LAPACK 3.12.0 LAPACK: Linear Algebra PACKage
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cgesvj.f
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1*> \brief <b> CGESVJ </b>
2*
3* =========== DOCUMENTATION ===========
4*
5* Online html documentation available at
6* http://www.netlib.org/lapack/explore-html/
7*
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15*> [TXT]</a>
16*> \endhtmlonly
17*
18* Definition:
19* ===========
20*
21* SUBROUTINE CGESVJ( JOBA, JOBU, JOBV, M, N, A, LDA, SVA, MV, V,
22* LDV, CWORK, LWORK, RWORK, LRWORK, INFO )
23*
24* .. Scalar Arguments ..
25* INTEGER INFO, LDA, LDV, LWORK, LRWORK, M, MV, N
26* CHARACTER*1 JOBA, JOBU, JOBV
27* ..
28* .. Array Arguments ..
29* COMPLEX A( LDA, * ), V( LDV, * ), CWORK( LWORK )
30* REAL RWORK( LRWORK ), SVA( N )
31* ..
32*
33*
34*> \par Purpose:
35* =============
36*>
37*> \verbatim
38*>
39*> CGESVJ computes the singular value decomposition (SVD) of a complex
40*> M-by-N matrix A, where M >= N. The SVD of A is written as
41*> [++] [xx] [x0] [xx]
42*> A = U * SIGMA * V^*, [++] = [xx] * [ox] * [xx]
43*> [++] [xx]
44*> where SIGMA is an N-by-N diagonal matrix, U is an M-by-N orthonormal
45*> matrix, and V is an N-by-N unitary matrix. The diagonal elements
46*> of SIGMA are the singular values of A. The columns of U and V are the
47*> left and the right singular vectors of A, respectively.
48*> \endverbatim
49*
50* Arguments:
51* ==========
52*
53*> \param[in] JOBA
54*> \verbatim
55*> JOBA is CHARACTER*1
56*> Specifies the structure of A.
57*> = 'L': The input matrix A is lower triangular;
58*> = 'U': The input matrix A is upper triangular;
59*> = 'G': The input matrix A is general M-by-N matrix, M >= N.
60*> \endverbatim
61*>
62*> \param[in] JOBU
63*> \verbatim
64*> JOBU is CHARACTER*1
65*> Specifies whether to compute the left singular vectors
66*> (columns of U):
67*> = 'U' or 'F': The left singular vectors corresponding to the nonzero
68*> singular values are computed and returned in the leading
69*> columns of A. See more details in the description of A.
70*> The default numerical orthogonality threshold is set to
71*> approximately TOL=CTOL*EPS, CTOL=SQRT(M), EPS=SLAMCH('E').
72*> = 'C': Analogous to JOBU='U', except that user can control the
73*> level of numerical orthogonality of the computed left
74*> singular vectors. TOL can be set to TOL = CTOL*EPS, where
75*> CTOL is given on input in the array WORK.
76*> No CTOL smaller than ONE is allowed. CTOL greater
77*> than 1 / EPS is meaningless. The option 'C'
78*> can be used if M*EPS is satisfactory orthogonality
79*> of the computed left singular vectors, so CTOL=M could
80*> save few sweeps of Jacobi rotations.
81*> See the descriptions of A and WORK(1).
82*> = 'N': The matrix U is not computed. However, see the
83*> description of A.
84*> \endverbatim
85*>
86*> \param[in] JOBV
87*> \verbatim
88*> JOBV is CHARACTER*1
89*> Specifies whether to compute the right singular vectors, that
90*> is, the matrix V:
91*> = 'V' or 'J': the matrix V is computed and returned in the array V
92*> = 'A': the Jacobi rotations are applied to the MV-by-N
93*> array V. In other words, the right singular vector
94*> matrix V is not computed explicitly; instead it is
95*> applied to an MV-by-N matrix initially stored in the
96*> first MV rows of V.
97*> = 'N': the matrix V is not computed and the array V is not
98*> referenced
99*> \endverbatim
100*>
101*> \param[in] M
102*> \verbatim
103*> M is INTEGER
104*> The number of rows of the input matrix A. 1/SLAMCH('E') > M >= 0.
105*> \endverbatim
106*>
107*> \param[in] N
108*> \verbatim
109*> N is INTEGER
110*> The number of columns of the input matrix A.
111*> M >= N >= 0.
112*> \endverbatim
113*>
114*> \param[in,out] A
115*> \verbatim
116*> A is COMPLEX array, dimension (LDA,N)
117*> On entry, the M-by-N matrix A.
118*> On exit,
119*> If JOBU = 'U' .OR. JOBU = 'C':
120*> If INFO = 0 :
121*> RANKA orthonormal columns of U are returned in the
122*> leading RANKA columns of the array A. Here RANKA <= N
123*> is the number of computed singular values of A that are
124*> above the underflow threshold SLAMCH('S'). The singular
125*> vectors corresponding to underflowed or zero singular
126*> values are not computed. The value of RANKA is returned
127*> in the array RWORK as RANKA=NINT(RWORK(2)). Also see the
128*> descriptions of SVA and RWORK. The computed columns of U
129*> are mutually numerically orthogonal up to approximately
130*> TOL=SQRT(M)*EPS (default); or TOL=CTOL*EPS (JOBU = 'C'),
131*> see the description of JOBU.
132*> If INFO > 0,
133*> the procedure CGESVJ did not converge in the given number
134*> of iterations (sweeps). In that case, the computed
135*> columns of U may not be orthogonal up to TOL. The output
136*> U (stored in A), SIGMA (given by the computed singular
137*> values in SVA(1:N)) and V is still a decomposition of the
138*> input matrix A in the sense that the residual
139*> || A - SCALE * U * SIGMA * V^* ||_2 / ||A||_2 is small.
140*> If JOBU = 'N':
141*> If INFO = 0 :
142*> Note that the left singular vectors are 'for free' in the
143*> one-sided Jacobi SVD algorithm. However, if only the
144*> singular values are needed, the level of numerical
145*> orthogonality of U is not an issue and iterations are
146*> stopped when the columns of the iterated matrix are
147*> numerically orthogonal up to approximately M*EPS. Thus,
148*> on exit, A contains the columns of U scaled with the
149*> corresponding singular values.
150*> If INFO > 0 :
151*> the procedure CGESVJ did not converge in the given number
152*> of iterations (sweeps).
153*> \endverbatim
154*>
155*> \param[in] LDA
156*> \verbatim
157*> LDA is INTEGER
158*> The leading dimension of the array A. LDA >= max(1,M).
159*> \endverbatim
160*>
161*> \param[out] SVA
162*> \verbatim
163*> SVA is REAL array, dimension (N)
164*> On exit,
165*> If INFO = 0 :
166*> depending on the value SCALE = RWORK(1), we have:
167*> If SCALE = ONE:
168*> SVA(1:N) contains the computed singular values of A.
169*> During the computation SVA contains the Euclidean column
170*> norms of the iterated matrices in the array A.
171*> If SCALE .NE. ONE:
172*> The singular values of A are SCALE*SVA(1:N), and this
173*> factored representation is due to the fact that some of the
174*> singular values of A might underflow or overflow.
175*>
176*> If INFO > 0 :
177*> the procedure CGESVJ did not converge in the given number of
178*> iterations (sweeps) and SCALE*SVA(1:N) may not be accurate.
179*> \endverbatim
180*>
181*> \param[in] MV
182*> \verbatim
183*> MV is INTEGER
184*> If JOBV = 'A', then the product of Jacobi rotations in CGESVJ
185*> is applied to the first MV rows of V. See the description of JOBV.
186*> \endverbatim
187*>
188*> \param[in,out] V
189*> \verbatim
190*> V is COMPLEX array, dimension (LDV,N)
191*> If JOBV = 'V', then V contains on exit the N-by-N matrix of
192*> the right singular vectors;
193*> If JOBV = 'A', then V contains the product of the computed right
194*> singular vector matrix and the initial matrix in
195*> the array V.
