LAPACK 3.11.0
LAPACK: Linear Algebra PACKage
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stgsja.f
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1*> \brief \b STGSJA
2*
3* =========== DOCUMENTATION ===========
4*
5* Online html documentation available at
6* http://www.netlib.org/lapack/explore-html/
7*
8*> \htmlonly
9*> Download STGSJA + dependencies
10*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.tgz?format=tgz&filename=/lapack/lapack_routine/stgsja.f">
11*> [TGZ]</a>
12*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.zip?format=zip&filename=/lapack/lapack_routine/stgsja.f">
13*> [ZIP]</a>
14*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.txt?format=txt&filename=/lapack/lapack_routine/stgsja.f">
15*> [TXT]</a>
16*> \endhtmlonly
17*
18* Definition:
19* ===========
20*
21* SUBROUTINE STGSJA( JOBU, JOBV, JOBQ, M, P, N, K, L, A, LDA, B,
22* LDB, TOLA, TOLB, ALPHA, BETA, U, LDU, V, LDV,
23* Q, LDQ, WORK, NCYCLE, INFO )
24*
25* .. Scalar Arguments ..
26* CHARACTER JOBQ, JOBU, JOBV
27* INTEGER INFO, K, L, LDA, LDB, LDQ, LDU, LDV, M, N,
28* $ NCYCLE, P
29* REAL TOLA, TOLB
30* ..
31* .. Array Arguments ..
32* REAL A( LDA, * ), ALPHA( * ), B( LDB, * ),
33* $ BETA( * ), Q( LDQ, * ), U( LDU, * ),
34* $ V( LDV, * ), WORK( * )
35* ..
36*
37*
38*> \par Purpose:
39* =============
40*>
41*> \verbatim
42*>
43*> STGSJA computes the generalized singular value decomposition (GSVD)
44*> of two real upper triangular (or trapezoidal) matrices A and B.
45*>
46*> On entry, it is assumed that matrices A and B have the following
47*> forms, which may be obtained by the preprocessing subroutine SGGSVP
48*> from a general M-by-N matrix A and P-by-N matrix B:
49*>
50*> N-K-L K L
51*> A = K ( 0 A12 A13 ) if M-K-L >= 0;
52*> L ( 0 0 A23 )
53*> M-K-L ( 0 0 0 )
54*>
55*> N-K-L K L
56*> A = K ( 0 A12 A13 ) if M-K-L < 0;
57*> M-K ( 0 0 A23 )
58*>
59*> N-K-L K L
60*> B = L ( 0 0 B13 )
61*> P-L ( 0 0 0 )
62*>
63*> where the K-by-K matrix A12 and L-by-L matrix B13 are nonsingular
64*> upper triangular; A23 is L-by-L upper triangular if M-K-L >= 0,
65*> otherwise A23 is (M-K)-by-L upper trapezoidal.
66*>
67*> On exit,
68*>
69*> U**T *A*Q = D1*( 0 R ), V**T *B*Q = D2*( 0 R ),
70*>
71*> where U, V and Q are orthogonal matrices.
72*> R is a nonsingular upper triangular matrix, and D1 and D2 are
73*> ``diagonal'' matrices, which are of the following structures:
74*>
75*> If M-K-L >= 0,
76*>
77*> K L
78*> D1 = K ( I 0 )
79*> L ( 0 C )
80*> M-K-L ( 0 0 )
81*>
82*> K L
83*> D2 = L ( 0 S )
84*> P-L ( 0 0 )
85*>
86*> N-K-L K L
87*> ( 0 R ) = K ( 0 R11 R12 ) K
88*> L ( 0 0 R22 ) L
89*>
90*> where
91*>
92*> C = diag( ALPHA(K+1), ... , ALPHA(K+L) ),
93*> S = diag( BETA(K+1), ... , BETA(K+L) ),
94*> C**2 + S**2 = I.
95*>
96*> R is stored in A(1:K+L,N-K-L+1:N) on exit.
97*>
98*> If M-K-L < 0,
99*>
100*> K M-K K+L-M
101*> D1 = K ( I 0 0 )
102*> M-K ( 0 C 0 )
103*>
104*> K M-K K+L-M
105*> D2 = M-K ( 0 S 0 )
106*> K+L-M ( 0 0 I )
107*> P-L ( 0 0 0 )
108*>
109*> N-K-L K M-K K+L-M
110*> ( 0 R ) = K ( 0 R11 R12 R13 )
111*> M-K ( 0 0 R22 R23 )
112*> K+L-M ( 0 0 0 R33 )
113*>
114*> where
115*> C = diag( ALPHA(K+1), ... , ALPHA(M) ),
116*> S = diag( BETA(K+1), ... , BETA(M) ),
117*> C**2 + S**2 = I.
