1: *> \brief \b DBBCSD
2: *
3: * =========== DOCUMENTATION ===========
4: *
5: * Online html documentation available at
6: * http://www.netlib.org/lapack/explore-html/
7: *
8: *> \htmlonly
9: *> Download DBBCSD + dependencies
10: *> <a href="http://www.netlib.org/cgi-bin/netlibfiles.tgz?format=tgz&filename=/lapack/lapack_routine/dbbcsd.f">
11: *> [TGZ]</a>
12: *> <a href="http://www.netlib.org/cgi-bin/netlibfiles.zip?format=zip&filename=/lapack/lapack_routine/dbbcsd.f">
13: *> [ZIP]</a>
14: *> <a href="http://www.netlib.org/cgi-bin/netlibfiles.txt?format=txt&filename=/lapack/lapack_routine/dbbcsd.f">
15: *> [TXT]</a>
16: *> \endhtmlonly
17: *
18: * Definition:
19: * ===========
20: *
21: * SUBROUTINE DBBCSD( JOBU1, JOBU2, JOBV1T, JOBV2T, TRANS, M, P, Q,
22: * THETA, PHI, U1, LDU1, U2, LDU2, V1T, LDV1T,
23: * V2T, LDV2T, B11D, B11E, B12D, B12E, B21D, B21E,
24: * B22D, B22E, WORK, LWORK, INFO )
25: *
26: * .. Scalar Arguments ..
27: * CHARACTER JOBU1, JOBU2, JOBV1T, JOBV2T, TRANS
28: * INTEGER INFO, LDU1, LDU2, LDV1T, LDV2T, LWORK, M, P, Q
29: * ..
30: * .. Array Arguments ..
31: * DOUBLE PRECISION B11D( * ), B11E( * ), B12D( * ), B12E( * ),
32: * $ B21D( * ), B21E( * ), B22D( * ), B22E( * ),
33: * $ PHI( * ), THETA( * ), WORK( * )
34: * DOUBLE PRECISION U1( LDU1, * ), U2( LDU2, * ), V1T( LDV1T, * ),
35: * $ V2T( LDV2T, * )
36: * ..
37: *
38: *
39: *> \par Purpose:
40: * =============
41: *>
42: *> \verbatim
43: *>
44: *> DBBCSD computes the CS decomposition of an orthogonal matrix in
45: *> bidiagonal-block form,
46: *>
47: *>
48: *> [ B11 | B12 0 0 ]
49: *> [ 0 | 0 -I 0 ]
50: *> X = [----------------]
51: *> [ B21 | B22 0 0 ]
52: *> [ 0 | 0 0 I ]
53: *>
54: *> [ C | -S 0 0 ]
55: *> [ U1 | ] [ 0 | 0 -I 0 ] [ V1 | ]**T
56: *> = [---------] [---------------] [---------] .
57: *> [ | U2 ] [ S | C 0 0 ] [ | V2 ]
58: *> [ 0 | 0 0 I ]
59: *>
60: *> X is M-by-M, its top-left block is P-by-Q, and Q must be no larger
61: *> than P, M-P, or M-Q. (If Q is not the smallest index, then X must be
62: *> transposed and/or permuted. This can be done in constant time using
63: *> the TRANS and SIGNS options. See DORCSD for details.)
64: *>
65: *> The bidiagonal matrices B11, B12, B21, and B22 are represented
66: *> implicitly by angles THETA(1:Q) and PHI(1:Q-1).
67: *>
68: *> The orthogonal matrices U1, U2, V1T, and V2T are input/output.
69: *> The input matrices are pre- or post-multiplied by the appropriate
70: *> singular vector matrices.
71: *> \endverbatim
72: *
73: * Arguments:
74: * ==========
75: *
76: *> \param[in] JOBU1
77: *> \verbatim
78: *> JOBU1 is CHARACTER
79: *> = 'Y': U1 is updated;
80: *> otherwise: U1 is not updated.
81: *> \endverbatim
82: *>
83: *> \param[in] JOBU2
84: *> \verbatim
85: *> JOBU2 is CHARACTER
86: *> = 'Y': U2 is updated;
87: *> otherwise: U2 is not updated.
88: *> \endverbatim
89: *>
90: *> \param[in] JOBV1T
91: *> \verbatim
92: *> JOBV1T is CHARACTER
93: *> = 'Y': V1T is updated;
94: *> otherwise: V1T is not updated.
95: *> \endverbatim
96: *>
97: *> \param[in] JOBV2T
98: *> \verbatim
99: *> JOBV2T is CHARACTER
100: *> = 'Y': V2T is updated;
101: *> otherwise: V2T is not updated.
102: *> \endverbatim
103: *>
104: *> \param[in] TRANS
105: *> \verbatim
106: *> TRANS is CHARACTER
107: *> = 'T': X, U1, U2, V1T, and V2T are stored in row-major
108: *> order;
109: *> otherwise: X, U1, U2, V1T, and V2T are stored in column-
110: *> major order.
111: *> \endverbatim
112: *>
113: *> \param[in] M
114: *> \verbatim
115: *> M is INTEGER
116: *> The number of rows and columns in X, the orthogonal matrix in
117: *> bidiagonal-block form.
118: *> \endverbatim
119: *>
120: *> \param[in] P
121: *> \verbatim
122: *> P is INTEGER
123: *> The number of rows in the top-left block of X. 0 <= P <= M.
124: *> \endverbatim
125: *>
126: *> \param[in] Q
127: *> \verbatim
128: *> Q is INTEGER
129: *> The number of columns in the top-left block of X.
130: *> 0 <= Q <= MIN(P,M-P,M-Q).
131: *> \endverbatim
132: *>
133: *> \param[in,out] THETA
134: *> \verbatim
135: *> THETA is DOUBLE PRECISION array, dimension (Q)
136: *> On entry, the angles THETA(1),...,THETA(Q) that, along with
137: *> PHI(1), ...,PHI(Q-1), define the matrix in bidiagonal-block
138: *> form. On exit, the angles whose cosines and sines define the
139: *> diagonal blocks in the CS decomposition.
140: *> \endverbatim
141: *>
142: *> \param[in,out] PHI
143: *> \verbatim
144: *> PHI is DOUBLE PRECISION array, dimension (Q-1)
145: *> The angles PHI(1),...,PHI(Q-1) that, along with THETA(1),...,
146: *> THETA(Q), define the matrix in bidiagonal-block form.
147: *> \endverbatim
148: *>
149: *> \param[in,out] U1
150: *> \verbatim
151: *> U1 is DOUBLE PRECISION array, dimension (LDU1,P)
152: *> On entry, an LDU1-by-P matrix. On exit, U1 is postmultiplied
153: *> by the left singular vector matrix common to [ B11 ; 0 ] and
154: *> [ B12 0 0 ; 0 -I 0 0 ].
155: *> \endverbatim
156: *>
157: *> \param[in] LDU1
158: *> \verbatim
159: *> LDU1 is INTEGER
160: *> The leading dimension of the array U1.