196*> If JOBV = 'N', then V is not referenced.
197*> \endverbatim
198*>
199*> \param[in] LDV
200*> \verbatim
201*> LDV is INTEGER
202*> The leading dimension of the array V, LDV >= 1.
203*> If JOBV = 'V', then LDV >= max(1,N).
204*> If JOBV = 'A', then LDV >= max(1,MV) .
205*> \endverbatim
206*>
207*> \param[in,out] CWORK
208*> \verbatim
209*> CWORK is COMPLEX array, dimension (max(1,LWORK))
210*> Used as workspace.
211*> If on entry LWORK = -1, then a workspace query is assumed and
212*> no computation is done; CWORK(1) is set to the minial (and optimal)
213*> length of CWORK.
214*> \endverbatim
215*>
216*> \param[in] LWORK
217*> \verbatim
218*> LWORK is INTEGER.
219*> Length of CWORK, LWORK >= M+N.
220*> \endverbatim
221*>
222*> \param[in,out] RWORK
223*> \verbatim
224*> RWORK is REAL array, dimension (max(6,LRWORK))
225*> On entry,
226*> If JOBU = 'C' :
227*> RWORK(1) = CTOL, where CTOL defines the threshold for convergence.
228*> The process stops if all columns of A are mutually
229*> orthogonal up to CTOL*EPS, EPS=SLAMCH('E').
230*> It is required that CTOL >= ONE, i.e. it is not
231*> allowed to force the routine to obtain orthogonality
232*> below EPSILON.
233*> On exit,
234*> RWORK(1) = SCALE is the scaling factor such that SCALE*SVA(1:N)
235*> are the computed singular values of A.
236*> (See description of SVA().)
237*> RWORK(2) = NINT(RWORK(2)) is the number of the computed nonzero
238*> singular values.
239*> RWORK(3) = NINT(RWORK(3)) is the number of the computed singular
240*> values that are larger than the underflow threshold.
241*> RWORK(4) = NINT(RWORK(4)) is the number of sweeps of Jacobi
242*> rotations needed for numerical convergence.
243*> RWORK(5) = max_{i.NE.j} |COS(A(:,i),A(:,j))| in the last sweep.
244*> This is useful information in cases when CGESVJ did
245*> not converge, as it can be used to estimate whether
246*> the output is still useful and for post festum analysis.
247*> RWORK(6) = the largest absolute value over all sines of the
248*> Jacobi rotation angles in the last sweep. It can be
249*> useful for a post festum analysis.
250*> If on entry LRWORK = -1, then a workspace query is assumed and
251*> no computation is done; RWORK(1) is set to the minial (and optimal)
252*> length of RWORK.
253*> \endverbatim
254*>
255*> \param[in] LRWORK
256*> \verbatim
257*> LRWORK is INTEGER
258*> Length of RWORK, LRWORK >= MAX(6,N).
259*> \endverbatim
260*>
261*> \param[out] INFO
262*> \verbatim
263*> INFO is INTEGER
264*> = 0: successful exit.
265*> < 0: if INFO = -i, then the i-th argument had an illegal value
266*> > 0: CGESVJ did not converge in the maximal allowed number
267*> (NSWEEP=30) of sweeps. The output may still be useful.
268*> See the description of RWORK.
269*> \endverbatim
270*>
271* Authors:
272* ========
273*
274*> \author Univ. of Tennessee
275*> \author Univ. of California Berkeley
276*> \author Univ. of Colorado Denver
277*> \author NAG Ltd.
278*
279*> \ingroup gesvj
280*
281*> \par Further Details:
282* =====================
283*>
284*> \verbatim
285*>
286*> The orthogonal N-by-N matrix V is obtained as a product of Jacobi plane
287*> rotations. In the case of underflow of the tangent of the Jacobi angle, a
288*> modified Jacobi transformation of Drmac [3] is used. Pivot strategy uses
289*> column interchanges of de Rijk [1]. The relative accuracy of the computed
290*> singular values and the accuracy of the computed singular vectors (in
291*> angle metric) is as guaranteed by the theory of Demmel and Veselic [2].
292*> The condition number that determines the accuracy in the full rank case
293*> is essentially min_{D=diag} kappa(A*D), where kappa(.) is the
294*> spectral condition number. The best performance of this Jacobi SVD
295*> procedure is achieved if used in an accelerated version of Drmac and
296*> Veselic [4,5], and it is the kernel routine in the SIGMA library [6].
297*> Some tuning parameters (marked with [TP]) are available for the
298*> implementer.
299*> The computational range for the nonzero singular values is the machine
300*> number interval ( UNDERFLOW , OVERFLOW ). In extreme cases, even
301*> denormalized singular values can be computed with the corresponding
302*> gradual loss of accurate digits.
303*> \endverbatim
304*
305*> \par Contributor:
306* ==================
307*>
308*> \verbatim
309*>
310*> ============
311*>
312*> Zlatko Drmac (Zagreb, Croatia)
313*>
314*> \endverbatim
315*
316*> \par References:
317* ================
318*>
319*> \verbatim
320*>
321*> [1] P. P. M. De Rijk: A one-sided Jacobi algorithm for computing the
322*> singular value decomposition on a vector computer.
323*> SIAM J. Sci. Stat. Comp., Vol. 10 (1998), pp. 359-371.
324*> [2] J. Demmel and K. Veselic: Jacobi method is more accurate than QR.
325*> [3] Z. Drmac: Implementation of Jacobi rotations for accurate singular
326*> value computation in floating point arithmetic.
327*> SIAM J. Sci. Comp., Vol. 18 (1997), pp. 1200-1222.
328*> [4] Z. Drmac and K. Veselic: New fast and accurate Jacobi SVD algorithm I.
329*> SIAM J. Matrix Anal. Appl. Vol. 35, No. 2 (2008), pp. 1322-1342.
330*> LAPACK Working note 169.
331*> [5] Z. Drmac and K. Veselic: New fast and accurate Jacobi SVD algorithm II.
332*> SIAM J. Matrix Anal. Appl. Vol. 35, No. 2 (2008), pp. 1343-1362.
333*> LAPACK Working note 170.
334*> [6] Z. Drmac: SIGMA - mathematical software library for accurate SVD, PSV,
335*> QSVD, (H,K)-SVD computations.
336*> Department of Mathematics, University of Zagreb, 2008, 2015.
337*> \endverbatim
338*
339*> \par Bugs, examples and comments:
340* =================================
341*>
342*> \verbatim
343*> ===========================
344*> Please report all bugs and send interesting test examples and comments to
345*> drmac@math.hr. Thank you.
346*> \endverbatim
347*>
348* =====================================================================
349 SUBROUTINE cgesvj( JOBA, JOBU, JOBV, M, N, A, LDA, SVA, MV, V,
350 \$ LDV, CWORK, LWORK, RWORK, LRWORK, INFO )
351*
352* -- LAPACK computational routine --
353* -- LAPACK is a software package provided by Univ. of Tennessee, --
354* -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
355*
356 IMPLICIT NONE
357* .. Scalar Arguments ..
358 INTEGER INFO, LDA, LDV, LWORK, LRWORK, M, MV, N
359 CHARACTER*1 JOBA, JOBU, JOBV
360* ..
361* .. Array Arguments ..
362 COMPLEX A( LDA, * ), V( LDV, * ), CWORK( LWORK )
363 REAL RWORK( LRWORK ), SVA( N )
364* ..
365*
366* =====================================================================
367*
368* .. Local Parameters ..