118*>
119*> R = ( R11 R12 R13 ) is stored in A(1:M, N-K-L+1:N) and R33 is stored
120*> ( 0 R22 R23 )
121*> in B(M-K+1:L,N+M-K-L+1:N) on exit.
122*>
123*> The computation of the orthogonal transformation matrices U, V or Q
124*> is optional. These matrices may either be formed explicitly, or they
125*> may be postmultiplied into input matrices U1, V1, or Q1.
126*> \endverbatim
127*
128* Arguments:
129* ==========
130*
131*> \param[in] JOBU
132*> \verbatim
133*> JOBU is CHARACTER*1
134*> = 'U': U must contain an orthogonal matrix U1 on entry, and
135*> the product U1*U is returned;
136*> = 'I': U is initialized to the unit matrix, and the
137*> orthogonal matrix U is returned;
138*> = 'N': U is not computed.
139*> \endverbatim
140*>
141*> \param[in] JOBV
142*> \verbatim
143*> JOBV is CHARACTER*1
144*> = 'V': V must contain an orthogonal matrix V1 on entry, and
145*> the product V1*V is returned;
146*> = 'I': V is initialized to the unit matrix, and the
147*> orthogonal matrix V is returned;
148*> = 'N': V is not computed.
149*> \endverbatim
150*>
151*> \param[in] JOBQ
152*> \verbatim
153*> JOBQ is CHARACTER*1
154*> = 'Q': Q must contain an orthogonal matrix Q1 on entry, and
155*> the product Q1*Q is returned;
156*> = 'I': Q is initialized to the unit matrix, and the
157*> orthogonal matrix Q is returned;
158*> = 'N': Q is not computed.
159*> \endverbatim
160*>
161*> \param[in] M
162*> \verbatim
163*> M is INTEGER
164*> The number of rows of the matrix A. M >= 0.
165*> \endverbatim
166*>
167*> \param[in] P
168*> \verbatim
169*> P is INTEGER
170*> The number of rows of the matrix B. P >= 0.
171*> \endverbatim
172*>
173*> \param[in] N
174*> \verbatim
175*> N is INTEGER
176*> The number of columns of the matrices A and B. N >= 0.
177*> \endverbatim
178*>
179*> \param[in] K
180*> \verbatim
181*> K is INTEGER
182*> \endverbatim
183*>
184*> \param[in] L
185*> \verbatim
186*> L is INTEGER
187*>
188*> K and L specify the subblocks in the input matrices A and B:
189*> A23 = A(K+1:MIN(K+L,M),N-L+1:N) and B13 = B(1:L,N-L+1:N)
190*> of A and B, whose GSVD is going to be computed by STGSJA.
191*> See Further Details.
192*> \endverbatim
193*>
194*> \param[in,out] A
195*> \verbatim
196*> A is REAL array, dimension (LDA,N)
197*> On entry, the M-by-N matrix A.
198*> On exit, A(N-K+1:N,1:MIN(K+L,M) ) contains the triangular
199*> matrix R or part of R. See Purpose for details.
200*> \endverbatim
201*>
202*> \param[in] LDA
203*> \verbatim
204*> LDA is INTEGER
205*> The leading dimension of the array A. LDA >= max(1,M).
206*> \endverbatim
207*>
208*> \param[in,out] B
209*> \verbatim
210*> B is REAL array, dimension (LDB,N)
211*> On entry, the P-by-N matrix B.
212*> On exit, if necessary, B(M-K+1:L,N+M-K-L+1:N) contains
213*> a part of R. See Purpose for details.
214*> \endverbatim
215*>
216*> \param[in] LDB
217*> \verbatim
218*> LDB is INTEGER
219*> The leading dimension of the array B. LDB >= max(1,P).
220*> \endverbatim
221*>
222*> \param[in] TOLA
223*> \verbatim
224*> TOLA is REAL
225*> \endverbatim
226*>
227*> \param[in] TOLB
228*> \verbatim
229*> TOLB is REAL
230*>
231*> TOLA and TOLB are the convergence criteria for the Jacobi-
232*> Kogbetliantz iteration procedure. Generally, they are the
233*> same as used in the preprocessing step, say
234*> TOLA = max(M,N)*norm(A)*MACHEPS,
235*> TOLB = max(P,N)*norm(B)*MACHEPS.
