1 SUBROUTINE DGRQTS( M, P, N, A, AF, Q, R, LDA, TAUA, B, BF, Z, T,
2 $ BWK, LDB, TAUB, WORK, LWORK, RWORK, RESULT )
3 *
4 * -- LAPACK test routine (version 3.1) --
5 * Univ. of Tennessee, Univ. of California Berkeley and NAG Ltd..
6 * November 2006
7 *
8 * .. Scalar Arguments ..
9 INTEGER LDA, LDB, LWORK, M, N, P
10 * ..
11 * .. Array Arguments ..
12 DOUBLE PRECISION A( LDA, * ), AF( LDA, * ), B( LDB, * ),
13 $ BF( LDB, * ), BWK( LDB, * ), Q( LDA, * ),
14 $ R( LDA, * ), RESULT( 4 ), RWORK( * ),
15 $ T( LDB, * ), TAUA( * ), TAUB( * ),
16 $ WORK( LWORK ), Z( LDB, * )
17 * ..
18 *
19 * Purpose
20 * =======
21 *
22 * DGRQTS tests DGGRQF, which computes the GRQ factorization of an
23 * M-by-N matrix A and a P-by-N matrix B: A = R*Q and B = Z*T*Q.
24 *
25 * Arguments
26 * =========
27 *
28 * M (input) INTEGER
29 * The number of rows of the matrix A. M >= 0.
30 *
31 * P (input) INTEGER
32 * The number of rows of the matrix B. P >= 0.
33 *
34 * N (input) INTEGER
35 * The number of columns of the matrices A and B. N >= 0.
36 *
37 * A (input) DOUBLE PRECISION array, dimension (LDA,N)
38 * The M-by-N matrix A.
39 *
40 * AF (output) DOUBLE PRECISION array, dimension (LDA,N)
41 * Details of the GRQ factorization of A and B, as returned
42 * by DGGRQF, see SGGRQF for further details.
43 *
44 * Q (output) DOUBLE PRECISION array, dimension (LDA,N)
45 * The N-by-N orthogonal matrix Q.
46 *
47 * R (workspace) DOUBLE PRECISION array, dimension (LDA,MAX(M,N))
48 *
49 * LDA (input) INTEGER
50 * The leading dimension of the arrays A, AF, R and Q.
51 * LDA >= max(M,N).
52 *
53 * TAUA (output) DOUBLE PRECISION array, dimension (min(M,N))
54 * The scalar factors of the elementary reflectors, as returned
55 * by DGGQRC.
56 *
57 * B (input) DOUBLE PRECISION array, dimension (LDB,N)
58 * On entry, the P-by-N matrix A.
59 *
60 * BF (output) DOUBLE PRECISION array, dimension (LDB,N)
61 * Details of the GQR factorization of A and B, as returned
62 * by DGGRQF, see SGGRQF for further details.
63 *
64 * Z (output) DOUBLE PRECISION array, dimension (LDB,P)
65 * The P-by-P orthogonal matrix Z.
66 *
67 * T (workspace) DOUBLE PRECISION array, dimension (LDB,max(P,N))
68 *
69 * BWK (workspace) DOUBLE PRECISION array, dimension (LDB,N)
70 *
71 * LDB (input) INTEGER
72 * The leading dimension of the arrays B, BF, Z and T.
73 * LDB >= max(P,N).
74 *
75 * TAUB (output) DOUBLE PRECISION array, dimension (min(P,N))
76 * The scalar factors of the elementary reflectors, as returned
77 * by DGGRQF.
78 *
79 * WORK (workspace) DOUBLE PRECISION array, dimension (LWORK)
80 *
81 * LWORK (input) INTEGER
82 * The dimension of the array WORK, LWORK >= max(M,P,N)**2.
83 *
84 * RWORK (workspace) DOUBLE PRECISION array, dimension (M)
85 *
86 * RESULT (output) DOUBLE PRECISION array, dimension (4)
87 * The test ratios:
88 * RESULT(1) = norm( R - A*Q' ) / ( MAX(M,N)*norm(A)*ULP)
89 * RESULT(2) = norm( T*Q - Z'*B ) / (MAX(P,N)*norm(B)*ULP)
90 * RESULT(3) = norm( I - Q'*Q ) / ( N*ULP )
91 * RESULT(4) = norm( I - Z'*Z ) / ( P*ULP )
92 *
93 * =====================================================================
94 *
95 * .. Parameters ..
