Annotation of rpl/lapack/lapack/dhgeqz.f, revision 1.21
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: *
285: *> \ingroup doubleGEcomputational
286: *
287: *> \par Further Details:
288: * =====================
289: *>
290: *> \verbatim
291: *>
292: *> Iteration counters:
293: *>
294: *> JITER -- counts iterations.
295: *> IITER -- counts iterations run since ILAST was last
296: *> changed. This is therefore reset only when a 1-by-1 or
297: *> 2-by-2 block deflates off the bottom.
298: *> \endverbatim
299: *>
300: * =====================================================================
1.1 bertrand 301: SUBROUTINE DHGEQZ( JOB, COMPQ, COMPZ, N, ILO, IHI, H, LDH, T, LDT,
302: $ ALPHAR, ALPHAI, BETA, Q, LDQ, Z, LDZ, WORK,
303: $ LWORK, INFO )
304: *
1.21 ! bertrand 305: * -- LAPACK computational routine --
1.1 bertrand 306: * -- LAPACK is a software package provided by Univ. of Tennessee, --
307: * -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
308: *
309: * .. Scalar Arguments ..
310: CHARACTER COMPQ, COMPZ, JOB
311: INTEGER IHI, ILO, INFO, LDH, LDQ, LDT, LDZ, LWORK, N
312: * ..
313: * .. Array Arguments ..
314: DOUBLE PRECISION ALPHAI( * ), ALPHAR( * ), BETA( * ),
315: $ H( LDH, * ), Q( LDQ, * ), T( LDT, * ),
316: $ WORK( * ), Z( LDZ, * )
317: * ..
318: *
319: * =====================================================================
320: *
321: * .. Parameters ..
322: * $ SAFETY = 1.0E+0 )
323: DOUBLE PRECISION HALF, ZERO, ONE, SAFETY
324: PARAMETER ( HALF = 0.5D+0, ZERO = 0.0D+0, ONE = 1.0D+0,
325: $ SAFETY = 1.0D+2 )
326: * ..
327: * .. Local Scalars ..
328: LOGICAL ILAZR2, ILAZRO, ILPIVT, ILQ, ILSCHR, ILZ,
329: $ LQUERY
330: INTEGER ICOMPQ, ICOMPZ, IFIRST, IFRSTM, IITER, ILAST,
331: $ ILASTM, IN, ISCHUR, ISTART, J, JC, JCH, JITER,
332: $ JR, MAXIT
333: DOUBLE PRECISION A11, A12, A1I, A1R, A21, A22, A2I, A2R, AD11,
334: $ AD11L, AD12, AD12L, AD21, AD21L, AD22, AD22L,
335: $ AD32L, AN, ANORM, ASCALE, ATOL, B11, B1A, B1I,
336: $ B1R, B22, B2A, B2I, B2R, BN, BNORM, BSCALE,
337: $ BTOL, C, C11I, C11R, C12, C21, C22I, C22R, CL,
338: $ CQ, CR, CZ, ESHIFT, S, S1, S1INV, S2, SAFMAX,
339: $ SAFMIN, SCALE, SL, SQI, SQR, SR, SZI, SZR, T1,
340: $ TAU, TEMP, TEMP2, TEMPI, TEMPR, U1, U12, U12L,
341: $ U2, ULP, VS, W11, W12, W21, W22, WABS, WI, WR,
342: $ WR2
343: * ..
344: * .. Local Arrays ..
345: DOUBLE PRECISION V( 3 )
346: * ..
347: * .. External Functions ..
348: LOGICAL LSAME
349: DOUBLE PRECISION DLAMCH, DLANHS, DLAPY2, DLAPY3
350: EXTERNAL LSAME, DLAMCH, DLANHS, DLAPY2, DLAPY3
351: * ..
352: * .. External Subroutines ..
353: EXTERNAL DLAG2, DLARFG, DLARTG, DLASET, DLASV2, DROT,
354: $ XERBLA
355: * ..
356: * .. Intrinsic Functions ..
357: INTRINSIC ABS, DBLE, MAX, MIN, SQRT
358: * ..
359: * .. Executable Statements ..
360: *
361: * Decode JOB, COMPQ, COMPZ
362: *
363: IF( LSAME( JOB, 'E' ) ) THEN
364: ILSCHR = .FALSE.
365: ISCHUR = 1
366: ELSE IF( LSAME( JOB, 'S' ) ) THEN
367: ILSCHR = .TRUE.
368: ISCHUR = 2
369: ELSE
370: ISCHUR = 0
371: END IF
372: *
373: IF( LSAME( COMPQ, 'N' ) ) THEN
374: ILQ = .FALSE.
375: ICOMPQ = 1
376: ELSE IF( LSAME( COMPQ, 'V' ) ) THEN
377: ILQ = .TRUE.
378: ICOMPQ = 2
379: ELSE IF( LSAME( COMPQ, 'I' ) ) THEN
380: ILQ = .TRUE.
381: ICOMPQ = 3
382: ELSE
383: ICOMPQ = 0
384: END IF
385: *
386: IF( LSAME( COMPZ, 'N' ) ) THEN
387: ILZ = .FALSE.
388: ICOMPZ = 1
389: ELSE IF( LSAME( COMPZ, 'V' ) ) THEN
390: ILZ = .TRUE.
391: ICOMPZ = 2
392: ELSE IF( LSAME( COMPZ, 'I' ) ) THEN
393: ILZ = .TRUE.
