 LAPACK  3.8.0 LAPACK: Linear Algebra PACKage

## ◆ dcsdts()

 subroutine dcsdts ( integer M, integer P, integer Q, double precision, dimension( ldx, * ) X, double precision, dimension( ldx, * ) XF, integer LDX, double precision, dimension( ldu1, * ) U1, integer LDU1, double precision, dimension( ldu2, * ) U2, integer LDU2, double precision, dimension( ldv1t, * ) V1T, integer LDV1T, double precision, dimension( ldv2t, * ) V2T, integer LDV2T, double precision, dimension( * ) THETA, integer, dimension( * ) IWORK, double precision, dimension( lwork ) WORK, integer LWORK, double precision, dimension( * ) RWORK, double precision, dimension( 15 ) RESULT )

DCSDTS

Purpose:
``` DCSDTS tests DORCSD, 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 DORCSD2BY1, 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 DOUBLE PRECISION array, dimension (LDX,M) The M-by-M matrix X.``` [out] XF ``` XF is DOUBLE PRECISION array, dimension (LDX,M) Details of the CSD of X, as returned by DORCSD; see DORCSD 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 DOUBLE PRECISION 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 DOUBLE PRECISION 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 DOUBLE PRECISION 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 DOUBLE PRECISION 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 DOUBLE PRECISION 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 DORCSD for details.``` [out] IWORK ` IWORK is INTEGER array, dimension (M)` [out] WORK ` WORK is DOUBLE PRECISION array, dimension (LWORK)` [in] LWORK ``` LWORK is INTEGER The dimension of the array WORK``` [out] RWORK ` RWORK is DOUBLE PRECISION array` [out] RESULT ``` RESULT is DOUBLE PRECISION 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 ). )```
Date
December 2016

Definition at line 231 of file dcsdts.f.

231 *
232 * -- LAPACK test routine (version 3.7.0) --
233 * -- LAPACK is a software package provided by Univ. of Tennessee, --
234 * -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
235 * December 2016
236 *
237 * .. Scalar Arguments ..
238  INTEGER ldx, ldu1, ldu2, ldv1t, ldv2t, lwork, m, p, q
239 * ..
240 * .. Array Arguments ..
241  INTEGER iwork( * )
242  DOUBLE PRECISION result( 15 ), rwork( * ), theta( * )
243  DOUBLE PRECISION u1( ldu1, * ), u2( ldu2, * ), v1t( ldv1t, * ),
244  \$ v2t( ldv2t, * ), work( lwork ), x( ldx, * ),
245  \$ xf( ldx, * )
246 * ..
247 *
248 * =====================================================================
249 *
250 * .. Parameters ..
251  DOUBLE PRECISION piover2, realone, realzero
252  parameter( piover2 = 1.57079632679489662d0,
253  \$ realone = 1.0d0, realzero = 0.0d0 )
254  DOUBLE PRECISION zero, one
255  parameter( zero = 0.0d0, one = 1.0d0 )
256 * ..
257 * .. Local Scalars ..
258  INTEGER i, info, r
259  DOUBLE PRECISION eps2, resid, ulp, ulpinv
260 * ..
261 * .. External Functions ..
262  DOUBLE PRECISION dlamch, dlange, dlansy
263  EXTERNAL dlamch, dlange, dlansy
264 * ..
265 * .. External Subroutines ..
266  EXTERNAL dgemm, dlacpy, dlaset, dorcsd, dorcsd2by1,
267  \$ dsyrk
268 * ..
269 * .. Intrinsic Functions ..
270  INTRINSIC cos, dble, max, min, sin
271 * ..
272 * .. Executable Statements ..
273 *
274  ulp = dlamch( 'Precision' )
275  ulpinv = realone / ulp
276 *
277 * The first half of the routine checks the 2-by-2 CSD
278 *
279  CALL dlaset( 'Full', m, m, zero, one, work, ldx )
280  CALL dsyrk( 'Upper', 'Conjugate transpose', m, m, -one, x, ldx,
281  \$ one, work, ldx )
282  IF (m.GT.0) THEN
283  eps2 = max( ulp,
284  \$ dlange( '1', m, m, work, ldx, rwork ) / dble( m ) )
285  ELSE
286  eps2 = ulp
287  END IF
288  r = min( p, m-p, q, m-q )
289 *
290 * Copy the matrix X to the array XF.
