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
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