394: ICOMPZ = 3
395: ELSE
396: ICOMPZ = 0
397: END IF
398: *
399: * Check Argument Values
400: *
401: INFO = 0
402: WORK( 1 ) = MAX( 1, N )
403: LQUERY = ( LWORK.EQ.-1 )
404: IF( ISCHUR.EQ.0 ) THEN
405: INFO = -1
406: ELSE IF( ICOMPQ.EQ.0 ) THEN
407: INFO = -2
408: ELSE IF( ICOMPZ.EQ.0 ) THEN
409: INFO = -3
410: ELSE IF( N.LT.0 ) THEN
411: INFO = -4
412: ELSE IF( ILO.LT.1 ) THEN
413: INFO = -5
414: ELSE IF( IHI.GT.N .OR. IHI.LT.ILO-1 ) THEN
415: INFO = -6
416: ELSE IF( LDH.LT.N ) THEN
417: INFO = -8
418: ELSE IF( LDT.LT.N ) THEN
419: INFO = -10
420: ELSE IF( LDQ.LT.1 .OR. ( ILQ .AND. LDQ.LT.N ) ) THEN
421: INFO = -15
422: ELSE IF( LDZ.LT.1 .OR. ( ILZ .AND. LDZ.LT.N ) ) THEN
423: INFO = -17
424: ELSE IF( LWORK.LT.MAX( 1, N ) .AND. .NOT.LQUERY ) THEN
425: INFO = -19
426: END IF
427: IF( INFO.NE.0 ) THEN
428: CALL XERBLA( 'DHGEQZ', -INFO )
429: RETURN
430: ELSE IF( LQUERY ) THEN
431: RETURN
432: END IF
433: *
434: * Quick return if possible
435: *
436: IF( N.LE.0 ) THEN
437: WORK( 1 ) = DBLE( 1 )
438: RETURN
439: END IF
440: *
441: * Initialize Q and Z
442: *
443: IF( ICOMPQ.EQ.3 )
444: $ CALL DLASET( 'Full', N, N, ZERO, ONE, Q, LDQ )
445: IF( ICOMPZ.EQ.3 )
446: $ CALL DLASET( 'Full', N, N, ZERO, ONE, Z, LDZ )
447: *
448: * Machine Constants
449: *
450: IN = IHI + 1 - ILO
451: SAFMIN = DLAMCH( 'S' )
452: SAFMAX = ONE / SAFMIN
453: ULP = DLAMCH( 'E' )*DLAMCH( 'B' )
454: ANORM = DLANHS( 'F', IN, H( ILO, ILO ), LDH, WORK )
455: BNORM = DLANHS( 'F', IN, T( ILO, ILO ), LDT, WORK )
456: ATOL = MAX( SAFMIN, ULP*ANORM )
457: BTOL = MAX( SAFMIN, ULP*BNORM )
458: ASCALE = ONE / MAX( SAFMIN, ANORM )
459: BSCALE = ONE / MAX( SAFMIN, BNORM )
460: *
461: * Set Eigenvalues IHI+1:N
462: *
463: DO 30 J = IHI + 1, N
464: IF( T( J, J ).LT.ZERO ) THEN
465: IF( ILSCHR ) THEN
466: DO 10 JR = 1, J
467: H( JR, J ) = -H( JR, J )
468: T( JR, J ) = -T( JR, J )
469: 10 CONTINUE
470: ELSE
471: H( J, J ) = -H( J, J )
472: T( J, J ) = -T( J, J )
473: END IF
474: IF( ILZ ) THEN
475: DO 20 JR = 1, N
476: Z( JR, J ) = -Z( JR, J )
477: 20 CONTINUE
478: END IF
479: END IF
480: ALPHAR( J ) = H( J, J )
481: ALPHAI( J ) = ZERO
482: BETA( J ) = T( J, J )
483: 30 CONTINUE
484: *
485: * If IHI < ILO, skip QZ steps
486: *
487: IF( IHI.LT.ILO )
488: $ GO TO 380
489: *
490: * MAIN QZ ITERATION LOOP
491: *
492: * Initialize dynamic indices
493: *
494: * Eigenvalues ILAST+1:N have been found.
495: * Column operations modify rows IFRSTM:whatever.
496: * Row operations modify columns whatever:ILASTM.
497: *
498: * If only eigenvalues are being computed, then
499: * IFRSTM is the row of the last splitting row above row ILAST;
500: * this is always at least ILO.
501: * IITER counts iterations since the last eigenvalue was found,
502: * to tell when to use an extraordinary shift.
503: * MAXIT is the maximum number of QZ sweeps allowed.
504: *
505: ILAST = IHI
506: IF( ILSCHR ) THEN
507: IFRSTM = 1
508: ILASTM = N
509: ELSE
510: IFRSTM = ILO
511: ILASTM = IHI
512: END IF
513: IITER = 0
514: ESHIFT = ZERO
515: MAXIT = 30*( IHI-ILO+1 )
516: *
517: DO 360 JITER = 1, MAXIT
518: *
519: * Split the matrix if possible.
520: *
521: * Two tests:
522: * 1: H(j,j-1)=0 or j=ILO
523: * 2: T(j,j)=0
524: *
525: IF( ILAST.EQ.ILO ) THEN
526: *
527: * Special case: j=ILAST
528: *
529: GO TO 80
530: ELSE
1.21 ! bertrand 531: IF( ABS( H( ILAST, ILAST-1 ) ).LE.MAX( SAFMIN, ULP*(
! 532: $ ABS( H( ILAST, ILAST ) ) + ABS( H( ILAST-1, ILAST-1 ) )
! 533: $ ) ) ) THEN
1.1 bertrand 534: H( ILAST, ILAST-1 ) = ZERO
535: GO TO 80
536: END IF
537: END IF
538: *
539: IF( ABS( T( ILAST, ILAST ) ).LE.BTOL ) THEN
540: T( ILAST, ILAST ) = ZERO
541: GO TO 70
542: END IF
543: *
544: * General case: j<ILAST
545: *
546: DO 60 J = ILAST - 1, ILO, -1
547: *
548: * Test 1: for H(j,j-1)=0 or j=ILO
549: *
550: IF( J.EQ.ILO ) THEN
551: ILAZRO = .TRUE.
552: ELSE
1.21 ! bertrand 553: IF( ABS( H( J, J-1 ) ).LE.MAX( SAFMIN, ULP*(
! 554: $ ABS( H( J, J ) ) + ABS( H( J-1, J-1 ) )
! 555: $ ) ) ) THEN
1.1 bertrand 556: H( J, J-1 ) = ZERO
557: ILAZRO = .TRUE.
558: ELSE
559: ILAZRO = .FALSE.
560: END IF
561: END IF
562: *
563: * Test 2: for T(j,j)=0
564: *
565: IF( ABS( T( J, J ) ).LT.BTOL ) THEN
566: T( J, J ) = ZERO
567: *
568: * Test 1a: Check for 2 consecutive small subdiagonals in A
569: *
570: ILAZR2 = .FALSE.
