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

 subroutine zgrqts ( integer M, integer P, integer N, complex*16, dimension( lda, * ) A, complex*16, dimension( lda, * ) AF, complex*16, dimension( lda, * ) Q, complex*16, dimension( lda, * ) R, integer LDA, complex*16, dimension( * ) TAUA, complex*16, dimension( ldb, * ) B, complex*16, dimension( ldb, * ) BF, complex*16, dimension( ldb, * ) Z, complex*16, dimension( ldb, * ) T, complex*16, dimension( ldb, * ) BWK, integer LDB, complex*16, dimension( * ) TAUB, complex*16, dimension( lwork ) WORK, integer LWORK, double precision, dimension( * ) RWORK, double precision, dimension( 4 ) RESULT )

ZGRQTS

Purpose:
``` ZGRQTS tests ZGGRQF, 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 COMPLEX*16 array, dimension (LDA,N) The M-by-N matrix A.``` [out] AF ``` AF is COMPLEX*16 array, dimension (LDA,N) Details of the GRQ factorization of A and B, as returned by ZGGRQF, see CGGRQF for further details.``` [out] Q ``` Q is COMPLEX*16 array, dimension (LDA,N) The N-by-N unitary matrix Q.``` [out] R ` R is COMPLEX*16 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 COMPLEX*16 array, dimension (min(M,N)) The scalar factors of the elementary reflectors, as returned by DGGQRC.``` [in] B ``` B is COMPLEX*16 array, dimension (LDB,N) On entry, the P-by-N matrix A.``` [out] BF ``` BF is COMPLEX*16 array, dimension (LDB,N) Details of the GQR factorization of A and B, as returned by ZGGRQF, see CGGRQF for further details.``` [out] Z ``` Z is DOUBLE PRECISION array, dimension (LDB,P) The P-by-P unitary matrix Z.``` [out] T ` T is COMPLEX*16 array, dimension (LDB,max(P,N))` [out] BWK ` BWK is COMPLEX*16 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 COMPLEX*16 array, dimension (min(P,N)) The scalar factors of the elementary reflectors, as returned by DGGRQF.``` [out] WORK ` WORK is COMPLEX*16 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 zgrqts.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 RESULT( 4 ), RWORK( * )
186 COMPLEX*16 A( LDA, * ), AF( LDA, * ), B( LDB, * ),
187 \$ BF( LDB, * ), BWK( LDB, * ), Q( LDA, * ),
188 \$ R( LDA, * ), 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 COMPLEX*16 CZERO, CONE
198 parameter( czero = ( 0.0d+0, 0.0d+0 ),
199 \$ cone = ( 1.0d+0, 0.0d+0 ) )
200 COMPLEX*16 CROGUE
201 parameter( crogue = ( -1.0d+10, 0.0d+0 ) )
202* ..
203* .. Local Scalars ..
204 INTEGER INFO
205 DOUBLE PRECISION ANORM, BNORM, RESID, ULP, UNFL
206* ..
207* .. External Functions ..
208 DOUBLE PRECISION DLAMCH, ZLANGE, ZLANHE
209 EXTERNAL dlamch, zlange, zlanhe
210* ..
211* .. External Subroutines ..
212 EXTERNAL zgemm, zggrqf, zherk, zlacpy, zlaset, zungqr,
213 \$ zungrq
214* ..
215* .. Intrinsic Functions ..
216 INTRINSIC dble, max, min
217* ..
218* .. Executable Statements ..
219*
220 ulp = dlamch( 'Precision' )
221 unfl = dlamch( 'Safe minimum' )
222*
223* Copy the matrix A to the array AF.
224*
225 CALL zlacpy( 'Full', m, n, a, lda, af, lda )
226 CALL zlacpy( 'Full', p, n, b, ldb, bf, ldb )
227*
228 anorm = max( zlange( '1', m, n, a, lda, rwork ), unfl )
229 bnorm = max( zlange( '1', p, n, b, ldb, rwork ), unfl )
230*
231* Factorize the matrices A and B in the arrays AF and BF.
232*
233 CALL zggrqf( m, p, n, af, lda, taua, bf, ldb, taub, work, lwork,
234 \$ info )
235*
236* Generate the N-by-N matrix Q
237*
238 CALL zlaset( 'Full', n, n, crogue, crogue, q, lda )
239 IF( m.LE.n ) THEN
240 IF( m.GT.0 .AND. m.LT.n )
241 \$ CALL zlacpy( 'Full', m, n-m, af, lda, q( n-m+1, 1 ), lda )
242 IF( m.GT.1 )
243 \$ CALL zlacpy( 'Lower', m-1, m-1, af( 2, n-m+1 ), lda,
244 \$ q( n-m+2, n-m+1 ), lda )
245 ELSE
246 IF( n.GT.1 )
247 \$ CALL zlacpy( 'Lower', n-1, n-1, af( m-n+2, 1 ), lda,
248 \$ q( 2, 1 ), lda )
