LAPACK 3.11.0
LAPACK: Linear Algebra PACKage
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zgrqts.f
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1*> \brief \b ZGRQTS
2*
3* =========== DOCUMENTATION ===========
4*
5* Online html documentation available at
6* http://www.netlib.org/lapack/explore-html/
7*
8* Definition:
9* ===========
10*
11* SUBROUTINE ZGRQTS( M, P, N, A, AF, Q, R, LDA, TAUA, B, BF, Z, T,
12* BWK, LDB, TAUB, WORK, LWORK, RWORK, RESULT )
13*
14* .. Scalar Arguments ..
15* INTEGER LDA, LDB, LWORK, M, N, P
16* ..
17* .. Array Arguments ..
18* DOUBLE PRECISION RESULT( 4 ), RWORK( * )
19* COMPLEX*16 A( LDA, * ), AF( LDA, * ), B( LDB, * ),
20* $ BF( LDB, * ), BWK( LDB, * ), Q( LDA, * ),
21* $ R( LDA, * ), T( LDB, * ), TAUA( * ), TAUB( * ),
22* $ WORK( LWORK ), Z( LDB, * )
23* ..
24*
25*
26*> \par Purpose:
27* =============
28*>
29*> \verbatim
30*>
31*> ZGRQTS tests ZGGRQF, which computes the GRQ factorization of an
32*> M-by-N matrix A and a P-by-N matrix B: A = R*Q and B = Z*T*Q.
33*> \endverbatim
34*
35* Arguments:
36* ==========
37*
38*> \param[in] M
39*> \verbatim
40*> M is INTEGER
41*> The number of rows of the matrix A. M >= 0.
42*> \endverbatim
43*>
44*> \param[in] P
45*> \verbatim
46*> P is INTEGER
47*> The number of rows of the matrix B. P >= 0.
48*> \endverbatim
49*>
50*> \param[in] N
51*> \verbatim
52*> N is INTEGER
53*> The number of columns of the matrices A and B. N >= 0.
54*> \endverbatim
55*>
56*> \param[in] A
57*> \verbatim
58*> A is COMPLEX*16 array, dimension (LDA,N)
59*> The M-by-N matrix A.
60*> \endverbatim
61*>
62*> \param[out] AF
63*> \verbatim
64*> AF is COMPLEX*16 array, dimension (LDA,N)
65*> Details of the GRQ factorization of A and B, as returned
66*> by ZGGRQF, see CGGRQF for further details.
67*> \endverbatim
68*>
69*> \param[out] Q
70*> \verbatim
71*> Q is COMPLEX*16 array, dimension (LDA,N)
72*> The N-by-N unitary matrix Q.
73*> \endverbatim
74*>
75*> \param[out] R
76*> \verbatim
77*> R is COMPLEX*16 array, dimension (LDA,MAX(M,N))
78*> \endverbatim
79*>
80*> \param[in] LDA
81*> \verbatim
82*> LDA is INTEGER
83*> The leading dimension of the arrays A, AF, R and Q.
84*> LDA >= max(M,N).
85*> \endverbatim
86*>
87*> \param[out] TAUA
88*> \verbatim
89*> TAUA is COMPLEX*16 array, dimension (min(M,N))
90*> The scalar factors of the elementary reflectors, as returned
91*> by DGGQRC.
92*> \endverbatim
93*>
94*> \param[in] B
95*> \verbatim
96*> B is COMPLEX*16 array, dimension (LDB,N)
97*> On entry, the P-by-N matrix A.
98*> \endverbatim
99*>
100*> \param[out] BF
101*> \verbatim
102*> BF is COMPLEX*16 array, dimension (LDB,N)
103*> Details of the GQR factorization of A and B, as returned
104*> by ZGGRQF, see CGGRQF for further details.
105*> \endverbatim
106*>
107*> \param[out] Z
108*> \verbatim
109*> Z is DOUBLE PRECISION array, dimension (LDB,P)
110*> The P-by-P unitary matrix Z.
111*> \endverbatim
112*>
113*> \param[out] T
114*> \verbatim
115*> T is COMPLEX*16 array, dimension (LDB,max(P,N))
116*> \endverbatim
117*>
118*> \param[out] BWK
119*> \verbatim
120*> BWK is COMPLEX*16 array, dimension (LDB,N)
121*> \endverbatim
122*>
123*> \param[in] LDB
124*> \verbatim
125*> LDB is INTEGER
126*> The leading dimension of the arrays B, BF, Z and T.
127*> LDB >= max(P,N).
128*> \endverbatim
129*>
130*> \param[out] TAUB
131*> \verbatim
132*> TAUB is COMPLEX*16 array, dimension (min(P,N))
133*> The scalar factors of the elementary reflectors, as returned
134*> by DGGRQF.
135*> \endverbatim
136*>
137*> \param[out] WORK
138*> \verbatim
139*> WORK is COMPLEX*16 array, dimension (LWORK)
140*> \endverbatim
141*>
142*> \param[in] LWORK
143*> \verbatim
144*> LWORK is INTEGER
145*> The dimension of the array WORK, LWORK >= max(M,P,N)**2.
146*> \endverbatim
147*>
148*> \param[out] RWORK
149*> \verbatim
150*> RWORK is DOUBLE PRECISION array, dimension (M)
151*> \endverbatim
152*>
153*> \param[out] RESULT
154*> \verbatim
155*> RESULT is DOUBLE PRECISION array, dimension (4)
156*> The test ratios:
157*> RESULT(1) = norm( R - A*Q' ) / ( MAX(M,N)*norm(A)*ULP)
158*> RESULT(2) = norm( T*Q - Z'*B ) / (MAX(P,N)*norm(B)*ULP)
159*> RESULT(3) = norm( I - Q'*Q ) / ( N*ULP )
160*> RESULT(4) = norm( I - Z'*Z ) / ( P*ULP )
161*> \endverbatim
162*
163* Authors:
164* ========
165*
166*> \author Univ. of Tennessee
167*> \author Univ. of California Berkeley
168*> \author Univ. of Colorado Denver
169*> \author NAG Ltd.
170*
171*> \ingroup complex16_eig
172*
173* =====================================================================
174 SUBROUTINE zgrqts( M, P, N, A, AF, Q, R, LDA, TAUA, B, BF, Z, T,
175 $ BWK, LDB, TAUB, WORK, LWORK, RWORK, RESULT )
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*
334 END
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
subroutine zgrqts(M, P, N, A, AF, Q, R, LDA, TAUA, B, BF, Z, T, BWK, LDB, TAUB, WORK, LWORK, RWORK, RESULT)
ZGRQTS
Definition: zgrqts.f:176
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