291 *
292  CALL dlacpy( 'Full', m, m, x, ldx, xf, ldx )
293 *
294 * Compute the CSD
295 *
296  CALL dorcsd( 'Y', 'Y', 'Y', 'Y', 'N', 'D', m, p, q, xf(1,1), ldx,
297  \$ xf(1,q+1), ldx, xf(p+1,1), ldx, xf(p+1,q+1), ldx,
298  \$ theta, u1, ldu1, u2, ldu2, v1t, ldv1t, v2t, ldv2t,
299  \$ work, lwork, iwork, info )
300 *
301 * Compute XF := diag(U1,U2)'*X*diag(V1,V2) - [D11 D12; D21 D22]
302 *
303  CALL dlacpy( 'Full', m, m, x, ldx, xf, ldx )
304 *
305  CALL dgemm( 'No transpose', 'Conjugate transpose', p, q, q, one,
306  \$ xf, ldx, v1t, ldv1t, zero, work, ldx )
307 *
308  CALL dgemm( 'Conjugate transpose', 'No transpose', p, q, p, one,
309  \$ u1, ldu1, work, ldx, zero, xf, ldx )
310 *
311  DO i = 1, min(p,q)-r
312  xf(i,i) = xf(i,i) - one
313  END DO
314  DO i = 1, r
315  xf(min(p,q)-r+i,min(p,q)-r+i) =
316  \$ xf(min(p,q)-r+i,min(p,q)-r+i) - cos(theta(i))
317  END DO
318 *
319  CALL dgemm( 'No transpose', 'Conjugate transpose', p, m-q, m-q,
320  \$ one, xf(1,q+1), ldx, v2t, ldv2t, zero, work, ldx )
321 *
322  CALL dgemm( 'Conjugate transpose', 'No transpose', p, m-q, p,
323  \$ one, u1, ldu1, work, ldx, zero, xf(1,q+1), ldx )
324 *
325  DO i = 1, min(p,m-q)-r
326  xf(p-i+1,m-i+1) = xf(p-i+1,m-i+1) + one
327  END DO
328  DO i = 1, r
329  xf(p-(min(p,m-q)-r)+1-i,m-(min(p,m-q)-r)+1-i) =
330  \$ xf(p-(min(p,m-q)-r)+1-i,m-(min(p,m-q)-r)+1-i) +
331  \$ sin(theta(r-i+1))
332  END DO
333 *
334  CALL dgemm( 'No transpose', 'Conjugate transpose', m-p, q, q, one,
335  \$ xf(p+1,1), ldx, v1t, ldv1t, zero, work, ldx )
336 *
337  CALL dgemm( 'Conjugate transpose', 'No transpose', m-p, q, m-p,
338  \$ one, u2, ldu2, work, ldx, zero, xf(p+1,1), ldx )
339 *
340  DO i = 1, min(m-p,q)-r
341  xf(m-i+1,q-i+1) = xf(m-i+1,q-i+1) - one
342  END DO
343  DO i = 1, r
344  xf(m-(min(m-p,q)-r)+1-i,q-(min(m-p,q)-r)+1-i) =
345  \$ xf(m-(min(m-p,q)-r)+1-i,q-(min(m-p,q)-r)+1-i) -
346  \$ sin(theta(r-i+1))
347  END DO
348 *
349  CALL dgemm( 'No transpose', 'Conjugate transpose', m-p, m-q, m-q,
350  \$ one, xf(p+1,q+1), ldx, v2t, ldv2t, zero, work, ldx )
351 *
352  CALL dgemm( 'Conjugate transpose', 'No transpose', m-p, m-q, m-p,
353  \$ one, u2, ldu2, work, ldx, zero, xf(p+1,q+1), ldx )
354 *
355  DO i = 1, min(m-p,m-q)-r
356  xf(p+i,q+i) = xf(p+i,q+i) - one
357  END DO
358  DO i = 1, r
359  xf(p+(min(m-p,m-q)-r)+i,q+(min(m-p,m-q)-r)+i) =
360  \$ xf(p+(min(m-p,m-q)-r)+i,q+(min(m-p,m-q)-r)+i) -
361  \$ cos(theta(i))
362  END DO
363 *
364 * Compute norm( U1'*X11*V1 - D11 ) / ( MAX(1,P,Q)*EPS2 ) .