369 REAL ZERO, HALF, ONE
370 parameter( zero = 0.0e0, half = 0.5e0, one = 1.0e0)
371 COMPLEX CZERO, CONE
372 parameter( czero = (0.0e0, 0.0e0), cone = (1.0e0, 0.0e0) )
373 INTEGER NSWEEP
374 parameter( nsweep = 30 )
375* ..
376* .. Local Scalars ..
377 COMPLEX AAPQ, OMPQ
378 REAL AAPP, AAPP0, AAPQ1, AAQQ, APOAQ, AQOAP, BIG,
379 \$ bigtheta, cs, ctol, epsln, mxaapq,
380 \$ mxsinj, rootbig, rooteps, rootsfmin, roottol,
381 \$ skl, sfmin, small, sn, t, temp1, theta, thsign, tol
382 INTEGER BLSKIP, EMPTSW, i, ibr, IERR, igl, IJBLSK, ir1,
383 \$ iswrot, jbc, jgl, kbl, lkahead, mvl, n2, n34,
384 \$ n4, nbl, notrot, p, pskipped, q, rowskip, swband
385 LOGICAL APPLV, GOSCALE, LOWER, LQUERY, LSVEC, NOSCALE, ROTOK,
386 \$ rsvec, uctol, upper
387* ..
388* ..
389* .. Intrinsic Functions ..
390 INTRINSIC abs, max, min, conjg, real, sign, sqrt
391* ..
392* .. External Functions ..
393* ..
394* from BLAS
395 REAL SCNRM2
396 COMPLEX CDOTC
397 EXTERNAL cdotc, scnrm2
398 INTEGER ISAMAX
399 EXTERNAL isamax
400* from LAPACK
401 REAL SLAMCH
402 EXTERNAL slamch
403 LOGICAL LSAME
404 EXTERNAL lsame
405* ..
406* .. External Subroutines ..
407* ..
408* from BLAS
409 EXTERNAL ccopy, crot, csscal, cswap, caxpy
410* from LAPACK
411 EXTERNAL clascl, claset, classq, slascl, xerbla
412 EXTERNAL cgsvj0, cgsvj1
413* ..
414* .. Executable Statements ..
415*
416* Test the input arguments
417*
418 lsvec = lsame( jobu, 'U' ) .OR. lsame( jobu, 'F' )
419 uctol = lsame( jobu, 'C' )
420 rsvec = lsame( jobv, 'V' ) .OR. lsame( jobv, 'J' )
421 applv = lsame( jobv, 'A' )
422 upper = lsame( joba, 'U' )
423 lower = lsame( joba, 'L' )
424*
425 lquery = ( lwork .EQ. -1 ) .OR. ( lrwork .EQ. -1 )
426 IF( .NOT.( upper .OR. lower .OR. lsame( joba, 'G' ) ) ) THEN
427 info = -1
428 ELSE IF( .NOT.( lsvec .OR. uctol .OR. lsame( jobu, 'N' ) ) ) THEN
429 info = -2
430 ELSE IF( .NOT.( rsvec .OR. applv .OR. lsame( jobv, 'N' ) ) ) THEN
431 info = -3
432 ELSE IF( m.LT.0 ) THEN
433 info = -4
434 ELSE IF( ( n.LT.0 ) .OR. ( n.GT.m ) ) THEN
435 info = -5
436 ELSE IF( lda.LT.m ) THEN
437 info = -7
438 ELSE IF( mv.LT.0 ) THEN
439 info = -9
440 ELSE IF( ( rsvec .AND. ( ldv.LT.n ) ) .OR.
441 \$ ( applv .AND. ( ldv.LT.mv ) ) ) THEN
442 info = -11
443 ELSE IF( uctol .AND. ( rwork( 1 ).LE.one ) ) THEN
444 info = -12
445 ELSE IF( lwork.LT.( m+n ) .AND. ( .NOT.lquery ) ) THEN
446 info = -13
447 ELSE IF( lrwork.LT.max( n, 6 ) .AND. ( .NOT.lquery ) ) THEN
448 info = -15
449 ELSE
450 info = 0
451 END IF
452*
453* #:(
454 IF( info.NE.0 ) THEN
455 CALL xerbla( 'CGESVJ', -info )
456 RETURN
457 ELSE IF ( lquery ) THEN
458 cwork(1) = m + n
459 rwork(1) = max( n, 6 )
460 RETURN
461 END IF
462*
463* #:) Quick return for void matrix
464*
465 IF( ( m.EQ.0 ) .OR. ( n.EQ.0 ) )RETURN
466*
467* Set numerical parameters
468* The stopping criterion for Jacobi rotations is
469*
470* max_{i<>j}|A(:,i)^* * A(:,j)| / (||A(:,i)||*||A(:,j)||) < CTOL*EPS
471*
472* where EPS is the round-off and CTOL is defined as follows:
473*
474 IF( uctol ) THEN
475* ... user controlled
476 ctol = rwork( 1 )
477 ELSE
478* ... default
479 IF( lsvec .OR. rsvec .OR. applv ) THEN
480 ctol = sqrt( real( m ) )
481 ELSE
482 ctol = real( m )
483 END IF
484 END IF
485* ... and the machine dependent parameters are
486*[!] (Make sure that SLAMCH() works properly on the target machine.)
487*
488 epsln = slamch( 'Epsilon' )
489 rooteps = sqrt( epsln )
490 sfmin = slamch( 'SafeMinimum' )
491 rootsfmin = sqrt( sfmin )
492 small = sfmin / epsln
493* BIG = SLAMCH( 'Overflow' )
494 big = one / sfmin
495 rootbig = one / rootsfmin
496* LARGE = BIG / SQRT( REAL( M*N ) )
497 bigtheta = one / rooteps
498*
499 tol = ctol*epsln
500 roottol = sqrt( tol )
501*
502 IF( real( m )*epsln.GE.one ) THEN
503 info = -4
504 CALL xerbla( 'CGESVJ', -info )
505 RETURN
506 END IF
507*
508* Initialize the right singular vector matrix.
509*
510 IF( rsvec ) THEN
511 mvl = n
512 CALL claset( 'A', mvl, n, czero, cone, v, ldv )
513 ELSE IF( applv ) THEN
514 mvl = mv
515 END IF
516 rsvec = rsvec .OR. applv
517*
518* Initialize SVA( 1:N ) = ( ||A e_i||_2, i = 1:N )
519*(!) If necessary, scale A to protect the largest singular value
520* from overflow. It is possible that saving the largest singular
521* value destroys the information about the small ones.
522* This initial scaling is almost minimal in the sense that the
523* goal is to make sure that no column norm overflows, and that
524* SQRT(N)*max_i SVA(i) does not overflow. If INFinite entries
525* in A are detected, the procedure returns with INFO=-6.
526*
527 skl = one / sqrt( real( m )*real( n ) )
528 noscale = .true.
529 goscale = .true.
530*
531 IF( lower ) THEN
532* the input matrix is M-by-N lower triangular (trapezoidal)
533 DO 1874 p = 1, n
534 aapp = zero
535 aaqq = one
536 CALL classq( m-p+1, a( p, p ), 1, aapp, aaqq )
537 IF( aapp.GT.big ) THEN
538 info = -6
539 CALL xerbla( 'CGESVJ', -info )
540 RETURN
541 END IF
542 aaqq = sqrt( aaqq )
543 IF( ( aapp.LT.( big / aaqq ) ) .AND. noscale ) THEN
544 sva( p ) = aapp*aaqq
545 ELSE
546 noscale = .false.
547 sva( p ) = aapp*( aaqq*skl )
548 IF( goscale ) THEN
549 goscale = .false.
