3 * =========== DOCUMENTATION ===========
5 * Online html documentation available at
6 * http://www.netlib.org/lapack/explore-html/
9 *> Download CHGEQZ + dependencies
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12 *> <a href="http://www.netlib.org/cgi-bin/netlibfiles.zip?format=zip&filename=/lapack/lapack_routine/chgeqz.f">
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21 * SUBROUTINE CHGEQZ( JOB, COMPQ, COMPZ, N, ILO, IHI, H, LDH, T, LDT,
22 * ALPHA, BETA, Q, LDQ, Z, LDZ, WORK, LWORK,
25 * .. Scalar Arguments ..
26 * CHARACTER COMPQ, COMPZ, JOB
27 * INTEGER IHI, ILO, INFO, LDH, LDQ, LDT, LDZ, LWORK, N
29 * .. Array Arguments ..
31 * COMPLEX ALPHA( * ), BETA( * ), H( LDH, * ),
32 * $ Q( LDQ, * ), T( LDT, * ), WORK( * ),
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):
48 *> A = Q1*H*Z1**H, B = Q1*T*Z1**H,
50 *> as computed by CGGHRD.
52 *> If JOB='S', then the Hessenberg-triangular pair (H,T) is
53 *> also reduced to generalized Schur form,
55 *> H = Q*S*Z**H, T = Q*P*Z**H,
57 *> where Q and Z are unitary matrices and S and P are upper triangular.
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):
67 *> A = (Q1*Q)*S*(Z1*Z)**H, B = (Q1*Q)*P*(Z1*Z)**H.
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)
74 *> and if alpha is nonzero, mu = beta / alpha is an eigenvalue of the
75 *> alternate form of the GNEP
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),
81 *> Ref: C.B. Moler & G.W. Stewart, "An Algorithm for Generalized Matrix
82 *> Eigenvalue Problems", SIAM J. Numer. Anal., 10(1973),
92 *> = 'E': Compute eigenvalues only;
93 *> = 'S': Computer eigenvalues and the Schur form.
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.
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.
119 *> The order of the matrices H, T, Q, and Z. N >= 0.
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.
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.
149 *> The leading dimension of the array H. LDH >= max( 1, N ).
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.
165 *> The leading dimension of the array T. LDT >= max( 1, N ).
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
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.
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.
192 *> Q is COMPLEX array, dimension (LDQ, N)
193 *> On entry, if COMPQ = 'V', the unitary matrix Q1 used in the
194 *> reduction of (A,B) to generalized Hessenberg form.
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
197 *> left Schur vectors of (A,B).
198 *> Not referenced if COMPQ = 'N'.
204 *> The leading dimension of the array Q. LDQ >= 1.
205 *> If COMPQ='V' or 'I', then LDQ >= N.
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'.
222 *> The leading dimension of the array Z. LDZ >= 1.
223 *> If COMPZ='V' or 'I', then LDZ >= N.
228 *> WORK is COMPLEX array, dimension (MAX(1,LWORK))
229 *> On exit, if INFO >= 0, WORK(1) returns the optimal LWORK.
235 *> The dimension of the array WORK. LWORK >= max(1,N).
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.
245 *> RWORK is REAL array, dimension (N)
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.
264 *> \author Univ. of Tennessee
265 *> \author Univ. of California Berkeley
266 *> \author Univ. of Colorado Denver
271 *> \ingroup complexGEcomputational
273 *> \par Further Details:
274 * =====================
278 *> We assume that complex ABS works as long as its value is less than
282 * =====================================================================
283 SUBROUTINE CHGEQZ( JOB, COMPQ, COMPZ, N, ILO, IHI, H, LDH, T, LDT,
284 $ ALPHA, BETA, Q, LDQ, Z, LDZ, WORK, LWORK,
287 * -- LAPACK computational routine (version 3.6.1) --
288 * -- LAPACK is a software package provided by Univ. of Tennessee, --
289 * -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
292 * .. Scalar Arguments ..
293 CHARACTER COMPQ, COMPZ, JOB
294 INTEGER IHI, ILO, INFO, LDH, LDQ, LDT, LDZ, LWORK, N
296 * .. Array Arguments ..
298 COMPLEX ALPHA( * ), BETA( * ), H( LDH, * ),
299 $ Q( LDQ, * ), T( LDT, * ), WORK( * ),
303 * =====================================================================
307 PARAMETER ( CZERO = ( 0.0E+0, 0.0E+0 ),
308 $ CONE = ( 1.0E+0, 0.0E+0 ) )
310 PARAMETER ( ZERO = 0.0E+0, ONE = 1.0E+0 )
312 PARAMETER ( HALF = 0.5E+0 )
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,
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,
325 * .. External Functions ..
328 EXTERNAL LSAME, CLANHS, SLAMCH
330 * .. External Subroutines ..
331 EXTERNAL CLARTG, CLASET, CROT, CSCAL, XERBLA
333 * .. Intrinsic Functions ..
334 INTRINSIC ABS, AIMAG, CMPLX, CONJG, MAX, MIN, REAL, SQRT
336 * .. Statement Functions ..
