LAPACK  3.10.0
LAPACK: Linear Algebra PACKage
sgrqts.f
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1 *> \brief \b SGRQTS
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 SGRQTS( 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, P, N
16 * ..
17 * .. Array Arguments ..
18 * REAL A( LDA, * ), AF( LDA, * ), R( LDA, * ),
19 * $ Q( LDA, * ),
20 * $ B( LDB, * ), BF( LDB, * ), T( LDB, * ),
21 * $ Z( LDB, * ), BWK( LDB, * ),
22 * $ TAUA( * ), TAUB( * ),
23 * $ RESULT( 4 ), RWORK( * ), WORK( LWORK )
24 * ..
25 *
26 *
27 *> \par Purpose:
28 * =============
29 *>
30 *> \verbatim
31 *>
32 *> SGRQTS tests SGGRQF, which computes the GRQ factorization of an
33 *> M-by-N matrix A and a P-by-N matrix B: A = R*Q and B = Z*T*Q.
34 *> \endverbatim
35 *
36 * Arguments:
37 * ==========
38 *
39 *> \param[in] M
40 *> \verbatim
41 *> M is INTEGER
42 *> The number of rows of the matrix A. M >= 0.
43 *> \endverbatim
44 *>
45 *> \param[in] P
46 *> \verbatim
47 *> P is INTEGER
48 *> The number of rows of the matrix B. P >= 0.
49 *> \endverbatim
50 *>
51 *> \param[in] N
52 *> \verbatim
53 *> N is INTEGER
54 *> The number of columns of the matrices A and B. N >= 0.
55 *> \endverbatim
56 *>
57 *> \param[in] A
58 *> \verbatim
59 *> A is REAL array, dimension (LDA,N)
60 *> The M-by-N matrix A.
61 *> \endverbatim
62 *>
63 *> \param[out] AF
64 *> \verbatim
65 *> AF is REAL array, dimension (LDA,N)
66 *> Details of the GRQ factorization of A and B, as returned
67 *> by SGGRQF, see SGGRQF for further details.
68 *> \endverbatim
69 *>
70 *> \param[out] Q
71 *> \verbatim
72 *> Q is REAL array, dimension (LDA,N)
73 *> The N-by-N orthogonal matrix Q.
74 *> \endverbatim
75 *>
76 *> \param[out] R
77 *> \verbatim
78 *> R is REAL array, dimension (LDA,MAX(M,N))
79 *> \endverbatim
80 *>
81 *> \param[in] LDA
82 *> \verbatim
83 *> LDA is INTEGER
84 *> The leading dimension of the arrays A, AF, R and Q.
85 *> LDA >= max(M,N).
86 *> \endverbatim
87 *>
88 *> \param[out] TAUA
89 *> \verbatim
90 *> TAUA is REAL array, dimension (min(M,N))
91 *> The scalar factors of the elementary reflectors, as returned
92 *> by SGGQRC.
93 *> \endverbatim
94 *>
95 *> \param[in] B
96 *> \verbatim
97 *> B is REAL array, dimension (LDB,N)
98 *> On entry, the P-by-N matrix A.
99 *> \endverbatim
100 *>
101 *> \param[out] BF
102 *> \verbatim
103 *> BF is REAL array, dimension (LDB,N)
104 *> Details of the GQR factorization of A and B, as returned
105 *> by SGGRQF, see SGGRQF for further details.
106 *> \endverbatim
107 *>
108 *> \param[out] Z
109 *> \verbatim
110 *> Z is REAL array, dimension (LDB,P)
111 *> The P-by-P orthogonal matrix Z.
112 *> \endverbatim
113 *>
114 *> \param[out] T
115 *> \verbatim
116 *> T is REAL array, dimension (LDB,max(P,N))
117 *> \endverbatim
118 *>
119 *> \param[out] BWK
120 *> \verbatim
121 *> BWK is REAL array, dimension (LDB,N)
122 *> \endverbatim
123 *>
124 *> \param[in] LDB
125 *> \verbatim
126 *> LDB is INTEGER
127 *> The leading dimension of the arrays B, BF, Z and T.
128 *> LDB >= max(P,N).
129 *> \endverbatim
130 *>
131 *> \param[out] TAUB
132 *> \verbatim
133 *> TAUB is REAL array, dimension (min(P,N))
134 *> The scalar factors of the elementary reflectors, as returned
135 *> by SGGRQF.
136 *> \endverbatim
137 *>
138 *> \param[out] WORK
139 *> \verbatim
140 *> WORK is REAL array, dimension (LWORK)
141 *> \endverbatim
142 *>
143 *> \param[in] LWORK
144 *> \verbatim
145 *> LWORK is INTEGER
146 *> The dimension of the array WORK, LWORK >= max(M,P,N)**2.
