1: SUBROUTINE DGSVJ0( JOBV, M, N, A, LDA, D, SVA, MV, V, LDV, EPS,
2: $ SFMIN, TOL, NSWEEP, WORK, LWORK, INFO )
3: *
4: * -- LAPACK routine (version 3.3.1) --
5: *
6: * -- Contributed by Zlatko Drmac of the University of Zagreb and --
7: * -- Kresimir Veselic of the Fernuniversitaet Hagen --
8: * -- April 2011 --
9: *
10: * -- LAPACK is a software package provided by Univ. of Tennessee, --
11: * -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
12: *
13: * This routine is also part of SIGMA (version 1.23, October 23. 2008.)
14: * SIGMA is a library of algorithms for highly accurate algorithms for
15: * computation of SVD, PSVD, QSVD, (H,K)-SVD, and for solution of the
16: * eigenvalue problems Hx = lambda M x, H M x = lambda x with H, M > 0.
17: *
18: IMPLICIT NONE
19: * ..
20: * .. Scalar Arguments ..
21: INTEGER INFO, LDA, LDV, LWORK, M, MV, N, NSWEEP
22: DOUBLE PRECISION EPS, SFMIN, TOL
23: CHARACTER*1 JOBV
24: * ..
25: * .. Array Arguments ..
26: DOUBLE PRECISION A( LDA, * ), SVA( N ), D( N ), V( LDV, * ),
27: $ WORK( LWORK )
28: * ..
29: *
30: * Purpose
31: * =======
32: *
33: * DGSVJ0 is called from DGESVJ as a pre-processor and that is its main
34: * purpose. It applies Jacobi rotations in the same way as DGESVJ does, but
35: * it does not check convergence (stopping criterion). Few tuning
36: * parameters (marked by [TP]) are available for the implementer.
37: *
38: * Further Details
39: * ~~~~~~~~~~~~~~~
40: * DGSVJ0 is used just to enable SGESVJ to call a simplified version of
41: * itself to work on a submatrix of the original matrix.
42: *
43: * Contributors
44: * ~~~~~~~~~~~~
45: * Zlatko Drmac (Zagreb, Croatia) and Kresimir Veselic (Hagen, Germany)
46: *
47: * Bugs, Examples and Comments
48: * ~~~~~~~~~~~~~~~~~~~~~~~~~~~
49: * Please report all bugs and send interesting test examples and comments to
50: * drmac@math.hr. Thank you.
51: *
52: * Arguments
53: * =========
54: *
55: * JOBV (input) CHARACTER*1
56: * Specifies whether the output from this procedure is used
57: * to compute the matrix V:
58: * = 'V': the product of the Jacobi rotations is accumulated
59: * by postmulyiplying the N-by-N array V.
60: * (See the description of V.)
61: * = 'A': the product of the Jacobi rotations is accumulated
62: * by postmulyiplying the MV-by-N array V.
63: * (See the descriptions of MV and V.)
64: * = 'N': the Jacobi rotations are not accumulated.
65: *
66: * M (input) INTEGER
67: * The number of rows of the input matrix A. M >= 0.
68: *
69: * N (input) INTEGER
70: * The number of columns of the input matrix A.
71: * M >= N >= 0.
72: *
73: * A (input/output) DOUBLE PRECISION array, dimension (LDA,N)
74: * On entry, M-by-N matrix A, such that A*diag(D) represents
75: * the input matrix.
76: * On exit,
77: * A_onexit * D_onexit represents the input matrix A*diag(D)
78: * post-multiplied by a sequence of Jacobi rotations, where the
79: * rotation threshold and the total number of sweeps are given in
80: * TOL and NSWEEP, respectively.
81: * (See the descriptions of D, TOL and NSWEEP.)
82: *
83: * LDA (input) INTEGER
84: * The leading dimension of the array A. LDA >= max(1,M).
85: *
86: * D (input/workspace/output) DOUBLE PRECISION array, dimension (N)
87: * The array D accumulates the scaling factors from the fast scaled
88: * Jacobi rotations.
89: * On entry, A*diag(D) represents the input matrix.
90: * On exit, A_onexit*diag(D_onexit) represents the input matrix
91: * post-multiplied by a sequence of Jacobi rotations, where the
92: * rotation threshold and the total number of sweeps are given in
93: * TOL and NSWEEP, respectively.
94: * (See the descriptions of A, TOL and NSWEEP.)
95: *
96: * SVA (input/workspace/output) DOUBLE PRECISION array, dimension (N)
97: * On entry, SVA contains the Euclidean norms of the columns of
98: * the matrix A*diag(D).
99: * On exit, SVA contains the Euclidean norms of the columns of
100: * the matrix onexit*diag(D_onexit).
101: *
102: * MV (input) INTEGER
103: * If JOBV .EQ. 'A', then MV rows of V are post-multipled by a
104: * sequence of Jacobi rotations.
105: * If JOBV = 'N', then MV is not referenced.
106: *
107: * V (input/output) DOUBLE PRECISION array, dimension (LDV,N)
108: * If JOBV .EQ. 'V' then N rows of V are post-multipled by a
109: * sequence of Jacobi rotations.
110: * If JOBV .EQ. 'A' then MV rows of V are post-multipled by a
111: * sequence of Jacobi rotations.
112: * If JOBV = 'N', then V is not referenced.
113: *
114: * LDV (input) INTEGER
115: * The leading dimension of the array V, LDV >= 1.