550 DO 1873 q = 1, p - 1
551 sva( q ) = sva( q )*skl
552 1873 CONTINUE
553 END IF
554 END IF
555 1874 CONTINUE
556 ELSE IF( upper ) THEN
557* the input matrix is M-by-N upper triangular (trapezoidal)
558 DO 2874 p = 1, n
559 aapp = zero
560 aaqq = one
561 CALL classq( p, a( 1, p ), 1, aapp, aaqq )
562 IF( aapp.GT.big ) THEN
563 info = -6
564 CALL xerbla( 'CGESVJ', -info )
565 RETURN
566 END IF
567 aaqq = sqrt( aaqq )
568 IF( ( aapp.LT.( big / aaqq ) ) .AND. noscale ) THEN
569 sva( p ) = aapp*aaqq
570 ELSE
571 noscale = .false.
572 sva( p ) = aapp*( aaqq*skl )
573 IF( goscale ) THEN
574 goscale = .false.
575 DO 2873 q = 1, p - 1
576 sva( q ) = sva( q )*skl
577 2873 CONTINUE
578 END IF
579 END IF
580 2874 CONTINUE
581 ELSE
582* the input matrix is M-by-N general dense
583 DO 3874 p = 1, n
584 aapp = zero
585 aaqq = one
586 CALL classq( m, a( 1, p ), 1, aapp, aaqq )
587 IF( aapp.GT.big ) THEN
588 info = -6
589 CALL xerbla( 'CGESVJ', -info )
590 RETURN
591 END IF
592 aaqq = sqrt( aaqq )
593 IF( ( aapp.LT.( big / aaqq ) ) .AND. noscale ) THEN
594 sva( p ) = aapp*aaqq
595 ELSE
596 noscale = .false.
597 sva( p ) = aapp*( aaqq*skl )
598 IF( goscale ) THEN
599 goscale = .false.
600 DO 3873 q = 1, p - 1
601 sva( q ) = sva( q )*skl
602 3873 CONTINUE
603 END IF
604 END IF
605 3874 CONTINUE
606 END IF
607*
608 IF( noscale )skl = one
609*
610* Move the smaller part of the spectrum from the underflow threshold
611*(!) Start by determining the position of the nonzero entries of the
612* array SVA() relative to ( SFMIN, BIG ).
613*
614 aapp = zero
615 aaqq = big
616 DO 4781 p = 1, n
617 IF( sva( p ).NE.zero )aaqq = min( aaqq, sva( p ) )
618 aapp = max( aapp, sva( p ) )
619 4781 CONTINUE
620*
621* #:) Quick return for zero matrix
622*
623 IF( aapp.EQ.zero ) THEN
624 IF( lsvec )CALL claset( 'G', m, n, czero, cone, a, lda )
625 rwork( 1 ) = one
626 rwork( 2 ) = zero
627 rwork( 3 ) = zero
628 rwork( 4 ) = zero
629 rwork( 5 ) = zero
630 rwork( 6 ) = zero
631 RETURN
632 END IF
633*
634* #:) Quick return for one-column matrix
635*
636 IF( n.EQ.1 ) THEN
637 IF( lsvec )CALL clascl( 'G', 0, 0, sva( 1 ), skl, m, 1,
638 \$ a( 1, 1 ), lda, ierr )
639 rwork( 1 ) = one / skl
640 IF( sva( 1 ).GE.sfmin ) THEN
641 rwork( 2 ) = one
642 ELSE
643 rwork( 2 ) = zero
644 END IF
645 rwork( 3 ) = zero
646 rwork( 4 ) = zero
647 rwork( 5 ) = zero
648 rwork( 6 ) = zero
649 RETURN
650 END IF
651*
652* Protect small singular values from underflow, and try to
653* avoid underflows/overflows in computing Jacobi rotations.
654*
655 sn = sqrt( sfmin / epsln )
656 temp1 = sqrt( big / real( n ) )
657 IF( ( aapp.LE.sn ) .OR. ( aaqq.GE.temp1 ) .OR.
658 \$ ( ( sn.LE.aaqq ) .AND. ( aapp.LE.temp1 ) ) ) THEN
659 temp1 = min( big, temp1 / aapp )
660* AAQQ = AAQQ*TEMP1
661* AAPP = AAPP*TEMP1
662 ELSE IF( ( aaqq.LE.sn ) .AND. ( aapp.LE.temp1 ) ) THEN
663 temp1 = min( sn / aaqq, big / ( aapp*sqrt( real( n ) ) ) )
664* AAQQ = AAQQ*TEMP1
665* AAPP = AAPP*TEMP1
666 ELSE IF( ( aaqq.GE.sn ) .AND. ( aapp.GE.temp1 ) ) THEN
667 temp1 = max( sn / aaqq, temp1 / aapp )
668* AAQQ = AAQQ*TEMP1
669* AAPP = AAPP*TEMP1
670 ELSE IF( ( aaqq.LE.sn ) .AND. ( aapp.GE.temp1 ) ) THEN
671 temp1 = min( sn / aaqq, big / ( sqrt( real( n ) )*aapp ) )
672* AAQQ = AAQQ*TEMP1
673* AAPP = AAPP*TEMP1
674 ELSE
675 temp1 = one
676 END IF
677*
678* Scale, if necessary
679*
680 IF( temp1.NE.one ) THEN
681 CALL slascl( 'G', 0, 0, one, temp1, n, 1, sva, n, ierr )
682 END IF
683 skl = temp1*skl
684 IF( skl.NE.one ) THEN
685 CALL clascl( joba, 0, 0, one, skl, m, n, a, lda, ierr )
686 skl = one / skl
687 END IF
688*
689* Row-cyclic Jacobi SVD algorithm with column pivoting
690*
691 emptsw = ( n*( n-1 ) ) / 2
692 notrot = 0
693
694 DO 1868 q = 1, n
695 cwork( q ) = cone
696 1868 CONTINUE
697*
698*
699*
700 swband = 3
701*[TP] SWBAND is a tuning parameter [TP]. It is meaningful and effective
702* if CGESVJ is used as a computational routine in the preconditioned
703* Jacobi SVD algorithm CGEJSV. For sweeps i=1:SWBAND the procedure
704* works on pivots inside a band-like region around the diagonal.
705* The boundaries are determined dynamically, based on the number of
706* pivots above a threshold.
707*
708 kbl = min( 8, n )
709*[TP] KBL is a tuning parameter that defines the tile size in the
710* tiling of the p-q loops of pivot pairs. In general, an optimal
711* value of KBL depends on the matrix dimensions and on the
712* parameters of the computer's memory.
713*
714 nbl = n / kbl
715 IF( ( nbl*kbl ).NE.n )nbl = nbl + 1
716*
717 blskip = kbl**2
718*[TP] BLKSKIP is a tuning parameter that depends on SWBAND and KBL.
719*
720 rowskip = min( 5, kbl )
721*[TP] ROWSKIP is a tuning parameter.
722*
724*[TP] LKAHEAD is a tuning parameter.
725*
726* Quasi block transformations, using the lower (upper) triangular
727* structure of the input matrix. The quasi-block-cycling usually
728* invokes cubic convergence. Big part of this cycle is done inside
729* canonical subspaces of dimensions less than M.
730*
731 IF( ( lower .OR. upper ) .AND. ( n.GT.max( 64, 4*kbl ) ) ) THEN
732*[TP] The number of partition levels and the actual partition are
733* tuning parameters.
734 n4 = n / 4
735 n2 = n / 2
736 n34 = 3*n4
737 IF( applv ) THEN
738 q = 0
739 ELSE
740 q = 1
741 END IF
742*
743 IF( lower ) THEN
744*
745* This works very well on lower triangular matrices, in particular
746* in the framework of the preconditioned Jacobi SVD (xGEJSV).