571: IF( .NOT.ILAZRO ) THEN
572: TEMP = ABS( H( J, J-1 ) )
573: TEMP2 = ABS( H( J, J ) )
574: TEMPR = MAX( TEMP, TEMP2 )
575: IF( TEMPR.LT.ONE .AND. TEMPR.NE.ZERO ) THEN
576: TEMP = TEMP / TEMPR
577: TEMP2 = TEMP2 / TEMPR
578: END IF
579: IF( TEMP*( ASCALE*ABS( H( J+1, J ) ) ).LE.TEMP2*
580: $ ( ASCALE*ATOL ) )ILAZR2 = .TRUE.
581: END IF
582: *
583: * If both tests pass (1 & 2), i.e., the leading diagonal
584: * element of B in the block is zero, split a 1x1 block off
585: * at the top. (I.e., at the J-th row/column) The leading
586: * diagonal element of the remainder can also be zero, so
587: * this may have to be done repeatedly.
588: *
589: IF( ILAZRO .OR. ILAZR2 ) THEN
590: DO 40 JCH = J, ILAST - 1
591: TEMP = H( JCH, JCH )
592: CALL DLARTG( TEMP, H( JCH+1, JCH ), C, S,
593: $ H( JCH, JCH ) )
594: H( JCH+1, JCH ) = ZERO
595: CALL DROT( ILASTM-JCH, H( JCH, JCH+1 ), LDH,
596: $ H( JCH+1, JCH+1 ), LDH, C, S )
597: CALL DROT( ILASTM-JCH, T( JCH, JCH+1 ), LDT,
598: $ T( JCH+1, JCH+1 ), LDT, C, S )
599: IF( ILQ )
600: $ CALL DROT( N, Q( 1, JCH ), 1, Q( 1, JCH+1 ), 1,
601: $ C, S )
602: IF( ILAZR2 )
603: $ H( JCH, JCH-1 ) = H( JCH, JCH-1 )*C
604: ILAZR2 = .FALSE.
605: IF( ABS( T( JCH+1, JCH+1 ) ).GE.BTOL ) THEN
606: IF( JCH+1.GE.ILAST ) THEN
607: GO TO 80
608: ELSE
609: IFIRST = JCH + 1
610: GO TO 110
611: END IF
612: END IF
613: T( JCH+1, JCH+1 ) = ZERO
614: 40 CONTINUE
615: GO TO 70
616: ELSE
617: *
618: * Only test 2 passed -- chase the zero to T(ILAST,ILAST)
619: * Then process as in the case T(ILAST,ILAST)=0
620: *
621: DO 50 JCH = J, ILAST - 1
622: TEMP = T( JCH, JCH+1 )
623: CALL DLARTG( TEMP, T( JCH+1, JCH+1 ), C, S,
624: $ T( JCH, JCH+1 ) )
625: T( JCH+1, JCH+1 ) = ZERO
626: IF( JCH.LT.ILASTM-1 )
627: $ CALL DROT( ILASTM-JCH-1, T( JCH, JCH+2 ), LDT,
628: $ T( JCH+1, JCH+2 ), LDT, C, S )
629: CALL DROT( ILASTM-JCH+2, H( JCH, JCH-1 ), LDH,
630: $ H( JCH+1, JCH-1 ), LDH, C, S )
631: IF( ILQ )
632: $ CALL DROT( N, Q( 1, JCH ), 1, Q( 1, JCH+1 ), 1,
633: $ C, S )
634: TEMP = H( JCH+1, JCH )
635: CALL DLARTG( TEMP, H( JCH+1, JCH-1 ), C, S,
636: $ H( JCH+1, JCH ) )
637: H( JCH+1, JCH-1 ) = ZERO
638: CALL DROT( JCH+1-IFRSTM, H( IFRSTM, JCH ), 1,
639: $ H( IFRSTM, JCH-1 ), 1, C, S )
640: CALL DROT( JCH-IFRSTM, T( IFRSTM, JCH ), 1,
641: $ T( IFRSTM, JCH-1 ), 1, C, S )
642: IF( ILZ )
643: $ CALL DROT( N, Z( 1, JCH ), 1, Z( 1, JCH-1 ), 1,
644: $ C, S )
645: 50 CONTINUE
646: GO TO 70
647: END IF
648: ELSE IF( ILAZRO ) THEN
649: *
650: * Only test 1 passed -- work on J:ILAST
651: *
652: IFIRST = J
653: GO TO 110
654: END IF
655: *
656: * Neither test passed -- try next J
657: *
658: 60 CONTINUE
659: *
660: * (Drop-through is "impossible")
661: *
662: INFO = N + 1
663: GO TO 420
664: *
665: * T(ILAST,ILAST)=0 -- clear H(ILAST,ILAST-1) to split off a
666: * 1x1 block.
667: *
668: 70 CONTINUE
669: TEMP = H( ILAST, ILAST )
670: CALL DLARTG( TEMP, H( ILAST, ILAST-1 ), C, S,
671: $ H( ILAST, ILAST ) )
672: H( ILAST, ILAST-1 ) = ZERO
673: CALL DROT( ILAST-IFRSTM, H( IFRSTM, ILAST ), 1,
674: $ H( IFRSTM, ILAST-1 ), 1, C, S )
675: CALL DROT( ILAST-IFRSTM, T( IFRSTM, ILAST ), 1,
676: $ T( IFRSTM, ILAST-1 ), 1, C, S )
677: IF( ILZ )
678: $ CALL DROT( N, Z( 1, ILAST ), 1, Z( 1, ILAST-1 ), 1, C, S )
679: *
680: * H(ILAST,ILAST-1)=0 -- Standardize B, set ALPHAR, ALPHAI,
681: * and BETA
682: *
683: 80 CONTINUE
684: IF( T( ILAST, ILAST ).LT.ZERO ) THEN
685: IF( ILSCHR ) THEN
686: DO 90 J = IFRSTM, ILAST
687: H( J, ILAST ) = -H( J, ILAST )
688: T( J, ILAST ) = -T( J, ILAST )
689: 90 CONTINUE
690: ELSE
691: H( ILAST, ILAST ) = -H( ILAST, ILAST )
692: T( ILAST, ILAST ) = -T( ILAST, ILAST )
693: END IF
694: IF( ILZ ) THEN
695: DO 100 J = 1, N
696: Z( J, ILAST ) = -Z( J, ILAST )
697: 100 CONTINUE
698: END IF
699: END IF
700: ALPHAR( ILAST ) = H( ILAST, ILAST )
701: ALPHAI( ILAST ) = ZERO
702: BETA( ILAST ) = T( ILAST, ILAST )
703: *
704: * Go to next block -- exit if finished.
