 LAPACK 3.11.0 LAPACK: Linear Algebra PACKage
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## ◆ dgqrts()

 subroutine dgqrts ( integer N, integer M, integer P, double precision, dimension( lda, * ) A, double precision, dimension( lda, * ) AF, double precision, dimension( lda, * ) Q, double precision, dimension( lda, * ) R, integer LDA, double precision, dimension( * ) TAUA, double precision, dimension( ldb, * ) B, double precision, dimension( ldb, * ) BF, double precision, dimension( ldb, * ) Z, double precision, dimension( ldb, * ) T, double precision, dimension( ldb, * ) BWK, integer LDB, double precision, dimension( * ) TAUB, double precision, dimension( lwork ) WORK, integer LWORK, double precision, dimension( * ) RWORK, double precision, dimension( 4 ) RESULT )

DGQRTS

Purpose:
``` DGQRTS tests DGGQRF, which computes the GQR factorization of an
N-by-M matrix A and a N-by-P matrix B: A = Q*R and B = Q*T*Z.```
Parameters
 [in] N ``` N is INTEGER The number of rows of the matrices A and B. N >= 0.``` [in] M ``` M is INTEGER The number of columns of the matrix A. M >= 0.``` [in] P ``` P is INTEGER The number of columns of the matrix B. P >= 0.``` [in] A ``` A is DOUBLE PRECISION array, dimension (LDA,M) The N-by-M matrix A.``` [out] AF ``` AF is DOUBLE PRECISION array, dimension (LDA,N) Details of the GQR factorization of A and B, as returned by DGGQRF, see SGGQRF for further details.``` [out] Q ``` Q is DOUBLE PRECISION array, dimension (LDA,N) The M-by-M orthogonal matrix Q.``` [out] R ` R is DOUBLE PRECISION array, dimension (LDA,MAX(M,N))` [in] LDA ``` LDA is INTEGER The leading dimension of the arrays A, AF, R and Q. LDA >= max(M,N).``` [out] TAUA ``` TAUA is DOUBLE PRECISION array, dimension (min(M,N)) The scalar factors of the elementary reflectors, as returned by DGGQRF.``` [in] B ``` B is DOUBLE PRECISION array, dimension (LDB,P) On entry, the N-by-P matrix A.``` [out] BF ``` BF is DOUBLE PRECISION array, dimension (LDB,N) Details of the GQR factorization of A and B, as returned by DGGQRF, see SGGQRF for further details.``` [out] Z ``` Z is DOUBLE PRECISION array, dimension (LDB,P) The P-by-P orthogonal matrix Z.``` [out] T ` T is DOUBLE PRECISION array, dimension (LDB,max(P,N))` [out] BWK ` BWK is DOUBLE PRECISION array, dimension (LDB,N)` [in] LDB ``` LDB is INTEGER The leading dimension of the arrays B, BF, Z and T. LDB >= max(P,N).``` [out] TAUB ``` TAUB is DOUBLE PRECISION array, dimension (min(P,N)) The scalar factors of the elementary reflectors, as returned by DGGRQF.``` [out] WORK ` WORK is DOUBLE PRECISION array, dimension (LWORK)` [in] LWORK ``` LWORK is INTEGER The dimension of the array WORK, LWORK >= max(N,M,P)**2.``` [out] RWORK ` RWORK is DOUBLE PRECISION array, dimension (max(N,M,P))` [out] RESULT ``` RESULT is DOUBLE PRECISION array, dimension (4) The test ratios: RESULT(1) = norm( R - Q'*A ) / ( MAX(M,N)*norm(A)*ULP) RESULT(2) = norm( T*Z - Q'*B ) / (MAX(P,N)*norm(B)*ULP) RESULT(3) = norm( I - Q'*Q ) / ( M*ULP ) RESULT(4) = norm( I - Z'*Z ) / ( P*ULP )```

Definition at line 174 of file dgqrts.f.

176*
177* -- LAPACK test routine --
178* -- LAPACK is a software package provided by Univ. of Tennessee, --
179* -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
180*
181* .. Scalar Arguments ..