147 *> \endverbatim
148 *>
149 *> \param[out] RWORK
150 *> \verbatim
151 *> RWORK is REAL array, dimension (M)
152 *> \endverbatim
153 *>
154 *> \param[out] RESULT
155 *> \verbatim
156 *> RESULT is REAL array, dimension (4)
157 *> The test ratios:
158 *> RESULT(1) = norm( R - A*Q' ) / ( MAX(M,N)*norm(A)*ULP)
159 *> RESULT(2) = norm( T*Q - Z'*B ) / (MAX(P,N)*norm(B)*ULP)
160 *> RESULT(3) = norm( I - Q'*Q ) / ( N*ULP )
161 *> RESULT(4) = norm( I - Z'*Z ) / ( P*ULP )
162 *> \endverbatim
163 *
164 * Authors:
165 * ========
166 *
167 *> \author Univ. of Tennessee
168 *> \author Univ. of California Berkeley
169 *> \author Univ. of Colorado Denver
170 *> \author NAG Ltd.
171 *
172 *> \ingroup single_eig
173 *
174 * =====================================================================
175  SUBROUTINE sgrqts( M, P, N, A, AF, Q, R, LDA, TAUA, B, BF, Z, T,
176  $ BWK, LDB, TAUB, WORK, LWORK, RWORK, RESULT )
177 *
178 * -- LAPACK test routine --
179 * -- LAPACK is a software package provided by Univ. of Tennessee, --
180 * -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
181 *
182 * .. Scalar Arguments ..
183  INTEGER LDA, LDB, LWORK, M, P, N
184 * ..
185 * .. Array Arguments ..
186  REAL A( LDA, * ), AF( LDA, * ), R( LDA, * ),
187  $ q( lda, * ),
188  $ b( ldb, * ), bf( ldb, * ), t( ldb, * ),
189  $ z( ldb, * ), bwk( ldb, * ),
190  $ taua( * ), taub( * ),
191  $ result( 4 ), rwork( * ), work( lwork )
192 * ..
193 *
194 * =====================================================================
195 *
196 * .. Parameters ..
197  REAL ZERO, ONE
198  parameter( zero = 0.0e+0, one = 1.0e+0 )
199  REAL ROGUE
200  parameter( rogue = -1.0e+10 )
201 * ..
202 * .. Local Scalars ..
203  INTEGER INFO
204  REAL ANORM, BNORM, ULP, UNFL, RESID
205 * ..
206 * .. External Functions ..
207  REAL SLAMCH, SLANGE, SLANSY
208  EXTERNAL slamch, slange, slansy
209 * ..
210 * .. External Subroutines ..
211  EXTERNAL sgemm, sggrqf, slacpy, slaset, sorgqr,
212  $ sorgrq, ssyrk
213 * ..
214 * .. Intrinsic Functions ..
215  INTRINSIC max, min, real
216 * ..
217 * .. Executable Statements ..
218 *
219  ulp = slamch( 'Precision' )
220  unfl = slamch( 'Safe minimum' )
221 *
222 * Copy the matrix A to the array AF.
223 *
224  CALL slacpy( 'Full', m, n, a, lda, af, lda )
225  CALL slacpy( 'Full', p, n, b, ldb, bf, ldb )
226 *
227  anorm = max( slange( '1', m, n, a, lda, rwork ), unfl )
228  bnorm = max( slange( '1', p, n, b, ldb, rwork ), unfl )
229 *
230 * Factorize the matrices A and B in the arrays AF and BF.
231 *
232  CALL sggrqf( m, p, n, af, lda, taua, bf, ldb, taub, work,
233  $ lwork, info )
234 *
235 * Generate the N-by-N matrix Q
236 *
237  CALL slaset( 'Full', n, n, rogue, rogue, q, lda )
238  IF( m.LE.n ) THEN
239  IF( m.GT.0 .AND. m.LT.n )
240  $ CALL slacpy( 'Full', m, n-m, af, lda, q( n-m+1, 1 ), lda )
241  IF( m.GT.1 )
242  $ CALL slacpy( 'Lower', m-1, m-1, af( 2, n-m+1 ), lda,
243  $ q( n-m+2, n-m+1 ), lda )
244  ELSE
245  IF( n.GT.1 )
246  $ CALL slacpy( 'Lower', n-1, n-1, af( m-n+2, 1 ), lda,
247  $ q( 2, 1 ), lda )
248  END IF
249  CALL sorgrq( n, n, min( m, n ), q, lda, taua, work, lwork, info )
250 *
251 * Generate the P-by-P matrix Z
252 *
253  CALL slaset( 'Full', p, p, rogue, rogue, z, ldb )
254  IF( p.GT.1 )
255  $ CALL slacpy( 'Lower', p-1, n, bf( 2,1 ), ldb, z( 2,1 ), ldb )
256  CALL sorgqr( p, p, min( p,n ), z, ldb, taub, work, lwork, info )
257 *
258 * Copy R
259 *
260  CALL slaset( 'Full', m, n, zero, zero, r, lda )
261  IF( m.LE.n )THEN
262  CALL slacpy( 'Upper', m, m, af( 1, n-m+1 ), lda, r( 1, n-m+1 ),
263  $ lda )
264  ELSE
265  CALL slacpy( 'Full', m-n, n, af, lda, r, lda )
266  CALL slacpy( 'Upper', n, n, af( m-n+1, 1 ), lda, r( m-n+1, 1 ),
267  $ lda )
268  END IF
269 *
270 * Copy T
271 *
272  CALL slaset( 'Full', p, n, zero, zero, t, ldb )
273  CALL slacpy( 'Upper', p, n, bf, ldb, t, ldb )
274 *
275 * Compute R - A*Q'
276 *
277  CALL sgemm( 'No transpose', 'Transpose', m, n, n, -one, a, lda, q,
278  $ lda, one, r, lda )
279 *
280 * Compute norm( R - A*Q' ) / ( MAX(M,N)*norm(A)*ULP ) .
