( ilast-ifrstm, h( ifrstm, ilast ), 1,
650  $ h( ifrstm, ilast-1 ), 1, c, s )
651  CALL crot( ilast-ifrstm, t( ifrstm, ilast ), 1,
652  $ t( ifrstm, ilast-1 ), 1, c, s )
653  IF( ilz )
654  $ CALL crot( n, z( 1, ilast ), 1, z( 1, ilast-1 ), 1, c, s )
655 *
656 * H(ILAST,ILAST-1)=0 -- Standardize B, set ALPHA and BETA
657 *
658  60 CONTINUE
659  absb = abs( t( ilast, ilast ) )
660  IF( absb.GT.safmin ) THEN
661  signbc = conjg( t( ilast, ilast ) / absb )
662  t( ilast, ilast ) = absb
663  IF( ilschr ) THEN
664  CALL cscal( ilast-ifrstm, signbc, t( ifrstm, ilast ), 1 )
665  CALL cscal( ilast+1-ifrstm, signbc, h( ifrstm, ilast ),
666  $ 1 )
667  ELSE
668  h( ilast, ilast ) = h( ilast, ilast )*signbc
669  END IF
670  IF( ilz )
671  $ CALL cscal( n, signbc, z( 1, ilast ), 1 )
672  ELSE
673  t( ilast, ilast ) = czero
674  END IF
675  alpha( ilast ) = h( ilast, ilast )
676  beta( ilast ) = t( ilast, ilast )
677 *
678 * Go to next block -- exit if finished.
679 *
680  ilast = ilast - 1
681  IF( ilast.LT.ilo )
682  $ go to 190
683 *
684 * Reset counters
685 *
686  iiter = 0
687  eshift = czero
688  IF( .NOT.ilschr ) THEN
689  ilastm = ilast
690  IF( ifrstm.GT.ilast )
691  $ ifrstm = ilo
692  END IF
693  go to 160
694 *
695 * QZ step
696 *
697 * This iteration only involves rows/columns IFIRST:ILAST. We
698 * assume IFIRST < ILAST, and that the diagonal of B is non-zero.
699 *
700  70 CONTINUE
701  iiter = iiter + 1
702  IF( .NOT.ilschr ) THEN
703  ifrstm = ifirst
704  END IF
705 *
706 * Compute the Shift.
707 *
708 * At this point, IFIRST < ILAST, and the diagonal elements of
709 * T(IFIRST:ILAST,IFIRST,ILAST) are larger than BTOL (in
710 * magnitude)
711 *
712  IF( ( iiter / 10 )*10.NE.iiter ) THEN
713 *
714 * The Wilkinson shift (AEP p.512), i.e., the eigenvalue of
715 * the bottom-right 2x2 block of A inv(B) which is nearest to
716 * the bottom-right element.
717 *
718 * We factor B as U*D, where U has unit diagonals, and
719 * compute (A*inv(D))*inv(U).
720 *
721  u12 = ( bscale*t( ilast-1, ilast ) ) /
722  $ ( bscale*t( ilast, ilast ) )
723  ad11 = ( ascale*h( ilast-1, ilast-1 ) ) /
724  $ ( bscale*t( ilast-1, ilast-1 ) )
725  ad21 = ( ascale*h( ilast, ilast-1 ) ) /
726  $ ( bscale*t( ilast-1, ilast-1 ) )
727  ad12 = ( ascale*h( ilast-1, ilast ) ) /
728  $ ( bscale*t( ilast, ilast ) )
729  ad22 = ( ascale*h( ilast, ilast ) ) /
730  $ ( bscale*t( ilast, ilast ) )
731  abi22 = ad22 - u12*ad21
732 *
733  t1 = half*( ad11+abi22 )
734  rtdisc = sqrt( t1**2+ad12*ad21-ad11*ad22 )
735  temp = REAL( t1-abi22 )*REAL( RTDISC ) +
736  $ aimag( t1-abi22 )*aimag( rtdisc )
737  IF( temp.LE.zero ) THEN
738  shift = t1 + rtdisc
739  ELSE
740  shift = t1 - rtdisc
741  END IF
742  ELSE
743 *
744 * Exceptional shift. Chosen for no particularly good reason.
745 *
746  eshift = eshift + (ascale*h(ilast,ilast-1))/
747  $ (bscale*t(ilast-1,ilast-1))
748  shift = eshift
749  END IF
750 *
751 * Now check for two consecutive small subdiagonals.