116: * If JOBV = 'V', LDV .GE. N.
117: * If JOBV = 'A', LDV .GE. MV.
118: *
119: * EPS (input) DOUBLE PRECISION
120: * EPS = DLAMCH('Epsilon')
121: *
122: * SFMIN (input) DOUBLE PRECISION
123: * SFMIN = DLAMCH('Safe Minimum')
124: *
125: * TOL (input) DOUBLE PRECISION
126: * TOL is the threshold for Jacobi rotations. For a pair
127: * A(:,p), A(:,q) of pivot columns, the Jacobi rotation is
128: * applied only if DABS(COS(angle(A(:,p),A(:,q)))) .GT. TOL.
129: *
130: * NSWEEP (input) INTEGER
131: * NSWEEP is the number of sweeps of Jacobi rotations to be
132: * performed.
133: *
134: * WORK (workspace) DOUBLE PRECISION array, dimension (LWORK)
135: *
136: * LWORK (input) INTEGER
137: * LWORK is the dimension of WORK. LWORK .GE. M.
138: *
139: * INFO (output) INTEGER
140: * = 0 : successful exit.
141: * < 0 : if INFO = -i, then the i-th argument had an illegal value
142: *
143: * =====================================================================
144: *
145: * .. Local Parameters ..
146: DOUBLE PRECISION ZERO, HALF, ONE, TWO
147: PARAMETER ( ZERO = 0.0D0, HALF = 0.5D0, ONE = 1.0D0,
148: $ TWO = 2.0D0 )
149: * ..
150: * .. Local Scalars ..
151: DOUBLE PRECISION AAPP, AAPP0, AAPQ, AAQQ, APOAQ, AQOAP, BIG,
152: $ BIGTHETA, CS, MXAAPQ, MXSINJ, ROOTBIG, ROOTEPS,
153: $ ROOTSFMIN, ROOTTOL, SMALL, SN, T, TEMP1, THETA,
154: $ THSIGN
155: INTEGER BLSKIP, EMPTSW, i, ibr, IERR, igl, IJBLSK, ir1,
156: $ ISWROT, jbc, jgl, KBL, LKAHEAD, MVL, NBL,
157: $ NOTROT, p, PSKIPPED, q, ROWSKIP, SWBAND
158: LOGICAL APPLV, ROTOK, RSVEC
159: * ..
160: * .. Local Arrays ..
161: DOUBLE PRECISION FASTR( 5 )
162: * ..
163: * .. Intrinsic Functions ..
164: INTRINSIC DABS, DMAX1, DBLE, MIN0, DSIGN, DSQRT
165: * ..
166: * .. External Functions ..
167: DOUBLE PRECISION DDOT, DNRM2
168: INTEGER IDAMAX
169: LOGICAL LSAME
170: EXTERNAL IDAMAX, LSAME, DDOT, DNRM2
171: * ..
172: * .. External Subroutines ..
173: EXTERNAL DAXPY, DCOPY, DLASCL, DLASSQ, DROTM, DSWAP
174: * ..
175: * .. Executable Statements ..
176: *
177: * Test the input parameters.
178: *
179: APPLV = LSAME( JOBV, 'A' )
180: RSVEC = LSAME( JOBV, 'V' )
181: IF( .NOT.( RSVEC .OR. APPLV .OR. LSAME( JOBV, 'N' ) ) ) THEN
182: INFO = -1
183: ELSE IF( M.LT.0 ) THEN
184: INFO = -2
185: ELSE IF( ( N.LT.0 ) .OR. ( N.GT.M ) ) THEN
186: INFO = -3
187: ELSE IF( LDA.LT.M ) THEN
188: INFO = -5
189: ELSE IF( ( RSVEC.OR.APPLV ) .AND. ( MV.LT.0 ) ) THEN
190: INFO = -8
191: ELSE IF( ( RSVEC.AND.( LDV.LT.N ) ).OR.
192: $ ( APPLV.AND.( LDV.LT.MV ) ) ) THEN
193: INFO = -10
194: ELSE IF( TOL.LE.EPS ) THEN
195: INFO = -13
196: ELSE IF( NSWEEP.LT.0 ) THEN
197: INFO = -14
198: ELSE IF( LWORK.LT.M ) THEN
199: INFO = -16
200: ELSE
201: INFO = 0
202: END IF
203: *
204: * #:(
205: IF( INFO.NE.0 ) THEN
206: CALL XERBLA( 'DGSVJ0', -INFO )
207: RETURN
208: END IF
209: *
210: IF( RSVEC ) THEN
211: MVL = N
212: ELSE IF( APPLV ) THEN
213: MVL = MV
214: END IF
215: RSVEC = RSVEC .OR. APPLV
216:
217: ROOTEPS = DSQRT( EPS )
218: ROOTSFMIN = DSQRT( SFMIN )
219: SMALL = SFMIN / EPS
220: BIG = ONE / SFMIN
221: ROOTBIG = ONE / ROOTSFMIN
222: BIGTHETA = ONE / ROOTEPS
223: ROOTTOL = DSQRT( TOL )
224: *
225: * -#- Row-cyclic Jacobi SVD algorithm with column pivoting -#-
226: *
227: EMPTSW = ( N*( N-1 ) ) / 2
228: NOTROT = 0
229: FASTR( 1 ) = ZERO
230: *
231: * -#- Row-cyclic pivot strategy with de Rijk's pivoting -#-
232: *
233:
234: SWBAND = 0
235: *[TP] SWBAND is a tuning parameter. It is meaningful and effective
236: * if SGESVJ is used as a computational routine in the preconditioned
237: * Jacobi SVD algorithm SGESVJ. For sweeps i=1:SWBAND the procedure
238: * ......
