LAPACK 3.12.0
LAPACK: Linear Algebra PACKage
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dgrqts.f
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1*> \brief \b DGRQTS
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 DGRQTS( 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 A( LDA, * ), AF( LDA, * ), B( LDB, * ),
19* $ BF( LDB, * ), BWK( LDB, * ), Q( LDA, * ),
20* $ R( LDA, * ), RESULT( 4 ), RWORK( * ),
21* $ T( LDB, * ), TAUA( * ), TAUB( * ),
22* $ WORK( LWORK ), Z( LDB, * )
23* ..
24*
25*
26*> \par Purpose:
27* =============
28*>
29*> \verbatim
30*>
31*> DGRQTS tests DGGRQF, 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 DOUBLE PRECISION array, dimension (LDA,N)
59*> The M-by-N matrix A.
60*> \endverbatim
61*>
62*> \param[out] AF
63*> \verbatim
64*> AF is DOUBLE PRECISION array, dimension (LDA,N)
65*> Details of the GRQ factorization of A and B, as returned
66*> by DGGRQF, see SGGRQF for further details.
67*> \endverbatim
68*>
69*> \param[out] Q
70*> \verbatim
71*> Q is DOUBLE PRECISION array, dimension (LDA,N)
72*> The N-by-N orthogonal matrix Q.
73*> \endverbatim
74*>
75*> \param[out] R
76*> \verbatim
77*> R is DOUBLE PRECISION 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 DOUBLE PRECISION 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 DOUBLE PRECISION 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 DOUBLE PRECISION array, dimension (LDB,N)
103*> Details of the GQR factorization of A and B, as returned
104*> by DGGRQF, see SGGRQF 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 orthogonal matrix Z.
111*> \endverbatim
112*>
113*> \param[out] T
114*> \verbatim
115*> T is DOUBLE PRECISION array, dimension (LDB,max(P,N))
116*> \endverbatim
117*>
118*> \param[out] BWK
119*> \verbatim
120*> BWK is DOUBLE PRECISION 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 DOUBLE PRECISION 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 DOUBLE PRECISION 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 double_eig
172*
173* =====================================================================
174 SUBROUTINE dgrqts( 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 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*
331 END
subroutine dgrqts(m, p, n, a, af, q, r, lda, taua, b, bf, z, t, bwk, ldb, taub, work, lwork, rwork, result)
DGRQTS
Definition dgrqts.f:176
subroutine dgemm(transa, transb, m, n, k, alpha, a, lda, b, ldb, beta, c, ldc)
DGEMM
Definition dgemm.f:188
subroutine dggrqf(m, p, n, a, lda, taua, b, ldb, taub, work, lwork, info)
DGGRQF
Definition dggrqf.f:214
subroutine dsyrk(uplo, trans, n, k, alpha, a, lda, beta, c, ldc)
DSYRK
Definition dsyrk.f:169
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 dorgqr(m, n, k, a, lda, tau, work, lwork, info)
DORGQR
Definition dorgqr.f:128
subroutine dorgrq(m, n, k, a, lda, tau, work, lwork, info)
DORGRQ
Definition dorgrq.f:128