705: *
706: ILAST = ILAST - 1
707: IF( ILAST.LT.ILO )
708: $ GO TO 380
709: *
710: * Reset counters
711: *
712: IITER = 0
713: ESHIFT = ZERO
714: IF( .NOT.ILSCHR ) THEN
715: ILASTM = ILAST
716: IF( IFRSTM.GT.ILAST )
717: $ IFRSTM = ILO
718: END IF
719: GO TO 350
720: *
721: * QZ step
722: *
723: * This iteration only involves rows/columns IFIRST:ILAST. We
724: * assume IFIRST < ILAST, and that the diagonal of B is non-zero.
725: *
726: 110 CONTINUE
727: IITER = IITER + 1
728: IF( .NOT.ILSCHR ) THEN
729: IFRSTM = IFIRST
730: END IF
731: *
732: * Compute single shifts.
733: *
734: * At this point, IFIRST < ILAST, and the diagonal elements of
735: * T(IFIRST:ILAST,IFIRST,ILAST) are larger than BTOL (in
736: * magnitude)
737: *
738: IF( ( IITER / 10 )*10.EQ.IITER ) THEN
739: *
740: * Exceptional shift. Chosen for no particularly good reason.
741: * (Single shift only.)
742: *
1.14 bertrand 743: IF( ( DBLE( MAXIT )*SAFMIN )*ABS( H( ILAST, ILAST-1 ) ).LT.
1.1 bertrand 744: $ ABS( T( ILAST-1, ILAST-1 ) ) ) THEN
1.14 bertrand 745: ESHIFT = H( ILAST, ILAST-1 ) /
1.1 bertrand 746: $ T( ILAST-1, ILAST-1 )
747: ELSE
748: ESHIFT = ESHIFT + ONE / ( SAFMIN*DBLE( MAXIT ) )
749: END IF
750: S1 = ONE
751: WR = ESHIFT
752: *
753: ELSE
754: *
755: * Shifts based on the generalized eigenvalues of the
756: * bottom-right 2x2 block of A and B. The first eigenvalue
757: * returned by DLAG2 is the Wilkinson shift (AEP p.512),
758: *
759: CALL DLAG2( H( ILAST-1, ILAST-1 ), LDH,
760: $ T( ILAST-1, ILAST-1 ), LDT, SAFMIN*SAFETY, S1,
761: $ S2, WR, WR2, WI )
762: *
1.14 bertrand 763: IF ( ABS( (WR/S1)*T( ILAST, ILAST ) - H( ILAST, ILAST ) )
1.18 bertrand 764: $ .GT. ABS( (WR2/S2)*T( ILAST, ILAST )
1.14 bertrand 765: $ - H( ILAST, ILAST ) ) ) THEN
766: TEMP = WR
767: WR = WR2
768: WR2 = TEMP
769: TEMP = S1
770: S1 = S2
771: S2 = TEMP
772: END IF
1.1 bertrand 773: TEMP = MAX( S1, SAFMIN*MAX( ONE, ABS( WR ), ABS( WI ) ) )
774: IF( WI.NE.ZERO )
775: $ GO TO 200
776: END IF
777: *
778: * Fiddle with shift to avoid overflow
779: *
780: TEMP = MIN( ASCALE, ONE )*( HALF*SAFMAX )
781: IF( S1.GT.TEMP ) THEN
782: SCALE = TEMP / S1
783: ELSE
784: SCALE = ONE
785: END IF
786: *
787: TEMP = MIN( BSCALE, ONE )*( HALF*SAFMAX )
788: IF( ABS( WR ).GT.TEMP )
789: $ SCALE = MIN( SCALE, TEMP / ABS( WR ) )
790: S1 = SCALE*S1
791: WR = SCALE*WR
792: *
793: * Now check for two consecutive small subdiagonals.
794: *
795: DO 120 J = ILAST - 1, IFIRST + 1, -1
796: ISTART = J
797: TEMP = ABS( S1*H( J, J-1 ) )
798: TEMP2 = ABS( S1*H( J, J )-WR*T( J, J ) )
799: TEMPR = MAX( TEMP, TEMP2 )
800: IF( TEMPR.LT.ONE .AND. TEMPR.NE.ZERO ) THEN
801: TEMP = TEMP / TEMPR
802: TEMP2 = TEMP2 / TEMPR
803: END IF
804: IF( ABS( ( ASCALE*H( J+1, J ) )*TEMP ).LE.( ASCALE*ATOL )*
805: $ TEMP2 )GO TO 130
806: 120 CONTINUE
807: *
808: ISTART = IFIRST
809: 130 CONTINUE
810: *
811: * Do an implicit single-shift QZ sweep.