747* The idea is simple:
748* [+ 0 0 0] Note that Jacobi transformations of [0 0]
749* [+ + 0 0] [0 0]
750* [+ + x 0] actually work on [x 0] [x 0]
751* [+ + x x] [x x]. [x x]
752*
753 CALL cgsvj0( jobv, m-n34, n-n34, a( n34+1, n34+1 ), lda,
754 \$ cwork( n34+1 ), sva( n34+1 ), mvl,
755 \$ v( n34*q+1, n34+1 ), ldv, epsln, sfmin, tol,
756 \$ 2, cwork( n+1 ), lwork-n, ierr )
757
758 CALL cgsvj0( jobv, m-n2, n34-n2, a( n2+1, n2+1 ), lda,
759 \$ cwork( n2+1 ), sva( n2+1 ), mvl,
760 \$ v( n2*q+1, n2+1 ), ldv, epsln, sfmin, tol, 2,
761 \$ cwork( n+1 ), lwork-n, ierr )
762
763 CALL cgsvj1( jobv, m-n2, n-n2, n4, a( n2+1, n2+1 ), lda,
764 \$ cwork( n2+1 ), sva( n2+1 ), mvl,
765 \$ v( n2*q+1, n2+1 ), ldv, epsln, sfmin, tol, 1,
766 \$ cwork( n+1 ), lwork-n, ierr )
767*
768 CALL cgsvj0( jobv, m-n4, n2-n4, a( n4+1, n4+1 ), lda,
769 \$ cwork( n4+1 ), sva( n4+1 ), mvl,
770 \$ v( n4*q+1, n4+1 ), ldv, epsln, sfmin, tol, 1,
771 \$ cwork( n+1 ), lwork-n, ierr )
772*
773 CALL cgsvj0( jobv, m, n4, a, lda, cwork, sva, mvl, v, ldv,
774 \$ epsln, sfmin, tol, 1, cwork( n+1 ), lwork-n,
775 \$ ierr )
776*
777 CALL cgsvj1( jobv, m, n2, n4, a, lda, cwork, sva, mvl, v,
778 \$ ldv, epsln, sfmin, tol, 1, cwork( n+1 ),
779 \$ lwork-n, ierr )
780*
781*
782 ELSE IF( upper ) THEN
783*
784*
785 CALL cgsvj0( jobv, n4, n4, a, lda, cwork, sva, mvl, v, ldv,
786 \$ epsln, sfmin, tol, 2, cwork( n+1 ), lwork-n,
787 \$ ierr )
788*
789 CALL cgsvj0( jobv, n2, n4, a( 1, n4+1 ), lda, cwork( n4+1 ),
790 \$ sva( n4+1 ), mvl, v( n4*q+1, n4+1 ), ldv,
791 \$ epsln, sfmin, tol, 1, cwork( n+1 ), lwork-n,
792 \$ ierr )
793*
794 CALL cgsvj1( jobv, n2, n2, n4, a, lda, cwork, sva, mvl, v,
795 \$ ldv, epsln, sfmin, tol, 1, cwork( n+1 ),
796 \$ lwork-n, ierr )
797*
798 CALL cgsvj0( jobv, n2+n4, n4, a( 1, n2+1 ), lda,
799 \$ cwork( n2+1 ), sva( n2+1 ), mvl,
800 \$ v( n2*q+1, n2+1 ), ldv, epsln, sfmin, tol, 1,
801 \$ cwork( n+1 ), lwork-n, ierr )
802
803 END IF
804*
805 END IF
806*
807* .. Row-cyclic pivot strategy with de Rijk's pivoting ..
808*
809 DO 1993 i = 1, nsweep
810*
811* .. go go go ...
812*
813 mxaapq = zero
814 mxsinj = zero
815 iswrot = 0
816*
817 notrot = 0
818 pskipped = 0
819*
820* Each sweep is unrolled using KBL-by-KBL tiles over the pivot pairs
821* 1 <= p < q <= N. This is the first step toward a blocked implementation
822* of the rotations. New implementation, based on block transformations,
823* is under development.
824*
825 DO 2000 ibr = 1, nbl
826*
827 igl = ( ibr-1 )*kbl + 1
828*
829 DO 1002 ir1 = 0, min( lkahead, nbl-ibr )
830*
831 igl = igl + ir1*kbl
832*
833 DO 2001 p = igl, min( igl+kbl-1, n-1 )
834*
835* .. de Rijk's pivoting
836*
837 q = isamax( n-p+1, sva( p ), 1 ) + p - 1
838 IF( p.NE.q ) THEN
839 CALL cswap( m, a( 1, p ), 1, a( 1, q ), 1 )
840 IF( rsvec )CALL cswap( mvl, v( 1, p ), 1,
841 \$ v( 1, q ), 1 )
842 temp1 = sva( p )
843 sva( p ) = sva( q )
844 sva( q ) = temp1
845 aapq = cwork(p)
846 cwork(p) = cwork(q)
847 cwork(q) = aapq
848 END IF
849*
850 IF( ir1.EQ.0 ) THEN
851*
852* Column norms are periodically updated by explicit
853* norm computation.
854*[!] Caveat:
855* Unfortunately, some BLAS implementations compute SCNRM2(M,A(1,p),1)
856* as SQRT(S=CDOTC(M,A(1,p),1,A(1,p),1)), which may cause the result to
857* overflow for ||A(:,p)||_2 > SQRT(overflow_threshold), and to
858* underflow for ||A(:,p)||_2 < SQRT(underflow_threshold).
859* Hence, SCNRM2 cannot be trusted, not even in the case when
860* the true norm is far from the under(over)flow boundaries.
861* If properly implemented SCNRM2 is available, the IF-THEN-ELSE-END IF
862* below should be replaced with "AAPP = SCNRM2( M, A(1,p), 1 )".
863*
864 IF( ( sva( p ).LT.rootbig ) .AND.
865 \$ ( sva( p ).GT.rootsfmin ) ) THEN
866 sva( p ) = scnrm2( m, a( 1, p ), 1 )
867 ELSE
868 temp1 = zero
869 aapp = one
870 CALL classq( m, a( 1, p ), 1, temp1, aapp )
871 sva( p ) = temp1*sqrt( aapp )
872 END IF
873 aapp = sva( p )
874 ELSE
875 aapp = sva( p )
876 END IF
877*
878 IF( aapp.GT.zero ) THEN
879*
880 pskipped = 0
881*
882 DO 2002 q = p + 1, min( igl+kbl-1, n )
883*
884 aaqq = sva( q )
885*
886 IF( aaqq.GT.zero ) THEN
887*
888 aapp0 = aapp
889 IF( aaqq.GE.one ) THEN
890 rotok = ( small*aapp ).LE.aaqq
891 IF( aapp.LT.( big / aaqq ) ) THEN
892 aapq = ( cdotc( m, a( 1, p ), 1,
893 \$ a( 1, q ), 1 ) / aaqq ) / aapp
894 ELSE
895 CALL ccopy( m, a( 1, p ), 1,
896 \$ cwork(n+1), 1 )
897 CALL clascl( 'G', 0, 0, aapp, one,
898 \$ m, 1, cwork(n+1), lda, ierr )
899 aapq = cdotc( m, cwork(n+1), 1,
900 \$ a( 1, q ), 1 ) / aaqq
901 END IF
902 ELSE
903 rotok = aapp.LE.( aaqq / small )
904 IF( aapp.GT.( small / aaqq ) ) THEN
905 aapq = ( cdotc( m, a( 1, p ), 1,
906 \$ a( 1, q ), 1 ) / aapp ) / aaqq
907 ELSE
908 CALL ccopy( m, a( 1, q ), 1,
909 \$ cwork(n+1), 1 )
910 CALL clascl( 'G', 0, 0, aaqq,
911 \$ one, m, 1,
912 \$ cwork(n+1), lda, ierr )
913 aapq = cdotc( m, a(1, p ), 1,
914 \$ cwork(n+1), 1 ) / aapp
915 END IF
916 END IF
917*
918* AAPQ = AAPQ * CONJG( CWORK(p) ) * CWORK(q)
919 aapq1 = -abs(aapq)
920 mxaapq = max( mxaapq, -aapq1 )
921*
922* TO rotate or NOT to rotate, THAT is the question ...
