Annotation of rpl/lapack/lapack/dhgeqz.f, revision 1.18
1.9 bertrand 1: *> \brief \b DHGEQZ
2: *
3: * =========== DOCUMENTATION ===========
4: *
1.18 ! bertrand 5: * Online html documentation available at
! 6: * http://www.netlib.org/lapack/explore-html/
1.9 bertrand 7: *
8: *> \htmlonly
1.18 ! bertrand 9: *> Download DHGEQZ + dependencies
! 10: *> <a href="http://www.netlib.org/cgi-bin/netlibfiles.tgz?format=tgz&filename=/lapack/lapack_routine/dhgeqz.f">
! 11: *> [TGZ]</a>
! 12: *> <a href="http://www.netlib.org/cgi-bin/netlibfiles.zip?format=zip&filename=/lapack/lapack_routine/dhgeqz.f">
! 13: *> [ZIP]</a>
! 14: *> <a href="http://www.netlib.org/cgi-bin/netlibfiles.txt?format=txt&filename=/lapack/lapack_routine/dhgeqz.f">
1.9 bertrand 15: *> [TXT]</a>
1.18 ! bertrand 16: *> \endhtmlonly
1.9 bertrand 17: *
18: * Definition:
19: * ===========
20: *
21: * SUBROUTINE DHGEQZ( JOB, COMPQ, COMPZ, N, ILO, IHI, H, LDH, T, LDT,
22: * ALPHAR, ALPHAI, BETA, Q, LDQ, Z, LDZ, WORK,
23: * LWORK, INFO )
1.18 ! bertrand 24: *
1.9 bertrand 25: * .. Scalar Arguments ..
26: * CHARACTER COMPQ, COMPZ, JOB
27: * INTEGER IHI, ILO, INFO, LDH, LDQ, LDT, LDZ, LWORK, N
28: * ..
29: * .. Array Arguments ..
30: * DOUBLE PRECISION ALPHAI( * ), ALPHAR( * ), BETA( * ),
31: * $ H( LDH, * ), Q( LDQ, * ), T( LDT, * ),
32: * $ WORK( * ), Z( LDZ, * )
33: * ..
1.18 ! bertrand 34: *
1.9 bertrand 35: *
36: *> \par Purpose:
37: * =============
38: *>
39: *> \verbatim
40: *>
41: *> DHGEQZ computes the eigenvalues of a real matrix pair (H,T),
42: *> where H is an upper Hessenberg matrix and T is upper triangular,
43: *> using the double-shift QZ method.
44: *> Matrix pairs of this type are produced by the reduction to
45: *> generalized upper Hessenberg form of a real matrix pair (A,B):
46: *>
47: *> A = Q1*H*Z1**T, B = Q1*T*Z1**T,
48: *>
49: *> as computed by DGGHRD.
50: *>
51: *> If JOB='S', then the Hessenberg-triangular pair (H,T) is
52: *> also reduced to generalized Schur form,
1.18 ! bertrand 53: *>
1.9 bertrand 54: *> H = Q*S*Z**T, T = Q*P*Z**T,
1.18 ! bertrand 55: *>
1.9 bertrand 56: *> where Q and Z are orthogonal matrices, P is an upper triangular
57: *> matrix, and S is a quasi-triangular matrix with 1-by-1 and 2-by-2
58: *> diagonal blocks.
59: *>
60: *> The 1-by-1 blocks correspond to real eigenvalues of the matrix pair
61: *> (H,T) and the 2-by-2 blocks correspond to complex conjugate pairs of
62: *> eigenvalues.
63: *>
64: *> Additionally, the 2-by-2 upper triangular diagonal blocks of P
65: *> corresponding to 2-by-2 blocks of S are reduced to positive diagonal
66: *> form, i.e., if S(j+1,j) is non-zero, then P(j+1,j) = P(j,j+1) = 0,
67: *> P(j,j) > 0, and P(j+1,j+1) > 0.
68: *>
69: *> Optionally, the orthogonal matrix Q from the generalized Schur
70: *> factorization may be postmultiplied into an input matrix Q1, and the
71: *> orthogonal matrix Z may be postmultiplied into an input matrix Z1.
72: *> If Q1 and Z1 are the orthogonal matrices from DGGHRD that reduced
73: *> the matrix pair (A,B) to generalized upper Hessenberg form, then the
74: *> output matrices Q1*Q and Z1*Z are the orthogonal factors from the
75: *> generalized Schur factorization of (A,B):
76: *>
77: *> A = (Q1*Q)*S*(Z1*Z)**T, B = (Q1*Q)*P*(Z1*Z)**T.
1.18 ! bertrand 78: *>
1.9 bertrand 79: *> To avoid overflow, eigenvalues of the matrix pair (H,T) (equivalently,
80: *> of (A,B)) are computed as a pair of values (alpha,beta), where alpha is
81: *> complex and beta real.
82: *> If beta is nonzero, lambda = alpha / beta is an eigenvalue of the
83: *> generalized nonsymmetric eigenvalue problem (GNEP)
84: *> A*x = lambda*B*x
85: *> and if alpha is nonzero, mu = beta / alpha is an eigenvalue of the
86: *> alternate form of the GNEP
87: *> mu*A*y = B*y.
88: *> Real eigenvalues can be read directly from the generalized Schur
1.18 ! bertrand 89: *> form:
1.9 bertrand 90: *> alpha = S(i,i), beta = P(i,i).
91: *>
92: *> Ref: C.B. Moler & G.W. Stewart, "An Algorithm for Generalized Matrix
93: *> Eigenvalue Problems", SIAM J. Numer. Anal., 10(1973),
94: *> pp. 241--256.
95: *> \endverbatim
96: *
97: * Arguments:
98: * ==========
99: *
100: *> \param[in] JOB
101: *> \verbatim
102: *> JOB is CHARACTER*1
103: *> = 'E': Compute eigenvalues only;
1.18 ! bertrand 104: *> = 'S': Compute eigenvalues and the Schur form.
1.9 bertrand 105: *> \endverbatim
106: *>
107: *> \param[in] COMPQ
108: *> \verbatim
109: *> COMPQ is CHARACTER*1
110: *> = 'N': Left Schur vectors (Q) are not computed;
111: *> = 'I': Q is initialized to the unit matrix and the matrix Q
112: *> of left Schur vectors of (H,T) is returned;
113: *> = 'V': Q must contain an orthogonal matrix Q1 on entry and
114: *> the product Q1*Q is returned.
115: *> \endverbatim
116: *>
117: *> \param[in] COMPZ
118: *> \verbatim
119: *> COMPZ is CHARACTER*1
120: *> = 'N': Right Schur vectors (Z) are not computed;
121: *> = 'I': Z is initialized to the unit matrix and the matrix Z
122: *> of right Schur vectors of (H,T) is returned;
123: *> = 'V': Z must contain an orthogonal matrix Z1 on entry and
124: *> the product Z1*Z is returned.
125: *> \endverbatim
126: *>
127: *> \param[in] N
128: *> \verbatim
129: *> N is INTEGER
130: *> The order of the matrices H, T, Q, and Z. N >= 0.
131: *> \endverbatim
132: *>
133: *> \param[in] ILO
134: *> \verbatim
135: *> ILO is INTEGER
136: *> \endverbatim
137: *>
138: *> \param[in] IHI
139: *> \verbatim
140: *> IHI is INTEGER
141: *> ILO and IHI mark the rows and columns of H which are in
142: *> Hessenberg form. It is assumed that A is already upper
143: *> triangular in rows and columns 1:ILO-1 and IHI+1:N.
144: *> If N > 0, 1 <= ILO <= IHI <= N; if N = 0, ILO=1 and IHI=0.
145: *> \endverbatim
146: *>
147: *> \param[in,out] H
148: *> \verbatim
149: *> H is DOUBLE PRECISION array, dimension (LDH, N)
150: *> On entry, the N-by-N upper Hessenberg matrix H.
