LAPACK  3.10.1 LAPACK: Linear Algebra PACKage
sgqrts.f
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1 *> \brief \b SGQRTS
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 SGQRTS( N, M, P, 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, * ), B( LDB, * ), BF( LDB, * ),
20 * \$ T( LDB, * ), Z( LDB, * ), BWK( LDB, * ),
21 * \$ TAUA( * ), TAUB( * ), RESULT( 4 ),
22 * \$ RWORK( * ), WORK( LWORK )
23 * ..
24 *
25 *
26 *> \par Purpose:
27 * =============
28 *>
29 *> \verbatim
30 *>
31 *> SGQRTS tests SGGQRF, which computes the GQR factorization of an
32 *> N-by-M matrix A and a N-by-P matrix B: A = Q*R and B = Q*T*Z.
33 *> \endverbatim
34 *
35 * Arguments:
36 * ==========
37 *
38 *> \param[in] N
39 *> \verbatim
40 *> N is INTEGER
41 *> The number of rows of the matrices A and B. N >= 0.
42 *> \endverbatim
43 *>
44 *> \param[in] M
45 *> \verbatim
46 *> M is INTEGER
47 *> The number of columns of the matrix A. M >= 0.
48 *> \endverbatim
49 *>
50 *> \param[in] P
51 *> \verbatim
52 *> P is INTEGER
53 *> The number of columns of the matrix B. P >= 0.
54 *> \endverbatim
55 *>
56 *> \param[in] A
57 *> \verbatim
58 *> A is REAL array, dimension (LDA,M)
59 *> The N-by-M matrix A.
60 *> \endverbatim
61 *>
62 *> \param[out] AF
63 *> \verbatim
64 *> AF is REAL array, dimension (LDA,N)
65 *> Details of the GQR factorization of A and B, as returned
66 *> by SGGQRF, see SGGQRF for further details.
67 *> \endverbatim
68 *>
69 *> \param[out] Q
70 *> \verbatim
71 *> Q is REAL array, dimension (LDA,N)
72 *> The M-by-M orthogonal matrix Q.
73 *> \endverbatim
74 *>
75 *> \param[out] R
76 *> \verbatim
77 *> R is REAL 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 REAL array, dimension (min(M,N))
90 *> The scalar factors of the elementary reflectors, as returned
91 *> by SGGQRF.
92 *> \endverbatim
93 *>
94 *> \param[in] B
95 *> \verbatim
96 *> B is REAL array, dimension (LDB,P)
97 *> On entry, the N-by-P matrix A.
98 *> \endverbatim
99 *>
100 *> \param[out] BF
101 *> \verbatim
102 *> BF is REAL array, dimension (LDB,N)
103 *> Details of the GQR factorization of A and B, as returned
104 *> by SGGQRF, see SGGQRF for further details.
105 *> \endverbatim
106 *>
107 *> \param[out] Z
108 *> \verbatim
109 *> Z is REAL array, dimension (LDB,P)
110 *> The P-by-P orthogonal matrix Z.
111 *> \endverbatim
112 *>
113 *> \param[out] T
114 *> \verbatim
115 *> T is REAL array, dimension (LDB,max(P,N))
116 *> \endverbatim
117 *>
118 *> \param[out] BWK
119 *> \verbatim
120 *> BWK is REAL 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 REAL array, dimension (min(P,N))
133 *> The scalar factors of the elementary reflectors, as returned
134 *> by SGGRQF.
135 *> \endverbatim
136 *>
137 *> \param[out] WORK
138 *> \verbatim
139 *> WORK is REAL 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(N,M,P)**2.
146 *> \endverbatim
147 *>
148 *> \param[out] RWORK
149 *> \verbatim
150 *> RWORK is REAL array, dimension (max(N,M,P))
151 *> \endverbatim
152 *>
153 *> \param[out] RESULT
154 *> \verbatim
155 *> RESULT is REAL array, dimension (4)
156 *> The test ratios:
157 *> RESULT(1) = norm( R - Q'*A ) / ( MAX(M,N)*norm(A)*ULP)
158 *> RESULT(2) = norm( T*Z - Q'*B ) / (MAX(P,N)*norm(B)*ULP)
159 *> RESULT(3) = norm( I - Q'*Q ) / ( M*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 single_eig
172 *
173 * =====================================================================
174  SUBROUTINE sgqrts( N, M, P, 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, P, N
183 * ..
184 * .. Array Arguments ..
185  REAL A( LDA, * ), AF( LDA, * ), R( LDA, * ),
186  \$ q( lda, * ), b( ldb, * ), bf( ldb, * ),
187  \$ t( ldb, * ), z( ldb, * ), bwk( ldb, * ),
188  \$ taua( * ), taub( * ), result( 4 ),
189  \$ rwork( * ), work( lwork )
190 * ..
191 *
192 * =====================================================================
193 *
194 * .. Parameters ..
195  REAL ZERO, ONE
196  parameter( zero = 0.0e+0, one = 1.0e+0 )
197  REAL ROGUE
198  parameter( rogue = -1.0e+10 )
199 * ..