236*> \endverbatim
237*>
238*> \param[out] ALPHA
239*> \verbatim
240*> ALPHA is REAL array, dimension (N)
241*> \endverbatim
242*>
243*> \param[out] BETA
244*> \verbatim
245*> BETA is REAL array, dimension (N)
246*>
247*> On exit, ALPHA and BETA contain the generalized singular
248*> value pairs of A and B;
249*> ALPHA(1:K) = 1,
250*> BETA(1:K) = 0,
251*> and if M-K-L >= 0,
252*> ALPHA(K+1:K+L) = diag(C),
253*> BETA(K+1:K+L) = diag(S),
254*> or if M-K-L < 0,
255*> ALPHA(K+1:M)= C, ALPHA(M+1:K+L)= 0
256*> BETA(K+1:M) = S, BETA(M+1:K+L) = 1.
257*> Furthermore, if K+L < N,
258*> ALPHA(K+L+1:N) = 0 and
259*> BETA(K+L+1:N) = 0.
260*> \endverbatim
261*>
262*> \param[in,out] U
263*> \verbatim
264*> U is REAL array, dimension (LDU,M)
265*> On entry, if JOBU = 'U', U must contain a matrix U1 (usually
266*> the orthogonal matrix returned by SGGSVP).
267*> On exit,
268*> if JOBU = 'I', U contains the orthogonal matrix U;
269*> if JOBU = 'U', U contains the product U1*U.
270*> If JOBU = 'N', U is not referenced.
271*> \endverbatim
272*>
273*> \param[in] LDU
274*> \verbatim
275*> LDU is INTEGER
276*> The leading dimension of the array U. LDU >= max(1,M) if
277*> JOBU = 'U'; LDU >= 1 otherwise.
278*> \endverbatim
279*>
280*> \param[in,out] V
281*> \verbatim
282*> V is REAL array, dimension (LDV,P)
283*> On entry, if JOBV = 'V', V must contain a matrix V1 (usually
284*> the orthogonal matrix returned by SGGSVP).
285*> On exit,
286*> if JOBV = 'I', V contains the orthogonal matrix V;
287*> if JOBV = 'V', V contains the product V1*V.
288*> If JOBV = 'N', V is not referenced.
289*> \endverbatim
290*>
291*> \param[in] LDV
292*> \verbatim
293*> LDV is INTEGER
294*> The leading dimension of the array V. LDV >= max(1,P) if
295*> JOBV = 'V'; LDV >= 1 otherwise.
296*> \endverbatim
297*>
298*> \param[in,out] Q
299*> \verbatim
300*> Q is REAL array, dimension (LDQ,N)
301*> On entry, if JOBQ = 'Q', Q must contain a matrix Q1 (usually
302*> the orthogonal matrix returned by SGGSVP).
303*> On exit,
304*> if JOBQ = 'I', Q contains the orthogonal matrix Q;
305*> if JOBQ = 'Q', Q contains the product Q1*Q.
306*> If JOBQ = 'N', Q is not referenced.
307*> \endverbatim
308*>
309*> \param[in] LDQ
310*> \verbatim
311*> LDQ is INTEGER
312*> The leading dimension of the array Q. LDQ >= max(1,N) if
313*> JOBQ = 'Q'; LDQ >= 1 otherwise.
314*> \endverbatim
315*>
316*> \param[out] WORK
317*> \verbatim
318*> WORK is REAL array, dimension (2*N)
319*> \endverbatim
320*>
321*> \param[out] NCYCLE
322*> \verbatim
323*> NCYCLE is INTEGER
324*> The number of cycles required for convergence.
325*> \endverbatim
326*>
327*> \param[out] INFO
328*> \verbatim
329*> INFO is INTEGER
330*> = 0: successful exit
331*> < 0: if INFO = -i, the i-th argument had an illegal value.
332*> = 1: the procedure does not converge after MAXIT cycles.
333*> \endverbatim
334*>
335*> \verbatim
336*> Internal Parameters
337*> ===================
338*>
339*> MAXIT INTEGER
340*> MAXIT specifies the total loops that the iterative procedure
341*> may take. If after MAXIT cycles, the routine fails to
342*> converge, we return INFO = 1.