151: *> On exit, if JOB = 'S', H contains the upper quasi-triangular
152: *> matrix S from the generalized Schur factorization.
153: *> If JOB = 'E', the diagonal blocks of H match those of S, but
154: *> the rest of H is unspecified.
155: *> \endverbatim
156: *>
157: *> \param[in] LDH
158: *> \verbatim
159: *> LDH is INTEGER
160: *> The leading dimension of the array H. LDH >= max( 1, N ).
161: *> \endverbatim
162: *>
163: *> \param[in,out] T
164: *> \verbatim
165: *> T is DOUBLE PRECISION array, dimension (LDT, N)
166: *> On entry, the N-by-N upper triangular matrix T.
167: *> On exit, if JOB = 'S', T contains the upper triangular
168: *> matrix P from the generalized Schur factorization;
169: *> 2-by-2 diagonal blocks of P corresponding to 2-by-2 blocks of S
170: *> are reduced to positive diagonal form, i.e., if H(j+1,j) is
171: *> non-zero, then T(j+1,j) = T(j,j+1) = 0, T(j,j) > 0, and
172: *> T(j+1,j+1) > 0.
173: *> If JOB = 'E', the diagonal blocks of T match those of P, but
174: *> the rest of T is unspecified.
175: *> \endverbatim
176: *>
177: *> \param[in] LDT
178: *> \verbatim
179: *> LDT is INTEGER
180: *> The leading dimension of the array T. LDT >= max( 1, N ).
181: *> \endverbatim
182: *>
183: *> \param[out] ALPHAR
184: *> \verbatim
185: *> ALPHAR is DOUBLE PRECISION array, dimension (N)
186: *> The real parts of each scalar alpha defining an eigenvalue
187: *> of GNEP.
188: *> \endverbatim
189: *>
190: *> \param[out] ALPHAI
191: *> \verbatim
192: *> ALPHAI is DOUBLE PRECISION array, dimension (N)
193: *> The imaginary parts of each scalar alpha defining an
194: *> eigenvalue of GNEP.
195: *> If ALPHAI(j) is zero, then the j-th eigenvalue is real; if
196: *> positive, then the j-th and (j+1)-st eigenvalues are a
197: *> complex conjugate pair, with ALPHAI(j+1) = -ALPHAI(j).
198: *> \endverbatim
199: *>
200: *> \param[out] BETA
201: *> \verbatim
202: *> BETA is DOUBLE PRECISION array, dimension (N)
203: *> The scalars beta that define the eigenvalues of GNEP.
204: *> Together, the quantities alpha = (ALPHAR(j),ALPHAI(j)) and
205: *> beta = BETA(j) represent the j-th eigenvalue of the matrix
206: *> pair (A,B), in one of the forms lambda = alpha/beta or
207: *> mu = beta/alpha. Since either lambda or mu may overflow,
208: *> they should not, in general, be computed.
209: *> \endverbatim
210: *>
211: *> \param[in,out] Q
212: *> \verbatim
213: *> Q is DOUBLE PRECISION array, dimension (LDQ, N)
1.16 bertrand 214: *> On entry, if COMPQ = 'V', the orthogonal matrix Q1 used in
1.9 bertrand 215: *> the reduction of (A,B) to generalized Hessenberg form.
1.16 bertrand 216: *> On exit, if COMPQ = 'I', the orthogonal matrix of left Schur
217: *> vectors of (H,T), and if COMPQ = 'V', the orthogonal matrix
1.9 bertrand 218: *> of left Schur vectors of (A,B).
1.16 bertrand 219: *> Not referenced if COMPQ = 'N'.
1.9 bertrand 220: *> \endverbatim
221: *>
222: *> \param[in] LDQ
223: *> \verbatim
224: *> LDQ is INTEGER
225: *> The leading dimension of the array Q. LDQ >= 1.
226: *> If COMPQ='V' or 'I', then LDQ >= N.
227: *> \endverbatim
228: *>
229: *> \param[in,out] Z
230: *> \verbatim
231: *> Z is DOUBLE PRECISION array, dimension (LDZ, N)
232: *> On entry, if COMPZ = 'V', the orthogonal matrix Z1 used in
233: *> the reduction of (A,B) to generalized Hessenberg form.
234: *> On exit, if COMPZ = 'I', the orthogonal matrix of
235: *> right Schur vectors of (H,T), and if COMPZ = 'V', the
236: *> orthogonal matrix of right Schur vectors of (A,B).
237: *> Not referenced if COMPZ = 'N'.
238: *> \endverbatim
239: *>
240: *> \param[in] LDZ
241: *> \verbatim
242: *> LDZ is INTEGER
243: *> The leading dimension of the array Z. LDZ >= 1.
244: *> If COMPZ='V' or 'I', then LDZ >= N.
245: *> \endverbatim
246: *>
247: *> \param[out] WORK
248: *> \verbatim
249: *> WORK is DOUBLE PRECISION array, dimension (MAX(1,LWORK))
250: *> On exit, if INFO >= 0, WORK(1) returns the optimal LWORK.
251: *> \endverbatim
252: *>
253: *> \param[in] LWORK
254: *> \verbatim
255: *> LWORK is INTEGER
256: *> The dimension of the array WORK. LWORK >= max(1,N).
257: *>
258: *> If LWORK = -1, then a workspace query is assumed; the routine
259: *> only calculates the optimal size of the WORK array, returns
260: *> this value as the first entry of the WORK array, and no error
261: *> message related to LWORK is issued by XERBLA.
262: *> \endverbatim
263: *>
264: *> \param[out] INFO
265: *> \verbatim
266: *> INFO is INTEGER
267: *> = 0: successful exit
268: *> < 0: if INFO = -i, the i-th argument had an illegal value
269: *> = 1,...,N: the QZ iteration did not converge. (H,T) is not
270: *> in Schur form, but ALPHAR(i), ALPHAI(i), and
271: *> BETA(i), i=INFO+1,...,N should be correct.
272: *> = N+1,...,2*N: the shift calculation failed. (H,T) is not
273: *> in Schur form, but ALPHAR(i), ALPHAI(i), and
274: *> BETA(i), i=INFO-N+1,...,N should be correct.
275: *> \endverbatim
276: *
277: * Authors:
278: * ========
279: *
1.18 ! bertrand 280: *> \author Univ. of Tennessee
! 281: *> \author Univ. of California Berkeley
! 282: *> \author Univ. of Colorado Denver
! 283: *> \author NAG Ltd.
1.9 bertrand 284: *
1.16 bertrand 285: *> \date June 2016
1.9 bertrand 286: *
287: *> \ingroup doubleGEcomputational
288: *
289: *> \par Further Details:
290: * =====================
291: *>
292: *> \verbatim
293: *>
294: *> Iteration counters:
295: *>
296: *> JITER -- counts iterations.
297: *> IITER -- counts iterations run since ILAST was last
298: *> changed. This is therefore reset only when a 1-by-1 or
299: *> 2-by-2 block deflates off the bottom.
300: *> \endverbatim
301: *>
302: * =====================================================================
1.1 bertrand 303: SUBROUTINE DHGEQZ( JOB, COMPQ, COMPZ, N, ILO, IHI, H, LDH, T, LDT,
304: $ ALPHAR, ALPHAI, BETA, Q, LDQ, Z, LDZ, WORK,
305: $ LWORK, INFO )
306: *
1.18 ! bertrand 307: * -- LAPACK computational routine (version 3.7.0) --
1.1 bertrand 308: * -- LAPACK is a software package provided by Univ. of Tennessee, --
309: * -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
1.16 bertrand 310: * June 2016
1.1 bertrand 311: *
312: * .. Scalar Arguments ..
313: CHARACTER COMPQ, COMPZ, JOB
314: INTEGER IHI, ILO, INFO, LDH, LDQ, LDT, LDZ, LWORK, N
315: * ..