239:
240: KBL = MIN0( 8, N )
241: *[TP] KBL is a tuning parameter that defines the tile size in the
242: * tiling of the p-q loops of pivot pairs. In general, an optimal
243: * value of KBL depends on the matrix dimensions and on the
244: * parameters of the computer's memory.
245: *
246: NBL = N / KBL
247: IF( ( NBL*KBL ).NE.N )NBL = NBL + 1
248:
249: BLSKIP = ( KBL**2 ) + 1
250: *[TP] BLKSKIP is a tuning parameter that depends on SWBAND and KBL.
251:
252: ROWSKIP = MIN0( 5, KBL )
253: *[TP] ROWSKIP is a tuning parameter.
254:
255: LKAHEAD = 1
256: *[TP] LKAHEAD is a tuning parameter.
257: SWBAND = 0
258: PSKIPPED = 0
259: *
260: DO 1993 i = 1, NSWEEP
261: * .. go go go ...
262: *
263: MXAAPQ = ZERO
264: MXSINJ = ZERO
265: ISWROT = 0
266: *
267: NOTROT = 0
268: PSKIPPED = 0
269: *
270: DO 2000 ibr = 1, NBL
271:
272: igl = ( ibr-1 )*KBL + 1
273: *
274: DO 1002 ir1 = 0, MIN0( LKAHEAD, NBL-ibr )
275: *
276: igl = igl + ir1*KBL
277: *
278: DO 2001 p = igl, MIN0( igl+KBL-1, N-1 )
279:
280: * .. de Rijk's pivoting
281: q = IDAMAX( N-p+1, SVA( p ), 1 ) + p - 1
282: IF( p.NE.q ) THEN
283: CALL DSWAP( M, A( 1, p ), 1, A( 1, q ), 1 )
284: IF( RSVEC )CALL DSWAP( MVL, V( 1, p ), 1,
285: $ V( 1, q ), 1 )
286: TEMP1 = SVA( p )
287: SVA( p ) = SVA( q )
288: SVA( q ) = TEMP1
289: TEMP1 = D( p )
290: D( p ) = D( q )
291: D( q ) = TEMP1
292: END IF
293: *
294: IF( ir1.EQ.0 ) THEN
295: *
296: * Column norms are periodically updated by explicit
297: * norm computation.
298: * Caveat:
299: * Some BLAS implementations compute DNRM2(M,A(1,p),1)
300: * as DSQRT(DDOT(M,A(1,p),1,A(1,p),1)), which may result in
301: * overflow for ||A(:,p)||_2 > DSQRT(overflow_threshold), and
302: * undeflow for ||A(:,p)||_2 < DSQRT(underflow_threshold).
303: * Hence, DNRM2 cannot be trusted, not even in the case when
304: * the true norm is far from the under(over)flow boundaries.
305: * If properly implemented DNRM2 is available, the IF-THEN-ELSE
306: * below should read "AAPP = DNRM2( M, A(1,p), 1 ) * D(p)".
307: *
308: IF( ( SVA( p ).LT.ROOTBIG ) .AND.
309: $ ( SVA( p ).GT.ROOTSFMIN ) ) THEN
310: SVA( p ) = DNRM2( M, A( 1, p ), 1 )*D( p )
311: ELSE
312: TEMP1 = ZERO
313: AAPP = ONE
314: CALL DLASSQ( M, A( 1, p ), 1, TEMP1, AAPP )
315: SVA( p ) = TEMP1*DSQRT( AAPP )*D( p )
316: END IF
317: AAPP = SVA( p )
318: ELSE
319: AAPP = SVA( p )
320: END IF
321:
322: *
323: IF( AAPP.GT.ZERO ) THEN
324: *
325: PSKIPPED = 0
326: *
327: DO 2002 q = p + 1, MIN0( igl+KBL-1, N )
328: *
329: AAQQ = SVA( q )
330:
331: IF( AAQQ.GT.ZERO ) THEN
332: *
333: AAPP0 = AAPP
334: IF( AAQQ.GE.ONE ) THEN
335: ROTOK = ( SMALL*AAPP ).LE.AAQQ
336: IF( AAPP.LT.( BIG / AAQQ ) ) THEN
337: AAPQ = ( DDOT( M, A( 1, p ), 1, A( 1,
338: $ q ), 1 )*D( p )*D( q ) / AAQQ )
339: $ / AAPP
340: ELSE
341: CALL DCOPY( M, A( 1, p ), 1, WORK, 1 )
342: CALL DLASCL( 'G', 0, 0, AAPP, D( p ),
343: $ M, 1, WORK, LDA, IERR )
344: AAPQ = DDOT( M, WORK, 1, A( 1, q ),
345: $ 1 )*D( q ) / AAQQ
346: END IF
347: ELSE
348: ROTOK = AAPP.LE.( AAQQ / SMALL )
349: IF( AAPP.GT.( SMALL / AAQQ ) ) THEN
350: AAPQ = ( DDOT( M, A( 1, p ), 1, A( 1,
351: $ q ), 1 )*D( p )*D( q ) / AAQQ )
352: $ / AAPP
353: ELSE
354: CALL DCOPY( M, A( 1, q ), 1, WORK, 1 )
355: CALL DLASCL( 'G', 0, 0, AAQQ, D( q ),
356: $ M, 1, WORK, LDA, IERR )
357: AAPQ = DDOT( M, WORK, 1, A( 1, p ),
358: $ 1 )*D( p ) / AAPP
359: END IF
360: END IF
361: *
362: MXAAPQ = DMAX1( MXAAPQ, DABS( AAPQ ) )
363: *
364: * TO rotate or NOT to rotate, THAT is the question ...