281 *
282  resid = slange( '1', m, n, r, lda, rwork )
283  IF( anorm.GT.zero ) THEN
284  result( 1 ) = ( ( resid / real(max(1,m,n) ) ) / anorm ) / ulp
285  ELSE
286  result( 1 ) = zero
287  END IF
288 *
289 * Compute T*Q - Z'*B
290 *
291  CALL sgemm( 'Transpose', 'No transpose', p, n, p, one, z, ldb, b,
292  $ ldb, zero, bwk, ldb )
293  CALL sgemm( 'No transpose', 'No transpose', p, n, n, one, t, ldb,
294  $ q, lda, -one, bwk, ldb )
295 *
296 * Compute norm( T*Q - Z'*B ) / ( MAX(P,N)*norm(A)*ULP ) .
297 *
298  resid = slange( '1', p, n, bwk, ldb, rwork )
299  IF( bnorm.GT.zero ) THEN
300  result( 2 ) = ( ( resid / real( max( 1,p,m ) ) )/bnorm ) / ulp
301  ELSE
302  result( 2 ) = zero
303  END IF
304 *
305 * Compute I - Q*Q'
306 *
307  CALL slaset( 'Full', n, n, zero, one, r, lda )
308  CALL ssyrk( 'Upper', 'No Transpose', n, n, -one, q, lda, one, r,
309  $ lda )
310 *
311 * Compute norm( I - Q'*Q ) / ( N * ULP ) .
312 *
313  resid = slansy( '1', 'Upper', n, r, lda, rwork )
314  result( 3 ) = ( resid / real( max( 1,n ) ) ) / ulp
315 *
316 * Compute I - Z'*Z
317 *
318  CALL slaset( 'Full', p, p, zero, one, t, ldb )
319  CALL ssyrk( 'Upper', 'Transpose', p, p, -one, z, ldb, one, t,
320  $ ldb )
321 *
322 * Compute norm( I - Z'*Z ) / ( P*ULP ) .
323 *
324  resid = slansy( '1', 'Upper', p, t, ldb, rwork )
325  result( 4 ) = ( resid / real( max( 1,p ) ) ) / ulp
326 *
327  RETURN
328 *
329 * End of SGRQTS
330 *
331  END
subroutine slaset(UPLO, M, N, ALPHA, BETA, A, LDA)
SLASET initializes the off-diagonal elements and the diagonal elements of a matrix to given values.
Definition: slaset.f:110
subroutine slacpy(UPLO, M, N, A, LDA, B, LDB)
SLACPY copies all or part of one two-dimensional array to another.
Definition: slacpy.f:103
subroutine sorgrq(M, N, K, A, LDA, TAU, WORK, LWORK, INFO)
SORGRQ
Definition: sorgrq.f:128
subroutine sorgqr(M, N, K, A, LDA, TAU, WORK, LWORK, INFO)
SORGQR
Definition: sorgqr.f:128
subroutine sggrqf(M, P, N, A, LDA, TAUA, B, LDB, TAUB, WORK, LWORK, INFO)
SGGRQF
Definition: sggrqf.f:214
subroutine ssyrk(UPLO, TRANS, N, K, ALPHA, A, LDA, BETA, C, LDC)
SSYRK
Definition: ssyrk.f:169
subroutine sgemm(TRANSA, TRANSB, M, N, K, ALPHA, A, LDA, B, LDB, BETA, C, LDC)
SGEMM
Definition: sgemm.f:187
subroutine sgrqts(M, P, N, A, AF, Q, R, LDA, TAUA, B, BF, Z, T, BWK, LDB, TAUB, WORK, LWORK, RWORK, RESULT)
SGRQTS
Definition: sgrqts.f:177