LAPACK 3.12.0 LAPACK: Linear Algebra PACKage
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## ◆ scsdts()

 subroutine scsdts ( integer m, integer p, integer q, real, dimension( ldx, * ) x, real, dimension( ldx, * ) xf, integer ldx, real, dimension( ldu1, * ) u1, integer ldu1, real, dimension( ldu2, * ) u2, integer ldu2, real, dimension( ldv1t, * ) v1t, integer ldv1t, real, dimension( ldv2t, * ) v2t, integer ldv2t, real, dimension( * ) theta, integer, dimension( * ) iwork, real, dimension( lwork ) work, integer lwork, real, dimension( * ) rwork, real, dimension( 15 ) result )

SCSDTS

Purpose:
``` SCSDTS tests SORCSD, which, given an M-by-M partitioned orthogonal
matrix X,
Q  M-Q
X = [ X11 X12 ] P   ,
[ X21 X22 ] M-P

computes the CSD

[ U1    ]**T * [ X11 X12 ] * [ V1    ]
[    U2 ]      [ X21 X22 ]   [    V2 ]

[  I  0  0 |  0  0  0 ]
[  0  C  0 |  0 -S  0 ]
[  0  0  0 |  0  0 -I ]
= [---------------------] = [ D11 D12 ] .
[  0  0  0 |  I  0  0 ]   [ D21 D22 ]
[  0  S  0 |  0  C  0 ]
[  0  0  I |  0  0  0 ]

and also SORCSD2BY1, which, given
Q
[ X11 ] P   ,
[ X21 ] M-P

computes the 2-by-1 CSD

[  I  0  0 ]
[  0  C  0 ]
[  0  0  0 ]
[ U1    ]**T * [ X11 ] * V1 = [----------] = [ D11 ] ,
[    U2 ]      [ X21 ]        [  0  0  0 ]   [ D21 ]
[  0  S  0 ]
[  0  0  I ]```
Parameters
 [in] M ``` M is INTEGER The number of rows of the matrix X. M >= 0.``` [in] P ``` P is INTEGER The number of rows of the matrix X11. P >= 0.``` [in] Q ``` Q is INTEGER The number of columns of the matrix X11. Q >= 0.``` [in] X ``` X is REAL array, dimension (LDX,M) The M-by-M matrix X.``` [out] XF ``` XF is REAL array, dimension (LDX,M) Details of the CSD of X, as returned by SORCSD; see SORCSD for further details.``` [in] LDX ``` LDX is INTEGER The leading dimension of the arrays X and XF. LDX >= max( 1,M ).``` [out] U1 ``` U1 is REAL array, dimension(LDU1,P) The P-by-P orthogonal matrix U1.``` [in] LDU1 ``` LDU1 is INTEGER The leading dimension of the array U1. LDU >= max(1,P).``` [out] U2 ``` U2 is REAL array, dimension(LDU2,M-P) The (M-P)-by-(M-P) orthogonal matrix U2.``` [in] LDU2 ``` LDU2 is INTEGER The leading dimension of the array U2. LDU >= max(1,M-P).``` [out] V1T ``` V1T is REAL array, dimension(LDV1T,Q) The Q-by-Q orthogonal matrix V1T.``` [in] LDV1T ``` LDV1T is INTEGER The leading dimension of the array V1T. LDV1T >= max(1,Q).``` [out] V2T ``` V2T is REAL array, dimension(LDV2T,M-Q) The (M-Q)-by-(M-Q) orthogonal matrix V2T.``` [in] LDV2T ``` LDV2T is INTEGER The leading dimension of the array V2T. LDV2T >= max(1,M-Q).``` [out] THETA ``` THETA is REAL array, dimension MIN(P,M-P,Q,M-Q) The CS values of X; the essentially diagonal matrices C and S are constructed from THETA; see subroutine SORCSD for details.