365: *
366: IF( DABS( AAPQ ).GT.TOL ) THEN
367: *
368: * .. rotate
369: * ROTATED = ROTATED + ONE
370: *
371: IF( ir1.EQ.0 ) THEN
372: NOTROT = 0
373: PSKIPPED = 0
374: ISWROT = ISWROT + 1
375: END IF
376: *
377: IF( ROTOK ) THEN
378: *
379: AQOAP = AAQQ / AAPP
380: APOAQ = AAPP / AAQQ
381: THETA = -HALF*DABS( AQOAP-APOAQ )/AAPQ
382: *
383: IF( DABS( THETA ).GT.BIGTHETA ) THEN
384: *
385: T = HALF / THETA
386: FASTR( 3 ) = T*D( p ) / D( q )
387: FASTR( 4 ) = -T*D( q ) / D( p )
388: CALL DROTM( M, A( 1, p ), 1,
389: $ A( 1, q ), 1, FASTR )
390: IF( RSVEC )CALL DROTM( MVL,
391: $ V( 1, p ), 1,
392: $ V( 1, q ), 1,
393: $ FASTR )
394: SVA( q ) = AAQQ*DSQRT( DMAX1( ZERO,
395: $ ONE+T*APOAQ*AAPQ ) )
396: AAPP = AAPP*DSQRT( DMAX1( ZERO,
397: $ ONE-T*AQOAP*AAPQ ) )
398: MXSINJ = DMAX1( MXSINJ, DABS( T ) )
399: *
400: ELSE
401: *
402: * .. choose correct signum for THETA and rotate
403: *
404: THSIGN = -DSIGN( ONE, AAPQ )
405: T = ONE / ( THETA+THSIGN*
406: $ DSQRT( ONE+THETA*THETA ) )
407: CS = DSQRT( ONE / ( ONE+T*T ) )
408: SN = T*CS
409: *
410: MXSINJ = DMAX1( MXSINJ, DABS( SN ) )
411: SVA( q ) = AAQQ*DSQRT( DMAX1( ZERO,
412: $ ONE+T*APOAQ*AAPQ ) )
413: AAPP = AAPP*DSQRT( DMAX1( ZERO,
414: $ ONE-T*AQOAP*AAPQ ) )
415: *
416: APOAQ = D( p ) / D( q )
417: AQOAP = D( q ) / D( p )
418: IF( D( p ).GE.ONE ) THEN
419: IF( D( q ).GE.ONE ) THEN
420: FASTR( 3 ) = T*APOAQ
421: FASTR( 4 ) = -T*AQOAP
422: D( p ) = D( p )*CS
423: D( q ) = D( q )*CS
424: CALL DROTM( M, A( 1, p ), 1,
425: $ A( 1, q ), 1,
426: $ FASTR )
427: IF( RSVEC )CALL DROTM( MVL,
428: $ V( 1, p ), 1, V( 1, q ),
429: $ 1, FASTR )
430: ELSE
431: CALL DAXPY( M, -T*AQOAP,
432: $ A( 1, q ), 1,
433: $ A( 1, p ), 1 )
434: CALL DAXPY( M, CS*SN*APOAQ,
435: $ A( 1, p ), 1,
436: $ A( 1, q ), 1 )
437: D( p ) = D( p )*CS
438: D( q ) = D( q ) / CS
439: IF( RSVEC ) THEN
440: CALL DAXPY( MVL, -T*AQOAP,
441: $ V( 1, q ), 1,
442: $ V( 1, p ), 1 )
443: CALL DAXPY( MVL,
444: $ CS*SN*APOAQ,
445: $ V( 1, p ), 1,
446: $ V( 1, q ), 1 )
447: END IF
448: END IF
449: ELSE
450: IF( D( q ).GE.ONE ) THEN
451: CALL DAXPY( M, T*APOAQ,
452: $ A( 1, p ), 1,
453: $ A( 1, q ), 1 )
454: CALL DAXPY( M, -CS*SN*AQOAP,
455: $ A( 1, q ), 1,
456: $ A( 1, p ), 1 )
457: D( p ) = D( p ) / CS
458: D( q ) = D( q )*CS
459: IF( RSVEC ) THEN
460: CALL DAXPY( MVL, T*APOAQ,
461: $ V( 1, p ), 1,
462: $ V( 1, q ), 1 )
463: CALL DAXPY( MVL,
464: $ -CS*SN*AQOAP,
465: $ V( 1, q ), 1,
466: $ V( 1, p ), 1 )
467: END IF
468: ELSE
469: IF( D( p ).GE.D( q ) ) THEN
470: CALL DAXPY( M, -T*AQOAP,
471: $ A( 1, q ), 1,
472: $ A( 1, p ), 1 )
473: CALL DAXPY( M, CS*SN*APOAQ,
474: $ A( 1, p ), 1,
475: $ A( 1, q ), 1 )
476: D( p ) = D( p )*CS
477: D( q ) = D( q ) / CS
478: IF( RSVEC ) THEN
479: CALL DAXPY( MVL,
480: $ -T*AQOAP,
481: $ V( 1, q ), 1,
482: $ V( 1, p ), 1 )
483: CALL DAXPY( MVL,
