1 SUBROUTINE ZHGEQZ( JOB, COMPQ, COMPZ, N, ILO, IHI, H, LDH, T, LDT,
2 $ ALPHA, BETA, Q, LDQ, Z, LDZ, WORK, LWORK,
3 $ RWORK, INFO )
4 *
5 * -- LAPACK routine (version 3.2) --
6 * -- LAPACK is a software package provided by Univ. of Tennessee, --
7 * -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
8 * November 2006
9 *
10 * .. Scalar Arguments ..
11 CHARACTER COMPQ, COMPZ, JOB
12 INTEGER IHI, ILO, INFO, LDH, LDQ, LDT, LDZ, LWORK, N
13 * ..
14 * .. Array Arguments ..
15 DOUBLE PRECISION RWORK( * )
16 COMPLEX*16 ALPHA( * ), BETA( * ), H( LDH, * ),
17 $ Q( LDQ, * ), T( LDT, * ), WORK( * ),
18 $ Z( LDZ, * )
19 * ..
20 *
21 * Purpose
22 * =======
23 *
24 * ZHGEQZ computes the eigenvalues of a complex matrix pair (H,T),
25 * where H is an upper Hessenberg matrix and T is upper triangular,
26 * using the single-shift QZ method.
27 * Matrix pairs of this type are produced by the reduction to
28 * generalized upper Hessenberg form of a complex matrix pair (A,B):
29 *
30 * A = Q1*H*Z1**H, B = Q1*T*Z1**H,
31 *
32 * as computed by ZGGHRD.
33 *
34 * If JOB='S', then the Hessenberg-triangular pair (H,T) is
35 * also reduced to generalized Schur form,
36 *
37 * H = Q*S*Z**H, T = Q*P*Z**H,
38 *
39 * where Q and Z are unitary matrices and S and P are upper triangular.
40 *
41 * Optionally, the unitary matrix Q from the generalized Schur
42 * factorization may be postmultiplied into an input matrix Q1, and the
43 * unitary matrix Z may be postmultiplied into an input matrix Z1.
44 * If Q1 and Z1 are the unitary matrices from ZGGHRD that reduced
45 * the matrix pair (A,B) to generalized Hessenberg form, then the output
46 * matrices Q1*Q and Z1*Z are the unitary factors from the generalized
47 * Schur factorization of (A,B):
48 *
49 * A = (Q1*Q)*S*(Z1*Z)**H, B = (Q1*Q)*P*(Z1*Z)**H.
50 *
51 * To avoid overflow, eigenvalues of the matrix pair (H,T)
52 * (equivalently, of (A,B)) are computed as a pair of complex values
53 * (alpha,beta). If beta is nonzero, lambda = alpha / beta is an
54 * eigenvalue of the generalized nonsymmetric eigenvalue problem (GNEP)
55 * A*x = lambda*B*x
56 * and if alpha is nonzero, mu = beta / alpha is an eigenvalue of the
57 * alternate form of the GNEP
58 * mu*A*y = B*y.
59 * The values of alpha and beta for the i-th eigenvalue can be read
60 * directly from the generalized Schur form: alpha = S(i,i),
61 * beta = P(i,i).
62 *
63 * Ref: C.B. Moler & G.W. Stewart, "An Algorithm for Generalized Matrix
64 * Eigenvalue Problems", SIAM J. Numer. Anal., 10(1973),
65 * pp. 241--256.
66 *
67 * Arguments
68 * =========
69 *
70 * JOB (input) CHARACTER*1
71 * = 'E': Compute eigenvalues only;
72 * = 'S': Computer eigenvalues and the Schur form.
73 *
74 * COMPQ (input) CHARACTER*1
75 * = 'N': Left Schur vectors (Q) are not computed;
76 * = 'I': Q is initialized to the unit matrix and the matrix Q
77 * of left Schur vectors of (H,T) is returned;
78 * = 'V': Q must contain a unitary matrix Q1 on entry and
79 * the product Q1*Q is returned.
80 *
81 * COMPZ (input) CHARACTER*1
82 * = 'N': Right Schur vectors (Z) are not computed;
83 * = 'I': Q is initialized to the unit matrix and the matrix Z
84 * of right Schur vectors of (H,T) is returned;
85 * = 'V': Z must contain a unitary matrix Z1 on entry and
86 * the product Z1*Z is returned.
87 *
88 * N (input) INTEGER
89 * The order of the matrices H, T, Q, and Z. N >= 0.
90 *
91 * ILO (input) INTEGER
92 * IHI (input) INTEGER
93 * ILO and IHI mark the rows and columns of H which are in
94 * Hessenberg form. It is assumed that A is already upper
95 * triangular in rows and columns 1:ILO-1 and IHI+1:N.
96 * If N > 0, 1 <= ILO <= IHI <= N; if N = 0, ILO=1 and IHI=0.