365 *
366  resid = dlange( '1', p, q, xf, ldx, rwork )
367  result( 1 ) = ( resid / dble(max(1,p,q)) ) / eps2
368 *
369 * Compute norm( U1'*X12*V2 - D12 ) / ( MAX(1,P,M-Q)*EPS2 ) .
370 *
371  resid = dlange( '1', p, m-q, xf(1,q+1), ldx, rwork )
372  result( 2 ) = ( resid / dble(max(1,p,m-q)) ) / eps2
373 *
374 * Compute norm( U2'*X21*V1 - D21 ) / ( MAX(1,M-P,Q)*EPS2 ) .
375 *
376  resid = dlange( '1', m-p, q, xf(p+1,1), ldx, rwork )
377  result( 3 ) = ( resid / dble(max(1,m-p,q)) ) / eps2
378 *
379 * Compute norm( U2'*X22*V2 - D22 ) / ( MAX(1,M-P,M-Q)*EPS2 ) .
380 *
381  resid = dlange( '1', m-p, m-q, xf(p+1,q+1), ldx, rwork )
382  result( 4 ) = ( resid / dble(max(1,m-p,m-q)) ) / eps2
383 *
384 * Compute I - U1'*U1
385 *
386  CALL dlaset( 'Full', p, p, zero, one, work, ldu1 )
387  CALL dsyrk( 'Upper', 'Conjugate transpose', p, p, -one, u1, ldu1,
388  \$ one, work, ldu1 )
389 *
390 * Compute norm( I - U'*U ) / ( MAX(1,P) * ULP ) .
391 *
392  resid = dlansy( '1', 'Upper', p, work, ldu1, rwork )
393  result( 5 ) = ( resid / dble(max(1,p)) ) / ulp
394 *
395 * Compute I - U2'*U2
396 *
397  CALL dlaset( 'Full', m-p, m-p, zero, one, work, ldu2 )
398  CALL dsyrk( 'Upper', 'Conjugate transpose', m-p, m-p, -one, u2,
399  \$ ldu2, one, work, ldu2 )
400 *
401 * Compute norm( I - U2'*U2 ) / ( MAX(1,M-P) * ULP ) .
402 *
403  resid = dlansy( '1', 'Upper', m-p, work, ldu2, rwork )
404  result( 6 ) = ( resid / dble(max(1,m-p)) ) / ulp
405 *
406 * Compute I - V1T*V1T'
407 *
408  CALL dlaset( 'Full', q, q, zero, one, work, ldv1t )
409  CALL dsyrk( 'Upper', 'No transpose', q, q, -one, v1t, ldv1t, one,
410  \$ work, ldv1t )
411 *
412 * Compute norm( I - V1T*V1T' ) / ( MAX(1,Q) * ULP ) .
413 *
414  resid = dlansy( '1', 'Upper', q, work, ldv1t, rwork )
415  result( 7 ) = ( resid / dble(max(1,q)) ) / ulp
416 *
417 * Compute I - V2T*V2T'
418 *
419  CALL dlaset( 'Full', m-q, m-q, zero, one, work, ldv2t )
420  CALL dsyrk( 'Upper', 'No transpose', m-q, m-q, -one, v2t, ldv2t,
421  \$ one, work, ldv2t )
422 *
423 * Compute norm( I - V2T*V2T' ) / ( MAX(1,M-Q) * ULP ) .