484: $ CS*SN*APOAQ,
485: $ V( 1, p ), 1,
486: $ V( 1, q ), 1 )
487: END IF
488: ELSE
489: CALL DAXPY( M, T*APOAQ,
490: $ A( 1, p ), 1,
491: $ A( 1, q ), 1 )
492: CALL DAXPY( M,
493: $ -CS*SN*AQOAP,
494: $ A( 1, q ), 1,
495: $ A( 1, p ), 1 )
496: D( p ) = D( p ) / CS
497: D( q ) = D( q )*CS
498: IF( RSVEC ) THEN
499: CALL DAXPY( MVL,
500: $ T*APOAQ, V( 1, p ),
501: $ 1, V( 1, q ), 1 )
502: CALL DAXPY( MVL,
503: $ -CS*SN*AQOAP,
504: $ V( 1, q ), 1,
505: $ V( 1, p ), 1 )
506: END IF
507: END IF
508: END IF
509: END IF
510: END IF
511: *
512: ELSE
513: * .. have to use modified Gram-Schmidt like transformation
514: CALL DCOPY( M, A( 1, p ), 1, WORK, 1 )
515: CALL DLASCL( 'G', 0, 0, AAPP, ONE, M,
516: $ 1, WORK, LDA, IERR )
517: CALL DLASCL( 'G', 0, 0, AAQQ, ONE, M,
518: $ 1, A( 1, q ), LDA, IERR )
519: TEMP1 = -AAPQ*D( p ) / D( q )
520: CALL DAXPY( M, TEMP1, WORK, 1,
521: $ A( 1, q ), 1 )
522: CALL DLASCL( 'G', 0, 0, ONE, AAQQ, M,
523: $ 1, A( 1, q ), LDA, IERR )
524: SVA( q ) = AAQQ*DSQRT( DMAX1( ZERO,
525: $ ONE-AAPQ*AAPQ ) )
526: MXSINJ = DMAX1( MXSINJ, SFMIN )
527: END IF
528: * END IF ROTOK THEN ... ELSE
529: *
530: * In the case of cancellation in updating SVA(q), SVA(p)
531: * recompute SVA(q), SVA(p).
532: IF( ( SVA( q ) / AAQQ )**2.LE.ROOTEPS )
533: $ THEN
534: IF( ( AAQQ.LT.ROOTBIG ) .AND.
535: $ ( AAQQ.GT.ROOTSFMIN ) ) THEN
536: SVA( q ) = DNRM2( M, A( 1, q ), 1 )*
537: $ D( q )
538: ELSE
539: T = ZERO
540: AAQQ = ONE
541: CALL DLASSQ( M, A( 1, q ), 1, T,
542: $ AAQQ )
543: SVA( q ) = T*DSQRT( AAQQ )*D( q )
544: END IF
545: END IF
546: IF( ( AAPP / AAPP0 ).LE.ROOTEPS ) THEN
547: IF( ( AAPP.LT.ROOTBIG ) .AND.
548: $ ( AAPP.GT.ROOTSFMIN ) ) THEN
549: AAPP = DNRM2( M, A( 1, p ), 1 )*
550: $ D( p )
551: ELSE
552: T = ZERO
553: AAPP = ONE
554: CALL DLASSQ( M, A( 1, p ), 1, T,
555: $ AAPP )
556: AAPP = T*DSQRT( AAPP )*D( p )
557: END IF
558: SVA( p ) = AAPP
559: END IF
560: *
561: ELSE
562: * A(:,p) and A(:,q) already numerically orthogonal
563: IF( ir1.EQ.0 )NOTROT = NOTROT + 1
564: PSKIPPED = PSKIPPED + 1
565: END IF
566: ELSE
567: * A(:,q) is zero column
568: IF( ir1.EQ.0 )NOTROT = NOTROT + 1
569: PSKIPPED = PSKIPPED + 1
570: END IF
571: *
572: IF( ( i.LE.SWBAND ) .AND.
573: $ ( PSKIPPED.GT.ROWSKIP ) ) THEN
574: IF( ir1.EQ.0 )AAPP = -AAPP
575: NOTROT = 0
576: GO TO 2103
577: END IF
578: *
579: 2002 CONTINUE
580: * END q-LOOP
581: *
582: 2103 CONTINUE
583: * bailed out of q-loop
584:
585: SVA( p ) = AAPP
586:
587: ELSE
588: SVA( p ) = AAPP
589: IF( ( ir1.EQ.0 ) .AND. ( AAPP.EQ.ZERO ) )
590: $ NOTROT = NOTROT + MIN0( igl+KBL-1, N ) - p
591: END IF
592: *
593: 2001 CONTINUE
594: * end of the p-loop
595: * end of doing the block ( ibr, ibr )
596: 1002 CONTINUE
597: * end of ir1-loop
598: *
599: *........................................................