96 DOUBLE PRECISION ZERO, ONE
97 PARAMETER ( ZERO = 0.0D+0, ONE = 1.0D+0 )
98 DOUBLE PRECISION ROGUE
99 PARAMETER ( ROGUE = -1.0D+10 )
100 * ..
101 * .. Local Scalars ..
102 INTEGER INFO
103 DOUBLE PRECISION ANORM, BNORM, RESID, ULP, UNFL
104 * ..
105 * .. External Functions ..
106 DOUBLE PRECISION DLAMCH, DLANGE, DLANSY
107 EXTERNAL DLAMCH, DLANGE, DLANSY
108 * ..
109 * .. External Subroutines ..
110 EXTERNAL DGEMM, DGGRQF, DLACPY, DLASET, DORGQR, DORGRQ,
111 $ DSYRK
112 * ..
113 * .. Intrinsic Functions ..
114 INTRINSIC DBLE, MAX, MIN
115 * ..
116 * .. Executable Statements ..
117 *
118 ULP = DLAMCH( 'Precision' )
119 UNFL = DLAMCH( 'Safe minimum' )
120 *
121 * Copy the matrix A to the array AF.
122 *
123 CALL DLACPY( 'Full', M, N, A, LDA, AF, LDA )
124 CALL DLACPY( 'Full', P, N, B, LDB, BF, LDB )
125 *
126 ANORM = MAX( DLANGE( '1', M, N, A, LDA, RWORK ), UNFL )
127 BNORM = MAX( DLANGE( '1', P, N, B, LDB, RWORK ), UNFL )
128 *
129 * Factorize the matrices A and B in the arrays AF and BF.
130 *
131 CALL DGGRQF( M, P, N, AF, LDA, TAUA, BF, LDB, TAUB, WORK, LWORK,
132 $ INFO )
133 *
134 * Generate the N-by-N matrix Q
135 *
136 CALL DLASET( 'Full', N, N, ROGUE, ROGUE, Q, LDA )
137 IF( M.LE.N ) THEN
138 IF( M.GT.0 .AND. M.LT.N )
139 $ CALL DLACPY( 'Full', M, N-M, AF, LDA, Q( N-M+1, 1 ), LDA )
140 IF( M.GT.1 )
141 $ CALL DLACPY( 'Lower', M-1, M-1, AF( 2, N-M+1 ), LDA,
142 $ Q( N-M+2, N-M+1 ), LDA )
143 ELSE
144 IF( N.GT.1 )
145 $ CALL DLACPY( 'Lower', N-1, N-1, AF( M-N+2, 1 ), LDA,
146 $ Q( 2, 1 ), LDA )
147 END IF
148 CALL DORGRQ( N, N, MIN( M, N ), Q, LDA, TAUA, WORK, LWORK, INFO )
149 *
150 * Generate the P-by-P matrix Z
151 *
152 CALL DLASET( 'Full', P, P, ROGUE, ROGUE, Z, LDB )
153 IF( P.GT.1 )
154 $ CALL DLACPY( 'Lower', P-1, N, BF( 2, 1 ), LDB, Z( 2, 1 ), LDB )
155 CALL DORGQR( P, P, MIN( P, N ), Z, LDB, TAUB, WORK, LWORK, INFO )
156 *
157 * Copy R
158 *
159 CALL DLASET( 'Full', M, N, ZERO, ZERO, R, LDA )
160 IF( M.LE.N ) THEN
161 CALL DLACPY( 'Upper', M, M, AF( 1, N-M+1 ), LDA, R( 1, N-M+1 ),
162 $ LDA )
163 ELSE
164 CALL DLACPY( 'Full', M-N, N, AF, LDA, R, LDA )
165 CALL DLACPY( 'Upper', N, N, AF( M-N+1, 1 ), LDA, R( M-N+1, 1 ),
166 $ LDA )
167 END IF
168 *
169 * Copy T
170 *
171 CALL DLASET( 'Full', P, N, ZERO, ZERO, T, LDB )
172 CALL DLACPY( 'Upper', P, N, BF, LDB, T, LDB )
173 *
174 * Compute R - A*Q'
175 *
176 CALL DGEMM( 'No transpose', 'Transpose', M, N, N, -ONE, A, LDA, Q,
177 $ LDA, ONE, R, LDA )
178 *
179 * Compute norm( R - A*Q' ) / ( MAX(M,N)*norm(A)*ULP ) .