``` [out] IWORK ` IWORK is INTEGER array, dimension (M)` [out] WORK ` WORK is REAL array, dimension (LWORK)` [in] LWORK ``` LWORK is INTEGER The dimension of the array WORK``` [out] RWORK ` RWORK is REAL array` [out] RESULT ``` RESULT is REAL array, dimension (15) The test ratios: First, the 2-by-2 CSD: RESULT(1) = norm( U1'*X11*V1 - D11 ) / ( MAX(1,P,Q)*EPS2 ) RESULT(2) = norm( U1'*X12*V2 - D12 ) / ( MAX(1,P,M-Q)*EPS2 ) RESULT(3) = norm( U2'*X21*V1 - D21 ) / ( MAX(1,M-P,Q)*EPS2 ) RESULT(4) = norm( U2'*X22*V2 - D22 ) / ( MAX(1,M-P,M-Q)*EPS2 ) RESULT(5) = norm( I - U1'*U1 ) / ( MAX(1,P)*ULP ) RESULT(6) = norm( I - U2'*U2 ) / ( MAX(1,M-P)*ULP ) RESULT(7) = norm( I - V1T'*V1T ) / ( MAX(1,Q)*ULP ) RESULT(8) = norm( I - V2T'*V2T ) / ( MAX(1,M-Q)*ULP ) RESULT(9) = 0 if THETA is in increasing order and all angles are in [0,pi/2]; = ULPINV otherwise. Then, the 2-by-1 CSD: RESULT(10) = norm( U1'*X11*V1 - D11 ) / ( MAX(1,P,Q)*EPS2 ) RESULT(11) = norm( U2'*X21*V1 - D21 ) / ( MAX(1,M-P,Q)*EPS2 ) RESULT(12) = norm( I - U1'*U1 ) / ( MAX(1,P)*ULP ) RESULT(13) = norm( I - U2'*U2 ) / ( MAX(1,M-P)*ULP ) RESULT(14) = norm( I - V1T'*V1T ) / ( MAX(1,Q)*ULP ) RESULT(15) = 0 if THETA is in increasing order and all angles are in [0,pi/2]; = ULPINV otherwise. ( EPS2 = MAX( norm( I - X'*X ) / M, ULP ). )```

Definition at line 226 of file scsdts.f.

229*
230* -- LAPACK test routine --
231* -- LAPACK is a software package provided by Univ. of Tennessee, --
232* -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
233*
234* .. Scalar Arguments ..
235 INTEGER LDX, LDU1, LDU2, LDV1T, LDV2T, LWORK, M, P, Q
236* ..
237* .. Array Arguments ..
238 INTEGER IWORK( * )
239 REAL RESULT( 15 ), RWORK( * ), THETA( * )
240 REAL U1( LDU1, * ), U2( LDU2, * ), V1T( LDV1T, * ),
241 \$ V2T( LDV2T, * ), WORK( LWORK ), X( LDX, * ),
242 \$ XF( LDX, * )
243* ..
244*
245* =====================================================================
246*
247* .. Parameters ..
248 REAL REALONE, REALZERO
249 parameter( realone = 1.0e0, realzero = 0.0e0 )
250 REAL ZERO, ONE
251 parameter( zero = 0.0e0, one = 1.0e0 )
252 REAL PIOVER2
253 parameter( piover2 = 1.57079632679489661923132169163975144210e0 )
254* ..
255* .. Local Scalars ..
256 INTEGER I, INFO, R
257 REAL EPS2, RESID, ULP, ULPINV
258* ..
259* .. External Functions ..
260 REAL SLAMCH, SLANGE, SLANSY
261 EXTERNAL slamch, slange, slansy
262* ..
263* .. External Subroutines ..
264 EXTERNAL sgemm, slacpy, slaset, sorcsd, sorcsd2by1,
265 \$ ssyrk
266* ..
267* .. Intrinsic Functions ..
268 INTRINSIC cos, max, min, real, sin
269* ..
270* .. Executable Statements ..