600: * ... go to the off diagonal blocks
601: *
602: igl = ( ibr-1 )*KBL + 1
603: *
604: DO 2010 jbc = ibr + 1, NBL
605: *
606: jgl = ( jbc-1 )*KBL + 1
607: *
608: * doing the block at ( ibr, jbc )
609: *
610: IJBLSK = 0
611: DO 2100 p = igl, MIN0( igl+KBL-1, N )
612: *
613: AAPP = SVA( p )
614: *
615: IF( AAPP.GT.ZERO ) THEN
616: *
617: PSKIPPED = 0
618: *
619: DO 2200 q = jgl, MIN0( jgl+KBL-1, N )
620: *
621: AAQQ = SVA( q )
622: *
623: IF( AAQQ.GT.ZERO ) THEN
624: AAPP0 = AAPP
625: *
626: * -#- M x 2 Jacobi SVD -#-
627: *
628: * -#- Safe Gram matrix computation -#-
629: *
630: IF( AAQQ.GE.ONE ) THEN
631: IF( AAPP.GE.AAQQ ) THEN
632: ROTOK = ( SMALL*AAPP ).LE.AAQQ
633: ELSE
634: ROTOK = ( SMALL*AAQQ ).LE.AAPP
635: END IF
636: IF( AAPP.LT.( BIG / AAQQ ) ) THEN
637: AAPQ = ( DDOT( M, A( 1, p ), 1, A( 1,
638: $ q ), 1 )*D( p )*D( q ) / AAQQ )
639: $ / AAPP
640: ELSE
641: CALL DCOPY( M, A( 1, p ), 1, WORK, 1 )
642: CALL DLASCL( 'G', 0, 0, AAPP, D( p ),
643: $ M, 1, WORK, LDA, IERR )
644: AAPQ = DDOT( M, WORK, 1, A( 1, q ),
645: $ 1 )*D( q ) / AAQQ
646: END IF
647: ELSE
648: IF( AAPP.GE.AAQQ ) THEN
649: ROTOK = AAPP.LE.( AAQQ / SMALL )
650: ELSE
651: ROTOK = AAQQ.LE.( AAPP / SMALL )
652: END IF
653: IF( AAPP.GT.( SMALL / AAQQ ) ) THEN
654: AAPQ = ( DDOT( M, A( 1, p ), 1, A( 1,
655: $ q ), 1 )*D( p )*D( q ) / AAQQ )
656: $ / AAPP
657: ELSE
658: CALL DCOPY( M, A( 1, q ), 1, WORK, 1 )
659: CALL DLASCL( 'G', 0, 0, AAQQ, D( q ),
660: $ M, 1, WORK, LDA, IERR )
661: AAPQ = DDOT( M, WORK, 1, A( 1, p ),
662: $ 1 )*D( p ) / AAPP
663: END IF
664: END IF
665: *
666: MXAAPQ = DMAX1( MXAAPQ, DABS( AAPQ ) )
667: *
668: * TO rotate or NOT to rotate, THAT is the question ...
669: *
670: IF( DABS( AAPQ ).GT.TOL ) THEN
671: NOTROT = 0
672: * ROTATED = ROTATED + 1
673: PSKIPPED = 0
674: ISWROT = ISWROT + 1
675: *
676: IF( ROTOK ) THEN
677: *
678: AQOAP = AAQQ / AAPP
679: APOAQ = AAPP / AAQQ
680: THETA = -HALF*DABS( AQOAP-APOAQ )/AAPQ
681: IF( AAQQ.GT.AAPP0 )THETA = -THETA
682: *
683: IF( DABS( THETA ).GT.BIGTHETA ) THEN
684: T = HALF / THETA
685: FASTR( 3 ) = T*D( p ) / D( q )
686: FASTR( 4 ) = -T*D( q ) / D( p )
687: CALL DROTM( M, A( 1, p ), 1,
688: $ A( 1, q ), 1, FASTR )
689: IF( RSVEC )CALL DROTM( MVL,
690: $ V( 1, p ), 1,
691: $ V( 1, q ), 1,
692: $ FASTR )
693: SVA( q ) = AAQQ*DSQRT( DMAX1( ZERO,
694: $ ONE+T*APOAQ*AAPQ ) )
695: AAPP = AAPP*DSQRT( DMAX1( ZERO,
696: $ ONE-T*AQOAP*AAPQ ) )
697: MXSINJ = DMAX1( MXSINJ, DABS( T ) )
698: ELSE
699: *
700: * .. choose correct signum for THETA and rotate
701: *
702: THSIGN = -DSIGN( ONE, AAPQ )
703: IF( AAQQ.GT.AAPP0 )THSIGN = -THSIGN
704: T = ONE / ( THETA+THSIGN*
705: $ DSQRT( ONE+THETA*THETA ) )
706: CS = DSQRT( ONE / ( ONE+T*T ) )
707: SN = T*CS
708: MXSINJ = DMAX1( MXSINJ, DABS( SN ) )
709: SVA( q ) = AAQQ*DSQRT( DMAX1( ZERO,
710: $ ONE+T*APOAQ*AAPQ ) )
711: AAPP = AAPP*DSQRT( DMAX1( ZERO,
712: $ ONE-T*AQOAP*AAPQ ) )
713: *
714: APOAQ = D( p ) / D( q )
715: AQOAP = D( q ) / D( p )
716: IF( D( p ).GE.ONE ) THEN
717: *
718: IF( D( q ).GE.ONE ) THEN
719: FASTR( 3 ) = T*APOAQ
720: FASTR( 4 ) = -T*AQOAP
721: D( p ) = D( p )*CS
722: D( q ) = D( q )*CS