812: *
813: * Initial Q
814: *
815: TEMP = S1*H( ISTART, ISTART ) - WR*T( ISTART, ISTART )
816: TEMP2 = S1*H( ISTART+1, ISTART )
817: CALL DLARTG( TEMP, TEMP2, C, S, TEMPR )
818: *
819: * Sweep
820: *
821: DO 190 J = ISTART, ILAST - 1
822: IF( J.GT.ISTART ) THEN
823: TEMP = H( J, J-1 )
824: CALL DLARTG( TEMP, H( J+1, J-1 ), C, S, H( J, J-1 ) )
825: H( J+1, J-1 ) = ZERO
826: END IF
827: *
828: DO 140 JC = J, ILASTM
829: TEMP = C*H( J, JC ) + S*H( J+1, JC )
830: H( J+1, JC ) = -S*H( J, JC ) + C*H( J+1, JC )
831: H( J, JC ) = TEMP
832: TEMP2 = C*T( J, JC ) + S*T( J+1, JC )
833: T( J+1, JC ) = -S*T( J, JC ) + C*T( J+1, JC )
834: T( J, JC ) = TEMP2
835: 140 CONTINUE
836: IF( ILQ ) THEN
837: DO 150 JR = 1, N
838: TEMP = C*Q( JR, J ) + S*Q( JR, J+1 )
839: Q( JR, J+1 ) = -S*Q( JR, J ) + C*Q( JR, J+1 )
840: Q( JR, J ) = TEMP
841: 150 CONTINUE
842: END IF
843: *
844: TEMP = T( J+1, J+1 )
845: CALL DLARTG( TEMP, T( J+1, J ), C, S, T( J+1, J+1 ) )
846: T( J+1, J ) = ZERO
847: *
848: DO 160 JR = IFRSTM, MIN( J+2, ILAST )
849: TEMP = C*H( JR, J+1 ) + S*H( JR, J )
850: H( JR, J ) = -S*H( JR, J+1 ) + C*H( JR, J )
851: H( JR, J+1 ) = TEMP
852: 160 CONTINUE
853: DO 170 JR = IFRSTM, J
854: TEMP = C*T( JR, J+1 ) + S*T( JR, J )
855: T( JR, J ) = -S*T( JR, J+1 ) + C*T( JR, J )
856: T( JR, J+1 ) = TEMP
857: 170 CONTINUE
858: IF( ILZ ) THEN
859: DO 180 JR = 1, N
860: TEMP = C*Z( JR, J+1 ) + S*Z( JR, J )
861: Z( JR, J ) = -S*Z( JR, J+1 ) + C*Z( JR, J )
862: Z( JR, J+1 ) = TEMP
863: 180 CONTINUE
864: END IF
865: 190 CONTINUE
866: *
867: GO TO 350
868: *
869: * Use Francis double-shift
870: *
871: * Note: the Francis double-shift should work with real shifts,
872: * but only if the block is at least 3x3.
873: * This code may break if this point is reached with
874: * a 2x2 block with real eigenvalues.
875: *
876: 200 CONTINUE
877: IF( IFIRST+1.EQ.ILAST ) THEN
878: *
879: * Special case -- 2x2 block with complex eigenvectors
880: *
881: * Step 1: Standardize, that is, rotate so that
882: *
883: * ( B11 0 )
884: * B = ( ) with B11 non-negative.
885: * ( 0 B22 )
886: *
887: CALL DLASV2( T( ILAST-1, ILAST-1 ), T( ILAST-1, ILAST ),
888: $ T( ILAST, ILAST ), B22, B11, SR, CR, SL, CL )
889: *
890: IF( B11.LT.ZERO ) THEN
891: CR = -CR
892: SR = -SR
893: B11 = -B11
894: B22 = -B22
895: END IF
896: *
897: CALL DROT( ILASTM+1-IFIRST, H( ILAST-1, ILAST-1 ), LDH,
898: $ H( ILAST, ILAST-1 ), LDH, CL, SL )
899: CALL DROT( ILAST+1-IFRSTM, H( IFRSTM, ILAST-1 ), 1,
900: $ H( IFRSTM, ILAST ), 1, CR, SR )
901: *
902: IF( ILAST.LT.ILASTM )
903: $ CALL DROT( ILASTM-ILAST, T( ILAST-1, ILAST+1 ), LDT,
904: $ T( ILAST, ILAST+1 ), LDT, CL, SL )
905: IF( IFRSTM.LT.ILAST-1 )
906: $ CALL DROT( IFIRST-IFRSTM, T( IFRSTM, ILAST-1 ), 1,
907: $ T( IFRSTM, ILAST ), 1, CR, SR )
908: *
909: IF( ILQ )
910: $ CALL DROT( N, Q( 1, ILAST-1 ), 1, Q( 1, ILAST ), 1, CL,
911: $ SL )
912: IF( ILZ )
913: $ CALL DROT( N, Z( 1, ILAST-1 ), 1, Z( 1, ILAST ), 1, CR,
914: $ SR )
915: *
916: T( ILAST-1, ILAST-1 ) = B11
917: T( ILAST-1, ILAST ) = ZERO
918: T( ILAST, ILAST-1 ) = ZERO
919: T( ILAST, ILAST ) = B22
920: *
921: * If B22 is negative, negate column ILAST
922: *
923: IF( B22.LT.ZERO ) THEN
924: DO 210 J = IFRSTM, ILAST
925: H( J, ILAST ) = -H( J, ILAST )
926: T( J, ILAST ) = -T( J, ILAST )
927: 210 CONTINUE
928: *
929: IF( ILZ ) THEN
930: DO 220 J = 1, N
931: Z( J, ILAST ) = -Z( J, ILAST )
932: 220 CONTINUE
933: END IF
1.11 bertrand 934: B22 = -B22
1.1 bertrand 935: END IF
936: *
937: * Step 2: Compute ALPHAR, ALPHAI, and BETA (see refs.)
938: *
939: * Recompute shift
940: *
941: CALL DLAG2( H( ILAST-1, ILAST-1 ), LDH,
942: $ T( ILAST-1, ILAST-1 ), LDT, SAFMIN*SAFETY, S1,
943: $ TEMP, WR, TEMP2, WI )
944: *
945: * If standardization has perturbed the shift onto real line,
946: * do another (real single-shift) QR step.