161: *> \endverbatim
162: *>
163: *> \param[in,out] U2
164: *> \verbatim
165: *> U2 is DOUBLE PRECISION array, dimension (LDU2,M-P)
166: *> On entry, an LDU2-by-(M-P) matrix. On exit, U2 is
167: *> postmultiplied by the left singular vector matrix common to
168: *> [ B21 ; 0 ] and [ B22 0 0 ; 0 0 I ].
169: *> \endverbatim
170: *>
171: *> \param[in] LDU2
172: *> \verbatim
173: *> LDU2 is INTEGER
174: *> The leading dimension of the array U2.
175: *> \endverbatim
176: *>
177: *> \param[in,out] V1T
178: *> \verbatim
179: *> V1T is DOUBLE PRECISION array, dimension (LDV1T,Q)
180: *> On entry, a LDV1T-by-Q matrix. On exit, V1T is premultiplied
181: *> by the transpose of the right singular vector
182: *> matrix common to [ B11 ; 0 ] and [ B21 ; 0 ].
183: *> \endverbatim
184: *>
185: *> \param[in] LDV1T
186: *> \verbatim
187: *> LDV1T is INTEGER
188: *> The leading dimension of the array V1T.
189: *> \endverbatim
190: *>
191: *> \param[in,out] V2T
192: *> \verbatim
193: *> V2T is DOUBLE PRECISION array, dimenison (LDV2T,M-Q)
194: *> On entry, a LDV2T-by-(M-Q) matrix. On exit, V2T is
195: *> premultiplied by the transpose of the right
196: *> singular vector matrix common to [ B12 0 0 ; 0 -I 0 ] and
197: *> [ B22 0 0 ; 0 0 I ].
198: *> \endverbatim
199: *>
200: *> \param[in] LDV2T
201: *> \verbatim
202: *> LDV2T is INTEGER
203: *> The leading dimension of the array V2T.
204: *> \endverbatim
205: *>
206: *> \param[out] B11D
207: *> \verbatim
208: *> B11D is DOUBLE PRECISION array, dimension (Q)
209: *> When DBBCSD converges, B11D contains the cosines of THETA(1),
210: *> ..., THETA(Q). If DBBCSD fails to converge, then B11D
211: *> contains the diagonal of the partially reduced top-left
212: *> block.
213: *> \endverbatim
214: *>
215: *> \param[out] B11E
216: *> \verbatim
217: *> B11E is DOUBLE PRECISION array, dimension (Q-1)
218: *> When DBBCSD converges, B11E contains zeros. If DBBCSD fails
219: *> to converge, then B11E contains the superdiagonal of the
220: *> partially reduced top-left block.
221: *> \endverbatim
222: *>
223: *> \param[out] B12D
224: *> \verbatim
225: *> B12D is DOUBLE PRECISION array, dimension (Q)
226: *> When DBBCSD converges, B12D contains the negative sines of
227: *> THETA(1), ..., THETA(Q). If DBBCSD fails to converge, then
228: *> B12D contains the diagonal of the partially reduced top-right
229: *> block.
230: *> \endverbatim
231: *>
232: *> \param[out] B12E
233: *> \verbatim
234: *> B12E is DOUBLE PRECISION array, dimension (Q-1)
235: *> When DBBCSD converges, B12E contains zeros. If DBBCSD fails
236: *> to converge, then B12E contains the subdiagonal of the
237: *> partially reduced top-right block.
238: *> \endverbatim
239: *>
240: *> \param[out] B21D
241: *> \verbatim
242: *> B21D is DOUBLE PRECISION array, dimension (Q)
243: *> When CBBCSD converges, B21D contains the negative sines of
244: *> THETA(1), ..., THETA(Q). If CBBCSD fails to converge, then
245: *> B21D contains the diagonal of the partially reduced bottom-left
246: *> block.
247: *> \endverbatim
248: *>
249: *> \param[out] B21E
250: *> \verbatim
251: *> B21E is DOUBLE PRECISION array, dimension (Q-1)
252: *> When CBBCSD converges, B21E contains zeros. If CBBCSD fails
253: *> to converge, then B21E contains the subdiagonal of the
254: *> partially reduced bottom-left block.
255: *> \endverbatim
256: *>
257: *> \param[out] B22D
258: *> \verbatim
259: *> B22D is DOUBLE PRECISION array, dimension (Q)
260: *> When CBBCSD converges, B22D contains the negative sines of
261: *> THETA(1), ..., THETA(Q). If CBBCSD fails to converge, then
262: *> B22D contains the diagonal of the partially reduced bottom-right
263: *> block.
264: *> \endverbatim
265: *>
266: *> \param[out] B22E
267: *> \verbatim
268: *> B22E is DOUBLE PRECISION array, dimension (Q-1)
269: *> When CBBCSD converges, B22E contains zeros. If CBBCSD fails
270: *> to converge, then B22E contains the subdiagonal of the
271: *> partially reduced bottom-right block.
272: *> \endverbatim
273: *>
274: *> \param[out] WORK
275: *> \verbatim
276: *> WORK is DOUBLE PRECISION array, dimension (MAX(1,LWORK))
277: *> On exit, if INFO = 0, WORK(1) returns the optimal LWORK.
278: *> \endverbatim
279: *>
280: *> \param[in] LWORK
281: *> \verbatim
282: *> LWORK is INTEGER
283: *> The dimension of the array WORK. LWORK >= MAX(1,8*Q).
284: *>
285: *> If LWORK = -1, then a workspace query is assumed; the
286: *> routine only calculates the optimal size of the WORK array,
287: *> returns this value as the first entry of the work array, and
288: *> no error message related to LWORK is issued by XERBLA.
289: *> \endverbatim
290: *>
291: *> \param[out] INFO
292: *> \verbatim
293: *> INFO is INTEGER
294: *> = 0: successful exit.
295: *> < 0: if INFO = -i, the i-th argument had an illegal value.
296: *> > 0: if DBBCSD did not converge, INFO specifies the number
297: *> of nonzero entries in PHI, and B11D, B11E, etc.,
298: *> contain the partially reduced matrix.
299: *> \endverbatim
300: *
301: *> \par Internal Parameters:
302: * =========================
303: *>
304: *> \verbatim
305: *> TOLMUL DOUBLE PRECISION, default = MAX(10,MIN(100,EPS**(-1/8)))
306: *> TOLMUL controls the convergence criterion of the QR loop.
307: *> Angles THETA(i), PHI(i) are rounded to 0 or PI/2 when they
308: *> are within TOLMUL*EPS of either bound.
309: *> \endverbatim
310: *
311: *> \par References:
312: * ================
313: *>
314: *> [1] Brian D. Sutton. Computing the complete CS decomposition. Numer.
315: *> Algorithms, 50(1):33-65, 2009.
316: *
317: * Authors:
318: * ========
319: *
320: *> \author Univ. of Tennessee
321: *> \author Univ. of California Berkeley
322: *> \author Univ. of Colorado Denver
323: *> \author NAG Ltd.