723: CALL DROTM( M, A( 1, p ), 1,
724: $ A( 1, q ), 1,
725: $ FASTR )
726: IF( RSVEC )CALL DROTM( MVL,
727: $ V( 1, p ), 1, V( 1, q ),
728: $ 1, FASTR )
729: ELSE
730: CALL DAXPY( M, -T*AQOAP,
731: $ A( 1, q ), 1,
732: $ A( 1, p ), 1 )
733: CALL DAXPY( M, CS*SN*APOAQ,
734: $ A( 1, p ), 1,
735: $ A( 1, q ), 1 )
736: IF( RSVEC ) THEN
737: CALL DAXPY( MVL, -T*AQOAP,
738: $ V( 1, q ), 1,
739: $ V( 1, p ), 1 )
740: CALL DAXPY( MVL,
741: $ CS*SN*APOAQ,
742: $ V( 1, p ), 1,
743: $ V( 1, q ), 1 )
744: END IF
745: D( p ) = D( p )*CS
746: D( q ) = D( q ) / CS
747: END IF
748: ELSE
749: IF( D( q ).GE.ONE ) THEN
750: CALL DAXPY( M, T*APOAQ,
751: $ A( 1, p ), 1,
752: $ A( 1, q ), 1 )
753: CALL DAXPY( M, -CS*SN*AQOAP,
754: $ A( 1, q ), 1,
755: $ A( 1, p ), 1 )
756: IF( RSVEC ) THEN
757: CALL DAXPY( MVL, T*APOAQ,
758: $ V( 1, p ), 1,
759: $ V( 1, q ), 1 )
760: CALL DAXPY( MVL,
761: $ -CS*SN*AQOAP,
762: $ V( 1, q ), 1,
763: $ V( 1, p ), 1 )
764: END IF
765: D( p ) = D( p ) / CS
766: D( q ) = D( q )*CS
767: ELSE
768: IF( D( p ).GE.D( q ) ) THEN
769: CALL DAXPY( M, -T*AQOAP,
770: $ A( 1, q ), 1,
771: $ A( 1, p ), 1 )
772: CALL DAXPY( M, CS*SN*APOAQ,
773: $ A( 1, p ), 1,
774: $ A( 1, q ), 1 )
775: D( p ) = D( p )*CS
776: D( q ) = D( q ) / CS
777: IF( RSVEC ) THEN
778: CALL DAXPY( MVL,
779: $ -T*AQOAP,
780: $ V( 1, q ), 1,
781: $ V( 1, p ), 1 )
782: CALL DAXPY( MVL,
783: $ CS*SN*APOAQ,
784: $ V( 1, p ), 1,
785: $ V( 1, q ), 1 )
786: END IF
787: ELSE
788: CALL DAXPY( M, T*APOAQ,
789: $ A( 1, p ), 1,
790: $ A( 1, q ), 1 )
791: CALL DAXPY( M,
792: $ -CS*SN*AQOAP,
793: $ A( 1, q ), 1,
794: $ A( 1, p ), 1 )
795: D( p ) = D( p ) / CS
796: D( q ) = D( q )*CS
797: IF( RSVEC ) THEN
798: CALL DAXPY( MVL,
799: $ T*APOAQ, V( 1, p ),
800: $ 1, V( 1, q ), 1 )
801: CALL DAXPY( MVL,
802: $ -CS*SN*AQOAP,
803: $ V( 1, q ), 1,
804: $ V( 1, p ), 1 )
805: END IF
806: END IF
807: END IF
808: END IF
809: END IF
810: *
811: ELSE
812: IF( AAPP.GT.AAQQ ) THEN
813: CALL DCOPY( M, A( 1, p ), 1, WORK,
814: $ 1 )
815: CALL DLASCL( 'G', 0, 0, AAPP, ONE,
816: $ M, 1, WORK, LDA, IERR )
817: CALL DLASCL( 'G', 0, 0, AAQQ, ONE,
818: $ M, 1, A( 1, q ), LDA,
819: $ IERR )
820: TEMP1 = -AAPQ*D( p ) / D( q )
821: CALL DAXPY( M, TEMP1, WORK, 1,
822: $ A( 1, q ), 1 )
823: CALL DLASCL( 'G', 0, 0, ONE, AAQQ,
824: $ M, 1, A( 1, q ), LDA,
825: $ IERR )
826: SVA( q ) = AAQQ*DSQRT( DMAX1( ZERO,
827: $ ONE-AAPQ*AAPQ ) )
828: MXSINJ = DMAX1( MXSINJ, SFMIN )
829: ELSE
830: CALL DCOPY( M, A( 1, q ), 1, WORK,
831: $ 1 )
832: CALL DLASCL( 'G', 0, 0, AAQQ, ONE,
833: $ M, 1, WORK, LDA, IERR )
834: CALL DLASCL( 'G', 0, 0, AAPP, ONE,
835: $ M, 1, A( 1, p ), LDA,
836: $ IERR )
837: TEMP1 = -AAPQ*D( q ) / D( p )
838: CALL DAXPY( M, TEMP1, WORK, 1,
839: $ A( 1, p ), 1 )
840: CALL DLASCL( 'G', 0, 0, ONE, AAPP,
841: $ M, 1, A( 1, p ), LDA,
842: $ IERR )
843: SVA( p ) = AAPP*DSQRT( DMAX1( ZERO,
844: $ ONE-AAPQ*AAPQ ) )
845: MXSINJ = DMAX1( MXSINJ, SFMIN )
846: END IF
847: END IF
848: * END IF ROTOK THEN ... ELSE
849: *
850: * In the case of cancellation in updating SVA(q)
851: * .. recompute SVA(q)
852: IF( ( SVA( q ) / AAQQ )**2.LE.ROOTEPS )
853: $ THEN
854: IF( ( AAQQ.LT.ROOTBIG ) .AND.