271*
272 ulp = slamch( 'Precision' )
273 ulpinv = realone / ulp
274*
275* The first half of the routine checks the 2-by-2 CSD
276*
277 CALL slaset( 'Full', m, m, zero, one, work, ldx )
278 CALL ssyrk( 'Upper', 'Conjugate transpose', m, m, -one, x, ldx,
279 \$ one, work, ldx )
280 IF (m.GT.0) THEN
281 eps2 = max( ulp,
282 \$ slange( '1', m, m, work, ldx, rwork ) / real( m ) )
283 ELSE
284 eps2 = ulp
285 END IF
286 r = min( p, m-p, q, m-q )
287*
288* Copy the matrix X to the array XF.
289*
290 CALL slacpy( 'Full', m, m, x, ldx, xf, ldx )
291*
292* Compute the CSD
293*
294 CALL sorcsd( 'Y', 'Y', 'Y', 'Y', 'N', 'D', m, p, q, xf(1,1), ldx,
295 \$ xf(1,q+1), ldx, xf(p+1,1), ldx, xf(p+1,q+1), ldx,
296 \$ theta, u1, ldu1, u2, ldu2, v1t, ldv1t, v2t, ldv2t,
297 \$ work, lwork, iwork, info )
298*
299* Compute XF := diag(U1,U2)'*X*diag(V1,V2) - [D11 D12; D21 D22]
300*
301 CALL slacpy( 'Full', m, m, x, ldx, xf, ldx )
302*
303 CALL sgemm( 'No transpose', 'Conjugate transpose', p, q, q, one,
304 \$ xf, ldx, v1t, ldv1t, zero, work, ldx )
305*
306 CALL sgemm( 'Conjugate transpose', 'No transpose', p, q, p, one,
307 \$ u1, ldu1, work, ldx, zero, xf, ldx )
308*
309 DO i = 1, min(p,q)-r
310 xf(i,i) = xf(i,i) - one
311 END DO
312 DO i = 1, r
313 xf(min(p,q)-r+i,min(p,q)-r+i) =
314 \$ xf(min(p,q)-r+i,min(p,q)-r+i) - cos(theta(i))
315 END DO
316*
317 CALL sgemm( 'No transpose', 'Conjugate transpose', p, m-q, m-q,
318 \$ one, xf(1,q+1), ldx, v2t, ldv2t, zero, work, ldx )
319*
320 CALL sgemm( 'Conjugate transpose', 'No transpose', p, m-q, p,
321 \$ one, u1, ldu1, work, ldx, zero, xf(1,q+1), ldx )
322*
323 DO i = 1, min(p,m-q)-r
324 xf(p-i+1,m-i+1) = xf(p-i+1,m-i+1) + one
325 END DO
326 DO i = 1, r
327 xf(p-(min(p,m-q)-r)+1-i,m-(min(p,m-q)-r)+1-i) =
328 \$ xf(p-(min(p,m-q)-r)+1-i,m-(min(p,m-q)-r)+1-i) +
329 \$ sin(theta(r-i+1))
330 END DO
331*
332 CALL sgemm( 'No transpose', 'Conjugate transpose', m-p, q, q, one,
333 \$ xf(p+1,1), ldx, v1t, ldv1t, zero, work, ldx )
334*
335 CALL sgemm( 'Conjugate transpose', 'No transpose', m-p, q, m-p,
336 \$ one, u2, ldu2, work, ldx, zero, xf(p+1,1), ldx )
337*
338 DO i = 1, min(m-p,q)-r
339 xf(m-i+1,q-i+1) = xf(m-i+1,q-i+1) - one
340 END DO
341 DO i = 1, r
342 xf(m-(min(m-p,q)-r)+1-i,q-(min(m-p,q)-r)+1-i) =
343 \$ xf(m-(min(m-p,q)-r)+1-i,q-(min(m-p,q)-r)+1-i) -
344 \$ sin(theta(r-i+1))
345 END DO
346*
347 CALL sgemm( 'No transpose', 'Conjugate transpose', m-p, m-q, m-q,
348 \$ one, xf(p+1,q+1), ldx, v2t, ldv2t, zero, work, ldx )
349*
350 CALL sgemm( 'Conjugate transpose', 'No transpose', m-p, m-q, m-p,
351 \$ one, u2, ldu2, work, ldx, zero, xf(p+1,q+1), ldx )
352*
353 DO i = 1, min(m-p,m-q)-r
354 xf(p+i,q+i) = xf(p+i,q+i) - one
355 END DO
356 DO i = 1, r
357 xf(p+(min(m-p,m-q)-r)+i,q+(min(m-p,m-q)-r)+i) =
358 \$ xf(p+(min(m-p,m-q)-r)+i,q+(min(m-p,m-q)-r)+i) -
359 \$ cos(theta(i))
360 END DO
361*
362* Compute norm( U1'*X11*V1 - D11 ) / ( MAX(1,P,Q)*EPS2 ) .