947: *
948: IF( WI.EQ.ZERO )
949: $ GO TO 350
950: S1INV = ONE / S1
951: *
952: * Do EISPACK (QZVAL) computation of alpha and beta
953: *
954: A11 = H( ILAST-1, ILAST-1 )
955: A21 = H( ILAST, ILAST-1 )
956: A12 = H( ILAST-1, ILAST )
957: A22 = H( ILAST, ILAST )
958: *
959: * Compute complex Givens rotation on right
960: * (Assume some element of C = (sA - wB) > unfl )
961: * __
962: * (sA - wB) ( CZ -SZ )
963: * ( SZ CZ )
964: *
965: C11R = S1*A11 - WR*B11
966: C11I = -WI*B11
967: C12 = S1*A12
968: C21 = S1*A21
969: C22R = S1*A22 - WR*B22
970: C22I = -WI*B22
971: *
972: IF( ABS( C11R )+ABS( C11I )+ABS( C12 ).GT.ABS( C21 )+
973: $ ABS( C22R )+ABS( C22I ) ) THEN
974: T1 = DLAPY3( C12, C11R, C11I )
975: CZ = C12 / T1
976: SZR = -C11R / T1
977: SZI = -C11I / T1
978: ELSE
979: CZ = DLAPY2( C22R, C22I )
980: IF( CZ.LE.SAFMIN ) THEN
981: CZ = ZERO
982: SZR = ONE
983: SZI = ZERO
984: ELSE
985: TEMPR = C22R / CZ
986: TEMPI = C22I / CZ
987: T1 = DLAPY2( CZ, C21 )
988: CZ = CZ / T1
989: SZR = -C21*TEMPR / T1
990: SZI = C21*TEMPI / T1
991: END IF
992: END IF
993: *
994: * Compute Givens rotation on left
995: *
996: * ( CQ SQ )
997: * ( __ ) A or B
998: * ( -SQ CQ )
999: *
1000: AN = ABS( A11 ) + ABS( A12 ) + ABS( A21 ) + ABS( A22 )
1001: BN = ABS( B11 ) + ABS( B22 )
1002: WABS = ABS( WR ) + ABS( WI )
1003: IF( S1*AN.GT.WABS*BN ) THEN
1004: CQ = CZ*B11
1005: SQR = SZR*B22
1006: SQI = -SZI*B22
1007: ELSE
1008: A1R = CZ*A11 + SZR*A12
1009: A1I = SZI*A12
1010: A2R = CZ*A21 + SZR*A22
1011: A2I = SZI*A22
1012: CQ = DLAPY2( A1R, A1I )
1013: IF( CQ.LE.SAFMIN ) THEN
1014: CQ = ZERO
1015: SQR = ONE
1016: SQI = ZERO
1017: ELSE
1018: TEMPR = A1R / CQ
1019: TEMPI = A1I / CQ
1020: SQR = TEMPR*A2R + TEMPI*A2I
1021: SQI = TEMPI*A2R - TEMPR*A2I
1022: END IF
1023: END IF
1024: T1 = DLAPY3( CQ, SQR, SQI )
1025: CQ = CQ / T1
1026: SQR = SQR / T1
1027: SQI = SQI / T1
1028: *
1029: * Compute diagonal elements of QBZ
1030: *
1031: TEMPR = SQR*SZR - SQI*SZI
1032: TEMPI = SQR*SZI + SQI*SZR
1033: B1R = CQ*CZ*B11 + TEMPR*B22
1034: B1I = TEMPI*B22
1035: B1A = DLAPY2( B1R, B1I )
1036: B2R = CQ*CZ*B22 + TEMPR*B11
1037: B2I = -TEMPI*B11
1038: B2A = DLAPY2( B2R, B2I )
1039: *
1040: * Normalize so beta > 0, and Im( alpha1 ) > 0
1041: *
1042: BETA( ILAST-1 ) = B1A
1043: BETA( ILAST ) = B2A
1044: ALPHAR( ILAST-1 ) = ( WR*B1A )*S1INV
1045: ALPHAI( ILAST-1 ) = ( WI*B1A )*S1INV
1046: ALPHAR( ILAST ) = ( WR*B2A )*S1INV
1047: ALPHAI( ILAST ) = -( WI*B2A )*S1INV
1048: *
1049: * Step 3: Go to next block -- exit if finished.
1050: *
1051: ILAST = IFIRST - 1
1052: IF( ILAST.LT.ILO )
1053: $ GO TO 380
1054: *
1055: * Reset counters
1056: *
1057: IITER = 0
1058: ESHIFT = ZERO
1059: IF( .NOT.ILSCHR ) THEN
1060: ILASTM = ILAST
1061: IF( IFRSTM.GT.ILAST )
1062: $ IFRSTM = ILO
1063: END IF
1064: GO TO 350
1065: ELSE
1066: *
1067: * Usual case: 3x3 or larger block, using Francis implicit
1068: * double-shift
1069: *
1070: * 2
1071: * Eigenvalue equation is w - c w + d = 0,
1072: *
1073: * -1 2 -1
1074: * so compute 1st column of (A B ) - c A B + d
1075: * using the formula in QZIT (from EISPACK)
1076: *
1077: * We assume that the block is at least 3x3
1078: *
1079: AD11 = ( ASCALE*H( ILAST-1, ILAST-1 ) ) /
1080: $ ( BSCALE*T( ILAST-1, ILAST-1 ) )
1081: AD21 = ( ASCALE*H( ILAST, ILAST-1 ) ) /
1082: $ ( BSCALE*T( ILAST-1, ILAST-1 ) )
1083: AD12 = ( ASCALE*H( ILAST-1, ILAST ) ) /
1084: $ ( BSCALE*T( ILAST, ILAST ) )
1085: AD22 = ( ASCALE*H( ILAST, ILAST ) ) /
1086: $ ( BSCALE*T( ILAST, ILAST ) )
1087: U12 = T( ILAST-1, ILAST ) / T( ILAST, ILAST )
1088: AD11L = ( ASCALE*H( IFIRST, IFIRST ) ) /
1089: $ ( BSCALE*T( IFIRST, IFIRST ) )
1090: AD21L = ( ASCALE*H( IFIRST+1, IFIRST ) ) /
1091: $ ( BSCALE*T( IFIRST, IFIRST ) )
1092: AD12L = ( ASCALE*H( IFIRST, IFIRST+1 ) ) /
1093: $ ( BSCALE*T( IFIRST+1, IFIRST+1 ) )
1094: AD22L = ( ASCALE*H( IFIRST+1, IFIRST+1 ) ) /
1095: $ ( BSCALE*T( IFIRST+1, IFIRST+1 ) )
1096: AD32L = ( ASCALE*H( IFIRST+2, IFIRST+1 ) ) /
1097: $ ( BSCALE*T( IFIRST+1, IFIRST+1 ) )
1098: U12L = T( IFIRST, IFIRST+1 ) / T( IFIRST+1, IFIRST+1 )
1099: *
1100: V( 1 ) = ( AD11-AD11L )*( AD22-AD11L ) - AD12*AD21 +
1101: $ AD21*U12*AD11L + ( AD12L-AD11L*U12L )*AD21L
1102: V( 2 ) = ( ( AD22L-AD11L )-AD21L*U12L-( AD11-AD11L )-
1103: $ ( AD22-AD11L )+AD21*U12 )*AD21L
1104: V( 3 ) = AD32L*AD21L
1105: *
1106: ISTART = IFIRST
1107: *
1108: CALL DLARFG( 3, V( 1 ), V( 2 ), 1, TAU )
1109: V( 1 ) = ONE
1110: *
1111: * Sweep
1112: *
1113: DO 290 J = ISTART, ILAST - 2
1114: *
1115: * All but last elements: use 3x3 Householder transforms.