343*> \endverbatim
344*
345* Authors:
346* ========
347*
348*> \author Univ. of Tennessee
349*> \author Univ. of California Berkeley
350*> \author Univ. of Colorado Denver
351*> \author NAG Ltd.
352*
353*> \ingroup realOTHERcomputational
354*
355*> \par Further Details:
356* =====================
357*>
358*> \verbatim
359*>
360*> STGSJA essentially uses a variant of Kogbetliantz algorithm to reduce
361*> min(L,M-K)-by-L triangular (or trapezoidal) matrix A23 and L-by-L
362*> matrix B13 to the form:
363*>
364*> U1**T *A13*Q1 = C1*R1; V1**T *B13*Q1 = S1*R1,
365*>
366*> where U1, V1 and Q1 are orthogonal matrix, and Z**T is the transpose
367*> of Z. C1 and S1 are diagonal matrices satisfying
368*>
369*> C1**2 + S1**2 = I,
370*>
371*> and R1 is an L-by-L nonsingular upper triangular matrix.
372*> \endverbatim
373*>
374* =====================================================================
375 SUBROUTINE stgsja( JOBU, JOBV, JOBQ, M, P, N, K, L, A, LDA, B,
376 $ LDB, TOLA, TOLB, ALPHA, BETA, U, LDU, V, LDV,
377 $ Q, LDQ, WORK, NCYCLE, INFO )
378*
379* -- LAPACK computational routine --
380* -- LAPACK is a software package provided by Univ. of Tennessee, --
381* -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
382*
383* .. Scalar Arguments ..
384 CHARACTER JOBQ, JOBU, JOBV
385 INTEGER INFO, K, L, LDA, LDB, LDQ, LDU, LDV, M, N,
386 $ ncycle, p
387 REAL TOLA, TOLB
388* ..
389* .. Array Arguments ..
390 REAL A( LDA, * ), ALPHA( * ), B( LDB, * ),
391 $ BETA( * ), Q( LDQ, * ), U( LDU, * ),
392 $ v( ldv, * ), work( * )
393* ..
394*
395* =====================================================================
396*
397* .. Parameters ..
398 INTEGER MAXIT
399 PARAMETER ( MAXIT = 40 )
400 REAL ZERO, ONE, HUGENUM
401 parameter( zero = 0.0e+0, one = 1.0e+0 )
402* ..
403* .. Local Scalars ..
404*
405 LOGICAL INITQ, INITU, INITV, UPPER, WANTQ, WANTU, WANTV
406 INTEGER I, J, KCYCLE
407 REAL A1, A2, A3, B1, B2, B3, CSQ, CSU, CSV, ERROR,
408 $ gamma, rwk, snq, snu, snv, ssmin
409* ..
410* .. External Functions ..
411 LOGICAL LSAME
412 EXTERNAL LSAME
413* ..
414* .. External Subroutines ..
415 EXTERNAL scopy, slags2, slapll, slartg, slaset, srot,
416 $ sscal, xerbla
417* ..
418* .. Intrinsic Functions ..
419 INTRINSIC abs, max, min, huge
420 parameter( hugenum = huge(zero) )
421* ..
422* .. Executable Statements ..
423*
424* Decode and test the input parameters
425*
426 initu = lsame( jobu, 'I' )
427 wantu = initu .OR. lsame( jobu, 'U' )
428*
429 initv = lsame( jobv, 'I' )
430 wantv = initv .OR. lsame( jobv, 'V' )
431*
432 initq = lsame( jobq, 'I' )
433 wantq = initq .OR. lsame( jobq, 'Q' )
434*
435 info = 0
436 IF( .NOT.( initu .OR. wantu .OR. lsame( jobu, 'N' ) ) ) THEN
437 info = -1
438 ELSE IF( .NOT.( initv .OR. wantv .OR. lsame( jobv, 'N' ) ) ) THEN
439 info = -2
440 ELSE IF( .NOT.( initq .OR. wantq .OR. lsame( jobq, 'N' ) ) ) THEN
441 info = -3
442 ELSE IF( m.LT.0 ) THEN
443 info = -4
444 ELSE IF( p.LT.0 ) THEN
445 info = -5
446 ELSE IF( n.LT.0 ) THEN
447 info = -6
448 ELSE IF( lda.LT.max( 1, m ) ) THEN
449 info = -10
450 ELSE IF( ldb.LT.max( 1, p ) ) THEN
451 info = -12
452 ELSE IF( ldu.LT.1 .OR. ( wantu .AND. ldu.LT.m ) ) THEN
453 info = -18
454 ELSE IF( ldv.LT.1 .OR. ( wantv .AND. ldv.LT.p ) ) THEN
455 info = -20
456 ELSE IF( ldq.LT.1 .OR. ( wantq .AND. ldq.LT.n ) ) THEN
457 info = -22
458 END IF
459 IF( info.NE.0 ) THEN
460 CALL xerbla( 'STGSJA', -info )
461 RETURN
462 END IF
463*
464* Initialize U, V and Q, if necessary
465*
466 IF( initu )
467 $ CALL slaset( 'Full', m, m, zero, one, u, ldu )
468 IF( initv )
469 $ CALL slaset( 'Full', p, p, zero, one, v, ldv )
470 IF( initq )
471 $ CALL slaset( 'Full', n, n, zero, one, q, ldq )
472*
473* Loop until convergence
474*
475 upper = .false.