339 * .. Statement Function definitions ..
340 ABS1( X ) = ABS( REAL( X ) ) + ABS( AIMAG( X ) )
342 * .. Executable Statements ..
344 * Decode JOB, COMPQ, COMPZ
346 IF( LSAME( JOB, 'E' ) ) THEN
349 ELSE IF( LSAME( JOB, 'S' ) ) THEN
356 IF( LSAME( COMPQ, 'N' ) ) THEN
359 ELSE IF( LSAME( COMPQ, 'V' ) ) THEN
362 ELSE IF( LSAME( COMPQ, 'I' ) ) THEN
369 IF( LSAME( COMPZ, 'N' ) ) THEN
372 ELSE IF( LSAME( COMPZ, 'V' ) ) THEN
375 ELSE IF( LSAME( COMPZ, 'I' ) ) THEN
382 * Check Argument Values
385 WORK( 1 ) = MAX( 1, N )
386 LQUERY = ( LWORK.EQ.-1 )
387 IF( ISCHUR.EQ.0 ) THEN
389 ELSE IF( ICOMPQ.EQ.0 ) THEN
391 ELSE IF( ICOMPZ.EQ.0 ) THEN
393 ELSE IF( N.LT.0 ) THEN
395 ELSE IF( ILO.LT.1 ) THEN
397 ELSE IF( IHI.GT.N .OR. IHI.LT.ILO-1 ) THEN
399 ELSE IF( LDH.LT.N ) THEN
401 ELSE IF( LDT.LT.N ) THEN
403 ELSE IF( LDQ.LT.1 .OR. ( ILQ .AND. LDQ.LT.N ) ) THEN
405 ELSE IF( LDZ.LT.1 .OR. ( ILZ .AND. LDZ.LT.N ) ) THEN
407 ELSE IF( LWORK.LT.MAX( 1, N ) .AND. .NOT.LQUERY ) THEN
411 CALL XERBLA( 'CHGEQZ', -INFO )
413 ELSE IF( LQUERY ) THEN
417 * Quick return if possible
419 * WORK( 1 ) = CMPLX( 1 )
421 WORK( 1 ) = CMPLX( 1 )
428 $ CALL CLASET( 'Full', N, N, CZERO, CONE, Q, LDQ )
430 $ CALL CLASET( 'Full', N, N, CZERO, CONE, Z, LDZ )
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 )
445 * Set Eigenvalues IHI+1:N
448 ABSB = ABS( T( J, J ) )
449 IF( ABSB.GT.SAFMIN ) THEN
450 SIGNBC = CONJG( T( J, J ) / ABSB )
453 CALL CSCAL( J-1, SIGNBC, T( 1, J ), 1 )
454 CALL CSCAL( J, SIGNBC, H( 1, J ), 1 )
456 CALL CSCAL( 1, SIGNBC, H( J, J ), 1 )
459 $ CALL CSCAL( N, SIGNBC, Z( 1, J ), 1 )
463 ALPHA( J ) = H( J, J )
464 BETA( J ) = T( J, J )
467 * If IHI < ILO, skip QZ steps
472 * MAIN QZ ITERATION LOOP
474 * Initialize dynamic indices
476 * Eigenvalues ILAST+1:N have been found.
477 * Column operations modify rows IFRSTM:whatever
478 * Row operations modify columns whatever:ILASTM
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.
497 MAXIT = 30*( IHI-ILO+1 )
499 DO 170 JITER = 1, MAXIT
501 * Check for too many iterations.
506 * Split the matrix if possible.
509 * 1: H(j,j-1)=0 or j=ILO
512 * Special case: j=ILAST
514 IF( ILAST.EQ.ILO ) THEN
517 IF( ABS1( H( ILAST, ILAST-1 ) ).LE.ATOL ) THEN
518 H( ILAST, ILAST-1 ) = CZERO
523 IF( ABS( T( ILAST, ILAST ) ).LE.BTOL ) THEN
524 T( ILAST, ILAST ) = CZERO
528 * General case: j<ILAST
530 DO 40 J = ILAST - 1, ILO, -1
532 * Test 1: for H(j,j-1)=0 or j=ILO
537 IF( ABS1( H( J, J-1 ) ).LE.ATOL ) THEN
545 * Test 2: for T(j,j)=0
547 IF( ABS( T( J, J ) ).LT.BTOL ) THEN
550 * Test 1a: Check for 2 consecutive small subdiagonals in A
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 ) )
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.