316: * .. Array Arguments ..
317: DOUBLE PRECISION ALPHAI( * ), ALPHAR( * ), BETA( * ),
318: $ H( LDH, * ), Q( LDQ, * ), T( LDT, * ),
319: $ WORK( * ), Z( LDZ, * )
320: * ..
321: *
322: * =====================================================================
323: *
324: * .. Parameters ..
325: * $ SAFETY = 1.0E+0 )
326: DOUBLE PRECISION HALF, ZERO, ONE, SAFETY
327: PARAMETER ( HALF = 0.5D+0, ZERO = 0.0D+0, ONE = 1.0D+0,
328: $ SAFETY = 1.0D+2 )
329: * ..
330: * .. Local Scalars ..
331: LOGICAL ILAZR2, ILAZRO, ILPIVT, ILQ, ILSCHR, ILZ,
332: $ LQUERY
333: INTEGER ICOMPQ, ICOMPZ, IFIRST, IFRSTM, IITER, ILAST,
334: $ ILASTM, IN, ISCHUR, ISTART, J, JC, JCH, JITER,
335: $ JR, MAXIT
336: DOUBLE PRECISION A11, A12, A1I, A1R, A21, A22, A2I, A2R, AD11,
337: $ AD11L, AD12, AD12L, AD21, AD21L, AD22, AD22L,
338: $ AD32L, AN, ANORM, ASCALE, ATOL, B11, B1A, B1I,
339: $ B1R, B22, B2A, B2I, B2R, BN, BNORM, BSCALE,
340: $ BTOL, C, C11I, C11R, C12, C21, C22I, C22R, CL,
341: $ CQ, CR, CZ, ESHIFT, S, S1, S1INV, S2, SAFMAX,
342: $ SAFMIN, SCALE, SL, SQI, SQR, SR, SZI, SZR, T1,
343: $ TAU, TEMP, TEMP2, TEMPI, TEMPR, U1, U12, U12L,
344: $ U2, ULP, VS, W11, W12, W21, W22, WABS, WI, WR,
345: $ WR2
346: * ..
347: * .. Local Arrays ..
348: DOUBLE PRECISION V( 3 )
349: * ..
350: * .. External Functions ..
351: LOGICAL LSAME
352: DOUBLE PRECISION DLAMCH, DLANHS, DLAPY2, DLAPY3
353: EXTERNAL LSAME, DLAMCH, DLANHS, DLAPY2, DLAPY3
354: * ..
355: * .. External Subroutines ..
356: EXTERNAL DLAG2, DLARFG, DLARTG, DLASET, DLASV2, DROT,
357: $ XERBLA
358: * ..
359: * .. Intrinsic Functions ..
360: INTRINSIC ABS, DBLE, MAX, MIN, SQRT
361: * ..
362: * .. Executable Statements ..
363: *
364: * Decode JOB, COMPQ, COMPZ
365: *
366: IF( LSAME( JOB, 'E' ) ) THEN
367: ILSCHR = .FALSE.
368: ISCHUR = 1
369: ELSE IF( LSAME( JOB, 'S' ) ) THEN
370: ILSCHR = .TRUE.
371: ISCHUR = 2
372: ELSE
373: ISCHUR = 0
374: END IF
375: *
376: IF( LSAME( COMPQ, 'N' ) ) THEN
377: ILQ = .FALSE.
378: ICOMPQ = 1
379: ELSE IF( LSAME( COMPQ, 'V' ) ) THEN
380: ILQ = .TRUE.
381: ICOMPQ = 2
382: ELSE IF( LSAME( COMPQ, 'I' ) ) THEN
383: ILQ = .TRUE.
384: ICOMPQ = 3
385: ELSE
386: ICOMPQ = 0
387: END IF
388: *
389: IF( LSAME( COMPZ, 'N' ) ) THEN
390: ILZ = .FALSE.
391: ICOMPZ = 1
392: ELSE IF( LSAME( COMPZ, 'V' ) ) THEN
393: ILZ = .TRUE.
394: ICOMPZ = 2
395: ELSE IF( LSAME( COMPZ, 'I' ) ) THEN
396: ILZ = .TRUE.
397: ICOMPZ = 3
398: ELSE
399: ICOMPZ = 0
400: END IF
401: *
402: * Check Argument Values
403: *
404: INFO = 0
405: WORK( 1 ) = MAX( 1, N )
406: LQUERY = ( LWORK.EQ.-1 )
407: IF( ISCHUR.EQ.0 ) THEN
408: INFO = -1
409: ELSE IF( ICOMPQ.EQ.0 ) THEN
410: INFO = -2
411: ELSE IF( ICOMPZ.EQ.0 ) THEN
412: INFO = -3
413: ELSE IF( N.LT.0 ) THEN
414: INFO = -4
415: ELSE IF( ILO.LT.1 ) THEN
416: INFO = -5
417: ELSE IF( IHI.GT.N .OR. IHI.LT.ILO-1 ) THEN
418: INFO = -6
419: ELSE IF( LDH.LT.N ) THEN
420: INFO = -8
421: ELSE IF( LDT.LT.N ) THEN
422: INFO = -10
423: ELSE IF( LDQ.LT.1 .OR. ( ILQ .AND. LDQ.LT.N ) ) THEN
424: INFO = -15
425: ELSE IF( LDZ.LT.1 .OR. ( ILZ .AND. LDZ.LT.N ) ) THEN
426: INFO = -17
427: ELSE IF( LWORK.LT.MAX( 1, N ) .AND. .NOT.LQUERY ) THEN
428: INFO = -19
429: END IF
430: IF( INFO.NE.0 ) THEN
431: CALL XERBLA( 'DHGEQZ', -INFO )
432: RETURN
433: ELSE IF( LQUERY ) THEN
434: RETURN
435: END IF
436: *
437: * Quick return if possible
438: *
439: IF( N.LE.0 ) THEN
440: WORK( 1 ) = DBLE( 1 )
441: RETURN
442: END IF
443: *
444: * Initialize Q and Z
445: *
446: IF( ICOMPQ.EQ.3 )
447: $ CALL DLASET( 'Full', N, N, ZERO, ONE, Q, LDQ )
448: IF( ICOMPZ.EQ.3 )
449: $ CALL DLASET( 'Full', N, N, ZERO, ONE, Z, LDZ )
450: *
451: * Machine Constants
452: *
453: IN = IHI + 1 - ILO
454: SAFMIN = DLAMCH( 'S' )
455: SAFMAX = ONE / SAFMIN
456: ULP = DLAMCH( 'E' )*DLAMCH( 'B' )
457: ANORM = DLANHS( 'F', IN, H( ILO, ILO ), LDH, WORK )
458: BNORM = DLANHS( 'F', IN, T( ILO, ILO ), LDT, WORK )
459: ATOL = MAX( SAFMIN, ULP*ANORM )
460: BTOL = MAX( SAFMIN, ULP*BNORM )
461: ASCALE = ONE / MAX( SAFMIN, ANORM )
462: BSCALE = ONE / MAX( SAFMIN, BNORM )
463: *
464: * Set Eigenvalues IHI+1:N
465: *
466: DO 30 J = IHI + 1, N
467: IF( T( J, J ).LT.ZERO ) THEN
468: IF( ILSCHR ) THEN
469: DO 10 JR = 1, J
470: H( JR, J ) = -H( JR, J )
471: T( JR, J ) = -T( JR, J )
472: 10 CONTINUE
473: ELSE
474: H( J, J ) = -H( J, J )
475: T( J, J ) = -T( J, J )
476: END IF
477: IF( ILZ ) THEN
478: DO 20 JR = 1, N
479: Z( JR, J ) = -Z( JR, J )
480: 20 CONTINUE
481: END IF
482: END IF
483: ALPHAR( J ) = H( J, J )
484: ALPHAI( J ) = ZERO
485: BETA( J ) = T( J, J )
486: 30 CONTINUE
487: *
488: * If IHI < ILO, skip QZ steps
489: *
490: IF( IHI.LT.ILO )
491: $ GO TO 380
492: *
493: * MAIN QZ ITERATION LOOP
494: *
495: * Initialize dynamic indices
496: *
497: * Eigenvalues ILAST+1:N have been found.