180 *
181 RESID = DLANGE( '1', M, N, R, LDA, RWORK )
182 IF( ANORM.GT.ZERO ) THEN
183 RESULT( 1 ) = ( ( RESID / DBLE( MAX( 1, M, N ) ) ) / ANORM ) /
184 $ ULP
185 ELSE
186 RESULT( 1 ) = ZERO
187 END IF
188 *
189 * Compute T*Q - Z'*B
190 *
191 CALL DGEMM( 'Transpose', 'No transpose', P, N, P, ONE, Z, LDB, B,
192 $ LDB, ZERO, BWK, LDB )
193 CALL DGEMM( 'No transpose', 'No transpose', P, N, N, ONE, T, LDB,
194 $ Q, LDA, -ONE, BWK, LDB )
195 *
196 * Compute norm( T*Q - Z'*B ) / ( MAX(P,N)*norm(A)*ULP ) .
197 *
198 RESID = DLANGE( '1', P, N, BWK, LDB, RWORK )
199 IF( BNORM.GT.ZERO ) THEN
200 RESULT( 2 ) = ( ( RESID / DBLE( MAX( 1, P, M ) ) ) / BNORM ) /
201 $ ULP
202 ELSE
203 RESULT( 2 ) = ZERO
204 END IF
205 *
206 * Compute I - Q*Q'
207 *
208 CALL DLASET( 'Full', N, N, ZERO, ONE, R, LDA )
209 CALL DSYRK( 'Upper', 'No Transpose', N, N, -ONE, Q, LDA, ONE, R,
210 $ LDA )
211 *
212 * Compute norm( I - Q'*Q ) / ( N * ULP ) .
213 *
214 RESID = DLANSY( '1', 'Upper', N, R, LDA, RWORK )
215 RESULT( 3 ) = ( RESID / DBLE( MAX( 1, N ) ) ) / ULP
216 *
217 * Compute I - Z'*Z
218 *
219 CALL DLASET( 'Full', P, P, ZERO, ONE, T, LDB )
220 CALL DSYRK( 'Upper', 'Transpose', P, P, -ONE, Z, LDB, ONE, T,
221 $ LDB )
222 *
223 * Compute norm( I - Z'*Z ) / ( P*ULP ) .
224 *
225 RESID = DLANSY( '1', 'Upper', P, T, LDB, RWORK )
226 RESULT( 4 ) = ( RESID / DBLE( MAX( 1, P ) ) ) / ULP
227 *
228 RETURN
229 *
230 * End of DGRQTS
231 *
232 END
2 $ BWK, LDB, TAUB, WORK, LWORK, RWORK, RESULT )
3 *
4 * -- LAPACK test routine (version 3.1) --
5 * Univ. of Tennessee, Univ. of California Berkeley and NAG Ltd..
6 * November 2006
7 *
8 * .. Scalar Arguments ..
9 INTEGER LDA, LDB, LWORK, M, N, P
10 * ..
11 * .. Array Arguments ..
12 DOUBLE PRECISION A( LDA, * ), AF( LDA, * ), B( LDB, * ),
13 $ BF( LDB, * ), BWK( LDB, * ), Q( LDA, * ),
14 $ R( LDA, * ), RESULT( 4 ), RWORK( * ),
15 $ T( LDB, * ), TAUA( * ), TAUB( * ),
16 $ WORK( LWORK ), Z( LDB, * )
17 * ..
18 *
19 * Purpose
20 * =======
21 *
22 * DGRQTS tests DGGRQF, which computes the GRQ factorization of an
23 * M-by-N matrix A and a P-by-N matrix B: A = R*Q and B = Z*T*Q.
24 *
25 * Arguments
26 * =========
27 *
28 * M (input) INTEGER
29 * The number of rows of the matrix A. M >= 0.
30 *
31 * P (input) INTEGER
32 * The number of rows of the matrix B. P >= 0.
33 *
34 * N (input) INTEGER
35 * The number of columns of the matrices A and B. N >= 0.
36 *
37 * A (input) DOUBLE PRECISION array, dimension (LDA,N)
38 * The M-by-N matrix A.
39 *
40 * AF (output) DOUBLE PRECISION array, dimension (LDA,N)
41 * Details of the GRQ factorization of A and B, as returned
42 * by DGGRQF, see SGGRQF for further details.
43 *
44 * Q (output) DOUBLE PRECISION array, dimension (LDA,N)
45 * The N-by-N orthogonal matrix Q.
46 *
47 * R (workspace) DOUBLE PRECISION array, dimension (LDA,MAX(M,N))
48 *
49 * LDA (input) INTEGER
50 * The leading dimension of the arrays A, AF, R and Q.
51 * LDA >= max(M,N).
