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

 subroutine dgrqts ( integer M, integer P, integer N, 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 )

DGRQTS

Purpose:
``` DGRQTS tests DGGRQF, which computes the GRQ factorization of an
M-by-N matrix A and a P-by-N matrix B: A = R*Q and B = Z*T*Q.```
Parameters
 [in] M ``` M is INTEGER The number of rows of the matrix A. M >= 0.``` [in] P ``` P is INTEGER The number of rows of the matrix B. P >= 0.``` [in] N ``` N is INTEGER The number of columns of the matrices A and B. N >= 0.``` [in] A ``` A is DOUBLE PRECISION array, dimension (LDA,N) The M-by-N matrix A.``` [out] AF ``` AF is DOUBLE PRECISION array, dimension (LDA,N) Details of the GRQ factorization of A and B, as returned by DGGRQF, see SGGRQF for further details.``` [out] Q ``` Q is DOUBLE PRECISION array, dimension (LDA,N) The N-by-N 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 DGGQRC.``` [in] B ``` B is DOUBLE PRECISION array, dimension (LDB,N) On entry, the P-by-N matrix A.``` [out] BF ``` BF is DOUBLE PRECISION array, dimension (LDB,N) Details of the GQR factorization of A and B, as returned by DGGRQF, see SGGRQF 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(M,P,N)**2.``` [out] RWORK ` RWORK is DOUBLE PRECISION array, dimension (M)` [out] RESULT ``` RESULT is DOUBLE PRECISION array, dimension (4) The test ratios: RESULT(1) = norm( R - A*Q' ) / ( MAX(M,N)*norm(A)*ULP) RESULT(2) = norm( T*Q - Z'*B ) / (MAX(P,N)*norm(B)*ULP) RESULT(3) = norm( I - Q'*Q ) / ( N*ULP ) RESULT(4) = norm( I - Z'*Z ) / ( P*ULP )```

Definition at line 174 of file dgrqts.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, dggrqf, 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', m, n, a, lda, af, lda )
223 CALL dlacpy( 'Full', p, n, b, ldb, bf, ldb )
224*
225 anorm = max( dlange( '1', m, n, a, lda, rwork ), unfl )
226 bnorm = max( dlange( '1', p, n, b, ldb, rwork ), unfl )
227*
228* Factorize the matrices A and B in the arrays AF and BF.
229*
230 CALL dggrqf( m, p, n, 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 IF( m.LE.n ) THEN
237 IF( m.GT.0 .AND. m.LT.n )
238 \$ CALL dlacpy( 'Full', m, n-m, af, lda, q( n-m+1, 1 ), lda )
239 IF( m.GT.1 )
240 \$ CALL dlacpy( 'Lower', m-1, m-1, af( 2, n-m+1 ), lda,
241 \$ q( n-m+2, n-m+1 ), lda )
242 ELSE
243 IF( n.GT.1 )
244 \$ CALL dlacpy( 'Lower', n-1, n-1, af( m-n+2, 1 ), lda,
245 \$ q( 2, 1 ), lda )
246 END IF
247 CALL dorgrq( n, n, min( m, n ), q, lda, taua, work, lwork, info )
248*
249* Generate the P-by-P matrix Z
250*
251 CALL dlaset( 'Full', p, p, rogue, rogue, z, ldb )
252 IF( p.GT.1 )
253 \$ CALL dlacpy( 'Lower', p-1, n, bf( 2, 1 ), ldb, z( 2, 1 ), ldb )
254 CALL dorgqr( p, p, min( p, n ), z, ldb, taub, work, lwork, info )
255*
256* Copy R
257*
258 CALL dlaset( 'Full', m, n, zero, zero, r, lda )
259 IF( m.LE.n ) THEN
260 CALL dlacpy( 'Upper', m, m, af( 1, n-m+1 ), lda, r( 1, n-m+1 ),
261 \$ lda )
262 ELSE
263 CALL dlacpy( 'Full', m-n, n, af, lda, r, lda )
264 CALL dlacpy( 'Upper', n, n, af( m-n+1, 1 ), lda, r( m-n+1, 1 ),
265 \$ lda )
266 END IF
267*
268* Copy T
269*
270 CALL dlaset( 'Full', p, n, zero, zero, t, ldb )
271 CALL dlacpy( 'Upper', p, n, bf, ldb, t, ldb )
272*
273* Compute R - A*Q'
274*
275 CALL dgemm( 'No transpose', 'Transpose', m, n, n, -one, a, lda, q,
276 \$ lda, one, r, lda )
277*
278* Compute norm( R - A*Q' ) / ( MAX(M,N)*norm(A)*ULP ) .
279*
280 resid = dlange( '1', m, n, r, lda, rwork )
281 IF( anorm.GT.zero ) THEN
282 result( 1 ) = ( ( resid / dble( max( 1, m, n ) ) ) / anorm ) /
283 \$ ulp
284 ELSE
285 result( 1 ) = zero
286 END IF
287*
288* Compute T*Q - Z'*B
289*
290 CALL dgemm( 'Transpose', 'No transpose', p, n, p, one, z, ldb, b,
291 \$ ldb, zero, bwk, ldb )
292 CALL dgemm( 'No transpose', 'No transpose', p, n, n, one, t, ldb,
293 \$ q, lda, -one, bwk, ldb )
294*
295* Compute norm( T*Q - Z'*B ) / ( MAX(P,N)*norm(A)*ULP ) .
296*
297 resid = dlange( '1', p, n, bwk, ldb, rwork )
298 IF( bnorm.GT.zero ) THEN
299 result( 2 ) = ( ( resid / dble( max( 1, p, m ) ) ) / bnorm ) /
300 \$ ulp
301 ELSE
302 result( 2 ) = zero
303 END IF
304*
305* Compute I - Q*Q'
306*
307 CALL dlaset( 'Full', n, n, zero, one, r, lda )
308 CALL dsyrk( 'Upper', 'No Transpose', n, n, -one, q, lda, one, r,
309 \$ lda )
310*
311* Compute norm( I - Q'*Q ) / ( N * ULP ) .
312*
313 resid = dlansy( '1', 'Upper', n, r, lda, rwork )
314 result( 3 ) = ( resid / dble( max( 1, n ) ) ) / ulp
315*
316* Compute I - Z'*Z
317*
318 CALL dlaset( 'Full', p, p, zero, one, t, ldb )
319 CALL dsyrk( 'Upper', 'Transpose', p, p, -one, z, ldb, one, t,
320 \$ ldb )
321*
322* Compute norm( I - Z'*Z ) / ( P*ULP ) .
323*
324 resid = dlansy( '1', 'Upper', p, t, ldb, rwork )
325 result( 4 ) = ( resid / dble( max( 1, p ) ) ) / ulp
326*
327 RETURN
328*
329* End of DGRQTS
330*
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 dggrqf(M, P, N, A, LDA, TAUA, B, LDB, TAUB, WORK, LWORK, INFO)
DGGRQF
Definition: dggrqf.f:214
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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