498: * Column operations modify rows IFRSTM:whatever.
499: * Row operations modify columns whatever:ILASTM.
500: *
501: * If only eigenvalues are being computed, then
502: * IFRSTM is the row of the last splitting row above row ILAST;
503: * this is always at least ILO.
504: * IITER counts iterations since the last eigenvalue was found,
505: * to tell when to use an extraordinary shift.
506: * MAXIT is the maximum number of QZ sweeps allowed.
507: *
508: ILAST = IHI
509: IF( ILSCHR ) THEN
510: IFRSTM = 1
511: ILASTM = N
512: ELSE
513: IFRSTM = ILO
514: ILASTM = IHI
515: END IF
516: IITER = 0
517: ESHIFT = ZERO
518: MAXIT = 30*( IHI-ILO+1 )
519: *
520: DO 360 JITER = 1, MAXIT
521: *
522: * Split the matrix if possible.
523: *
524: * Two tests:
525: * 1: H(j,j-1)=0 or j=ILO
526: * 2: T(j,j)=0
527: *
528: IF( ILAST.EQ.ILO ) THEN
529: *
530: * Special case: j=ILAST
531: *
532: GO TO 80
533: ELSE
534: IF( ABS( H( ILAST, ILAST-1 ) ).LE.ATOL ) THEN
535: H( ILAST, ILAST-1 ) = ZERO
536: GO TO 80
537: END IF
538: END IF
539: *
540: IF( ABS( T( ILAST, ILAST ) ).LE.BTOL ) THEN
541: T( ILAST, ILAST ) = ZERO
542: GO TO 70
543: END IF
544: *
545: * General case: j<ILAST
546: *
547: DO 60 J = ILAST - 1, ILO, -1
548: *
549: * Test 1: for H(j,j-1)=0 or j=ILO
550: *
551: IF( J.EQ.ILO ) THEN
552: ILAZRO = .TRUE.
553: ELSE
554: IF( ABS( H( J, J-1 ) ).LE.ATOL ) THEN
555: H( J, J-1 ) = ZERO
556: ILAZRO = .TRUE.
557: ELSE
558: ILAZRO = .FALSE.
559: END IF
560: END IF
561: *
562: * Test 2: for T(j,j)=0
563: *
564: IF( ABS( T( J, J ) ).LT.BTOL ) THEN
565: T( J, J ) = ZERO
566: *
567: * Test 1a: Check for 2 consecutive small subdiagonals in A
568: *
569: ILAZR2 = .FALSE.
570: IF( .NOT.ILAZRO ) THEN
571: TEMP = ABS( H( J, J-1 ) )
572: TEMP2 = ABS( H( J, J ) )
573: TEMPR = MAX( TEMP, TEMP2 )
574: IF( TEMPR.LT.ONE .AND. TEMPR.NE.ZERO ) THEN
575: TEMP = TEMP / TEMPR
576: TEMP2 = TEMP2 / TEMPR
577: END IF
578: IF( TEMP*( ASCALE*ABS( H( J+1, J ) ) ).LE.TEMP2*
579: $ ( ASCALE*ATOL ) )ILAZR2 = .TRUE.
580: END IF
581: *
582: * If both tests pass (1 & 2), i.e., the leading diagonal
583: * element of B in the block is zero, split a 1x1 block off
584: * at the top. (I.e., at the J-th row/column) The leading
585: * diagonal element of the remainder can also be zero, so
586: * this may have to be done repeatedly.
587: *
588: IF( ILAZRO .OR. ILAZR2 ) THEN
589: DO 40 JCH = J, ILAST - 1
590: TEMP = H( JCH, JCH )
591: CALL DLARTG( TEMP, H( JCH+1, JCH ), C, S,
592: $ H( JCH, JCH ) )
593: H( JCH+1, JCH ) = ZERO
594: CALL DROT( ILASTM-JCH, H( JCH, JCH+1 ), LDH,
595: $ H( JCH+1, JCH+1 ), LDH, C, S )
596: CALL DROT( ILASTM-JCH, T( JCH, JCH+1 ), LDT,
597: $ T( JCH+1, JCH+1 ), LDT, C, S )
598: IF( ILQ )
599: $ CALL DROT( N, Q( 1, JCH ), 1, Q( 1, JCH+1 ), 1,
600: $ C, S )
601: IF( ILAZR2 )
602: $ H( JCH, JCH-1 ) = H( JCH, JCH-1 )*C
603: ILAZR2 = .FALSE.
604: IF( ABS( T( JCH+1, JCH+1 ) ).GE.BTOL ) THEN
605: IF( JCH+1.GE.ILAST ) THEN
606: GO TO 80
607: ELSE
608: IFIRST = JCH + 1
609: GO TO 110
610: END IF
611: END IF
612: T( JCH+1, JCH+1 ) = ZERO
613: 40 CONTINUE
614: GO TO 70
615: ELSE
616: *
617: * Only test 2 passed -- chase the zero to T(ILAST,ILAST)
618: * Then process as in the case T(ILAST,ILAST)=0
619: *
620: DO 50 JCH = J, ILAST - 1
621: TEMP = T( JCH, JCH+1 )
622: CALL DLARTG( TEMP, T( JCH+1, JCH+1 ), C, S,
623: $ T( JCH, JCH+1 ) )
624: T( JCH+1, JCH+1 ) = ZERO
625: IF( JCH.LT.ILASTM-1 )
626: $ CALL DROT( ILASTM-JCH-1, T( JCH, JCH+2 ), LDT,
627: $ T( JCH+1, JCH+2 ), LDT, C, S )
628: CALL DROT( ILASTM-JCH+2, H( JCH, JCH-1 ), LDH,
629: $ H( JCH+1, JCH-1 ), LDH, C, S )
630: IF( ILQ )
631: $ CALL DROT( N, Q( 1, JCH ), 1, Q( 1, JCH+1 ), 1,
632: $ C, S )
633: TEMP = H( JCH+1, JCH )
634: CALL DLARTG( TEMP, H( JCH+1, JCH-1 ), C, S,
635: $ H( JCH+1, JCH ) )
636: H( JCH+1, JCH-1 ) = ZERO
637: CALL DROT( JCH+1-IFRSTM, H( IFRSTM, JCH ), 1,
638: $ H( IFRSTM, JCH-1 ), 1, C, S )
639: CALL DROT( JCH-IFRSTM, T( IFRSTM, JCH ), 1,
640: $ T( IFRSTM, JCH-1 ), 1, C, S )
641: IF( ILZ )
642: $ CALL DROT( N, Z( 1, JCH ), 1, Z( 1, JCH-1 ), 1,
643: $ C, S )
644: 50 CONTINUE
645: GO TO 70
646: END IF
647: ELSE IF( ILAZRO ) THEN
648: *
649: * Only test 1 passed -- work on J:ILAST
650: *
651: IFIRST = J
652: GO TO 110
653: END IF
654: *
655: * Neither test passed -- try next J
656: *
657: 60 CONTINUE
658: *
659: * (Drop-through is "impossible")
660: *
661: INFO = N + 1
662: GO TO 420
663: *
664: * T(ILAST,ILAST)=0 -- clear H(ILAST,ILAST-1) to split off a
665: * 1x1 block.