52 *
53 * TAUA (output) DOUBLE PRECISION array, dimension (min(M,N))
54 * The scalar factors of the elementary reflectors, as returned
55 * by DGGQRC.
56 *
57 * B (input) DOUBLE PRECISION array, dimension (LDB,N)
58 * On entry, the P-by-N matrix A.
59 *
60 * BF (output) DOUBLE PRECISION array, dimension (LDB,N)
61 * Details of the GQR factorization of A and B, as returned
62 * by DGGRQF, see SGGRQF for further details.
63 *
64 * Z (output) DOUBLE PRECISION array, dimension (LDB,P)
65 * The P-by-P orthogonal matrix Z.
66 *
67 * T (workspace) DOUBLE PRECISION array, dimension (LDB,max(P,N))
68 *
69 * BWK (workspace) DOUBLE PRECISION array, dimension (LDB,N)
70 *
71 * LDB (input) INTEGER
72 * The leading dimension of the arrays B, BF, Z and T.
73 * LDB >= max(P,N).
74 *
75 * TAUB (output) DOUBLE PRECISION array, dimension (min(P,N))
76 * The scalar factors of the elementary reflectors, as returned
77 * by DGGRQF.
78 *
79 * WORK (workspace) DOUBLE PRECISION array, dimension (LWORK)
80 *
81 * LWORK (input) INTEGER
82 * The dimension of the array WORK, LWORK >= max(M,P,N)**2.
83 *
84 * RWORK (workspace) DOUBLE PRECISION array, dimension (M)
85 *
86 * RESULT (output) DOUBLE PRECISION array, dimension (4)
87 * The test ratios:
88 * RESULT(1) = norm( R - A*Q' ) / ( MAX(M,N)*norm(A)*ULP)
89 * RESULT(2) = norm( T*Q - Z'*B ) / (MAX(P,N)*norm(B)*ULP)
90 * RESULT(3) = norm( I - Q'*Q ) / ( N*ULP )
91 * RESULT(4) = norm( I - Z'*Z ) / ( P*ULP )
92 *
93 * =====================================================================
94 *
95 * .. Parameters ..
96 DOUBLE PRECISION ZERO, ONE
97 PARAMETER ( ZERO = 0.0D+0, ONE = 1.0D+0 )
98 DOUBLE PRECISION ROGUE
99 PARAMETER ( ROGUE = -1.0D+10 )
100 * ..
101 * .. Local Scalars ..
102 INTEGER INFO
103 DOUBLE PRECISION ANORM, BNORM, RESID, ULP, UNFL
104 * ..
105 * .. External Functions ..
106 DOUBLE PRECISION DLAMCH, DLANGE, DLANSY
107 EXTERNAL DLAMCH, DLANGE, DLANSY
108 * ..
109 * .. External Subroutines ..
110 EXTERNAL DGEMM, DGGRQF, DLACPY, DLASET, DORGQR, DORGRQ,
111 $ DSYRK
112 * ..
113 * .. Intrinsic Functions ..
114 INTRINSIC DBLE, MAX, MIN
115 * ..
116 * .. Executable Statements ..
117 *
118 ULP = DLAMCH( 'Precision' )
119 UNFL = DLAMCH( 'Safe minimum' )
120 *
121 * Copy the matrix A to the array AF.
122 *
123 CALL DLACPY( 'Full', M, N, A, LDA, AF, LDA )
124 CALL DLACPY( 'Full', P, N, B, LDB, BF, LDB )
125 *
126 ANORM = MAX( DLANGE( '1', M, N, A, LDA, RWORK ), UNFL )
127 BNORM = MAX( DLANGE( '1', P, N, B, LDB, RWORK ), UNFL )
128 *
129 * Factorize the matrices A and B in the arrays AF and BF.