324: *
325: *> \date November 2013
326: *
327: *> \ingroup doubleOTHERcomputational
328: *
329: * =====================================================================
330: SUBROUTINE DBBCSD( JOBU1, JOBU2, JOBV1T, JOBV2T, TRANS, M, P, Q,
331: $ THETA, PHI, U1, LDU1, U2, LDU2, V1T, LDV1T,
332: $ V2T, LDV2T, B11D, B11E, B12D, B12E, B21D, B21E,
333: $ B22D, B22E, WORK, LWORK, INFO )
334: *
335: * -- LAPACK computational routine (version 3.5.0) --
336: * -- LAPACK is a software package provided by Univ. of Tennessee, --
337: * -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
338: * November 2013
339: *
340: * .. Scalar Arguments ..
341: CHARACTER JOBU1, JOBU2, JOBV1T, JOBV2T, TRANS
342: INTEGER INFO, LDU1, LDU2, LDV1T, LDV2T, LWORK, M, P, Q
343: * ..
344: * .. Array Arguments ..
345: DOUBLE PRECISION B11D( * ), B11E( * ), B12D( * ), B12E( * ),
346: $ B21D( * ), B21E( * ), B22D( * ), B22E( * ),
347: $ PHI( * ), THETA( * ), WORK( * )
348: DOUBLE PRECISION U1( LDU1, * ), U2( LDU2, * ), V1T( LDV1T, * ),
349: $ V2T( LDV2T, * )
350: * ..
351: *
352: * ===================================================================
353: *
354: * .. Parameters ..
355: INTEGER MAXITR
356: PARAMETER ( MAXITR = 6 )
357: DOUBLE PRECISION HUNDRED, MEIGHTH, ONE, PIOVER2, TEN, ZERO
358: PARAMETER ( HUNDRED = 100.0D0, MEIGHTH = -0.125D0,
359: $ ONE = 1.0D0, PIOVER2 = 1.57079632679489662D0,
360: $ TEN = 10.0D0, ZERO = 0.0D0 )
361: DOUBLE PRECISION NEGONE
362: PARAMETER ( NEGONE = -1.0D0 )
363: * ..
364: * .. Local Scalars ..
365: LOGICAL COLMAJOR, LQUERY, RESTART11, RESTART12,
366: $ RESTART21, RESTART22, WANTU1, WANTU2, WANTV1T,
367: $ WANTV2T
368: INTEGER I, IMIN, IMAX, ITER, IU1CS, IU1SN, IU2CS,
369: $ IU2SN, IV1TCS, IV1TSN, IV2TCS, IV2TSN, J,
370: $ LWORKMIN, LWORKOPT, MAXIT, MINI
371: DOUBLE PRECISION B11BULGE, B12BULGE, B21BULGE, B22BULGE, DUMMY,
372: $ EPS, MU, NU, R, SIGMA11, SIGMA21,
373: $ TEMP, THETAMAX, THETAMIN, THRESH, TOL, TOLMUL,
374: $ UNFL, X1, X2, Y1, Y2
375: *
376: * .. External Subroutines ..
377: EXTERNAL DLASR, DSCAL, DSWAP, DLARTGP, DLARTGS, DLAS2,
378: $ XERBLA
379: * ..
380: * .. External Functions ..
381: DOUBLE PRECISION DLAMCH
382: LOGICAL LSAME
383: EXTERNAL LSAME, DLAMCH
384: * ..
385: * .. Intrinsic Functions ..
386: INTRINSIC ABS, ATAN2, COS, MAX, MIN, SIN, SQRT
387: * ..
388: * .. Executable Statements ..
389: *
390: * Test input arguments
391: *
392: INFO = 0
393: LQUERY = LWORK .EQ. -1
394: WANTU1 = LSAME( JOBU1, 'Y' )
395: WANTU2 = LSAME( JOBU2, 'Y' )
396: WANTV1T = LSAME( JOBV1T, 'Y' )
397: WANTV2T = LSAME( JOBV2T, 'Y' )
398: COLMAJOR = .NOT. LSAME( TRANS, 'T' )
399: *
400: IF( M .LT. 0 ) THEN
401: INFO = -6
402: ELSE IF( P .LT. 0 .OR. P .GT. M ) THEN
403: INFO = -7
404: ELSE IF( Q .LT. 0 .OR. Q .GT. M ) THEN
405: INFO = -8
406: ELSE IF( Q .GT. P .OR. Q .GT. M-P .OR. Q .GT. M-Q ) THEN
407: INFO = -8
408: ELSE IF( WANTU1 .AND. LDU1 .LT. P ) THEN
409: INFO = -12
410: ELSE IF( WANTU2 .AND. LDU2 .LT. M-P ) THEN
411: INFO = -14
412: ELSE IF( WANTV1T .AND. LDV1T .LT. Q ) THEN
413: INFO = -16
414: ELSE IF( WANTV2T .AND. LDV2T .LT. M-Q ) THEN
415: INFO = -18
416: END IF
417: *
418: * Quick return if Q = 0
419: *
420: IF( INFO .EQ. 0 .AND. Q .EQ. 0 ) THEN
421: LWORKMIN = 1
422: WORK(1) = LWORKMIN
423: RETURN
424: END IF
425: *
426: * Compute workspace
427: *
428: IF( INFO .EQ. 0 ) THEN
429: IU1CS = 1
430: IU1SN = IU1CS + Q
431: IU2CS = IU1SN + Q
432: IU2SN = IU2CS + Q
433: IV1TCS = IU2SN + Q
434: IV1TSN = IV1TCS + Q
435: IV2TCS = IV1TSN + Q
436: IV2TSN = IV2TCS + Q
437: LWORKOPT = IV2TSN + Q - 1
438: LWORKMIN = LWORKOPT
439: WORK(1) = LWORKOPT
440: IF( LWORK .LT. LWORKMIN .AND. .NOT. LQUERY ) THEN
441: INFO = -28
442: END IF
443: END IF
444: *
445: IF( INFO .NE. 0 ) THEN
446: CALL XERBLA( 'DBBCSD', -INFO )
447: RETURN