182 INTEGER LDA, LDB, LWORK, M, N, P
183* ..
184* .. Array Arguments ..
185 DOUBLE PRECISION A( LDA, * ), AF( LDA, * ), B( LDB, * ),
186 \$ BF( LDB, * ), BWK( LDB, * ), Q( LDA, * ),
187 \$ R( LDA, * ), RESULT( 4 ), RWORK( * ),
188 \$ T( LDB, * ), TAUA( * ), TAUB( * ),
189 \$ WORK( LWORK ), Z( LDB, * )
190* ..
191*
192* =====================================================================
193*
194* .. Parameters ..
195 DOUBLE PRECISION ZERO, ONE
196 parameter( zero = 0.0d+0, one = 1.0d+0 )
197 DOUBLE PRECISION ROGUE
198 parameter( rogue = -1.0d+10 )
199* ..
200* .. Local Scalars ..
201 INTEGER INFO
202 DOUBLE PRECISION ANORM, BNORM, RESID, ULP, UNFL
203* ..
204* .. External Functions ..
205 DOUBLE PRECISION DLAMCH, DLANGE, DLANSY
206 EXTERNAL dlamch, dlange, dlansy
207* ..
208* .. External Subroutines ..
209 EXTERNAL dgemm, dggqrf, dlacpy, dlaset, dorgqr, dorgrq,
210 \$ dsyrk
211* ..
212* .. Intrinsic Functions ..
213 INTRINSIC dble, max, min
214* ..
215* .. Executable Statements ..
216*
217 ulp = dlamch( 'Precision' )
218 unfl = dlamch( 'Safe minimum' )
219*
220* Copy the matrix A to the array AF.
221*
222 CALL dlacpy( 'Full', n, m, a, lda, af, lda )
223 CALL dlacpy( 'Full', n, p, b, ldb, bf, ldb )
224*
225 anorm = max( dlange( '1', n, m, a, lda, rwork ), unfl )
226 bnorm = max( dlange( '1', n, p, b, ldb, rwork ), unfl )
227*
228* Factorize the matrices A and B in the arrays AF and BF.
229*
230 CALL dggqrf( n, m, p, af, lda, taua, bf, ldb, taub, work, lwork,
231 \$ info )
232*
233* Generate the N-by-N matrix Q
234*
235 CALL dlaset( 'Full', n, n, rogue, rogue, q, lda )
236 CALL dlacpy( 'Lower', n-1, m, af( 2, 1 ), lda, q( 2, 1 ), lda )
237 CALL dorgqr( n, n, min( n, m ), q, lda, taua, work, lwork, info )
238*
239* Generate the P-by-P matrix Z
240*
241 CALL dlaset( 'Full', p, p, rogue, rogue, z, ldb )
242 IF( n.LE.p ) THEN
243 IF( n.GT.0 .AND. n.LT.p )
244 \$ CALL dlacpy( 'Full', n, p-n, bf, ldb, z( p-n+1, 1 ), ldb )
245 IF( n.GT.1 )
246 \$ CALL dlacpy( 'Lower', n-1, n-1, bf( 2, p-n+1 ), ldb,
247 \$ z( p-n+2, p-n+1 ), ldb )
248 ELSE
249 IF( p.GT.1 )
250 \$ CALL dlacpy( 'Lower', p-1, p-1, bf( n-p+2, 1 ), ldb,
251 \$ z( 2, 1 ), ldb )
252 END IF
253 CALL dorgrq( p, p, min( n, p ), z, ldb, taub, work, lwork, info )
254*
255* Copy R
256*
257 CALL dlaset( 'Full', n, m, zero, zero, r, lda )
258 CALL dlacpy( 'Upper', n, m, af, lda, r, lda )
259*
260* Copy T
261*
262 CALL dlaset( 'Full', n, p, zero, zero, t, ldb )
263 IF( n.LE.p ) THEN
264 CALL dlacpy( 'Upper', n, n, bf( 1, p-n+1 ), ldb, t( 1, p-n+1 ),
265 \$ ldb )
266 ELSE
267 CALL dlacpy( 'Full', n-p, p, bf, ldb, t, ldb )
268 CALL dlacpy( 'Upper', p, p, bf( n-p+1, 1 ), ldb, t( n-p+1, 1 ),
269 \$ ldb )
270 END IF
271*
272* Compute R - Q'*A
273*
274 CALL dgemm( 'Transpose', 'No transpose', n, m, n, -one, q, lda, a,
275 \$ lda, one, r, lda )
276*
277* Compute norm( R - Q'*A ) / ( MAX(M,N)*norm(A)*ULP ) .