1116: *
1117: * Zero (j-1)st column of A
1118: *
1119: IF( J.GT.ISTART ) THEN
1120: V( 1 ) = H( J, J-1 )
1121: V( 2 ) = H( J+1, J-1 )
1122: V( 3 ) = H( J+2, J-1 )
1123: *
1124: CALL DLARFG( 3, H( J, J-1 ), V( 2 ), 1, TAU )
1125: V( 1 ) = ONE
1126: H( J+1, J-1 ) = ZERO
1127: H( J+2, J-1 ) = ZERO
1128: END IF
1129: *
1130: DO 230 JC = J, ILASTM
1131: TEMP = TAU*( H( J, JC )+V( 2 )*H( J+1, JC )+V( 3 )*
1132: $ H( J+2, JC ) )
1133: H( J, JC ) = H( J, JC ) - TEMP
1134: H( J+1, JC ) = H( J+1, JC ) - TEMP*V( 2 )
1135: H( J+2, JC ) = H( J+2, JC ) - TEMP*V( 3 )
1136: TEMP2 = TAU*( T( J, JC )+V( 2 )*T( J+1, JC )+V( 3 )*
1137: $ T( J+2, JC ) )
1138: T( J, JC ) = T( J, JC ) - TEMP2
1139: T( J+1, JC ) = T( J+1, JC ) - TEMP2*V( 2 )
1140: T( J+2, JC ) = T( J+2, JC ) - TEMP2*V( 3 )
1141: 230 CONTINUE
1142: IF( ILQ ) THEN
1143: DO 240 JR = 1, N
1144: TEMP = TAU*( Q( JR, J )+V( 2 )*Q( JR, J+1 )+V( 3 )*
1145: $ Q( JR, J+2 ) )
1146: Q( JR, J ) = Q( JR, J ) - TEMP
1147: Q( JR, J+1 ) = Q( JR, J+1 ) - TEMP*V( 2 )
1148: Q( JR, J+2 ) = Q( JR, J+2 ) - TEMP*V( 3 )
1149: 240 CONTINUE
1150: END IF
1151: *
1152: * Zero j-th column of B (see DLAGBC for details)
1153: *
1154: * Swap rows to pivot
1155: *
1156: ILPIVT = .FALSE.
1157: TEMP = MAX( ABS( T( J+1, J+1 ) ), ABS( T( J+1, J+2 ) ) )
1158: TEMP2 = MAX( ABS( T( J+2, J+1 ) ), ABS( T( J+2, J+2 ) ) )
1159: IF( MAX( TEMP, TEMP2 ).LT.SAFMIN ) THEN
1160: SCALE = ZERO
1161: U1 = ONE
1162: U2 = ZERO
1163: GO TO 250
1164: ELSE IF( TEMP.GE.TEMP2 ) THEN
1165: W11 = T( J+1, J+1 )
1166: W21 = T( J+2, J+1 )
1167: W12 = T( J+1, J+2 )
1168: W22 = T( J+2, J+2 )
1169: U1 = T( J+1, J )
1170: U2 = T( J+2, J )
1171: ELSE
1172: W21 = T( J+1, J+1 )
1173: W11 = T( J+2, J+1 )
1174: W22 = T( J+1, J+2 )
1175: W12 = T( J+2, J+2 )
1176: U2 = T( J+1, J )
1177: U1 = T( J+2, J )
1178: END IF
1179: *
1180: * Swap columns if nec.
1181: *
1182: IF( ABS( W12 ).GT.ABS( W11 ) ) THEN
1183: ILPIVT = .TRUE.
1184: TEMP = W12
1185: TEMP2 = W22
1186: W12 = W11
1187: W22 = W21
1188: W11 = TEMP
1189: W21 = TEMP2
1190: END IF
1191: *
1192: * LU-factor
1193: *
1194: TEMP = W21 / W11
1195: U2 = U2 - TEMP*U1
1196: W22 = W22 - TEMP*W12
1197: W21 = ZERO
1198: *
1199: * Compute SCALE
1200: *
1201: SCALE = ONE
1202: IF( ABS( W22 ).LT.SAFMIN ) THEN
1203: SCALE = ZERO
1204: U2 = ONE
1205: U1 = -W12 / W11
1206: GO TO 250
1207: END IF
1208: IF( ABS( W22 ).LT.ABS( U2 ) )
1209: $ SCALE = ABS( W22 / U2 )
1210: IF( ABS( W11 ).LT.ABS( U1 ) )
1211: $ SCALE = MIN( SCALE, ABS( W11 / U1 ) )
1212: *
1213: * Solve
1214: *
1215: U2 = ( SCALE*U2 ) / W22
1216: U1 = ( SCALE*U1-W12*U2 ) / W11
1217: *
1218: 250 CONTINUE
1219: IF( ILPIVT ) THEN
1220: TEMP = U2
1221: U2 = U1
1222: U1 = TEMP
1223: END IF
1224: *
1225: * Compute Householder Vector
1226: *
1227: T1 = SQRT( SCALE**2+U1**2+U2**2 )
1228: TAU = ONE + SCALE / T1
1229: VS = -ONE / ( SCALE+T1 )
1230: V( 1 ) = ONE
1231: V( 2 ) = VS*U1
1232: V( 3 ) = VS*U2
1233: *
1234: * Apply transformations from the right.