666: *
667: 70 CONTINUE
668: TEMP = H( ILAST, ILAST )
669: CALL DLARTG( TEMP, H( ILAST, ILAST-1 ), C, S,
670: $ H( ILAST, ILAST ) )
671: H( ILAST, ILAST-1 ) = ZERO
672: CALL DROT( ILAST-IFRSTM, H( IFRSTM, ILAST ), 1,
673: $ H( IFRSTM, ILAST-1 ), 1, C, S )
674: CALL DROT( ILAST-IFRSTM, T( IFRSTM, ILAST ), 1,
675: $ T( IFRSTM, ILAST-1 ), 1, C, S )
676: IF( ILZ )
677: $ CALL DROT( N, Z( 1, ILAST ), 1, Z( 1, ILAST-1 ), 1, C, S )
678: *
679: * H(ILAST,ILAST-1)=0 -- Standardize B, set ALPHAR, ALPHAI,
680: * and BETA
681: *
682: 80 CONTINUE
683: IF( T( ILAST, ILAST ).LT.ZERO ) THEN
684: IF( ILSCHR ) THEN
685: DO 90 J = IFRSTM, ILAST
686: H( J, ILAST ) = -H( J, ILAST )
687: T( J, ILAST ) = -T( J, ILAST )
688: 90 CONTINUE
689: ELSE
690: H( ILAST, ILAST ) = -H( ILAST, ILAST )
691: T( ILAST, ILAST ) = -T( ILAST, ILAST )
692: END IF
693: IF( ILZ ) THEN
694: DO 100 J = 1, N
695: Z( J, ILAST ) = -Z( J, ILAST )
696: 100 CONTINUE
697: END IF
698: END IF
699: ALPHAR( ILAST ) = H( ILAST, ILAST )
700: ALPHAI( ILAST ) = ZERO
701: BETA( ILAST ) = T( ILAST, ILAST )
702: *
703: * Go to next block -- exit if finished.
704: *
705: ILAST = ILAST - 1
706: IF( ILAST.LT.ILO )
707: $ GO TO 380
708: *
709: * Reset counters
710: *
711: IITER = 0
712: ESHIFT = ZERO
713: IF( .NOT.ILSCHR ) THEN
714: ILASTM = ILAST
715: IF( IFRSTM.GT.ILAST )
716: $ IFRSTM = ILO
717: END IF
718: GO TO 350
719: *
720: * QZ step
721: *
722: * This iteration only involves rows/columns IFIRST:ILAST. We
723: * assume IFIRST < ILAST, and that the diagonal of B is non-zero.
724: *
725: 110 CONTINUE
726: IITER = IITER + 1
727: IF( .NOT.ILSCHR ) THEN
728: IFRSTM = IFIRST
729: END IF
730: *
731: * Compute single shifts.
732: *
733: * At this point, IFIRST < ILAST, and the diagonal elements of
734: * T(IFIRST:ILAST,IFIRST,ILAST) are larger than BTOL (in
735: * magnitude)
736: *
737: IF( ( IITER / 10 )*10.EQ.IITER ) THEN
738: *
739: * Exceptional shift. Chosen for no particularly good reason.
740: * (Single shift only.)
741: *
1.14 bertrand 742: IF( ( DBLE( MAXIT )*SAFMIN )*ABS( H( ILAST, ILAST-1 ) ).LT.
1.1 bertrand 743: $ ABS( T( ILAST-1, ILAST-1 ) ) ) THEN
1.14 bertrand 744: ESHIFT = H( ILAST, ILAST-1 ) /
1.1 bertrand 745: $ T( ILAST-1, ILAST-1 )
746: ELSE
747: ESHIFT = ESHIFT + ONE / ( SAFMIN*DBLE( MAXIT ) )
748: END IF
749: S1 = ONE
750: WR = ESHIFT
751: *
752: ELSE
753: *
754: * Shifts based on the generalized eigenvalues of the
755: * bottom-right 2x2 block of A and B. The first eigenvalue
756: * returned by DLAG2 is the Wilkinson shift (AEP p.512),
757: *
758: CALL DLAG2( H( ILAST-1, ILAST-1 ), LDH,
759: $ T( ILAST-1, ILAST-1 ), LDT, SAFMIN*SAFETY, S1,
760: $ S2, WR, WR2, WI )
761: *
1.14 bertrand 762: IF ( ABS( (WR/S1)*T( ILAST, ILAST ) - H( ILAST, ILAST ) )
1.18 ! bertrand 763: $ .GT. ABS( (WR2/S2)*T( ILAST, ILAST )
1.14 bertrand 764: $ - H( ILAST, ILAST ) ) ) THEN
765: TEMP = WR
766: WR = WR2
767: WR2 = TEMP
768: TEMP = S1
769: S1 = S2
770: S2 = TEMP
771: END IF
1.1 bertrand 772: TEMP = MAX( S1, SAFMIN*MAX( ONE, ABS( WR ), ABS( WI ) ) )
773: IF( WI.NE.ZERO )
774: $ GO TO 200
775: END IF
776: *
777: * Fiddle with shift to avoid overflow
778: *
779: TEMP = MIN( ASCALE, ONE )*( HALF*SAFMAX )
780: IF( S1.GT.TEMP ) THEN
781: SCALE = TEMP / S1
782: ELSE
783: SCALE = ONE
784: END IF
785: *
786: TEMP = MIN( BSCALE, ONE )*( HALF*SAFMAX )
787: IF( ABS( WR ).GT.TEMP )
788: $ SCALE = MIN( SCALE, TEMP / ABS( WR ) )
789: S1 = SCALE*S1
790: WR = SCALE*WR
791: *
792: * Now check for two consecutive small subdiagonals.
793: *
794: DO 120 J = ILAST - 1, IFIRST + 1, -1
795: ISTART = J
796: TEMP = ABS( S1*H( J, J-1 ) )
797: TEMP2 = ABS( S1*H( J, J )-WR*T( J, J ) )
798: TEMPR = MAX( TEMP, TEMP2 )
799: IF( TEMPR.LT.ONE .AND. TEMPR.NE.ZERO ) THEN
800: TEMP = TEMP / TEMPR
801: TEMP2 = TEMP2 / TEMPR
802: END IF
803: IF( ABS( ( ASCALE*H( J+1, J ) )*TEMP ).LE.( ASCALE*ATOL )*
804: $ TEMP2 )GO TO 130
805: 120 CONTINUE
806: *
807: ISTART = IFIRST
808: 130 CONTINUE
809: *
810: * Do an implicit single-shift QZ sweep.
811: *
812: * Initial Q
813: *
814: TEMP = S1*H( ISTART, ISTART ) - WR*T( ISTART, ISTART )
815: TEMP2 = S1*H( ISTART+1, ISTART )
816: CALL DLARTG( TEMP, TEMP2, C, S, TEMPR )
817: *
818: * Sweep
819: *
820: DO 190 J = ISTART, ILAST - 1
821: IF( J.GT.ISTART ) THEN
822: TEMP = H( J, J-1 )
823: CALL DLARTG( TEMP, H( J+1, J-1 ), C, S, H( J, J-1 ) )
824: H( J+1, J-1 ) = ZERO
825: END IF
826: *
827: DO 140 JC = J, ILASTM
828: TEMP = C*H( J, JC ) + S*H( J+1, JC )
829: H( J+1, JC ) = -S*H( J, JC ) + C*H( J+1, JC )
830: H( J, JC ) = TEMP
831: TEMP2 = C*T( J, JC ) + S*T( J+1, JC )
832: T( J+1, JC ) = -S*T( J, JC ) + C*T( J+1, JC )
833: T( J, JC ) = TEMP2
834: 140 CONTINUE
835: IF( ILQ ) THEN
836: DO 150 JR = 1, N
837: TEMP = C*Q( JR, J ) + S*Q( JR, J+1 )
838: Q( JR, J+1 ) = -S*Q( JR, J ) + C*Q( JR, J+1 )
839: Q( JR, J ) = TEMP
840: 150 CONTINUE
841: END IF
842: *
843: TEMP = T( J+1, J+1 )
844: CALL DLARTG( TEMP, T( J+1, J ), C, S, T( J+1, J+1 ) )
845: T( J+1, J ) = ZERO
846: *
847: DO 160 JR = IFRSTM, MIN( J+2, ILAST )
848: TEMP = C*H( JR, J+1 ) + S*H( JR, J )
849: H( JR, J ) = -S*H( JR, J+1 ) + C*H( JR, J )
850: H( JR, J+1 ) = TEMP
851: 160 CONTINUE
852: DO 170 JR = IFRSTM, J
853: TEMP = C*T( JR, J+1 ) + S*T( JR, J )
854: T( JR, J ) = -S*T( JR, J+1 ) + C*T( JR, J )
855: T( JR, J+1 ) = TEMP
856: 170 CONTINUE
857: IF( ILZ ) THEN
858: DO 180 JR = 1, N
859: TEMP = C*Z( JR, J+1 ) + S*Z( JR, J )
860: Z( JR, J ) = -S*Z( JR, J+1 ) + C*Z( JR, J )
861: Z( JR, J+1 ) = TEMP
862: 180 CONTINUE
863: END IF
864: 190 CONTINUE
865: *
866: GO TO 350
867: *
868: * Use Francis double-shift
869: *
870: * Note: the Francis double-shift should work with real shifts,
871: * but only if the block is at least 3x3.