752 *
753  DO 80 j = ilast - 1, ifirst + 1, -1
754  istart = j
755  ctemp = ascale*h( j, j ) - shift*( bscale*t( j, j ) )
756  temp = abs1( ctemp )
757  temp2 = ascale*abs1( h( j+1, j ) )
758  tempr = max( temp, temp2 )
759  IF( tempr.LT.one .AND. tempr.NE.zero ) THEN
760  temp = temp / tempr
761  temp2 = temp2 / tempr
762  END IF
763  IF( abs1( h( j, j-1 ) )*temp2.LE.temp*atol )
764  $ go to 90
765  80 CONTINUE
766 *
767  istart = ifirst
768  ctemp = ascale*h( ifirst, ifirst ) -
769  $ shift*( bscale*t( ifirst, ifirst ) )
770  90 CONTINUE
771 *
772 * Do an implicit-shift QZ sweep.
773 *
774 * Initial Q
775 *
776  ctemp2 = ascale*h( istart+1, istart )
777  CALL clartg( ctemp, ctemp2, c, s, ctemp3 )
778 *
779 * Sweep
780 *
781  DO 150 j = istart, ilast - 1
782  IF( j.GT.istart ) THEN
783  ctemp = h( j, j-1 )
784  CALL clartg( ctemp, h( j+1, j-1 ), c, s, h( j, j-1 ) )
785  h( j+1, j-1 ) = czero
786  END IF
787 *
788  DO 100 jc = j, ilastm
789  ctemp = c*h( j, jc ) + s*h( j+1, jc )
790  h( j+1, jc ) = -conjg( s )*h( j, jc ) + c*h( j+1, jc )
791  h( j, jc ) = ctemp
792  ctemp2 = c*t( j, jc ) + s*t( j+1, jc )
793  t( j+1, jc ) = -conjg( s )*t( j, jc ) + c*t( j+1, jc )
794  t( j, jc ) = ctemp2
795  100 CONTINUE
796  IF( ilq ) THEN
797  DO 110 jr = 1, n
798  ctemp = c*q( jr, j ) + conjg( s )*q( jr, j+1 )
799  q( jr, j+1 ) = -s*q( jr, j ) + c*q( jr, j+1 )
800  q( jr, j ) = ctemp
801  110 CONTINUE
802  END IF
803 *
804  ctemp = t( j+1, j+1 )
805  CALL clartg( ctemp, t( j+1, j ), c, s, t( j+1, j+1 ) )
806  t( j+1, j ) = czero
807 *
808  DO 120 jr = ifrstm, min( j+2, ilast )
809  ctemp = c*h( jr, j+1 ) + s*h( jr, j )
810  h( jr, j ) = -conjg( s )*h( jr, j+1 ) + c*h( jr, j )
811  h( jr, j+1 ) = ctemp
812  120 CONTINUE
813  DO 130 jr = ifrstm, j
814  ctemp = c*t( jr, j+1 ) + s*t( jr, j )
815  t( jr, j ) = -conjg( s )*t( jr, j+1 ) + c*t( jr, j )
816  t( jr, j+1 ) = ctemp
817  130 CONTINUE
818  IF( ilz ) THEN
819  DO 140 jr = 1, n
820  ctemp = c*z( jr, j+1 ) + s*z( jr, j )
821  z( jr, j ) = -conjg( s )*z( jr, j+1 ) + c*z( jr, j )
822  z( jr, j+1 ) = ctemp
823  140 CONTINUE
824  END IF
825  150 CONTINUE
826 *
827  160 CONTINUE
828 *
829  170 CONTINUE
830 *
831 * Drop-through = non-convergence
832 *
833  180 CONTINUE
834  info = ilast
835  go to 210
836 *
837 * Successful completion of all QZ steps
838 *
839  190 CONTINUE
840 *
841 * Set Eigenvalues 1:ILO-1
842 *
843  DO 200 j = 1, ilo - 1
844  absb = abs( t( j, j ) )
845  IF( absb.GT.safmin ) THEN
846  signbc = conjg( t( j, j ) / absb )
847  t( j, j ) = absb
848  IF( ilschr ) THEN
849  CALL cscal( j-1, signbc, t( 1, j ), 1 )
850  CALL cscal( j, signbc, h( 1, j ), 1 )
851  ELSE
852  h( j, j ) = h( j, j )*signbc
853  END IF
854  IF( ilz )
855  $ CALL cscal( n, signbc, z( 1, j ), 1 )
856  ELSE
857  t( j, j ) = czero
858  END IF
859  alpha( j ) = h( j, j )
860  beta( j ) = t( j, j )
861  200 CONTINUE
862 *
863 * Normal Termination
864 *
865  info = 0
866 *
867 * Exit (other than argument error) -- return optimal workspace size
868 *
869  210 CONTINUE
870  work( 1 ) = cmplx( n )
871  RETURN
872 *
873 * End of CHGEQZ
874 *
875  END