448: ELSE IF( LQUERY ) THEN
449: RETURN
450: END IF
451: *
452: * Get machine constants
453: *
454: EPS = DLAMCH( 'Epsilon' )
455: UNFL = DLAMCH( 'Safe minimum' )
456: TOLMUL = MAX( TEN, MIN( HUNDRED, EPS**MEIGHTH ) )
457: TOL = TOLMUL*EPS
458: THRESH = MAX( TOL, MAXITR*Q*Q*UNFL )
459: *
460: * Test for negligible sines or cosines
461: *
462: DO I = 1, Q
463: IF( THETA(I) .LT. THRESH ) THEN
464: THETA(I) = ZERO
465: ELSE IF( THETA(I) .GT. PIOVER2-THRESH ) THEN
466: THETA(I) = PIOVER2
467: END IF
468: END DO
469: DO I = 1, Q-1
470: IF( PHI(I) .LT. THRESH ) THEN
471: PHI(I) = ZERO
472: ELSE IF( PHI(I) .GT. PIOVER2-THRESH ) THEN
473: PHI(I) = PIOVER2
474: END IF
475: END DO
476: *
477: * Initial deflation
478: *
479: IMAX = Q
480: DO WHILE( IMAX .GT. 1 )
481: IF( PHI(IMAX-1) .NE. ZERO ) THEN
482: EXIT
483: END IF
484: IMAX = IMAX - 1
485: END DO
486: IMIN = IMAX - 1
487: IF ( IMIN .GT. 1 ) THEN
488: DO WHILE( PHI(IMIN-1) .NE. ZERO )
489: IMIN = IMIN - 1
490: IF ( IMIN .LE. 1 ) EXIT
491: END DO
492: END IF
493: *
494: * Initialize iteration counter
495: *
496: MAXIT = MAXITR*Q*Q
497: ITER = 0
498: *
499: * Begin main iteration loop
500: *
501: DO WHILE( IMAX .GT. 1 )
502: *
503: * Compute the matrix entries
504: *
505: B11D(IMIN) = COS( THETA(IMIN) )
506: B21D(IMIN) = -SIN( THETA(IMIN) )
507: DO I = IMIN, IMAX - 1
508: B11E(I) = -SIN( THETA(I) ) * SIN( PHI(I) )
509: B11D(I+1) = COS( THETA(I+1) ) * COS( PHI(I) )
510: B12D(I) = SIN( THETA(I) ) * COS( PHI(I) )
511: B12E(I) = COS( THETA(I+1) ) * SIN( PHI(I) )
512: B21E(I) = -COS( THETA(I) ) * SIN( PHI(I) )
513: B21D(I+1) = -SIN( THETA(I+1) ) * COS( PHI(I) )
514: B22D(I) = COS( THETA(I) ) * COS( PHI(I) )
515: B22E(I) = -SIN( THETA(I+1) ) * SIN( PHI(I) )
516: END DO
517: B12D(IMAX) = SIN( THETA(IMAX) )
518: B22D(IMAX) = COS( THETA(IMAX) )
519: *
520: * Abort if not converging; otherwise, increment ITER
521: *
522: IF( ITER .GT. MAXIT ) THEN
523: INFO = 0
524: DO I = 1, Q
525: IF( PHI(I) .NE. ZERO )
526: $ INFO = INFO + 1
527: END DO
528: RETURN
529: END IF
530: *
531: ITER = ITER + IMAX - IMIN
532: *
533: * Compute shifts
534: *
535: THETAMAX = THETA(IMIN)
536: THETAMIN = THETA(IMIN)
537: DO I = IMIN+1, IMAX
538: IF( THETA(I) > THETAMAX )
539: $ THETAMAX = THETA(I)
540: IF( THETA(I) < THETAMIN )
541: $ THETAMIN = THETA(I)
542: END DO
543: *
544: IF( THETAMAX .GT. PIOVER2 - THRESH ) THEN
545: *
546: * Zero on diagonals of B11 and B22; induce deflation with a
547: * zero shift
548: *
549: MU = ZERO
550: NU = ONE
551: *
552: ELSE IF( THETAMIN .LT. THRESH ) THEN
553: *
554: * Zero on diagonals of B12 and B22; induce deflation with a
555: * zero shift
556: *
557: MU = ONE
558: NU = ZERO
559: *
560: ELSE
561: *
562: * Compute shifts for B11 and B21 and use the lesser
563: *
564: CALL DLAS2( B11D(IMAX-1), B11E(IMAX-1), B11D(IMAX), SIGMA11,
565: $ DUMMY )
566: CALL DLAS2( B21D(IMAX-1), B21E(IMAX-1), B21D(IMAX), SIGMA21,
567: $ DUMMY )
568: *
569: IF( SIGMA11 .LE. SIGMA21 ) THEN
570: MU = SIGMA11
571: NU = SQRT( ONE - MU**2 )
572: IF( MU .LT. THRESH ) THEN
573: MU = ZERO
574: NU = ONE
575: END IF
576: ELSE
577: NU = SIGMA21
578: MU = SQRT( 1.0 - NU**2 )
579: IF( NU .LT. THRESH ) THEN
580: MU = ONE
581: NU = ZERO
582: END IF
583: END IF
584: END IF
585: *
586: * Rotate to produce bulges in B11 and B21
587: *
588: IF( MU .LE. NU ) THEN
589: CALL DLARTGS( B11D(IMIN), B11E(IMIN), MU,
590: $ WORK(IV1TCS+IMIN-1), WORK(IV1TSN+IMIN-1) )
591: ELSE
592: CALL DLARTGS( B21D(IMIN), B21E(IMIN), NU,
593: $ WORK(IV1TCS+IMIN-1), WORK(IV1TSN+IMIN-1) )
594: END IF
595: *
596: TEMP = WORK(IV1TCS+IMIN-1)*B11D(IMIN) +
597: $ WORK(IV1TSN+IMIN-1)*B11E(IMIN)
598: B11E(IMIN) = WORK(IV1TCS+IMIN-1)*B11E(IMIN) -
599: $ WORK(IV1TSN+IMIN-1)*B11D(IMIN)
600: B11D(IMIN) = TEMP
601: B11BULGE = WORK(IV1TSN+IMIN-1)*B11D(IMIN+1)