923*
924 IF( abs( aapq1 ).GT.tol ) THEN
925 ompq = aapq / abs(aapq)
926*
927* .. rotate
928*[RTD] ROTATED = ROTATED + ONE
929*
930 IF( ir1.EQ.0 ) THEN
931 notrot = 0
932 pskipped = 0
933 iswrot = iswrot + 1
934 END IF
935*
936 IF( rotok ) THEN
937*
938 aqoap = aaqq / aapp
939 apoaq = aapp / aaqq
940 theta = -half*abs( aqoap-apoaq )/aapq1
941*
942 IF( abs( theta ).GT.bigtheta ) THEN
943*
944 t = half / theta
945 cs = one
946
947 CALL crot( m, a(1,p), 1, a(1,q), 1,
948 \$ cs, conjg(ompq)*t )
949 IF ( rsvec ) THEN
950 CALL crot( mvl, v(1,p), 1,
951 \$ v(1,q), 1, cs, conjg(ompq)*t )
952 END IF
953
954 sva( q ) = aaqq*sqrt( max( zero,
955 \$ one+t*apoaq*aapq1 ) )
956 aapp = aapp*sqrt( max( zero,
957 \$ one-t*aqoap*aapq1 ) )
958 mxsinj = max( mxsinj, abs( t ) )
959*
960 ELSE
961*
962* .. choose correct signum for THETA and rotate
963*
964 thsign = -sign( one, aapq1 )
965 t = one / ( theta+thsign*
966 \$ sqrt( one+theta*theta ) )
967 cs = sqrt( one / ( one+t*t ) )
968 sn = t*cs
969*
970 mxsinj = max( mxsinj, abs( sn ) )
971 sva( q ) = aaqq*sqrt( max( zero,
972 \$ one+t*apoaq*aapq1 ) )
973 aapp = aapp*sqrt( max( zero,
974 \$ one-t*aqoap*aapq1 ) )
975*
976 CALL crot( m, a(1,p), 1, a(1,q), 1,
977 \$ cs, conjg(ompq)*sn )
978 IF ( rsvec ) THEN
979 CALL crot( mvl, v(1,p), 1,
980 \$ v(1,q), 1, cs, conjg(ompq)*sn )
981 END IF
982 END IF
983 cwork(p) = -cwork(q) * ompq
984*
985 ELSE
986* .. have to use modified Gram-Schmidt like transformation
987 CALL ccopy( m, a( 1, p ), 1,
988 \$ cwork(n+1), 1 )
989 CALL clascl( 'G', 0, 0, aapp, one, m,
990 \$ 1, cwork(n+1), lda,
991 \$ ierr )
992 CALL clascl( 'G', 0, 0, aaqq, one, m,
993 \$ 1, a( 1, q ), lda, ierr )
994 CALL caxpy( m, -aapq, cwork(n+1), 1,
995 \$ a( 1, q ), 1 )
996 CALL clascl( 'G', 0, 0, one, aaqq, m,
997 \$ 1, a( 1, q ), lda, ierr )
998 sva( q ) = aaqq*sqrt( max( zero,
999 \$ one-aapq1*aapq1 ) )
1000 mxsinj = max( mxsinj, sfmin )
1001 END IF
1002* END IF ROTOK THEN ... ELSE
1003*
1004* In the case of cancellation in updating SVA(q), SVA(p)
1005* recompute SVA(q), SVA(p).
1006*
1007 IF( ( sva( q ) / aaqq )**2.LE.rooteps )
1008 \$ THEN
1009 IF( ( aaqq.LT.rootbig ) .AND.
1010 \$ ( aaqq.GT.rootsfmin ) ) THEN
1011 sva( q ) = scnrm2( m, a( 1, q ), 1 )
1012 ELSE
1013 t = zero
1014 aaqq = one
1015 CALL classq( m, a( 1, q ), 1, t,
1016 \$ aaqq )
1017 sva( q ) = t*sqrt( aaqq )
1018 END IF
1019 END IF
1020 IF( ( aapp / aapp0 ).LE.rooteps ) THEN
1021 IF( ( aapp.LT.rootbig ) .AND.
1022 \$ ( aapp.GT.rootsfmin ) ) THEN
1023 aapp = scnrm2( m, a( 1, p ), 1 )
1024 ELSE
1025 t = zero
1026 aapp = one
1027 CALL classq( m, a( 1, p ), 1, t,
1028 \$ aapp )
1029 aapp = t*sqrt( aapp )
1030 END IF
1031 sva( p ) = aapp
1032 END IF
1033*
1034 ELSE
1035* A(:,p) and A(:,q) already numerically orthogonal
1036 IF( ir1.EQ.0 )notrot = notrot + 1
1037*[RTD] SKIPPED = SKIPPED + 1
1038 pskipped = pskipped + 1
1039 END IF
1040 ELSE
1041* A(:,q) is zero column
1042 IF( ir1.EQ.0 )notrot = notrot + 1
1043 pskipped = pskipped + 1
1044 END IF
1045*
1046 IF( ( i.LE.swband ) .AND.
1047 \$ ( pskipped.GT.rowskip ) ) THEN
1048 IF( ir1.EQ.0 )aapp = -aapp
1049 notrot = 0
1050 GO TO 2103
1051 END IF
1052*
1053 2002 CONTINUE
1054* END q-LOOP
1055*
1056 2103 CONTINUE
1057* bailed out of q-loop
1058*
1059 sva( p ) = aapp
1060*
1061 ELSE
1062 sva( p ) = aapp
1063 IF( ( ir1.EQ.0 ) .AND. ( aapp.EQ.zero ) )
1064 \$ notrot = notrot + min( igl+kbl-1, n ) - p
1065 END IF
1066*
1067 2001 CONTINUE
1068* end of the p-loop
1069* end of doing the block ( ibr, ibr )
1070 1002 CONTINUE
1071* end of ir1-loop
1072*
1073* ... go to the off diagonal blocks
1074*
1075 igl = ( ibr-1 )*kbl + 1
1076*
1077 DO 2010 jbc = ibr + 1, nbl
1078*
1079 jgl = ( jbc-1 )*kbl + 1
1080*
1081* doing the block at ( ibr, jbc )
1082*
1083 ijblsk = 0
1084 DO 2100 p = igl, min( igl+kbl-1, n )
1085*
1086 aapp = sva( p )
1087 IF( aapp.GT.zero ) THEN
1088*
1089 pskipped = 0
1090*
1091 DO 2200 q = jgl, min( jgl+kbl-1, n )
1092*
1093 aaqq = sva( q )
1094 IF( aaqq.GT.zero ) THEN
1095 aapp0 = aapp
1096*
1097* .. M x 2 Jacobi SVD ..