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,
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 )
576 $ CALL CROT( N, Q( 1, JCH ), 1, Q( 1, JCH+1 ), 1,
579 $ H( JCH, JCH-1 ) = H( JCH, JCH-1 )*C
581 IF( ABS1( T( JCH+1, JCH+1 ) ).GE.BTOL ) THEN
582 IF( JCH+1.GE.ILAST ) THEN
589 T( JCH+1, JCH+1 ) = CZERO
594 * Only test 2 passed -- chase the zero to T(ILAST,ILAST)
595 * Then process as in the case T(ILAST,ILAST)=0
597 DO 30 JCH = J, ILAST - 1
598 CTEMP = T( JCH, JCH+1 )
599 CALL CLARTG( CTEMP, T( JCH+1, JCH+1 ), C, S,
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 )
608 $ CALL CROT( N, Q( 1, JCH ), 1, Q( 1, JCH+1 ), 1,
610 CTEMP = H( JCH+1, JCH )
611 CALL CLARTG( CTEMP, H( JCH+1, JCH-1 ), C, S,
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 )
619 $ CALL CROT( N, Z( 1, JCH ), 1, Z( 1, JCH-1 ), 1,
624 ELSE IF( ILAZRO ) THEN
626 * Only test 1 passed -- work on J:ILAST
632 * Neither test passed -- try next J
636 * (Drop-through is "impossible")
641 * T(ILAST,ILAST)=0 -- clear H(ILAST,ILAST-1) to split off a
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 )
654 $ CALL CROT( N, Z( 1, ILAST ), 1, Z( 1, ILAST-1 ), 1, C, S )
656 * H(ILAST,ILAST-1)=0 -- Standardize B, set ALPHA and BETA
659 ABSB = ABS( T( ILAST, ILAST ) )
660 IF( ABSB.GT.SAFMIN ) THEN
661 SIGNBC = CONJG( T( ILAST, ILAST ) / ABSB )
662 T( ILAST, ILAST ) = ABSB
664 CALL CSCAL( ILAST-IFRSTM, SIGNBC, T( IFRSTM, ILAST ), 1 )
665 CALL CSCAL( ILAST+1-IFRSTM, SIGNBC, H( IFRSTM, ILAST ),
668 CALL CSCAL( 1, SIGNBC, H( ILAST, ILAST ), 1 )
671 $ CALL CSCAL( N, SIGNBC, Z( 1, ILAST ), 1 )
673 T( ILAST, ILAST ) = CZERO
675 ALPHA( ILAST ) = H( ILAST, ILAST )
676 BETA( ILAST ) = T( ILAST, ILAST )
678 * Go to next block -- exit if finished.
688 IF( .NOT.ILSCHR ) THEN
690 IF( IFRSTM.GT.ILAST )
697 * This iteration only involves rows/columns IFIRST:ILAST. We
698 * assume IFIRST < ILAST, and that the diagonal of B is non-zero.
702 IF( .NOT.ILSCHR ) THEN
708 * At this point, IFIRST < ILAST, and the diagonal elements of
709 * T(IFIRST:ILAST,IFIRST,ILAST) are larger than BTOL (in
712 IF( ( IITER / 10 )*10.NE.IITER ) THEN
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.
718 * We factor B as U*D, where U has unit diagonals, and
719 * compute (A*inv(D))*inv(U).
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
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
744 * Exceptional shift. Chosen for no particularly good reason.
746 ESHIFT = ESHIFT + (ASCALE*H(ILAST,ILAST-1))/
747 $ (BSCALE*T(ILAST-1,ILAST-1))
751 * Now check for two consecutive small subdiagonals.
753 DO 80 J = ILAST - 1, IFIRST + 1, -1
755 CTEMP = ASCALE*H( J, J ) - SHIFT*( BSCALE*T( J, J ) )
757 TEMP2 = ASCALE*ABS1( H( J+1, J ) )
758 TEMPR = MAX( TEMP, TEMP2 )
759 IF( TEMPR.LT.ONE .AND. TEMPR.NE.ZERO ) THEN
761 TEMP2 = TEMP2 / TEMPR
763 IF( ABS1( H( J, J-1 ) )*TEMP2.LE.TEMP*ATOL )
768 CTEMP = ASCALE*H( IFIRST, IFIRST ) -
769 $ SHIFT*( BSCALE*T( IFIRST, IFIRST ) )
772 * Do an implicit-shift QZ sweep.
776 CTEMP2 = ASCALE*H( ISTART+1, ISTART )
777 CALL CLARTG( CTEMP, CTEMP2, C, S, CTEMP3 )
781 DO 150 J = ISTART, ILAST - 1
782 IF( J.GT.ISTART ) THEN
784 CALL CLARTG( CTEMP, H( J+1, J-1 ), C, S, H( J, J-1 ) )
785 H( J+1, J-1 ) = CZERO
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 )
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 )
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 )
804 CTEMP = T( J+1, J+1 )
805 CALL CLARTG( CTEMP, T( J+1, J ), C, S, T( J+1, J+1 ) )
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 )
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 )
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 )
831 * Drop-through = non-convergence
837 * Successful completion of all QZ steps
841 * Set Eigenvalues 1:ILO-1
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 )
849 CALL CSCAL( J-1, SIGNBC, T( 1, J ), 1 )
850 CALL CSCAL( J, SIGNBC, H( 1, J ), 1 )
852 CALL CSCAL( 1, SIGNBC, H( J, J ), 1 )
855 $ CALL CSCAL( N, SIGNBC, Z( 1, J ), 1 )
859 ALPHA( J ) = H( J, J )
860 BETA( J ) = T( J, J )
867 * Exit (other than argument error) -- return optimal workspace size
870 WORK( 1 ) = CMPLX( N )