855: $ ( AAQQ.GT.ROOTSFMIN ) ) THEN
856: SVA( q ) = DNRM2( M, A( 1, q ), 1 )*
857: $ D( q )
858: ELSE
859: T = ZERO
860: AAQQ = ONE
861: CALL DLASSQ( M, A( 1, q ), 1, T,
862: $ AAQQ )
863: SVA( q ) = T*DSQRT( AAQQ )*D( q )
864: END IF
865: END IF
866: IF( ( AAPP / AAPP0 )**2.LE.ROOTEPS ) THEN
867: IF( ( AAPP.LT.ROOTBIG ) .AND.
868: $ ( AAPP.GT.ROOTSFMIN ) ) THEN
869: AAPP = DNRM2( M, A( 1, p ), 1 )*
870: $ D( p )
871: ELSE
872: T = ZERO
873: AAPP = ONE
874: CALL DLASSQ( M, A( 1, p ), 1, T,
875: $ AAPP )
876: AAPP = T*DSQRT( AAPP )*D( p )
877: END IF
878: SVA( p ) = AAPP
879: END IF
880: * end of OK rotation
881: ELSE
882: NOTROT = NOTROT + 1
883: PSKIPPED = PSKIPPED + 1
884: IJBLSK = IJBLSK + 1
885: END IF
886: ELSE
887: NOTROT = NOTROT + 1
888: PSKIPPED = PSKIPPED + 1
889: IJBLSK = IJBLSK + 1
890: END IF
891: *
892: IF( ( i.LE.SWBAND ) .AND. ( IJBLSK.GE.BLSKIP ) )
893: $ THEN
894: SVA( p ) = AAPP
895: NOTROT = 0
896: GO TO 2011
897: END IF
898: IF( ( i.LE.SWBAND ) .AND.
899: $ ( PSKIPPED.GT.ROWSKIP ) ) THEN
900: AAPP = -AAPP
901: NOTROT = 0
902: GO TO 2203
903: END IF
904: *
905: 2200 CONTINUE
906: * end of the q-loop
907: 2203 CONTINUE
908: *
909: SVA( p ) = AAPP
910: *
911: ELSE
912: IF( AAPP.EQ.ZERO )NOTROT = NOTROT +
913: $ MIN0( jgl+KBL-1, N ) - jgl + 1
914: IF( AAPP.LT.ZERO )NOTROT = 0
915: END IF
916:
917: 2100 CONTINUE
918: * end of the p-loop
919: 2010 CONTINUE
920: * end of the jbc-loop
921: 2011 CONTINUE
922: *2011 bailed out of the jbc-loop
923: DO 2012 p = igl, MIN0( igl+KBL-1, N )
924: SVA( p ) = DABS( SVA( p ) )
925: 2012 CONTINUE
926: *
927: 2000 CONTINUE
928: *2000 :: end of the ibr-loop
929: *
930: * .. update SVA(N)
931: IF( ( SVA( N ).LT.ROOTBIG ) .AND. ( SVA( N ).GT.ROOTSFMIN ) )
932: $ THEN
933: SVA( N ) = DNRM2( M, A( 1, N ), 1 )*D( N )
934: ELSE
935: T = ZERO
936: AAPP = ONE
937: CALL DLASSQ( M, A( 1, N ), 1, T, AAPP )
938: SVA( N ) = T*DSQRT( AAPP )*D( N )
939: END IF
940: *
941: * Additional steering devices
942: *
943: IF( ( i.LT.SWBAND ) .AND. ( ( MXAAPQ.LE.ROOTTOL ) .OR.
944: $ ( ISWROT.LE.N ) ) )SWBAND = i
945: *
946: IF( ( i.GT.SWBAND+1 ) .AND. ( MXAAPQ.LT.DBLE( N )*TOL ) .AND.
947: $ ( DBLE( N )*MXAAPQ*MXSINJ.LT.TOL ) ) THEN
948: GO TO 1994
949: END IF
950: *
951: IF( NOTROT.GE.EMPTSW )GO TO 1994
952:
953: 1993 CONTINUE
954: * end i=1:NSWEEP loop
955: * #:) Reaching this point means that the procedure has comleted the given
956: * number of iterations.
957: INFO = NSWEEP - 1
958: GO TO 1995
959: 1994 CONTINUE
960: * #:) Reaching this point means that during the i-th sweep all pivots were
961: * below the given tolerance, causing early exit.
962: *
963: INFO = 0
964: * #:) INFO = 0 confirms successful iterations.
965: 1995 CONTINUE
966: *
967: * Sort the vector D.
968: DO 5991 p = 1, N - 1
969: q = IDAMAX( N-p+1, SVA( p ), 1 ) + p - 1
970: IF( p.NE.q ) THEN
971: TEMP1 = SVA( p )
972: SVA( p ) = SVA( q )
973: SVA( q ) = TEMP1
974: TEMP1 = D( p )
975: D( p ) = D( q )
976: D( q ) = TEMP1
977: CALL DSWAP( M, A( 1, p ), 1, A( 1, q ), 1 )
978: IF( RSVEC )CALL DSWAP( MVL, V( 1, p ), 1, V( 1, q ), 1 )
979: END IF
980: 5991 CONTINUE
981: *
982: RETURN
983: * ..
984: * .. END OF DGSVJ0
985: * ..
986: END
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