1098*
1099* Safe Gram matrix computation
1100*
1101 IF( aaqq.GE.one ) THEN
1102 IF( aapp.GE.aaqq ) THEN
1103 rotok = ( small*aapp ).LE.aaqq
1104 ELSE
1105 rotok = ( small*aaqq ).LE.aapp
1106 END IF
1107 IF( aapp.LT.( big / aaqq ) ) THEN
1108 aapq = ( cdotc( m, a( 1, p ), 1,
1109 \$ a( 1, q ), 1 ) / aaqq ) / aapp
1110 ELSE
1111 CALL ccopy( m, a( 1, p ), 1,
1112 \$ cwork(n+1), 1 )
1113 CALL clascl( 'G', 0, 0, aapp,
1114 \$ one, m, 1,
1115 \$ cwork(n+1), lda, ierr )
1116 aapq = cdotc( m, cwork(n+1), 1,
1117 \$ a( 1, q ), 1 ) / aaqq
1118 END IF
1119 ELSE
1120 IF( aapp.GE.aaqq ) THEN
1121 rotok = aapp.LE.( aaqq / small )
1122 ELSE
1123 rotok = aaqq.LE.( aapp / small )
1124 END IF
1125 IF( aapp.GT.( small / aaqq ) ) THEN
1126 aapq = ( cdotc( m, a( 1, p ), 1,
1127 \$ a( 1, q ), 1 ) / max(aaqq,aapp) )
1128 \$ / min(aaqq,aapp)
1129 ELSE
1130 CALL ccopy( m, a( 1, q ), 1,
1131 \$ cwork(n+1), 1 )
1132 CALL clascl( 'G', 0, 0, aaqq,
1133 \$ one, m, 1,
1134 \$ cwork(n+1), lda, ierr )
1135 aapq = cdotc( m, a( 1, p ), 1,
1136 \$ cwork(n+1), 1 ) / aapp
1137 END IF
1138 END IF
1139*
1140* AAPQ = AAPQ * CONJG(CWORK(p))*CWORK(q)
1141 aapq1 = -abs(aapq)
1142 mxaapq = max( mxaapq, -aapq1 )
1143*
1144* TO rotate or NOT to rotate, THAT is the question ...
1145*
1146 IF( abs( aapq1 ).GT.tol ) THEN
1147 ompq = aapq / abs(aapq)
1148 notrot = 0
1149*[RTD] ROTATED = ROTATED + 1
1150 pskipped = 0
1151 iswrot = iswrot + 1
1152*
1153 IF( rotok ) THEN
1154*
1155 aqoap = aaqq / aapp
1156 apoaq = aapp / aaqq
1157 theta = -half*abs( aqoap-apoaq )/ aapq1
1158 IF( aaqq.GT.aapp0 )theta = -theta
1159*
1160 IF( abs( theta ).GT.bigtheta ) THEN
1161 t = half / theta
1162 cs = one
1163 CALL crot( m, a(1,p), 1, a(1,q), 1,
1164 \$ cs, conjg(ompq)*t )
1165 IF( rsvec ) THEN
1166 CALL crot( mvl, v(1,p), 1,
1167 \$ v(1,q), 1, cs, conjg(ompq)*t )
1168 END IF
1169 sva( q ) = aaqq*sqrt( max( zero,
1170 \$ one+t*apoaq*aapq1 ) )
1171 aapp = aapp*sqrt( max( zero,
1172 \$ one-t*aqoap*aapq1 ) )
1173 mxsinj = max( mxsinj, abs( t ) )
1174 ELSE
1175*
1176* .. choose correct signum for THETA and rotate
1177*
1178 thsign = -sign( one, aapq1 )
1179 IF( aaqq.GT.aapp0 )thsign = -thsign
1180 t = one / ( theta+thsign*
1181 \$ sqrt( one+theta*theta ) )
1182 cs = sqrt( one / ( one+t*t ) )
1183 sn = t*cs
1184 mxsinj = max( mxsinj, abs( sn ) )
1185 sva( q ) = aaqq*sqrt( max( zero,
1186 \$ one+t*apoaq*aapq1 ) )
1187 aapp = aapp*sqrt( max( zero,
1188 \$ one-t*aqoap*aapq1 ) )
1189*
1190 CALL crot( m, a(1,p), 1, a(1,q), 1,
1191 \$ cs, conjg(ompq)*sn )
1192 IF( rsvec ) THEN
1193 CALL crot( mvl, v(1,p), 1,
1194 \$ v(1,q), 1, cs, conjg(ompq)*sn )
1195 END IF
1196 END IF
1197 cwork(p) = -cwork(q) * ompq
1198*
1199 ELSE
1200* .. have to use modified Gram-Schmidt like transformation
1201 IF( aapp.GT.aaqq ) THEN
1202 CALL ccopy( m, a( 1, p ), 1,
1203 \$ cwork(n+1), 1 )
1204 CALL clascl( 'G', 0, 0, aapp, one,
1205 \$ m, 1, cwork(n+1),lda,
1206 \$ ierr )
1207 CALL clascl( 'G', 0, 0, aaqq, one,
1208 \$ m, 1, a( 1, q ), lda,
1209 \$ ierr )
1210 CALL caxpy( m, -aapq, cwork(n+1),
1211 \$ 1, a( 1, q ), 1 )
1212 CALL clascl( 'G', 0, 0, one, aaqq,
1213 \$ m, 1, a( 1, q ), lda,
1214 \$ ierr )
1215 sva( q ) = aaqq*sqrt( max( zero,
1216 \$ one-aapq1*aapq1 ) )
1217 mxsinj = max( mxsinj, sfmin )
1218 ELSE
1219 CALL ccopy( m, a( 1, q ), 1,
1220 \$ cwork(n+1), 1 )
1221 CALL clascl( 'G', 0, 0, aaqq, one,
1222 \$ m, 1, cwork(n+1),lda,
1223 \$ ierr )
1224 CALL clascl( 'G', 0, 0, aapp, one,
1225 \$ m, 1, a( 1, p ), lda,
1226 \$ ierr )
1227 CALL caxpy( m, -conjg(aapq),
1228 \$ cwork(n+1), 1, a( 1, p ), 1 )
1229 CALL clascl( 'G', 0, 0, one, aapp,
1230 \$ m, 1, a( 1, p ), lda,
1231 \$ ierr )
1232 sva( p ) = aapp*sqrt( max( zero,
1233 \$ one-aapq1*aapq1 ) )
1234 mxsinj = max( mxsinj, sfmin )
1235 END IF
1236 END IF
1237* END IF ROTOK THEN ... ELSE
1238*
1239* In the case of cancellation in updating SVA(q), SVA(p)
1240* .. recompute SVA(q), SVA(p)
1241 IF( ( sva( q ) / aaqq )**2.LE.rooteps )
1242 \$ THEN
1243 IF( ( aaqq.LT.rootbig ) .AND.
1244 \$ ( aaqq.GT.rootsfmin ) ) THEN
1245 sva( q ) = scnrm2( m, a( 1, q ), 1)
1246 ELSE
1247 t = zero
1248 aaqq = one
1249 CALL classq( m, a( 1, q ), 1, t,
1250 \$ aaqq )
1251 sva( q ) = t*sqrt( aaqq )
1252 END IF
1253 END IF
1254 IF( ( aapp / aapp0 )**2.LE.rooteps ) THEN
1255 IF( ( aapp.LT.rootbig ) .AND.
1256 \$ ( aapp.GT.rootsfmin ) ) THEN
1257 aapp = scnrm2( m, a( 1, p ), 1 )
1258 ELSE
1259 t = zero
1260 aapp = one
1261 CALL classq( m, a( 1, p ), 1, t,
1262 \$ aapp )
1263 aapp = t*sqrt( aapp )