872: * This code may break if this point is reached with
873: * a 2x2 block with real eigenvalues.
874: *
875: 200 CONTINUE
876: IF( IFIRST+1.EQ.ILAST ) THEN
877: *
878: * Special case -- 2x2 block with complex eigenvectors
879: *
880: * Step 1: Standardize, that is, rotate so that
881: *
882: * ( B11 0 )
883: * B = ( ) with B11 non-negative.
884: * ( 0 B22 )
885: *
886: CALL DLASV2( T( ILAST-1, ILAST-1 ), T( ILAST-1, ILAST ),
887: $ T( ILAST, ILAST ), B22, B11, SR, CR, SL, CL )
888: *
889: IF( B11.LT.ZERO ) THEN
890: CR = -CR
891: SR = -SR
892: B11 = -B11
893: B22 = -B22
894: END IF
895: *
896: CALL DROT( ILASTM+1-IFIRST, H( ILAST-1, ILAST-1 ), LDH,
897: $ H( ILAST, ILAST-1 ), LDH, CL, SL )
898: CALL DROT( ILAST+1-IFRSTM, H( IFRSTM, ILAST-1 ), 1,
899: $ H( IFRSTM, ILAST ), 1, CR, SR )
900: *
901: IF( ILAST.LT.ILASTM )
902: $ CALL DROT( ILASTM-ILAST, T( ILAST-1, ILAST+1 ), LDT,
903: $ T( ILAST, ILAST+1 ), LDT, CL, SL )
904: IF( IFRSTM.LT.ILAST-1 )
905: $ CALL DROT( IFIRST-IFRSTM, T( IFRSTM, ILAST-1 ), 1,
906: $ T( IFRSTM, ILAST ), 1, CR, SR )
907: *
908: IF( ILQ )
909: $ CALL DROT( N, Q( 1, ILAST-1 ), 1, Q( 1, ILAST ), 1, CL,
910: $ SL )
911: IF( ILZ )
912: $ CALL DROT( N, Z( 1, ILAST-1 ), 1, Z( 1, ILAST ), 1, CR,
913: $ SR )
914: *
915: T( ILAST-1, ILAST-1 ) = B11
916: T( ILAST-1, ILAST ) = ZERO
917: T( ILAST, ILAST-1 ) = ZERO
918: T( ILAST, ILAST ) = B22
919: *
920: * If B22 is negative, negate column ILAST
921: *
922: IF( B22.LT.ZERO ) THEN
923: DO 210 J = IFRSTM, ILAST
924: H( J, ILAST ) = -H( J, ILAST )
925: T( J, ILAST ) = -T( J, ILAST )
926: 210 CONTINUE
927: *
928: IF( ILZ ) THEN
929: DO 220 J = 1, N
930: Z( J, ILAST ) = -Z( J, ILAST )
931: 220 CONTINUE
932: END IF
1.11 bertrand 933: B22 = -B22
1.1 bertrand 934: END IF
935: *
936: * Step 2: Compute ALPHAR, ALPHAI, and BETA (see refs.)
937: *
938: * Recompute shift
939: *
940: CALL DLAG2( H( ILAST-1, ILAST-1 ), LDH,
941: $ T( ILAST-1, ILAST-1 ), LDT, SAFMIN*SAFETY, S1,
942: $ TEMP, WR, TEMP2, WI )
943: *
944: * If standardization has perturbed the shift onto real line,
945: * do another (real single-shift) QR step.
946: *
947: IF( WI.EQ.ZERO )
948: $ GO TO 350
949: S1INV = ONE / S1
950: *
951: * Do EISPACK (QZVAL) computation of alpha and beta
952: *
953: A11 = H( ILAST-1, ILAST-1 )
954: A21 = H( ILAST, ILAST-1 )
955: A12 = H( ILAST-1, ILAST )
956: A22 = H( ILAST, ILAST )
957: *
958: * Compute complex Givens rotation on right
959: * (Assume some element of C = (sA - wB) > unfl )
960: * __
961: * (sA - wB) ( CZ -SZ )
962: * ( SZ CZ )
963: *
964: C11R = S1*A11 - WR*B11
965: C11I = -WI*B11
966: C12 = S1*A12
967: C21 = S1*A21
968: C22R = S1*A22 - WR*B22
969: C22I = -WI*B22
970: *
971: IF( ABS( C11R )+ABS( C11I )+ABS( C12 ).GT.ABS( C21 )+
972: $ ABS( C22R )+ABS( C22I ) ) THEN
973: T1 = DLAPY3( C12, C11R, C11I )
974: CZ = C12 / T1
975: SZR = -C11R / T1
976: SZI = -C11I / T1
977: ELSE
978: CZ = DLAPY2( C22R, C22I )
979: IF( CZ.LE.SAFMIN ) THEN
980: CZ = ZERO
981: SZR = ONE
982: SZI = ZERO
983: ELSE
984: TEMPR = C22R / CZ
985: TEMPI = C22I / CZ
986: T1 = DLAPY2( CZ, C21 )
987: CZ = CZ / T1
988: SZR = -C21*TEMPR / T1
989: SZI = C21*TEMPI / T1
990: END IF
991: END IF
992: *
993: * Compute Givens rotation on left
994: *
995: * ( CQ SQ )
996: * ( __ ) A or B
997: * ( -SQ CQ )
998: *
999: AN = ABS( A11 ) + ABS( A12 ) + ABS( A21 ) + ABS( A22 )
1000: BN = ABS( B11 ) + ABS( B22 )
1001: WABS = ABS( WR ) + ABS( WI )
1002: IF( S1*AN.GT.WABS*BN ) THEN
1003: CQ = CZ*B11
1004: SQR = SZR*B22
1005: SQI = -SZI*B22
1006: ELSE
1007: A1R = CZ*A11 + SZR*A12
1008: A1I = SZI*A12
1009: A2R = CZ*A21 + SZR*A22
1010: A2I = SZI*A22
1011: CQ = DLAPY2( A1R, A1I )
1012: IF( CQ.LE.SAFMIN ) THEN
1013: CQ = ZERO
1014: SQR = ONE
1015: SQI = ZERO
1016: ELSE
1017: TEMPR = A1R / CQ
1018: TEMPI = A1I / CQ
1019: SQR = TEMPR*A2R + TEMPI*A2I
1020: SQI = TEMPI*A2R - TEMPR*A2I
1021: END IF
1022: END IF
1023: T1 = DLAPY3( CQ, SQR, SQI )
1024: CQ = CQ / T1
1025: SQR = SQR / T1
1026: SQI = SQI / T1
1027: *
1028: * Compute diagonal elements of QBZ
1029: *
1030: TEMPR = SQR*SZR - SQI*SZI
1031: TEMPI = SQR*SZI + SQI*SZR
1032: B1R = CQ*CZ*B11 + TEMPR*B22
1033: B1I = TEMPI*B22
1034: B1A = DLAPY2( B1R, B1I )
1035: B2R = CQ*CZ*B22 + TEMPR*B11
1036: B2I = -TEMPI*B11
1037: B2A = DLAPY2( B2R, B2I )
1038: *
1039: * Normalize so beta > 0, and Im( alpha1 ) > 0
1040: *
1041: BETA( ILAST-1 ) = B1A
1042: BETA( ILAST ) = B2A
1043: ALPHAR( ILAST-1 ) = ( WR*B1A )*S1INV
1044: ALPHAI( ILAST-1 ) = ( WI*B1A )*S1INV
1045: ALPHAR( ILAST ) = ( WR*B2A )*S1INV
1046: ALPHAI( ILAST ) = -( WI*B2A )*S1INV
1047: *
1048: * Step 3: Go to next block -- exit if finished.