363*
364 resid = slange( '1', p, q, xf, ldx, rwork )
365 result( 1 ) = ( resid / real(max(1,p,q)) ) / eps2
366*
367* Compute norm( U1'*X12*V2 - D12 ) / ( MAX(1,P,M-Q)*EPS2 ) .
368*
369 resid = slange( '1', p, m-q, xf(1,q+1), ldx, rwork )
370 result( 2 ) = ( resid / real(max(1,p,m-q)) ) / eps2
371*
372* Compute norm( U2'*X21*V1 - D21 ) / ( MAX(1,M-P,Q)*EPS2 ) .
373*
374 resid = slange( '1', m-p, q, xf(p+1,1), ldx, rwork )
375 result( 3 ) = ( resid / real(max(1,m-p,q)) ) / eps2
376*
377* Compute norm( U2'*X22*V2 - D22 ) / ( MAX(1,M-P,M-Q)*EPS2 ) .
378*
379 resid = slange( '1', m-p, m-q, xf(p+1,q+1), ldx, rwork )
380 result( 4 ) = ( resid / real(max(1,m-p,m-q)) ) / eps2
381*
382* Compute I - U1'*U1
383*
384 CALL slaset( 'Full', p, p, zero, one, work, ldu1 )
385 CALL ssyrk( 'Upper', 'Conjugate transpose', p, p, -one, u1, ldu1,
386 \$ one, work, ldu1 )
387*
388* Compute norm( I - U'*U ) / ( MAX(1,P) * ULP ) .
389*
390 resid = slansy( '1', 'Upper', p, work, ldu1, rwork )
391 result( 5 ) = ( resid / real(max(1,p)) ) / ulp
392*
393* Compute I - U2'*U2
394*
395 CALL slaset( 'Full', m-p, m-p, zero, one, work, ldu2 )
396 CALL ssyrk( 'Upper', 'Conjugate transpose', m-p, m-p, -one, u2,
397 \$ ldu2, one, work, ldu2 )
398*
399* Compute norm( I - U2'*U2 ) / ( MAX(1,M-P) * ULP ) .
400*
401 resid = slansy( '1', 'Upper', m-p, work, ldu2, rwork )
402 result( 6 ) = ( resid / real(max(1,m-p)) ) / ulp
403*
404* Compute I - V1T*V1T'
405*
406 CALL slaset( 'Full', q, q, zero, one, work, ldv1t )
407 CALL ssyrk( 'Upper', 'No transpose', q, q, -one, v1t, ldv1t, one,
408 \$ work, ldv1t )
409*
410* Compute norm( I - V1T*V1T' ) / ( MAX(1,Q) * ULP ) .
411*
412 resid = slansy( '1', 'Upper', q, work, ldv1t, rwork )
413 result( 7 ) = ( resid / real(max(1,q)) ) / ulp
414*
415* Compute I - V2T*V2T'
416*
417 CALL slaset( 'Full', m-q, m-q, zero, one, work, ldv2t )
418 CALL ssyrk( 'Upper', 'No transpose', m-q, m-q, -one, v2t, ldv2t,
419 \$ one, work, ldv2t )
420*
421* Compute norm( I - V2T*V2T' ) / ( MAX(1,M-Q) * ULP ) .