1235: *
1236: DO 260 JR = IFRSTM, MIN( J+3, ILAST )
1237: TEMP = TAU*( H( JR, J )+V( 2 )*H( JR, J+1 )+V( 3 )*
1238: $ H( JR, J+2 ) )
1239: H( JR, J ) = H( JR, J ) - TEMP
1240: H( JR, J+1 ) = H( JR, J+1 ) - TEMP*V( 2 )
1241: H( JR, J+2 ) = H( JR, J+2 ) - TEMP*V( 3 )
1242: 260 CONTINUE
1243: DO 270 JR = IFRSTM, J + 2
1244: TEMP = TAU*( T( JR, J )+V( 2 )*T( JR, J+1 )+V( 3 )*
1245: $ T( JR, J+2 ) )
1246: T( JR, J ) = T( JR, J ) - TEMP
1247: T( JR, J+1 ) = T( JR, J+1 ) - TEMP*V( 2 )
1248: T( JR, J+2 ) = T( JR, J+2 ) - TEMP*V( 3 )
1249: 270 CONTINUE
1250: IF( ILZ ) THEN
1251: DO 280 JR = 1, N
1252: TEMP = TAU*( Z( JR, J )+V( 2 )*Z( JR, J+1 )+V( 3 )*
1253: $ Z( JR, J+2 ) )
1254: Z( JR, J ) = Z( JR, J ) - TEMP
1255: Z( JR, J+1 ) = Z( JR, J+1 ) - TEMP*V( 2 )
1256: Z( JR, J+2 ) = Z( JR, J+2 ) - TEMP*V( 3 )
1257: 280 CONTINUE
1258: END IF
1259: T( J+1, J ) = ZERO
1260: T( J+2, J ) = ZERO
1261: 290 CONTINUE
1262: *
1263: * Last elements: Use Givens rotations
1264: *
1265: * Rotations from the left
1266: *
1267: J = ILAST - 1
1268: TEMP = H( J, J-1 )
1269: CALL DLARTG( TEMP, H( J+1, J-1 ), C, S, H( J, J-1 ) )
1270: H( J+1, J-1 ) = ZERO
1271: *
1272: DO 300 JC = J, ILASTM
1273: TEMP = C*H( J, JC ) + S*H( J+1, JC )
1274: H( J+1, JC ) = -S*H( J, JC ) + C*H( J+1, JC )
1275: H( J, JC ) = TEMP
1276: TEMP2 = C*T( J, JC ) + S*T( J+1, JC )
1277: T( J+1, JC ) = -S*T( J, JC ) + C*T( J+1, JC )
1278: T( J, JC ) = TEMP2
1279: 300 CONTINUE
1280: IF( ILQ ) THEN
1281: DO 310 JR = 1, N
1282: TEMP = C*Q( JR, J ) + S*Q( JR, J+1 )
1283: Q( JR, J+1 ) = -S*Q( JR, J ) + C*Q( JR, J+1 )
1284: Q( JR, J ) = TEMP
1285: 310 CONTINUE
1286: END IF
1287: *
1288: * Rotations from the right.
1289: *
1290: TEMP = T( J+1, J+1 )
1291: CALL DLARTG( TEMP, T( J+1, J ), C, S, T( J+1, J+1 ) )
1292: T( J+1, J ) = ZERO
1293: *
1294: DO 320 JR = IFRSTM, ILAST
1295: TEMP = C*H( JR, J+1 ) + S*H( JR, J )
1296: H( JR, J ) = -S*H( JR, J+1 ) + C*H( JR, J )
1297: H( JR, J+1 ) = TEMP
1298: 320 CONTINUE
1299: DO 330 JR = IFRSTM, ILAST - 1
1300: TEMP = C*T( JR, J+1 ) + S*T( JR, J )
1301: T( JR, J ) = -S*T( JR, J+1 ) + C*T( JR, J )
1302: T( JR, J+1 ) = TEMP
1303: 330 CONTINUE
1304: IF( ILZ ) THEN
1305: DO 340 JR = 1, N
1306: TEMP = C*Z( JR, J+1 ) + S*Z( JR, J )
1307: Z( JR, J ) = -S*Z( JR, J+1 ) + C*Z( JR, J )
1308: Z( JR, J+1 ) = TEMP
1309: 340 CONTINUE
1310: END IF
1311: *
1312: * End of Double-Shift code
1313: *
1314: END IF
1315: *
1316: GO TO 350
1317: *
1318: * End of iteration loop
1319: *
1320: 350 CONTINUE
1321: 360 CONTINUE
1322: *
1323: * Drop-through = non-convergence
1324: *
1325: INFO = ILAST
1326: GO TO 420
1327: *
1328: * Successful completion of all QZ steps
1329: *
1330: 380 CONTINUE
1331: *
1332: * Set Eigenvalues 1:ILO-1
1333: *
1334: DO 410 J = 1, ILO - 1
1335: IF( T( J, J ).LT.ZERO ) THEN
1336: IF( ILSCHR ) THEN
1337: DO 390 JR = 1, J
1338: H( JR, J ) = -H( JR, J )
1339: T( JR, J ) = -T( JR, J )
1340: 390 CONTINUE
1341: ELSE
1342: H( J, J ) = -H( J, J )
1343: T( J, J ) = -T( J, J )
1344: END IF
1345: IF( ILZ ) THEN
1346: DO 400 JR = 1, N
1347: Z( JR, J ) = -Z( JR, J )
1348: 400 CONTINUE
1349: END IF
1350: END IF
1351: ALPHAR( J ) = H( J, J )
1352: ALPHAI( J ) = ZERO
1353: BETA( J ) = T( J, J )
1354: 410 CONTINUE
1355: *
1356: * Normal Termination
1357: *
1358: INFO = 0
1359: *
1360: * Exit (other than argument error) -- return optimal workspace size
1361: *
1362: 420 CONTINUE
1363: WORK( 1 ) = DBLE( N )
1364: RETURN
1365: *
1366: * End of DHGEQZ
1367: *
1368: END
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