1049: *
1050: ILAST = IFIRST - 1
1051: IF( ILAST.LT.ILO )
1052: $ GO TO 380
1053: *
1054: * Reset counters
1055: *
1056: IITER = 0
1057: ESHIFT = ZERO
1058: IF( .NOT.ILSCHR ) THEN
1059: ILASTM = ILAST
1060: IF( IFRSTM.GT.ILAST )
1061: $ IFRSTM = ILO
1062: END IF
1063: GO TO 350
1064: ELSE
1065: *
1066: * Usual case: 3x3 or larger block, using Francis implicit
1067: * double-shift
1068: *
1069: * 2
1070: * Eigenvalue equation is w - c w + d = 0,
1071: *
1072: * -1 2 -1
1073: * so compute 1st column of (A B ) - c A B + d
1074: * using the formula in QZIT (from EISPACK)
1075: *
1076: * We assume that the block is at least 3x3
1077: *
1078: AD11 = ( ASCALE*H( ILAST-1, ILAST-1 ) ) /
1079: $ ( BSCALE*T( ILAST-1, ILAST-1 ) )
1080: AD21 = ( ASCALE*H( ILAST, ILAST-1 ) ) /
1081: $ ( BSCALE*T( ILAST-1, ILAST-1 ) )
1082: AD12 = ( ASCALE*H( ILAST-1, ILAST ) ) /
1083: $ ( BSCALE*T( ILAST, ILAST ) )
1084: AD22 = ( ASCALE*H( ILAST, ILAST ) ) /
1085: $ ( BSCALE*T( ILAST, ILAST ) )
1086: U12 = T( ILAST-1, ILAST ) / T( ILAST, ILAST )
1087: AD11L = ( ASCALE*H( IFIRST, IFIRST ) ) /
1088: $ ( BSCALE*T( IFIRST, IFIRST ) )
1089: AD21L = ( ASCALE*H( IFIRST+1, IFIRST ) ) /
1090: $ ( BSCALE*T( IFIRST, IFIRST ) )
1091: AD12L = ( ASCALE*H( IFIRST, IFIRST+1 ) ) /
1092: $ ( BSCALE*T( IFIRST+1, IFIRST+1 ) )
1093: AD22L = ( ASCALE*H( IFIRST+1, IFIRST+1 ) ) /
1094: $ ( BSCALE*T( IFIRST+1, IFIRST+1 ) )
1095: AD32L = ( ASCALE*H( IFIRST+2, IFIRST+1 ) ) /
1096: $ ( BSCALE*T( IFIRST+1, IFIRST+1 ) )
1097: U12L = T( IFIRST, IFIRST+1 ) / T( IFIRST+1, IFIRST+1 )
1098: *
1099: V( 1 ) = ( AD11-AD11L )*( AD22-AD11L ) - AD12*AD21 +
1100: $ AD21*U12*AD11L + ( AD12L-AD11L*U12L )*AD21L
1101: V( 2 ) = ( ( AD22L-AD11L )-AD21L*U12L-( AD11-AD11L )-
1102: $ ( AD22-AD11L )+AD21*U12 )*AD21L
1103: V( 3 ) = AD32L*AD21L
1104: *
1105: ISTART = IFIRST
1106: *
1107: CALL DLARFG( 3, V( 1 ), V( 2 ), 1, TAU )
1108: V( 1 ) = ONE
1109: *
1110: * Sweep
1111: *
1112: DO 290 J = ISTART, ILAST - 2
1113: *
1114: * All but last elements: use 3x3 Householder transforms.
1115: *
1116: * Zero (j-1)st column of A
1117: *
1118: IF( J.GT.ISTART ) THEN
1119: V( 1 ) = H( J, J-1 )
1120: V( 2 ) = H( J+1, J-1 )
1121: V( 3 ) = H( J+2, J-1 )
1122: *
1123: CALL DLARFG( 3, H( J, J-1 ), V( 2 ), 1, TAU )
1124: V( 1 ) = ONE
1125: H( J+1, J-1 ) = ZERO
1126: H( J+2, J-1 ) = ZERO
1127: END IF
1128: *
1129: DO 230 JC = J, ILASTM
1130: TEMP = TAU*( H( J, JC )+V( 2 )*H( J+1, JC )+V( 3 )*
1131: $ H( J+2, JC ) )
1132: H( J, JC ) = H( J, JC ) - TEMP
1133: H( J+1, JC ) = H( J+1, JC ) - TEMP*V( 2 )
1134: H( J+2, JC ) = H( J+2, JC ) - TEMP*V( 3 )
1135: TEMP2 = TAU*( T( J, JC )+V( 2 )*T( J+1, JC )+V( 3 )*
1136: $ T( J+2, JC ) )
1137: T( J, JC ) = T( J, JC ) - TEMP2
1138: T( J+1, JC ) = T( J+1, JC ) - TEMP2*V( 2 )
1139: T( J+2, JC ) = T( J+2, JC ) - TEMP2*V( 3 )
1140: 230 CONTINUE
1141: IF( ILQ ) THEN
1142: DO 240 JR = 1, N
1143: TEMP = TAU*( Q( JR, J )+V( 2 )*Q( JR, J+1 )+V( 3 )*
1144: $ Q( JR, J+2 ) )
1145: Q( JR, J ) = Q( JR, J ) - TEMP
1146: Q( JR, J+1 ) = Q( JR, J+1 ) - TEMP*V( 2 )
1147: Q( JR, J+2 ) = Q( JR, J+2 ) - TEMP*V( 3 )
1148: 240 CONTINUE
1149: END IF
1150: *
1151: * Zero j-th column of B (see DLAGBC for details)
1152: *
1153: * Swap rows to pivot
1154: *
1155: ILPIVT = .FALSE.
1156: TEMP = MAX( ABS( T( J+1, J+1 ) ), ABS( T( J+1, J+2 ) ) )
1157: TEMP2 = MAX( ABS( T( J+2, J+1 ) ), ABS( T( J+2, J+2 ) ) )
1158: IF( MAX( TEMP, TEMP2 ).LT.SAFMIN ) THEN
1159: SCALE = ZERO
1160: U1 = ONE
1161: U2 = ZERO
1162: GO TO 250
1163: ELSE IF( TEMP.GE.TEMP2 ) THEN
1164: W11 = T( J+1, J+1 )
1165: W21 = T( J+2, J+1 )
1166: W12 = T( J+1, J+2 )
1167: W22 = T( J+2, J+2 )
1168: U1 = T( J+1, J )
1169: U2 = T( J+2, J )
1170: ELSE
1171: W21 = T( J+1, J+1 )
1172: W11 = T( J+2, J+1 )
1173: W22 = T( J+1, J+2 )
1174: W12 = T( J+2, J+2 )
1175: U2 = T( J+1, J )
1176: U1 = T( J+2, J )
1177: END IF
1178: *
1179: * Swap columns if nec.