602: B11D(IMIN+1) = WORK(IV1TCS+IMIN-1)*B11D(IMIN+1)
603: TEMP = WORK(IV1TCS+IMIN-1)*B21D(IMIN) +
604: $ WORK(IV1TSN+IMIN-1)*B21E(IMIN)
605: B21E(IMIN) = WORK(IV1TCS+IMIN-1)*B21E(IMIN) -
606: $ WORK(IV1TSN+IMIN-1)*B21D(IMIN)
607: B21D(IMIN) = TEMP
608: B21BULGE = WORK(IV1TSN+IMIN-1)*B21D(IMIN+1)
609: B21D(IMIN+1) = WORK(IV1TCS+IMIN-1)*B21D(IMIN+1)
610: *
611: * Compute THETA(IMIN)
612: *
613: THETA( IMIN ) = ATAN2( SQRT( B21D(IMIN)**2+B21BULGE**2 ),
614: $ SQRT( B11D(IMIN)**2+B11BULGE**2 ) )
615: *
616: * Chase the bulges in B11(IMIN+1,IMIN) and B21(IMIN+1,IMIN)
617: *
618: IF( B11D(IMIN)**2+B11BULGE**2 .GT. THRESH**2 ) THEN
619: CALL DLARTGP( B11BULGE, B11D(IMIN), WORK(IU1SN+IMIN-1),
620: $ WORK(IU1CS+IMIN-1), R )
621: ELSE IF( MU .LE. NU ) THEN
622: CALL DLARTGS( B11E( IMIN ), B11D( IMIN + 1 ), MU,
623: $ WORK(IU1CS+IMIN-1), WORK(IU1SN+IMIN-1) )
624: ELSE
625: CALL DLARTGS( B12D( IMIN ), B12E( IMIN ), NU,
626: $ WORK(IU1CS+IMIN-1), WORK(IU1SN+IMIN-1) )
627: END IF
628: IF( B21D(IMIN)**2+B21BULGE**2 .GT. THRESH**2 ) THEN
629: CALL DLARTGP( B21BULGE, B21D(IMIN), WORK(IU2SN+IMIN-1),
630: $ WORK(IU2CS+IMIN-1), R )
631: ELSE IF( NU .LT. MU ) THEN
632: CALL DLARTGS( B21E( IMIN ), B21D( IMIN + 1 ), NU,
633: $ WORK(IU2CS+IMIN-1), WORK(IU2SN+IMIN-1) )
634: ELSE
635: CALL DLARTGS( B22D(IMIN), B22E(IMIN), MU,
636: $ WORK(IU2CS+IMIN-1), WORK(IU2SN+IMIN-1) )
637: END IF
638: WORK(IU2CS+IMIN-1) = -WORK(IU2CS+IMIN-1)
639: WORK(IU2SN+IMIN-1) = -WORK(IU2SN+IMIN-1)
640: *
641: TEMP = WORK(IU1CS+IMIN-1)*B11E(IMIN) +
642: $ WORK(IU1SN+IMIN-1)*B11D(IMIN+1)
643: B11D(IMIN+1) = WORK(IU1CS+IMIN-1)*B11D(IMIN+1) -
644: $ WORK(IU1SN+IMIN-1)*B11E(IMIN)
645: B11E(IMIN) = TEMP
646: IF( IMAX .GT. IMIN+1 ) THEN
647: B11BULGE = WORK(IU1SN+IMIN-1)*B11E(IMIN+1)
648: B11E(IMIN+1) = WORK(IU1CS+IMIN-1)*B11E(IMIN+1)
649: END IF
650: TEMP = WORK(IU1CS+IMIN-1)*B12D(IMIN) +
651: $ WORK(IU1SN+IMIN-1)*B12E(IMIN)
652: B12E(IMIN) = WORK(IU1CS+IMIN-1)*B12E(IMIN) -
653: $ WORK(IU1SN+IMIN-1)*B12D(IMIN)
654: B12D(IMIN) = TEMP
655: B12BULGE = WORK(IU1SN+IMIN-1)*B12D(IMIN+1)
656: B12D(IMIN+1) = WORK(IU1CS+IMIN-1)*B12D(IMIN+1)
657: TEMP = WORK(IU2CS+IMIN-1)*B21E(IMIN) +
658: $ WORK(IU2SN+IMIN-1)*B21D(IMIN+1)
659: B21D(IMIN+1) = WORK(IU2CS+IMIN-1)*B21D(IMIN+1) -
660: $ WORK(IU2SN+IMIN-1)*B21E(IMIN)
661: B21E(IMIN) = TEMP
662: IF( IMAX .GT. IMIN+1 ) THEN
663: B21BULGE = WORK(IU2SN+IMIN-1)*B21E(IMIN+1)
664: B21E(IMIN+1) = WORK(IU2CS+IMIN-1)*B21E(IMIN+1)
665: END IF
666: TEMP = WORK(IU2CS+IMIN-1)*B22D(IMIN) +
667: $ WORK(IU2SN+IMIN-1)*B22E(IMIN)
668: B22E(IMIN) = WORK(IU2CS+IMIN-1)*B22E(IMIN) -
669: $ WORK(IU2SN+IMIN-1)*B22D(IMIN)
670: B22D(IMIN) = TEMP
671: B22BULGE = WORK(IU2SN+IMIN-1)*B22D(IMIN+1)
672: B22D(IMIN+1) = WORK(IU2CS+IMIN-1)*B22D(IMIN+1)
673: *
674: * Inner loop: chase bulges from B11(IMIN,IMIN+2),
675: * B12(IMIN,IMIN+1), B21(IMIN,IMIN+2), and B22(IMIN,IMIN+1) to
676: * bottom-right
677: *
678: DO I = IMIN+1, IMAX-1
679: *
680: * Compute PHI(I-1)
681: *
682: X1 = SIN(THETA(I-1))*B11E(I-1) + COS(THETA(I-1))*B21E(I-1)
683: X2 = SIN(THETA(I-1))*B11BULGE + COS(THETA(I-1))*B21BULGE
684: Y1 = SIN(THETA(I-1))*B12D(I-1) + COS(THETA(I-1))*B22D(I-1)
685: Y2 = SIN(THETA(I-1))*B12BULGE + COS(THETA(I-1))*B22BULGE
686: *
687: PHI(I-1) = ATAN2( SQRT(X1**2+X2**2), SQRT(Y1**2+Y2**2) )
688: *
689: * Determine if there are bulges to chase or if a new direct
690: * summand has been reached
691: *
692: RESTART11 = B11E(I-1)**2 + B11BULGE**2 .LE. THRESH**2
693: RESTART21 = B21E(I-1)**2 + B21BULGE**2 .LE. THRESH**2
694: RESTART12 = B12D(I-1)**2 + B12BULGE**2 .LE. THRESH**2
695: RESTART22 = B22D(I-1)**2 + B22BULGE**2 .LE. THRESH**2
696: *
697: * If possible, chase bulges from B11(I-1,I+1), B12(I-1,I),
698: * B21(I-1,I+1), and B22(I-1,I). If necessary, restart bulge-
699: * chasing by applying the original shift again.