249 END IF
250 CALL zungrq( n, n, min( m, n ), q, lda, taua, work, lwork, info )
251*
252* Generate the P-by-P matrix Z
253*
254 CALL zlaset( 'Full', p, p, crogue, crogue, z, ldb )
255 IF( p.GT.1 )
256 \$ CALL zlacpy( 'Lower', p-1, n, bf( 2, 1 ), ldb, z( 2, 1 ), ldb )
257 CALL zungqr( p, p, min( p, n ), z, ldb, taub, work, lwork, info )
258*
259* Copy R
260*
261 CALL zlaset( 'Full', m, n, czero, czero, r, lda )
262 IF( m.LE.n ) THEN
263 CALL zlacpy( 'Upper', m, m, af( 1, n-m+1 ), lda, r( 1, n-m+1 ),
264 \$ lda )
265 ELSE
266 CALL zlacpy( 'Full', m-n, n, af, lda, r, lda )
267 CALL zlacpy( 'Upper', n, n, af( m-n+1, 1 ), lda, r( m-n+1, 1 ),
268 \$ lda )
269 END IF
270*
271* Copy T
272*
273 CALL zlaset( 'Full', p, n, czero, czero, t, ldb )
274 CALL zlacpy( 'Upper', p, n, bf, ldb, t, ldb )
275*
276* Compute R - A*Q'
277*
278 CALL zgemm( 'No transpose', 'Conjugate transpose', m, n, n, -cone,
279 \$ a, lda, q, lda, cone, r, lda )
280*
281* Compute norm( R - A*Q' ) / ( MAX(M,N)*norm(A)*ULP ) .
282*
283 resid = zlange( '1', m, n, r, lda, rwork )
284 IF( anorm.GT.zero ) THEN
285 result( 1 ) = ( ( resid / dble( max( 1, m, n ) ) ) / anorm ) /
286 \$ ulp
287 ELSE
288 result( 1 ) = zero
289 END IF
290*
291* Compute T*Q - Z'*B
292*
293 CALL zgemm( 'Conjugate transpose', 'No transpose', p, n, p, cone,
294 \$ z, ldb, b, ldb, czero, bwk, ldb )
295 CALL zgemm( 'No transpose', 'No transpose', p, n, n, cone, t, ldb,
296 \$ q, lda, -cone, bwk, ldb )
297*
298* Compute norm( T*Q - Z'*B ) / ( MAX(P,N)*norm(A)*ULP ) .
299*
300 resid = zlange( '1', p, n, bwk, ldb, rwork )
301 IF( bnorm.GT.zero ) THEN
302 result( 2 ) = ( ( resid / dble( max( 1, p, m ) ) ) / bnorm ) /
303 \$ ulp
304 ELSE
305 result( 2 ) = zero
306 END IF
307*
308* Compute I - Q*Q'
309*
310 CALL zlaset( 'Full', n, n, czero, cone, r, lda )
311 CALL zherk( 'Upper', 'No Transpose', n, n, -one, q, lda, one, r,
312 \$ lda )
313*
314* Compute norm( I - Q'*Q ) / ( N * ULP ) .
315*
316 resid = zlanhe( '1', 'Upper', n, r, lda, rwork )
317 result( 3 ) = ( resid / dble( max( 1, n ) ) ) / ulp
318*
319* Compute I - Z'*Z
320*
321 CALL zlaset( 'Full', p, p, czero, cone, t, ldb )
322 CALL zherk( 'Upper', 'Conjugate transpose', p, p, -one, z, ldb,
323 \$ one, t, ldb )
324*
325* Compute norm( I - Z'*Z ) / ( P*ULP ) .
326*
327 resid = zlanhe( '1', 'Upper', p, t, ldb, rwork )
328 result( 4 ) = ( resid / dble( max( 1, p ) ) ) / ulp
329*
330 RETURN
331*
332* End of ZGRQTS
333*
double precision function dlamch(CMACH)
DLAMCH
Definition: dlamch.f:69
subroutine zgemm(TRANSA, TRANSB, M, N, K, ALPHA, A, LDA, B, LDB, BETA, C, LDC)
ZGEMM
Definition: zgemm.f:187
subroutine zherk(UPLO, TRANS, N, K, ALPHA, A, LDA, BETA, C, LDC)
ZHERK
Definition: zherk.f:173
double precision function zlange(NORM, M, N, A, LDA, WORK)
ZLANGE returns the value of the 1-norm, Frobenius norm, infinity-norm, or the largest absolute value ...
Definition: zlange.f:115
double precision function zlanhe(NORM, UPLO, N, A, LDA, WORK)
ZLANHE returns the value of the 1-norm, or the Frobenius norm, or the infinity norm,...
Definition: zlanhe.f:124
subroutine zlacpy(UPLO, M, N, A, LDA, B, LDB)
ZLACPY copies all or part of one two-dimensional array to another.
Definition: zlacpy.f:103
subroutine zlaset(UPLO, M, N, ALPHA, BETA, A, LDA)
ZLASET initializes the off-diagonal elements and the diagonal elements of a matrix to given values.
Definition: zlaset.f:106
subroutine zungqr(M, N, K, A, LDA, TAU, WORK, LWORK, INFO)
ZUNGQR
Definition: zungqr.f:128
subroutine zungrq(M, N, K, A, LDA, TAU, WORK, LWORK, INFO)
ZUNGRQ
Definition: zungrq.f:128
subroutine zggrqf(M, P, N, A, LDA, TAUA, B, LDB, TAUB, WORK, LWORK, INFO)
ZGGRQF
Definition: zggrqf.f:214
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