422*
423 resid = slansy( '1', 'Upper', m-q, work, ldv2t, rwork )
424 result( 8 ) = ( resid / real(max(1,m-q)) ) / ulp
425*
426* Check sorting
427*
428 result( 9 ) = realzero
429 DO i = 1, r
430 IF( theta(i).LT.realzero .OR. theta(i).GT.piover2 ) THEN
431 result( 9 ) = ulpinv
432 END IF
433 IF( i.GT.1 ) THEN
434 IF ( theta(i).LT.theta(i-1) ) THEN
435 result( 9 ) = ulpinv
436 END IF
437 END IF
438 END DO
439*
440* The second half of the routine checks the 2-by-1 CSD
441*
442 CALL slaset( 'Full', q, q, zero, one, work, ldx )
443 CALL ssyrk( 'Upper', 'Conjugate transpose', q, m, -one, x, ldx,
444 \$ one, work, ldx )
445 IF (m.GT.0) THEN
446 eps2 = max( ulp,
447 \$ slange( '1', q, q, work, ldx, rwork ) / real( m ) )
448 ELSE
449 eps2 = ulp
450 END IF
451 r = min( p, m-p, q, m-q )
452*
453* Copy the matrix [X11;X21] to the array XF.
454*
455 CALL slacpy( 'Full', m, q, x, ldx, xf, ldx )
456*
457* Compute the CSD
458*
459 CALL sorcsd2by1( 'Y', 'Y', 'Y', m, p, q, xf(1,1), ldx, xf(p+1,1),
460 \$ ldx, theta, u1, ldu1, u2, ldu2, v1t, ldv1t, work,
461 \$ lwork, iwork, info )
462*
463* Compute [X11;X21] := diag(U1,U2)'*[X11;X21]*V1 - [D11;D21]
464*
465 CALL sgemm( 'No transpose', 'Conjugate transpose', p, q, q, one,
466 \$ x, ldx, v1t, ldv1t, zero, work, ldx )
467*
468 CALL sgemm( 'Conjugate transpose', 'No transpose', p, q, p, one,
469 \$ u1, ldu1, work, ldx, zero, x, ldx )
470*
471 DO i = 1, min(p,q)-r
472 x(i,i) = x(i,i) - one
473 END DO
474 DO i = 1, r
475 x(min(p,q)-r+i,min(p,q)-r+i) =
476 \$ x(min(p,q)-r+i,min(p,q)-r+i) - cos(theta(i))
477 END DO
478*
479 CALL sgemm( 'No transpose', 'Conjugate transpose', m-p, q, q, one,
480 \$ x(p+1,1), ldx, v1t, ldv1t, zero, work, ldx )
481*
482 CALL sgemm( 'Conjugate transpose', 'No transpose', m-p, q, m-p,
483 \$ one, u2, ldu2, work, ldx, zero, x(p+1,1), ldx )
484*
485 DO i = 1, min(m-p,q)-r
486 x(m-i+1,q-i+1) = x(m-i+1,q-i+1) - one
487 END DO
488 DO i = 1, r
489 x(m-(min(m-p,q)-r)+1-i,q-(min(m-p,q)-r)+1-i) =
490 \$ x(m-(min(m-p,q)-r)+1-i,q-(min(m-p,q)-r)+1-i) -
491 \$ sin(theta(r-i+1))
492 END DO
493*
494* Compute norm( U1'*X11*V1 - D11 ) / ( MAX(1,P,Q)*EPS2 ) .
495*
496 resid = slange( '1', p, q, x, ldx, rwork )
497 result( 10 ) = ( resid / real(max(1,p,q)) ) / eps2
498*
499* Compute norm( U2'*X21*V1 - D21 ) / ( MAX(1,M-P,Q)*EPS2 ) .