700: *
701: IF( .NOT. RESTART11 .AND. .NOT. RESTART21 ) THEN
702: CALL DLARTGP( X2, X1, WORK(IV1TSN+I-1), WORK(IV1TCS+I-1),
703: $ R )
704: ELSE IF( .NOT. RESTART11 .AND. RESTART21 ) THEN
705: CALL DLARTGP( B11BULGE, B11E(I-1), WORK(IV1TSN+I-1),
706: $ WORK(IV1TCS+I-1), R )
707: ELSE IF( RESTART11 .AND. .NOT. RESTART21 ) THEN
708: CALL DLARTGP( B21BULGE, B21E(I-1), WORK(IV1TSN+I-1),
709: $ WORK(IV1TCS+I-1), R )
710: ELSE IF( MU .LE. NU ) THEN
711: CALL DLARTGS( B11D(I), B11E(I), MU, WORK(IV1TCS+I-1),
712: $ WORK(IV1TSN+I-1) )
713: ELSE
714: CALL DLARTGS( B21D(I), B21E(I), NU, WORK(IV1TCS+I-1),
715: $ WORK(IV1TSN+I-1) )
716: END IF
717: WORK(IV1TCS+I-1) = -WORK(IV1TCS+I-1)
718: WORK(IV1TSN+I-1) = -WORK(IV1TSN+I-1)
719: IF( .NOT. RESTART12 .AND. .NOT. RESTART22 ) THEN
720: CALL DLARTGP( Y2, Y1, WORK(IV2TSN+I-1-1),
721: $ WORK(IV2TCS+I-1-1), R )
722: ELSE IF( .NOT. RESTART12 .AND. RESTART22 ) THEN
723: CALL DLARTGP( B12BULGE, B12D(I-1), WORK(IV2TSN+I-1-1),
724: $ WORK(IV2TCS+I-1-1), R )
725: ELSE IF( RESTART12 .AND. .NOT. RESTART22 ) THEN
726: CALL DLARTGP( B22BULGE, B22D(I-1), WORK(IV2TSN+I-1-1),
727: $ WORK(IV2TCS+I-1-1), R )
728: ELSE IF( NU .LT. MU ) THEN
729: CALL DLARTGS( B12E(I-1), B12D(I), NU, WORK(IV2TCS+I-1-1),
730: $ WORK(IV2TSN+I-1-1) )
731: ELSE
732: CALL DLARTGS( B22E(I-1), B22D(I), MU, WORK(IV2TCS+I-1-1),
733: $ WORK(IV2TSN+I-1-1) )
734: END IF
735: *
736: TEMP = WORK(IV1TCS+I-1)*B11D(I) + WORK(IV1TSN+I-1)*B11E(I)
737: B11E(I) = WORK(IV1TCS+I-1)*B11E(I) -
738: $ WORK(IV1TSN+I-1)*B11D(I)
739: B11D(I) = TEMP
740: B11BULGE = WORK(IV1TSN+I-1)*B11D(I+1)
741: B11D(I+1) = WORK(IV1TCS+I-1)*B11D(I+1)
742: TEMP = WORK(IV1TCS+I-1)*B21D(I) + WORK(IV1TSN+I-1)*B21E(I)
743: B21E(I) = WORK(IV1TCS+I-1)*B21E(I) -
744: $ WORK(IV1TSN+I-1)*B21D(I)
745: B21D(I) = TEMP
746: B21BULGE = WORK(IV1TSN+I-1)*B21D(I+1)
747: B21D(I+1) = WORK(IV1TCS+I-1)*B21D(I+1)
748: TEMP = WORK(IV2TCS+I-1-1)*B12E(I-1) +
749: $ WORK(IV2TSN+I-1-1)*B12D(I)
750: B12D(I) = WORK(IV2TCS+I-1-1)*B12D(I) -
751: $ WORK(IV2TSN+I-1-1)*B12E(I-1)
752: B12E(I-1) = TEMP
753: B12BULGE = WORK(IV2TSN+I-1-1)*B12E(I)
754: B12E(I) = WORK(IV2TCS+I-1-1)*B12E(I)
755: TEMP = WORK(IV2TCS+I-1-1)*B22E(I-1) +
756: $ WORK(IV2TSN+I-1-1)*B22D(I)
757: B22D(I) = WORK(IV2TCS+I-1-1)*B22D(I) -
758: $ WORK(IV2TSN+I-1-1)*B22E(I-1)
759: B22E(I-1) = TEMP
760: B22BULGE = WORK(IV2TSN+I-1-1)*B22E(I)
761: B22E(I) = WORK(IV2TCS+I-1-1)*B22E(I)
762: *
763: * Compute THETA(I)
764: *
765: X1 = COS(PHI(I-1))*B11D(I) + SIN(PHI(I-1))*B12E(I-1)
766: X2 = COS(PHI(I-1))*B11BULGE + SIN(PHI(I-1))*B12BULGE
767: Y1 = COS(PHI(I-1))*B21D(I) + SIN(PHI(I-1))*B22E(I-1)
768: Y2 = COS(PHI(I-1))*B21BULGE + SIN(PHI(I-1))*B22BULGE
769: *
770: THETA(I) = ATAN2( SQRT(Y1**2+Y2**2), SQRT(X1**2+X2**2) )
771: *
772: * Determine if there are bulges to chase or if a new direct
773: * summand has been reached
774: *
775: RESTART11 = B11D(I)**2 + B11BULGE**2 .LE. THRESH**2
776: RESTART12 = B12E(I-1)**2 + B12BULGE**2 .LE. THRESH**2
777: RESTART21 = B21D(I)**2 + B21BULGE**2 .LE. THRESH**2
778: RESTART22 = B22E(I-1)**2 + B22BULGE**2 .LE. THRESH**2
779: *
780: * If possible, chase bulges from B11(I+1,I), B12(I+1,I-1),
781: * B21(I+1,I), and B22(I+1,I-1). If necessary, restart bulge-
782: * chasing by applying the original shift again.
783: *
784: IF( .NOT. RESTART11 .AND. .NOT. RESTART12 ) THEN
785: CALL DLARTGP( X2, X1, WORK(IU1SN+I-1), WORK(IU1CS+I-1),
786: $ R )
787: ELSE IF( .NOT. RESTART11 .AND. RESTART12 ) THEN
788: CALL DLARTGP( B11BULGE, B11D(I), WORK(IU1SN+I-1),
789: $ WORK(IU1CS+I-1), R )
790: ELSE IF( RESTART11 .AND. .NOT. RESTART12 ) THEN
791: CALL DLARTGP( B12BULGE, B12E(I-1), WORK(IU1SN+I-1),
792: $ WORK(IU1CS+I-1), R )
793: ELSE IF( MU .LE. NU ) THEN
794: CALL DLARTGS( B11E(I), B11D(I+1), MU, WORK(IU1CS+I-1),
795: $ WORK(IU1SN+I-1) )
796: ELSE
797: CALL DLARTGS( B12D(I), B12E(I), NU, WORK(IU1CS+I-1),
798: $ WORK(IU1SN+I-1) )
799: END IF
800: IF( .NOT. RESTART21 .AND. .NOT. RESTART22 ) THEN
801: CALL DLARTGP( Y2, Y1, WORK(IU2SN+I-1), WORK(IU2CS+I-1),
802: $ R )
803: ELSE IF( .NOT. RESTART21 .AND. RESTART22 ) THEN