424 *
425  resid = dlansy( '1', 'Upper', m-q, work, ldv2t, rwork )
426  result( 8 ) = ( resid / dble(max(1,m-q)) ) / ulp
427 *
428 * Check sorting
429 *
430  result( 9 ) = realzero
431  DO i = 1, r
432  IF( theta(i).LT.realzero .OR. theta(i).GT.piover2 ) THEN
433  result( 9 ) = ulpinv
434  END IF
435  IF( i.GT.1 ) THEN
436  IF ( theta(i).LT.theta(i-1) ) THEN
437  result( 9 ) = ulpinv
438  END IF
439  END IF
440  END DO
441 *
442 * The second half of the routine checks the 2-by-1 CSD
443 *
444  CALL dlaset( 'Full', q, q, zero, one, work, ldx )
445  CALL dsyrk( 'Upper', 'Conjugate transpose', q, m, -one, x, ldx,
446  \$ one, work, ldx )
447  IF( m.GT.0 ) THEN
448  eps2 = max( ulp,
449  \$ dlange( '1', q, q, work, ldx, rwork ) / dble( m ) )
450  ELSE
451  eps2 = ulp
452  END IF
453  r = min( p, m-p, q, m-q )
454 *
455 * Copy the matrix [ X11; X21 ] to the array XF.
456 *
457  CALL dlacpy( 'Full', m, q, x, ldx, xf, ldx )
458 *
459 * Compute the CSD
460 *
461  CALL dorcsd2by1( 'Y', 'Y', 'Y', m, p, q, xf(1,1), ldx, xf(p+1,1),
462  \$ ldx, theta, u1, ldu1, u2, ldu2, v1t, ldv1t, work,
463  \$ lwork, iwork, info )
464 *
465 * Compute [X11;X21] := diag(U1,U2)'*[X11;X21]*V1 - [D11;D21]
466 *
467  CALL dgemm( 'No transpose', 'Conjugate transpose', p, q, q, one,
468  \$ x, ldx, v1t, ldv1t, zero, work, ldx )
469 *
470  CALL dgemm( 'Conjugate transpose', 'No transpose', p, q, p, one,
471  \$ u1, ldu1, work, ldx, zero, x, ldx )
472 *
473  DO i = 1, min(p,q)-r
474  x(i,i) = x(i,i) - one
475  END DO
476  DO i = 1, r
477  x(min(p,q)-r+i,min(p,q)-r+i) =
478  \$ x(min(p,q)-r+i,min(p,q)-r+i) - cos(theta(i))
479  END DO
480 *
481  CALL dgemm( 'No transpose', 'Conjugate transpose', m-p, q, q, one,
482  \$ x(p+1,1), ldx, v1t, ldv1t, zero, work, ldx )
483 *
484  CALL dgemm( 'Conjugate transpose', 'No transpose', m-p, q, m-p,
485  \$ one, u2, ldu2, work, ldx, zero, x(p+1,1), ldx )
486 *
487  DO i = 1, min(m-p,q)-r
488  x(m-i+1,q-i+1) = x(m-i+1,q-i+1) - one
489  END DO
490  DO i = 1, r
491  x(m-(min(m-p,q)-r)+1-i,q-(min(m-p,q)-r)+1-i) =
492  \$ x(m-(min(m-p,q)-r)+1-i,q-(min(m-p,q)-r)+1-i) -
493  \$ sin(theta(r-i+1))
494  END DO
495 *
496 * Compute norm( U1'*X11*V1 - D11 ) / ( MAX(1,P,Q)*EPS2 ) .
497 *
498  resid = dlange( '1', p, q, x, ldx, rwork )
499  result( 10 ) = ( resid / dble(max(1,p,q)) ) / eps2
500 *
501 * Compute norm( U2'*X21*V1 - D21 ) / ( MAX(1,M-P,Q)*EPS2 ) .