500*
501 resid = slange( '1', m-p, q, x(p+1,1), ldx, rwork )
502 result( 11 ) = ( resid / real(max(1,m-p,q)) ) / eps2
503*
504* Compute I - U1'*U1
505*
506 CALL slaset( 'Full', p, p, zero, one, work, ldu1 )
507 CALL ssyrk( 'Upper', 'Conjugate transpose', p, p, -one, u1, ldu1,
508 \$ one, work, ldu1 )
509*
510* Compute norm( I - U1'*U1 ) / ( MAX(1,P) * ULP ) .
511*
512 resid = slansy( '1', 'Upper', p, work, ldu1, rwork )
513 result( 12 ) = ( resid / real(max(1,p)) ) / ulp
514*
515* Compute I - U2'*U2
516*
517 CALL slaset( 'Full', m-p, m-p, zero, one, work, ldu2 )
518 CALL ssyrk( 'Upper', 'Conjugate transpose', m-p, m-p, -one, u2,
519 \$ ldu2, one, work, ldu2 )
520*
521* Compute norm( I - U2'*U2 ) / ( MAX(1,M-P) * ULP ) .
522*
523 resid = slansy( '1', 'Upper', m-p, work, ldu2, rwork )
524 result( 13 ) = ( resid / real(max(1,m-p)) ) / ulp
525*
526* Compute I - V1T*V1T'
527*
528 CALL slaset( 'Full', q, q, zero, one, work, ldv1t )
529 CALL ssyrk( 'Upper', 'No transpose', q, q, -one, v1t, ldv1t, one,
530 \$ work, ldv1t )
531*
532* Compute norm( I - V1T*V1T' ) / ( MAX(1,Q) * ULP ) .
533*
534 resid = slansy( '1', 'Upper', q, work, ldv1t, rwork )
535 result( 14 ) = ( resid / real(max(1,q)) ) / ulp
536*
537* Check sorting
538*
539 result( 15 ) = realzero
540 DO i = 1, r
541 IF( theta(i).LT.realzero .OR. theta(i).GT.piover2 ) THEN
542 result( 15 ) = ulpinv
543 END IF
544 IF( i.GT.1 ) THEN
545 IF ( theta(i).LT.theta(i-1) ) THEN
546 result( 15 ) = ulpinv
547 END IF
548 END IF
549 END DO
550*
551 RETURN
552*
553* End of SCSDTS
554*
subroutine sgemm(transa, transb, m, n, k, alpha, a, lda, b, ldb, beta, c, ldc)
SGEMM
Definition sgemm.f:188
subroutine ssyrk(uplo, trans, n, k, alpha, a, lda, beta, c, ldc)
SSYRK
Definition ssyrk.f:169
subroutine slacpy(uplo, m, n, a, lda, b, ldb)
SLACPY copies all or part of one two-dimensional array to another.
Definition slacpy.f:103
real function slamch(cmach)
SLAMCH
Definition slamch.f:68
real function slange(norm, m, n, a, lda, work)
SLANGE returns the value of the 1-norm, Frobenius norm, infinity-norm, or the largest absolute value ...
Definition slange.f:114
real function slansy(norm, uplo, n, a, lda, work)
SLANSY returns the value of the 1-norm, or the Frobenius norm, or the infinity norm,...
Definition slansy.f:122
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 sorcsd2by1(jobu1, jobu2, jobv1t, m, p, q, x11, ldx11, x21, ldx21, theta, u1, ldu1, u2, ldu2, v1t, ldv1t, work, lwork, iwork, info)
SORCSD2BY1
Definition sorcsd2by1.f:233
recursive subroutine sorcsd(jobu1, jobu2, jobv1t, jobv2t, trans, signs, m, p, q, x11, ldx11, x12, ldx12, x21, ldx21, x22, ldx22, theta, u1, ldu1, u2, ldu2, v1t, ldv1t, v2t, ldv2t, work, lwork, iwork, info)
SORCSD
Definition sorcsd.f:300
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