804: CALL DLARTGP( B21BULGE, B21D(I), WORK(IU2SN+I-1),
805: $ WORK(IU2CS+I-1), R )
806: ELSE IF( RESTART21 .AND. .NOT. RESTART22 ) THEN
807: CALL DLARTGP( B22BULGE, B22E(I-1), WORK(IU2SN+I-1),
808: $ WORK(IU2CS+I-1), R )
809: ELSE IF( NU .LT. MU ) THEN
810: CALL DLARTGS( B21E(I), B21E(I+1), NU, WORK(IU2CS+I-1),
811: $ WORK(IU2SN+I-1) )
812: ELSE
813: CALL DLARTGS( B22D(I), B22E(I), MU, WORK(IU2CS+I-1),
814: $ WORK(IU2SN+I-1) )
815: END IF
816: WORK(IU2CS+I-1) = -WORK(IU2CS+I-1)
817: WORK(IU2SN+I-1) = -WORK(IU2SN+I-1)
818: *
819: TEMP = WORK(IU1CS+I-1)*B11E(I) + WORK(IU1SN+I-1)*B11D(I+1)
820: B11D(I+1) = WORK(IU1CS+I-1)*B11D(I+1) -
821: $ WORK(IU1SN+I-1)*B11E(I)
822: B11E(I) = TEMP
823: IF( I .LT. IMAX - 1 ) THEN
824: B11BULGE = WORK(IU1SN+I-1)*B11E(I+1)
825: B11E(I+1) = WORK(IU1CS+I-1)*B11E(I+1)
826: END IF
827: TEMP = WORK(IU2CS+I-1)*B21E(I) + WORK(IU2SN+I-1)*B21D(I+1)
828: B21D(I+1) = WORK(IU2CS+I-1)*B21D(I+1) -
829: $ WORK(IU2SN+I-1)*B21E(I)
830: B21E(I) = TEMP
831: IF( I .LT. IMAX - 1 ) THEN
832: B21BULGE = WORK(IU2SN+I-1)*B21E(I+1)
833: B21E(I+1) = WORK(IU2CS+I-1)*B21E(I+1)
834: END IF
835: TEMP = WORK(IU1CS+I-1)*B12D(I) + WORK(IU1SN+I-1)*B12E(I)
836: B12E(I) = WORK(IU1CS+I-1)*B12E(I) - WORK(IU1SN+I-1)*B12D(I)
837: B12D(I) = TEMP
838: B12BULGE = WORK(IU1SN+I-1)*B12D(I+1)
839: B12D(I+1) = WORK(IU1CS+I-1)*B12D(I+1)
840: TEMP = WORK(IU2CS+I-1)*B22D(I) + WORK(IU2SN+I-1)*B22E(I)
841: B22E(I) = WORK(IU2CS+I-1)*B22E(I) - WORK(IU2SN+I-1)*B22D(I)
842: B22D(I) = TEMP
843: B22BULGE = WORK(IU2SN+I-1)*B22D(I+1)
844: B22D(I+1) = WORK(IU2CS+I-1)*B22D(I+1)
845: *
846: END DO
847: *
848: * Compute PHI(IMAX-1)
849: *
850: X1 = SIN(THETA(IMAX-1))*B11E(IMAX-1) +
851: $ COS(THETA(IMAX-1))*B21E(IMAX-1)
852: Y1 = SIN(THETA(IMAX-1))*B12D(IMAX-1) +
853: $ COS(THETA(IMAX-1))*B22D(IMAX-1)
854: Y2 = SIN(THETA(IMAX-1))*B12BULGE + COS(THETA(IMAX-1))*B22BULGE
855: *
856: PHI(IMAX-1) = ATAN2( ABS(X1), SQRT(Y1**2+Y2**2) )
857: *
858: * Chase bulges from B12(IMAX-1,IMAX) and B22(IMAX-1,IMAX)
859: *
860: RESTART12 = B12D(IMAX-1)**2 + B12BULGE**2 .LE. THRESH**2
861: RESTART22 = B22D(IMAX-1)**2 + B22BULGE**2 .LE. THRESH**2
862: *
863: IF( .NOT. RESTART12 .AND. .NOT. RESTART22 ) THEN
864: CALL DLARTGP( Y2, Y1, WORK(IV2TSN+IMAX-1-1),
865: $ WORK(IV2TCS+IMAX-1-1), R )
866: ELSE IF( .NOT. RESTART12 .AND. RESTART22 ) THEN
867: CALL DLARTGP( B12BULGE, B12D(IMAX-1), WORK(IV2TSN+IMAX-1-1),
868: $ WORK(IV2TCS+IMAX-1-1), R )
869: ELSE IF( RESTART12 .AND. .NOT. RESTART22 ) THEN
870: CALL DLARTGP( B22BULGE, B22D(IMAX-1), WORK(IV2TSN+IMAX-1-1),
871: $ WORK(IV2TCS+IMAX-1-1), R )
872: ELSE IF( NU .LT. MU ) THEN
873: CALL DLARTGS( B12E(IMAX-1), B12D(IMAX), NU,
874: $ WORK(IV2TCS+IMAX-1-1), WORK(IV2TSN+IMAX-1-1) )
875: ELSE
876: CALL DLARTGS( B22E(IMAX-1), B22D(IMAX), MU,
877: $ WORK(IV2TCS+IMAX-1-1), WORK(IV2TSN+IMAX-1-1) )
878: END IF
879: *
880: TEMP = WORK(IV2TCS+IMAX-1-1)*B12E(IMAX-1) +
881: $ WORK(IV2TSN+IMAX-1-1)*B12D(IMAX)
882: B12D(IMAX) = WORK(IV2TCS+IMAX-1-1)*B12D(IMAX) -
883: $ WORK(IV2TSN+IMAX-1-1)*B12E(IMAX-1)
884: B12E(IMAX-1) = TEMP
885: TEMP = WORK(IV2TCS+IMAX-1-1)*B22E(IMAX-1) +
886: $ WORK(IV2TSN+IMAX-1-1)*B22D(IMAX)
887: B22D(IMAX) = WORK(IV2TCS+IMAX-1-1)*B22D(IMAX) -
888: $ WORK(IV2TSN+IMAX-1-1)*B22E(IMAX-1)
889: B22E(IMAX-1) = TEMP
890: *
891: * Update singular vectors
892: *
893: IF( WANTU1 ) THEN
894: IF( COLMAJOR ) THEN
895: CALL DLASR( 'R', 'V', 'F', P, IMAX-IMIN+1,
896: $ WORK(IU1CS+IMIN-1), WORK(IU1SN+IMIN-1),
897: $ U1(1,IMIN), LDU1 )
898: ELSE
899: CALL DLASR( 'L', 'V', 'F', IMAX-IMIN+1, P,
900: $ WORK(IU1CS+IMIN-1), WORK(IU1SN+IMIN-1),
901: $ U1(IMIN,1), LDU1 )
902: END IF
903: END IF
904: IF( WANTU2 ) THEN
905: IF( COLMAJOR ) THEN
906: CALL DLASR( 'R', 'V', 'F', M-P, IMAX-IMIN+1,
907: $ WORK(IU2CS+IMIN-1), WORK(IU2SN+IMIN-1),
908: $ U2(1,IMIN), LDU2 )
909: ELSE
910: CALL DLASR( 'L', 'V', 'F', IMAX-IMIN+1, M-P,
911: $ WORK(IU2CS+IMIN-1), WORK(IU2SN+IMIN-1),
912: $ U2(IMIN,1), LDU2 )
913: END IF
914: END IF
915: IF( WANTV1T ) THEN
916: IF( COLMAJOR ) THEN
917: CALL DLASR( 'L', 'V', 'F', IMAX-IMIN+1, Q,
918: $ WORK(IV1TCS+IMIN-1), WORK(IV1TSN+IMIN-1),
919: $ V1T(IMIN,1), LDV1T )