502 *
503  resid = dlange( '1', m-p, q, x(p+1,1), ldx, rwork )
504  result( 11 ) = ( resid / dble(max(1,m-p,q)) ) / eps2
505 *
506 * Compute I - U1'*U1
507 *
508  CALL dlaset( 'Full', p, p, zero, one, work, ldu1 )
509  CALL dsyrk( 'Upper', 'Conjugate transpose', p, p, -one, u1, ldu1,
510  \$ one, work, ldu1 )
511 *
512 * Compute norm( I - U1'*U1 ) / ( MAX(1,P) * ULP ) .
513 *
514  resid = dlansy( '1', 'Upper', p, work, ldu1, rwork )
515  result( 12 ) = ( resid / dble(max(1,p)) ) / ulp
516 *
517 * Compute I - U2'*U2
518 *
519  CALL dlaset( 'Full', m-p, m-p, zero, one, work, ldu2 )
520  CALL dsyrk( 'Upper', 'Conjugate transpose', m-p, m-p, -one, u2,
521  \$ ldu2, one, work, ldu2 )
522 *
523 * Compute norm( I - U2'*U2 ) / ( MAX(1,M-P) * ULP ) .
524 *
525  resid = dlansy( '1', 'Upper', m-p, work, ldu2, rwork )
526  result( 13 ) = ( resid / dble(max(1,m-p)) ) / ulp
527 *
528 * Compute I - V1T*V1T'
529 *
530  CALL dlaset( 'Full', q, q, zero, one, work, ldv1t )
531  CALL dsyrk( 'Upper', 'No transpose', q, q, -one, v1t, ldv1t, one,
532  \$ work, ldv1t )
533 *
534 * Compute norm( I - V1T*V1T' ) / ( MAX(1,Q) * ULP ) .
535 *
536  resid = dlansy( '1', 'Upper', q, work, ldv1t, rwork )
537  result( 14 ) = ( resid / dble(max(1,q)) ) / ulp
538 *
539 * Check sorting
540 *
541  result( 15 ) = realzero
542  DO i = 1, r
543  IF( theta(i).LT.realzero .OR. theta(i).GT.piover2 ) THEN
544  result( 15 ) = ulpinv
545  END IF
546  IF( i.GT.1 ) THEN
547  IF ( theta(i).LT.theta(i-1) ) THEN
548  result( 15 ) = ulpinv
549  END IF
550  END IF
551  END DO
552 *
553  RETURN
554 *
555 * End of DCSDTS
556 *
subroutine dlacpy(UPLO, M, N, A, LDA, B, LDB)
DLACPY copies all or part of one two-dimensional array to another.
Definition: dlacpy.f:105
double precision function dlamch(CMACH)
DLAMCH
Definition: dlamch.f:65
double precision function dlansy(NORM, UPLO, N, A, LDA, WORK)
DLANSY returns the value of the 1-norm, or the Frobenius norm, or the infinity norm, or the element of largest absolute value of a real symmetric matrix.
Definition: dlansy.f:124
subroutine dlaset(UPLO, M, N, ALPHA, BETA, A, LDA)
DLASET initializes the off-diagonal elements and the diagonal elements of a matrix to given values...
Definition: dlaset.f:112
subroutine dgemm(TRANSA, TRANSB, M, N, K, ALPHA, A, LDA, B, LDB, BETA, C, LDC)
DGEMM
Definition: dgemm.f:189
subroutine dsyrk(UPLO, TRANS, N, K, ALPHA, A, LDA, BETA, C, LDC)
DSYRK
Definition: dsyrk.f:171
recursive subroutine dorcsd(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)
DORCSD
Definition: dorcsd.f:302
double precision function dlange(NORM, M, N, A, LDA, WORK)
DLANGE returns the value of the 1-norm, Frobenius norm, infinity-norm, or the largest absolute value ...
Definition: dlange.f:116
subroutine dorcsd2by1(JOBU1, JOBU2, JOBV1T, M, P, Q, X11, LDX11, X21, LDX21, THETA, U1, LDU1, U2, LDU2, V1T, LDV1T, WORK, LWORK, IWORK, INFO)
DORCSD2BY1
Definition: dorcsd2by1.f:235
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