1180: *
1181: IF( ABS( W12 ).GT.ABS( W11 ) ) THEN
1182: ILPIVT = .TRUE.
1183: TEMP = W12
1184: TEMP2 = W22
1185: W12 = W11
1186: W22 = W21
1187: W11 = TEMP
1188: W21 = TEMP2
1189: END IF
1190: *
1191: * LU-factor
1192: *
1193: TEMP = W21 / W11
1194: U2 = U2 - TEMP*U1
1195: W22 = W22 - TEMP*W12
1196: W21 = ZERO
1197: *
1198: * Compute SCALE
1199: *
1200: SCALE = ONE
1201: IF( ABS( W22 ).LT.SAFMIN ) THEN
1202: SCALE = ZERO
1203: U2 = ONE
1204: U1 = -W12 / W11
1205: GO TO 250
1206: END IF
1207: IF( ABS( W22 ).LT.ABS( U2 ) )
1208: $ SCALE = ABS( W22 / U2 )
1209: IF( ABS( W11 ).LT.ABS( U1 ) )
1210: $ SCALE = MIN( SCALE, ABS( W11 / U1 ) )
1211: *
1212: * Solve
1213: *
1214: U2 = ( SCALE*U2 ) / W22
1215: U1 = ( SCALE*U1-W12*U2 ) / W11
1216: *
1217: 250 CONTINUE
1218: IF( ILPIVT ) THEN
1219: TEMP = U2
1220: U2 = U1
1221: U1 = TEMP
1222: END IF
1223: *
1224: * Compute Householder Vector
1225: *
1226: T1 = SQRT( SCALE**2+U1**2+U2**2 )
1227: TAU = ONE + SCALE / T1
1228: VS = -ONE / ( SCALE+T1 )
1229: V( 1 ) = ONE
1230: V( 2 ) = VS*U1
1231: V( 3 ) = VS*U2
1232: *
1233: * Apply transformations from the right.
1234: *
1235: DO 260 JR = IFRSTM, MIN( J+3, ILAST )
1236: TEMP = TAU*( H( JR, J )+V( 2 )*H( JR, J+1 )+V( 3 )*
1237: $ H( JR, J+2 ) )
1238: H( JR, J ) = H( JR, J ) - TEMP
1239: H( JR, J+1 ) = H( JR, J+1 ) - TEMP*V( 2 )
1240: H( JR, J+2 ) = H( JR, J+2 ) - TEMP*V( 3 )
1241: 260 CONTINUE
1242: DO 270 JR = IFRSTM, J + 2
1243: TEMP = TAU*( T( JR, J )+V( 2 )*T( JR, J+1 )+V( 3 )*
1244: $ T( JR, J+2 ) )
1245: T( JR, J ) = T( JR, J ) - TEMP
1246: T( JR, J+1 ) = T( JR, J+1 ) - TEMP*V( 2 )
1247: T( JR, J+2 ) = T( JR, J+2 ) - TEMP*V( 3 )
1248: 270 CONTINUE
1249: IF( ILZ ) THEN
1250: DO 280 JR = 1, N
1251: TEMP = TAU*( Z( JR, J )+V( 2 )*Z( JR, J+1 )+V( 3 )*
1252: $ Z( JR, J+2 ) )
1253: Z( JR, J ) = Z( JR, J ) - TEMP
1254: Z( JR, J+1 ) = Z( JR, J+1 ) - TEMP*V( 2 )
1255: Z( JR, J+2 ) = Z( JR, J+2 ) - TEMP*V( 3 )
1256: 280 CONTINUE
1257: END IF
1258: T( J+1, J ) = ZERO
1259: T( J+2, J ) = ZERO
1260: 290 CONTINUE
1261: *
1262: * Last elements: Use Givens rotations
1263: *
1264: * Rotations from the left
1265: *
1266: J = ILAST - 1
1267: TEMP = H( J, J-1 )
1268: CALL DLARTG( TEMP, H( J+1, J-1 ), C, S, H( J, J-1 ) )
1269: H( J+1, J-1 ) = ZERO
1270: *
1271: DO 300 JC = J, ILASTM
1272: TEMP = C*H( J, JC ) + S*H( J+1, JC )
1273: H( J+1, JC ) = -S*H( J, JC ) + C*H( J+1, JC )
1274: H( J, JC ) = TEMP
1275: TEMP2 = C*T( J, JC ) + S*T( J+1, JC )
1276: T( J+1, JC ) = -S*T( J, JC ) + C*T( J+1, JC )
1277: T( J, JC ) = TEMP2
1278: 300 CONTINUE
1279: IF( ILQ ) THEN
1280: DO 310 JR = 1, N
1281: TEMP = C*Q( JR, J ) + S*Q( JR, J+1 )
1282: Q( JR, J+1 ) = -S*Q( JR, J ) + C*Q( JR, J+1 )
1283: Q( JR, J ) = TEMP
1284: 310 CONTINUE
1285: END IF
1286: *
1287: * Rotations from the right.
1288: *
1289: TEMP = T( J+1, J+1 )
1290: CALL DLARTG( TEMP, T( J+1, J ), C, S, T( J+1, J+1 ) )
1291: T( J+1, J ) = ZERO
1292: *
1293: DO 320 JR = IFRSTM, ILAST
1294: TEMP = C*H( JR, J+1 ) + S*H( JR, J )
1295: H( JR, J ) = -S*H( JR, J+1 ) + C*H( JR, J )
1296: H( JR, J+1 ) = TEMP
1297: 320 CONTINUE
1298: DO 330 JR = IFRSTM, ILAST - 1
1299: TEMP = C*T( JR, J+1 ) + S*T( JR, J )
1300: T( JR, J ) = -S*T( JR, J+1 ) + C*T( JR, J )
1301: T( JR, J+1 ) = TEMP
1302: 330 CONTINUE
1303: IF( ILZ ) THEN
1304: DO 340 JR = 1, N
1305: TEMP = C*Z( JR, J+1 ) + S*Z( JR, J )
1306: Z( JR, J ) = -S*Z( JR, J+1 ) + C*Z( JR, J )
1307: Z( JR, J+1 ) = TEMP
1308: 340 CONTINUE
1309: END IF
1310: *
1311: * End of Double-Shift code
1312: *
1313: END IF
1314: *
1315: GO TO 350
1316: *
1317: * End of iteration loop
1318: *
1319: 350 CONTINUE
1320: 360 CONTINUE
1321: *
1322: * Drop-through = non-convergence
1323: *
1324: INFO = ILAST
1325: GO TO 420
1326: *
1327: * Successful completion of all QZ steps
1328: *
1329: 380 CONTINUE
1330: *
1331: * Set Eigenvalues 1:ILO-1
1332: *
1333: DO 410 J = 1, ILO - 1
1334: IF( T( J, J ).LT.ZERO ) THEN
1335: IF( ILSCHR ) THEN
1336: DO 390 JR = 1, J
1337: H( JR, J ) = -H( JR, J )
1338: T( JR, J ) = -T( JR, J )
1339: 390 CONTINUE
1340: ELSE
1341: H( J, J ) = -H( J, J )
1342: T( J, J ) = -T( J, J )
1343: END IF
1344: IF( ILZ ) THEN
1345: DO 400 JR = 1, N
1346: Z( JR, J ) = -Z( JR, J )
1347: 400 CONTINUE
1348: END IF
1349: END IF
1350: ALPHAR( J ) = H( J, J )
1351: ALPHAI( J ) = ZERO
1352: BETA( J ) = T( J, J )
1353: 410 CONTINUE
1354: *
1355: * Normal Termination
1356: *
1357: INFO = 0
1358: *
1359: * Exit (other than argument error) -- return optimal workspace size
1360: *
1361: 420 CONTINUE
1362: WORK( 1 ) = DBLE( N )
1363: RETURN
1364: *
1365: * End of DHGEQZ
1366: *
1367: END
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