476 DO 40 kcycle = 1, maxit
477*
478 upper = .NOT.upper
479*
480 DO 20 i = 1, l - 1
481 DO 10 j = i + 1, l
482*
483 a1 = zero
484 a2 = zero
485 a3 = zero
486 IF( k+i.LE.m )
487 $ a1 = a( k+i, n-l+i )
488 IF( k+j.LE.m )
489 $ a3 = a( k+j, n-l+j )
490*
491 b1 = b( i, n-l+i )
492 b3 = b( j, n-l+j )
493*
494 IF( upper ) THEN
495 IF( k+i.LE.m )
496 $ a2 = a( k+i, n-l+j )
497 b2 = b( i, n-l+j )
498 ELSE
499 IF( k+j.LE.m )
500 $ a2 = a( k+j, n-l+i )
501 b2 = b( j, n-l+i )
502 END IF
503*
504 CALL slags2( upper, a1, a2, a3, b1, b2, b3, csu, snu,
505 $ csv, snv, csq, snq )
506*
507* Update (K+I)-th and (K+J)-th rows of matrix A: U**T *A
508*
509 IF( k+j.LE.m )
510 $ CALL srot( l, a( k+j, n-l+1 ), lda, a( k+i, n-l+1 ),
511 $ lda, csu, snu )
512*
513* Update I-th and J-th rows of matrix B: V**T *B
514*
515 CALL srot( l, b( j, n-l+1 ), ldb, b( i, n-l+1 ), ldb,
516 $ csv, snv )
517*
518* Update (N-L+I)-th and (N-L+J)-th columns of matrices
519* A and B: A*Q and B*Q
520*
521 CALL srot( min( k+l, m ), a( 1, n-l+j ), 1,
522 $ a( 1, n-l+i ), 1, csq, snq )
523*
524 CALL srot( l, b( 1, n-l+j ), 1, b( 1, n-l+i ), 1, csq,
525 $ snq )
526*
527 IF( upper ) THEN
528 IF( k+i.LE.m )
529 $ a( k+i, n-l+j ) = zero
530 b( i, n-l+j ) = zero
531 ELSE
532 IF( k+j.LE.m )
533 $ a( k+j, n-l+i ) = zero
534 b( j, n-l+i ) = zero
535 END IF
536*
537* Update orthogonal matrices U, V, Q, if desired.
538*
539 IF( wantu .AND. k+j.LE.m )
540 $ CALL srot( m, u( 1, k+j ), 1, u( 1, k+i ), 1, csu,
541 $ snu )
542*
543 IF( wantv )
544 $ CALL srot( p, v( 1, j ), 1, v( 1, i ), 1, csv, snv )
545*
546 IF( wantq )
547 $ CALL srot( n, q( 1, n-l+j ), 1, q( 1, n-l+i ), 1, csq,
548 $ snq )
549*
550 10 CONTINUE
551 20 CONTINUE
552*
553 IF( .NOT.upper ) THEN
554*
555* The matrices A13 and B13 were lower triangular at the start
556* of the cycle, and are now upper triangular.
557*
558* Convergence test: test the parallelism of the corresponding
559* rows of A and B.