97 *
98 * H (input/output) COMPLEX*16 array, dimension (LDH, N)
99 * 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
2 $ ALPHA, BETA, Q, LDQ, Z, LDZ, WORK, LWORK,
3 $ RWORK, INFO )
4 *
5 * -- LAPACK routine (version 3.2) --
6 * -- LAPACK is a software package provided by Univ. of Tennessee, --
7 * -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
8 * November 2006
9 *
10 * .. Scalar Arguments ..
11 CHARACTER COMPQ, COMPZ, JOB
12 INTEGER IHI, ILO, INFO, LDH, LDQ, LDT, LDZ, LWORK, N
13 * ..
14 * .. Array Arguments ..
15 DOUBLE PRECISION RWORK( * )
16 COMPLEX*16 ALPHA( * ), BETA( * ), H( LDH, * ),
17 $ Q( LDQ, * ), T( LDT, * ), WORK( * ),
18 $ Z( LDZ, * )
19 * ..
20 *
21 * Purpose
22 * =======
23 *
24 * ZHGEQZ computes the eigenvalues of a complex matrix pair (H,T),
25 * where H is an upper Hessenberg matrix and T is upper triangular,
26 * using the single-shift QZ method.
27 * Matrix pairs of this type are produced by the reduction to
28 * generalized upper Hessenberg form of a complex matrix pair (A,B):
29 *
30 * A = Q1*H*Z1**H, B = Q1*T*Z1**H,
31 *
32 * as computed by ZGGHRD.
33 *
34 * If JOB='S', then the Hessenberg-triangular pair (H,T) is
35 * also reduced to generalized Schur form,
36 *
37 * H = Q*S*Z**H, T = Q*P*Z**H,
38 *
39 * where Q and Z are unitary matrices and S and P are upper triangular.
40 *
41 * Optionally, the unitary matrix Q from the generalized Schur
42 * factorization may be postmultiplied into an input matrix Q1, and the
43 * unitary matrix Z may be postmultiplied into an input matrix Z1.
44 * If Q1 and Z1 are the unitary matrices from ZGGHRD that reduced
45 * the matrix pair (A,B) to generalized Hessenberg form, then the output
46 * matrices Q1*Q and Z1*Z are the unitary factors from the generalized
47 * Schur factorization of (A,B):
48 *
49 * A = (Q1*Q)*S*(Z1*Z)**H, B = (Q1*Q)*P*(Z1*Z)**H.
50 *
51 * To avoid overflow, eigenvalues of the matrix pair (H,T)
52 * (equivalently, of (A,B)) are computed as a pair of complex values
53 * (alpha,beta). If beta is nonzero, lambda = alpha / beta is an
54 * eigenvalue of the generalized nonsymmetric eigenvalue problem (GNEP)
55 * A*x = lambda*B*x
56 * and if alpha is nonzero, mu = beta / alpha is an eigenvalue of the
57 * alternate form of the GNEP
58 * mu*A*y = B*y.
59 * The values of alpha and beta for the i-th eigenvalue can be read
60 * directly from the generalized Schur form: alpha = S(i,i),
61 * beta = P(i,i).
62 *
63 * Ref: C.B. Moler & G.W. Stewart, "An Algorithm for Generalized Matrix
64 * Eigenvalue Problems", SIAM J. Numer. Anal., 10(1973),
65 * pp. 241--256.
66 *
67 * Arguments
68 * =========
69 *
70 * JOB (input) CHARACTER*1
71 * = 'E': Compute eigenvalues only;
72 * = 'S': Computer eigenvalues and the Schur form.
73 *
74 * COMPQ (input) CHARACTER*1
75 * = 'N': Left Schur vectors (Q) are not computed;
76 * = 'I': Q is initialized to the unit matrix and the matrix Q
77 * of left Schur vectors of (H,T) is returned;
78 * = 'V': Q must contain a unitary matrix Q1 on entry and
79 * the product Q1*Q is returned.
80 *
81 * COMPZ (input) CHARACTER*1
82 * = 'N': Right Schur vectors (Z) are not computed;
83 * = 'I': Q is initialized to the unit matrix and the matrix Z
84 * of right Schur vectors of (H,T) is returned;
85 * = 'V': Z must contain a unitary matrix Z1 on entry and
86 * the product Z1*Z is returned.
87 *
88 * N (input) INTEGER
89 * The order of the matrices H, T, Q, and Z. N >= 0.
90 *
91 * ILO (input) INTEGER
92 * IHI (input) INTEGER
93 * ILO and IHI mark the rows and columns of H which are in
94 * Hessenberg form. It is assumed that A is already upper
95 * triangular in rows and columns 1:ILO-1 and IHI+1:N.
96 * If N > 0, 1 <= ILO <= IHI <= N; if N = 0, ILO=1 and IHI=0.
97 *
98 * H (input/output) COMPLEX*16 array, dimension (LDH, N)
99 * 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