920: ELSE
921: CALL DLASR( 'R', 'V', 'F', Q, IMAX-IMIN+1,
922: $ WORK(IV1TCS+IMIN-1), WORK(IV1TSN+IMIN-1),
923: $ V1T(1,IMIN), LDV1T )
924: END IF
925: END IF
926: IF( WANTV2T ) THEN
927: IF( COLMAJOR ) THEN
928: CALL DLASR( 'L', 'V', 'F', IMAX-IMIN+1, M-Q,
929: $ WORK(IV2TCS+IMIN-1), WORK(IV2TSN+IMIN-1),
930: $ V2T(IMIN,1), LDV2T )
931: ELSE
932: CALL DLASR( 'R', 'V', 'F', M-Q, IMAX-IMIN+1,
933: $ WORK(IV2TCS+IMIN-1), WORK(IV2TSN+IMIN-1),
934: $ V2T(1,IMIN), LDV2T )
935: END IF
936: END IF
937: *
938: * Fix signs on B11(IMAX-1,IMAX) and B21(IMAX-1,IMAX)
939: *
940: IF( B11E(IMAX-1)+B21E(IMAX-1) .GT. 0 ) THEN
941: B11D(IMAX) = -B11D(IMAX)
942: B21D(IMAX) = -B21D(IMAX)
943: IF( WANTV1T ) THEN
944: IF( COLMAJOR ) THEN
945: CALL DSCAL( Q, NEGONE, V1T(IMAX,1), LDV1T )
946: ELSE
947: CALL DSCAL( Q, NEGONE, V1T(1,IMAX), 1 )
948: END IF
949: END IF
950: END IF
951: *
952: * Compute THETA(IMAX)
953: *
954: X1 = COS(PHI(IMAX-1))*B11D(IMAX) +
955: $ SIN(PHI(IMAX-1))*B12E(IMAX-1)
956: Y1 = COS(PHI(IMAX-1))*B21D(IMAX) +
957: $ SIN(PHI(IMAX-1))*B22E(IMAX-1)
958: *
959: THETA(IMAX) = ATAN2( ABS(Y1), ABS(X1) )
960: *
961: * Fix signs on B11(IMAX,IMAX), B12(IMAX,IMAX-1), B21(IMAX,IMAX),
962: * and B22(IMAX,IMAX-1)
963: *
964: IF( B11D(IMAX)+B12E(IMAX-1) .LT. 0 ) THEN
965: B12D(IMAX) = -B12D(IMAX)
966: IF( WANTU1 ) THEN
967: IF( COLMAJOR ) THEN
968: CALL DSCAL( P, NEGONE, U1(1,IMAX), 1 )
969: ELSE
970: CALL DSCAL( P, NEGONE, U1(IMAX,1), LDU1 )
971: END IF
972: END IF
973: END IF
974: IF( B21D(IMAX)+B22E(IMAX-1) .GT. 0 ) THEN
975: B22D(IMAX) = -B22D(IMAX)
976: IF( WANTU2 ) THEN
977: IF( COLMAJOR ) THEN
978: CALL DSCAL( M-P, NEGONE, U2(1,IMAX), 1 )
979: ELSE
980: CALL DSCAL( M-P, NEGONE, U2(IMAX,1), LDU2 )
981: END IF
982: END IF
983: END IF
984: *
985: * Fix signs on B12(IMAX,IMAX) and B22(IMAX,IMAX)
986: *
987: IF( B12D(IMAX)+B22D(IMAX) .LT. 0 ) THEN
988: IF( WANTV2T ) THEN
989: IF( COLMAJOR ) THEN
990: CALL DSCAL( M-Q, NEGONE, V2T(IMAX,1), LDV2T )
991: ELSE
992: CALL DSCAL( M-Q, NEGONE, V2T(1,IMAX), 1 )
993: END IF
994: END IF
995: END IF
996: *
997: * Test for negligible sines or cosines
998: *
999: DO I = IMIN, IMAX
1000: IF( THETA(I) .LT. THRESH ) THEN
1001: THETA(I) = ZERO
1002: ELSE IF( THETA(I) .GT. PIOVER2-THRESH ) THEN
1003: THETA(I) = PIOVER2
1004: END IF
1005: END DO
1006: DO I = IMIN, IMAX-1
1007: IF( PHI(I) .LT. THRESH ) THEN
1008: PHI(I) = ZERO
1009: ELSE IF( PHI(I) .GT. PIOVER2-THRESH ) THEN
1010: PHI(I) = PIOVER2
1011: END IF
1012: END DO
1013: *
1014: * Deflate
1015: *
1016: IF (IMAX .GT. 1) THEN
1017: DO WHILE( PHI(IMAX-1) .EQ. ZERO )
1018: IMAX = IMAX - 1
1019: IF (IMAX .LE. 1) EXIT
1020: END DO
1021: END IF
1022: IF( IMIN .GT. IMAX - 1 )
1023: $ IMIN = IMAX - 1
1024: IF (IMIN .GT. 1) THEN
1025: DO WHILE (PHI(IMIN-1) .NE. ZERO)
1026: IMIN = IMIN - 1
1027: IF (IMIN .LE. 1) EXIT
1028: END DO
1029: END IF
1030: *
1031: * Repeat main iteration loop
1032: *
1033: END DO
1034: *
1035: * Postprocessing: order THETA from least to greatest
1036: *
1037: DO I = 1, Q
1038: *
1039: MINI = I
1040: THETAMIN = THETA(I)
1041: DO J = I+1, Q
1042: IF( THETA(J) .LT. THETAMIN ) THEN
1043: MINI = J
1044: THETAMIN = THETA(J)
1045: END IF
1046: END DO
1047: *
1048: IF( MINI .NE. I ) THEN
1049: THETA(MINI) = THETA(I)
1050: THETA(I) = THETAMIN
1051: IF( COLMAJOR ) THEN
1052: IF( WANTU1 )
1053: $ CALL DSWAP( P, U1(1,I), 1, U1(1,MINI), 1 )
1054: IF( WANTU2 )
1055: $ CALL DSWAP( M-P, U2(1,I), 1, U2(1,MINI), 1 )
1056: IF( WANTV1T )
1057: $ CALL DSWAP( Q, V1T(I,1), LDV1T, V1T(MINI,1), LDV1T )
1058: IF( WANTV2T )
1059: $ CALL DSWAP( M-Q, V2T(I,1), LDV2T, V2T(MINI,1),
1060: $ LDV2T )
1061: ELSE
1062: IF( WANTU1 )
1063: $ CALL DSWAP( P, U1(I,1), LDU1, U1(MINI,1), LDU1 )
1064: IF( WANTU2 )
1065: $ CALL DSWAP( M-P, U2(I,1), LDU2, U2(MINI,1), LDU2 )
1066: IF( WANTV1T )
1067: $ CALL DSWAP( Q, V1T(1,I), 1, V1T(1,MINI), 1 )
1068: IF( WANTV2T )
1069: $ CALL DSWAP( M-Q, V2T(1,I), 1, V2T(1,MINI), 1 )
1070: END IF
1071: END IF
1072: *
1073: END DO
1074: *
1075: RETURN
1076: *
1077: * End of DBBCSD
1078: *
1079: END
1080:
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