130 *
131 CALL DGGRQF( M, P, N, AF, LDA, TAUA, BF, LDB, TAUB, WORK, LWORK,
132 $ INFO )
133 *
134 * Generate the N-by-N matrix Q
135 *
136 CALL DLASET( 'Full', N, N, ROGUE, ROGUE, Q, LDA )
137 IF( M.LE.N ) THEN
138 IF( M.GT.0 .AND. M.LT.N )
139 $ CALL DLACPY( 'Full', M, N-M, AF, LDA, Q( N-M+1, 1 ), LDA )
140 IF( M.GT.1 )
141 $ CALL DLACPY( 'Lower', M-1, M-1, AF( 2, N-M+1 ), LDA,
142 $ Q( N-M+2, N-M+1 ), LDA )
143 ELSE
144 IF( N.GT.1 )
145 $ CALL DLACPY( 'Lower', N-1, N-1, AF( M-N+2, 1 ), LDA,
146 $ Q( 2, 1 ), LDA )
147 END IF
148 CALL DORGRQ( N, N, MIN( M, N ), Q, LDA, TAUA, WORK, LWORK, INFO )
149 *
150 * Generate the P-by-P matrix Z
151 *
152 CALL DLASET( 'Full', P, P, ROGUE, ROGUE, Z, LDB )
153 IF( P.GT.1 )
154 $ CALL DLACPY( 'Lower', P-1, N, BF( 2, 1 ), LDB, Z( 2, 1 ), LDB )
155 CALL DORGQR( P, P, MIN( P, N ), Z, LDB, TAUB, WORK, LWORK, INFO )
156 *
157 * Copy R
158 *
159 CALL DLASET( 'Full', M, N, ZERO, ZERO, R, LDA )
160 IF( M.LE.N ) THEN
161 CALL DLACPY( 'Upper', M, M, AF( 1, N-M+1 ), LDA, R( 1, N-M+1 ),
162 $ LDA )
163 ELSE
164 CALL DLACPY( 'Full', M-N, N, AF, LDA, R, LDA )
165 CALL DLACPY( 'Upper', N, N, AF( M-N+1, 1 ), LDA, R( M-N+1, 1 ),
166 $ LDA )
167 END IF
168 *
169 * Copy T
170 *
171 CALL DLASET( 'Full', P, N, ZERO, ZERO, T, LDB )
172 CALL DLACPY( 'Upper', P, N, BF, LDB, T, LDB )
173 *
174 * Compute R - A*Q'
175 *
176 CALL DGEMM( 'No transpose', 'Transpose', M, N, N, -ONE, A, LDA, Q,
177 $ LDA, ONE, R, LDA )
178 *
179 * Compute norm( R - A*Q' ) / ( MAX(M,N)*norm(A)*ULP ) .
180 *
181 RESID = DLANGE( '1', M, N, R, LDA, RWORK )
182 IF( ANORM.GT.ZERO ) THEN
183 RESULT( 1 ) = ( ( RESID / DBLE( MAX( 1, M, N ) ) ) / ANORM ) /
184 $ ULP
185 ELSE
186 RESULT( 1 ) = ZERO
187 END IF
188 *
189 * Compute T*Q - Z'*B
190 *
191 CALL DGEMM( 'Transpose', 'No transpose', P, N, P, ONE, Z, LDB, B,
192 $ LDB, ZERO, BWK, LDB )
193 CALL DGEMM( 'No transpose', 'No transpose', P, N, N, ONE, T, LDB,
194 $ Q, LDA, -ONE, BWK, LDB )
195 *
196 * Compute norm( T*Q - Z'*B ) / ( MAX(P,N)*norm(A)*ULP ) .
197 *
198 RESID = DLANGE( '1', P, N, BWK, LDB, RWORK )
199 IF( BNORM.GT.ZERO ) THEN
200 RESULT( 2 ) = ( ( RESID / DBLE( MAX( 1, P, M ) ) ) / BNORM ) /
201 $ ULP
202 ELSE
203 RESULT( 2 ) = ZERO
204 END IF
205 *
206 * Compute I - Q*Q'
207 *
208 CALL DLASET( 'Full', N, N, ZERO, ONE, R, LDA )
209 CALL DSYRK( 'Upper', 'No Transpose', N, N, -ONE, Q, LDA, ONE, R,
210 $ LDA )
211 *
212 * Compute norm( I - Q'*Q ) / ( N * ULP ) .
213 *
214 RESID = DLANSY( '1', 'Upper', N, R, LDA, RWORK )
215 RESULT( 3 ) = ( RESID / DBLE( MAX( 1, N ) ) ) / ULP
216 *
217 * Compute I - Z'*Z
218 *
219 CALL DLASET( 'Full', P, P, ZERO, ONE, T, LDB )
220 CALL DSYRK( 'Upper', 'Transpose', P, P, -ONE, Z, LDB, ONE, T,
221 $ LDB )
222 *
223 * Compute norm( I - Z'*Z ) / ( P*ULP ) .
224 *
225 RESID = DLANSY( '1', 'Upper', P, T, LDB, RWORK )
226 RESULT( 4 ) = ( RESID / DBLE( MAX( 1, P ) ) ) / ULP
227 *
228 RETURN
229 *
230 * End of DGRQTS
231 *
232 END