278*
279 resid = dlange( '1', n, m, r, lda, rwork )
280 IF( anorm.GT.zero ) THEN
281 result( 1 ) = ( ( resid / dble( max( 1, m, n ) ) ) / anorm ) /
282 \$ ulp
283 ELSE
284 result( 1 ) = zero
285 END IF
286*
287* Compute T*Z - Q'*B
288*
289 CALL dgemm( 'No Transpose', 'No transpose', n, p, p, one, t, ldb,
290 \$ z, ldb, zero, bwk, ldb )
291 CALL dgemm( 'Transpose', 'No transpose', n, p, n, -one, q, lda, b,
292 \$ ldb, one, bwk, ldb )
293*
294* Compute norm( T*Z - Q'*B ) / ( MAX(P,N)*norm(A)*ULP ) .
295*
296 resid = dlange( '1', n, p, bwk, ldb, rwork )
297 IF( bnorm.GT.zero ) THEN
298 result( 2 ) = ( ( resid / dble( max( 1, p, n ) ) ) / bnorm ) /
299 \$ ulp
300 ELSE
301 result( 2 ) = zero
302 END IF
303*
304* Compute I - Q'*Q
305*
306 CALL dlaset( 'Full', n, n, zero, one, r, lda )
307 CALL dsyrk( 'Upper', 'Transpose', n, n, -one, q, lda, one, r,
308 \$ lda )
309*
310* Compute norm( I - Q'*Q ) / ( N * ULP ) .
311*
312 resid = dlansy( '1', 'Upper', n, r, lda, rwork )
313 result( 3 ) = ( resid / dble( max( 1, n ) ) ) / ulp
314*
315* Compute I - Z'*Z
316*
317 CALL dlaset( 'Full', p, p, zero, one, t, ldb )
318 CALL dsyrk( 'Upper', 'Transpose', p, p, -one, z, ldb, one, t,
319 \$ ldb )
320*
321* Compute norm( I - Z'*Z ) / ( P*ULP ) .
322*
323 resid = dlansy( '1', 'Upper', p, t, ldb, rwork )
324 result( 4 ) = ( resid / dble( max( 1, p ) ) ) / ulp
325*
326 RETURN
327*
328* End of DGQRTS
329*
double precision function dlamch(CMACH)
DLAMCH
Definition: dlamch.f:69
subroutine dlacpy(UPLO, M, N, A, LDA, B, LDB)
DLACPY copies all or part of one two-dimensional array to another.
Definition: dlacpy.f:103
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:110
subroutine dsyrk(UPLO, TRANS, N, K, ALPHA, A, LDA, BETA, C, LDC)
DSYRK
Definition: dsyrk.f:169
subroutine dgemm(TRANSA, TRANSB, M, N, K, ALPHA, A, LDA, B, LDB, BETA, C, LDC)
DGEMM
Definition: dgemm.f:187
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:114
subroutine dorgrq(M, N, K, A, LDA, TAU, WORK, LWORK, INFO)
DORGRQ
Definition: dorgrq.f:128
subroutine dorgqr(M, N, K, A, LDA, TAU, WORK, LWORK, INFO)
DORGQR
Definition: dorgqr.f:128
subroutine dggqrf(N, M, P, A, LDA, TAUA, B, LDB, TAUB, WORK, LWORK, INFO)
DGGQRF
Definition: dggqrf.f:215
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,...
Definition: dlansy.f:122
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