560*
561 error = zero
562 DO 30 i = 1, min( l, m-k )
563 CALL scopy( l-i+1, a( k+i, n-l+i ), lda, work, 1 )
564 CALL scopy( l-i+1, b( i, n-l+i ), ldb, work( l+1 ), 1 )
565 CALL slapll( l-i+1, work, 1, work( l+1 ), 1, ssmin )
566 error = max( error, ssmin )
567 30 CONTINUE
568*
569 IF( abs( error ).LE.min( tola, tolb ) )
570 $ GO TO 50
571 END IF
572*
573* End of cycle loop
574*
575 40 CONTINUE
576*
577* The algorithm has not converged after MAXIT cycles.
578*
579 info = 1
580 GO TO 100
581*
582 50 CONTINUE
583*
584* If ERROR <= MIN(TOLA,TOLB), then the algorithm has converged.
585* Compute the generalized singular value pairs (ALPHA, BETA), and
586* set the triangular matrix R to array A.
587*
588 DO 60 i = 1, k
589 alpha( i ) = one
590 beta( i ) = zero
591 60 CONTINUE
592*
593 DO 70 i = 1, min( l, m-k )
594*
595 a1 = a( k+i, n-l+i )
596 b1 = b( i, n-l+i )
597 gamma = b1 / a1
598*
599 IF( (gamma.LE.hugenum).AND.(gamma.GE.-hugenum) ) THEN
600*
601* change sign if necessary
602*
603 IF( gamma.LT.zero ) THEN
604 CALL sscal( l-i+1, -one, b( i, n-l+i ), ldb )
605 IF( wantv )
606 $ CALL sscal( p, -one, v( 1, i ), 1 )
607 END IF
608*
609 CALL slartg( abs( gamma ), one, beta( k+i ), alpha( k+i ),
610 $ rwk )
611*
612 IF( alpha( k+i ).GE.beta( k+i ) ) THEN
613 CALL sscal( l-i+1, one / alpha( k+i ), a( k+i, n-l+i ),
614 $ lda )
615 ELSE
616 CALL sscal( l-i+1, one / beta( k+i ), b( i, n-l+i ),
617 $ ldb )
618 CALL scopy( l-i+1, b( i, n-l+i ), ldb, a( k+i, n-l+i ),
619 $ lda )
620 END IF
621*
622 ELSE
623*
624 alpha( k+i ) = zero
625 beta( k+i ) = one
626 CALL scopy( l-i+1, b( i, n-l+i ), ldb, a( k+i, n-l+i ),
627 $ lda )
628*
629 END IF
630*
631 70 CONTINUE
632*
633* Post-assignment
634*
635 DO 80 i = m + 1, k + l
636 alpha( i ) = zero
637 beta( i ) = one
638 80 CONTINUE
639*
640 IF( k+l.LT.n ) THEN
641 DO 90 i = k + l + 1, n
642 alpha( i ) = zero
643 beta( i ) = zero
644 90 CONTINUE
645 END IF
646*
647 100 CONTINUE
648 ncycle = kcycle
649 RETURN
650*
651* End of STGSJA
652*
653 END
subroutine slaset(UPLO, M, N, ALPHA, BETA, A, LDA)
SLASET initializes the off-diagonal elements and the diagonal elements of a matrix to given values.
Definition: slaset.f:110
subroutine slartg(f, g, c, s, r)
SLARTG generates a plane rotation with real cosine and real sine.
Definition: slartg.f90:111
subroutine xerbla(SRNAME, INFO)
XERBLA
Definition: xerbla.f:60
subroutine slags2(UPPER, A1, A2, A3, B1, B2, B3, CSU, SNU, CSV, SNV, CSQ, SNQ)
SLAGS2 computes 2-by-2 orthogonal matrices U, V, and Q, and applies them to matrices A and B such tha...
Definition: slags2.f:152
subroutine slapll(N, X, INCX, Y, INCY, SSMIN)
SLAPLL measures the linear dependence of two vectors.
Definition: slapll.f:102
subroutine stgsja(JOBU, JOBV, JOBQ, M, P, N, K, L, A, LDA, B, LDB, TOLA, TOLB, ALPHA, BETA, U, LDU, V, LDV, Q, LDQ, WORK, NCYCLE, INFO)
STGSJA
Definition: stgsja.f:378
subroutine srot(N, SX, INCX, SY, INCY, C, S)
SROT
Definition: srot.f:92
subroutine scopy(N, SX, INCX, SY, INCY)
SCOPY
Definition: scopy.f:82
subroutine sscal(N, SA, SX, INCX)
SSCAL
Definition: sscal.f:79