1264 END IF
1265 sva( p ) = aapp
1266 END IF
1267* end of OK rotation
1268 ELSE
1269 notrot = notrot + 1
1270*[RTD] SKIPPED = SKIPPED + 1
1271 pskipped = pskipped + 1
1272 ijblsk = ijblsk + 1
1273 END IF
1274 ELSE
1275 notrot = notrot + 1
1276 pskipped = pskipped + 1
1277 ijblsk = ijblsk + 1
1278 END IF
1279*
1280 IF( ( i.LE.swband ) .AND. ( ijblsk.GE.blskip ) )
1281 \$ THEN
1282 sva( p ) = aapp
1283 notrot = 0
1284 GO TO 2011
1285 END IF
1286 IF( ( i.LE.swband ) .AND.
1287 \$ ( pskipped.GT.rowskip ) ) THEN
1288 aapp = -aapp
1289 notrot = 0
1290 GO TO 2203
1291 END IF
1292*
1293 2200 CONTINUE
1294* end of the q-loop
1295 2203 CONTINUE
1296*
1297 sva( p ) = aapp
1298*
1299 ELSE
1300*
1301 IF( aapp.EQ.zero )notrot = notrot +
1302 \$ min( jgl+kbl-1, n ) - jgl + 1
1303 IF( aapp.LT.zero )notrot = 0
1304*
1305 END IF
1306*
1307 2100 CONTINUE
1308* end of the p-loop
1309 2010 CONTINUE
1310* end of the jbc-loop
1311 2011 CONTINUE
1312*2011 bailed out of the jbc-loop
1313 DO 2012 p = igl, min( igl+kbl-1, n )
1314 sva( p ) = abs( sva( p ) )
1315 2012 CONTINUE
1316***
1317 2000 CONTINUE
1318*2000 :: end of the ibr-loop
1319*
1320* .. update SVA(N)
1321 IF( ( sva( n ).LT.rootbig ) .AND. ( sva( n ).GT.rootsfmin ) )
1322 \$ THEN
1323 sva( n ) = scnrm2( m, a( 1, n ), 1 )
1324 ELSE
1325 t = zero
1326 aapp = one
1327 CALL classq( m, a( 1, n ), 1, t, aapp )
1328 sva( n ) = t*sqrt( aapp )
1329 END IF
1330*
1332*
1333 IF( ( i.LT.swband ) .AND. ( ( mxaapq.LE.roottol ) .OR.
1334 \$ ( iswrot.LE.n ) ) )swband = i
1335*
1336 IF( ( i.GT.swband+1 ) .AND. ( mxaapq.LT.sqrt( real( n ) )*
1337 \$ tol ) .AND. ( real( n )*mxaapq*mxsinj.LT.tol ) ) THEN
1338 GO TO 1994
1339 END IF
1340*
1341 IF( notrot.GE.emptsw )GO TO 1994
1342*
1343 1993 CONTINUE
1344* end i=1:NSWEEP loop
1345*
1346* #:( Reaching this point means that the procedure has not converged.
1347 info = nsweep - 1
1348 GO TO 1995
1349*
1350 1994 CONTINUE
1351* #:) Reaching this point means numerical convergence after the i-th
1352* sweep.
1353*
1354 info = 0
1355* #:) INFO = 0 confirms successful iterations.
1356 1995 CONTINUE
1357*
1358* Sort the singular values and find how many are above
1359* the underflow threshold.
1360*
1361 n2 = 0
1362 n4 = 0
1363 DO 5991 p = 1, n - 1
1364 q = isamax( n-p+1, sva( p ), 1 ) + p - 1
1365 IF( p.NE.q ) THEN
1366 temp1 = sva( p )
1367 sva( p ) = sva( q )
1368 sva( q ) = temp1
1369 CALL cswap( m, a( 1, p ), 1, a( 1, q ), 1 )
1370 IF( rsvec )CALL cswap( mvl, v( 1, p ), 1, v( 1, q ), 1 )
1371 END IF
1372 IF( sva( p ).NE.zero ) THEN
1373 n4 = n4 + 1
1374 IF( sva( p )*skl.GT.sfmin )n2 = n2 + 1
1375 END IF
1376 5991 CONTINUE
1377 IF( sva( n ).NE.zero ) THEN
1378 n4 = n4 + 1
1379 IF( sva( n )*skl.GT.sfmin )n2 = n2 + 1
1380 END IF
1381*
1382* Normalize the left singular vectors.
1383*
1384 IF( lsvec .OR. uctol ) THEN
1385 DO 1998 p = 1, n4
1386* CALL CSSCAL( M, ONE / SVA( p ), A( 1, p ), 1 )
1387 CALL clascl( 'G',0,0, sva(p), one, m, 1, a(1,p), m, ierr )
1388 1998 CONTINUE
1389 END IF
1390*
1391* Scale the product of Jacobi rotations.
1392*
1393 IF( rsvec ) THEN
1394 DO 2399 p = 1, n
1395 temp1 = one / scnrm2( mvl, v( 1, p ), 1 )
1396 CALL csscal( mvl, temp1, v( 1, p ), 1 )
1397 2399 CONTINUE
1398 END IF
1399*
1400* Undo scaling, if necessary (and possible).
1401 IF( ( ( skl.GT.one ) .AND. ( sva( 1 ).LT.( big / skl ) ) )
1402 \$ .OR. ( ( skl.LT.one ) .AND. ( sva( max( n2, 1 ) ) .GT.
1403 \$ ( sfmin / skl ) ) ) ) THEN
1404 DO 2400 p = 1, n
1405 sva( p ) = skl*sva( p )
1406 2400 CONTINUE
1407 skl = one
1408 END IF
1409*
1410 rwork( 1 ) = skl
1411* The singular values of A are SKL*SVA(1:N). If SKL.NE.ONE
1412* then some of the singular values may overflow or underflow and
1413* the spectrum is given in this factored representation.
1414*
1415 rwork( 2 ) = real( n4 )
1416* N4 is the number of computed nonzero singular values of A.
1417*
1418 rwork( 3 ) = real( n2 )
1419* N2 is the number of singular values of A greater than SFMIN.
1420* If N2<N, SVA(N2:N) contains ZEROS and/or denormalized numbers
1421* that may carry some information.
1422*
1423 rwork( 4 ) = real( i )
1424* i is the index of the last sweep before declaring convergence.
1425*
1426 rwork( 5 ) = mxaapq
1427* MXAAPQ is the largest absolute value of scaled pivots in the
1428* last sweep
1429*
1430 rwork( 6 ) = mxsinj
1431* MXSINJ is the largest absolute value of the sines of Jacobi angles
1432* in the last sweep
1433*
1434 RETURN
1435* ..
1436* .. END OF CGESVJ
1437* ..
1438 END
1439*
subroutine xerbla(srname, info)
Definition cblat2.f:3285
subroutine caxpy(n, ca, cx, incx, cy, incy)
CAXPY
Definition caxpy.f:88
subroutine ccopy(n, cx, incx, cy, incy)
CCOPY
Definition ccopy.f:81
subroutine cgesvj(joba, jobu, jobv, m, n, a, lda, sva, mv, v, ldv, cwork, lwork, rwork, lrwork, info)
CGESVJ
Definition cgesvj.f:351
subroutine cgsvj0(jobv, m, n, a, lda, d, sva, mv, v, ldv, eps, sfmin, tol, nsweep, work, lwork, info)
CGSVJ0 pre-processor for the routine cgesvj.
Definition cgsvj0.f:218
subroutine cgsvj1(jobv, m, n, n1, a, lda, d, sva, mv, v, ldv, eps, sfmin, tol, nsweep, work, lwork, info)
CGSVJ1 pre-processor for the routine cgesvj, applies Jacobi rotations targeting only particular pivot...
Definition cgsvj1.f:236
subroutine clascl(type, kl, ku, cfrom, cto, m, n, a, lda, info)
CLASCL multiplies a general rectangular matrix by a real scalar defined as cto/cfrom.
Definition clascl.f:143
subroutine slascl(type, kl, ku, cfrom, cto, m, n, a, lda, info)
SLASCL multiplies a general rectangular matrix by a real scalar defined as cto/cfrom.
Definition slascl.f:143
subroutine claset(uplo, m, n, alpha, beta, a, lda)
CLASET initializes the off-diagonal elements and the diagonal elements of a matrix to given values.
Definition claset.f:106
subroutine classq(n, x, incx, scale, sumsq)
CLASSQ updates a sum of squares represented in scaled form.
Definition classq.f90:124
subroutine crot(n, cx, incx, cy, incy, c, s)
CROT applies a plane rotation with real cosine and complex sine to a pair of complex vectors.
Definition crot.f:103
subroutine csscal(n, sa, cx, incx)
CSSCAL
Definition csscal.f:78
subroutine cswap(n, cx, incx, cy, incy)
CSWAP
Definition cswap.f:81