200 * .. Local Scalars ..
201  INTEGER INFO
202  REAL ANORM, BNORM, ULP, UNFL, RESID
203 * ..
204 * .. External Functions ..
205  REAL SLAMCH, SLANGE, SLANSY
206  EXTERNAL slamch, slange, slansy
207 * ..
208 * .. External Subroutines ..
209  EXTERNAL sgemm, slacpy, slaset, sorgqr,
210  \$ sorgrq, ssyrk
211 * ..
212 * .. Intrinsic Functions ..
213  INTRINSIC max, min, real
214 * ..
215 * .. Executable Statements ..
216 *
217  ulp = slamch( 'Precision' )
218  unfl = slamch( 'Safe minimum' )
219 *
220 * Copy the matrix A to the array AF.
221 *
222  CALL slacpy( 'Full', n, m, a, lda, af, lda )
223  CALL slacpy( 'Full', n, p, b, ldb, bf, ldb )
224 *
225  anorm = max( slange( '1', n, m, a, lda, rwork ), unfl )
226  bnorm = max( slange( '1', n, p, b, ldb, rwork ), unfl )
227 *
228 * Factorize the matrices A and B in the arrays AF and BF.
229 *
230  CALL sggqrf( n, m, p, af, lda, taua, bf, ldb, taub, work,
231  \$ lwork, info )
232 *
233 * Generate the N-by-N matrix Q
234 *
235  CALL slaset( 'Full', n, n, rogue, rogue, q, lda )
236  CALL slacpy( 'Lower', n-1, m, af( 2,1 ), lda, q( 2,1 ), lda )
237  CALL sorgqr( n, n, min( n, m ), q, lda, taua, work, lwork, info )
238 *
239 * Generate the P-by-P matrix Z
240 *
241  CALL slaset( 'Full', p, p, rogue, rogue, z, ldb )
242  IF( n.LE.p ) THEN
243  IF( n.GT.0 .AND. n.LT.p )
244  \$ CALL slacpy( 'Full', n, p-n, bf, ldb, z( p-n+1, 1 ), ldb )
245  IF( n.GT.1 )
246  \$ CALL slacpy( 'Lower', n-1, n-1, bf( 2, p-n+1 ), ldb,
247  \$ z( p-n+2, p-n+1 ), ldb )
248  ELSE
249  IF( p.GT.1)
250  \$ CALL slacpy( 'Lower', p-1, p-1, bf( n-p+2, 1 ), ldb,
251  \$ z( 2, 1 ), ldb )
252  END IF
253  CALL sorgrq( p, p, min( n, p ), z, ldb, taub, work, lwork, info )
254 *
255 * Copy R
256 *
257  CALL slaset( 'Full', n, m, zero, zero, r, lda )
258  CALL slacpy( 'Upper', n, m, af, lda, r, lda )
259 *
260 * Copy T
261 *
262  CALL slaset( 'Full', n, p, zero, zero, t, ldb )
263  IF( n.LE.p ) THEN
264  CALL slacpy( 'Upper', n, n, bf( 1, p-n+1 ), ldb, t( 1, p-n+1 ),
265  \$ ldb )
266  ELSE
267  CALL slacpy( 'Full', n-p, p, bf, ldb, t, ldb )
268  CALL slacpy( 'Upper', p, p, bf( n-p+1, 1 ), ldb, t( n-p+1, 1 ),
269  \$ ldb )
270  END IF
271 *
272 * Compute R - Q'*A
273 *
274  CALL sgemm( 'Transpose', 'No transpose', n, m, n, -one, q, lda, a,
275  \$ lda, one, r, lda )
276 *
277 * Compute norm( R - Q'*A ) / ( MAX(M,N)*norm(A)*ULP ) .
278 *
279  resid = slange( '1', n, m, r, lda, rwork )
280  IF( anorm.GT.zero ) THEN
281  result( 1 ) = ( ( resid / real( max(1,m,n) ) ) / anorm ) / ulp
282  ELSE
283  result( 1 ) = zero
284  END IF
285 *
286 * Compute T*Z - Q'*B
287 *
288  CALL sgemm( 'No Transpose', 'No transpose', n, p, p, one, t, ldb,
289  \$ z, ldb, zero, bwk, ldb )
290  CALL sgemm( 'Transpose', 'No transpose', n, p, n, -one, q, lda,
291  \$ b, ldb, one, bwk, ldb )
292 *
293 * Compute norm( T*Z - Q'*B ) / ( MAX(P,N)*norm(A)*ULP ) .
294 *
295  resid = slange( '1', n, p, bwk, ldb, rwork )
296  IF( bnorm.GT.zero ) THEN
297  result( 2 ) = ( ( resid / real( max(1,p,n ) ) )/bnorm ) / ulp
298  ELSE
299  result( 2 ) = zero
300  END IF
301 *
302 * Compute I - Q'*Q
303 *
304  CALL slaset( 'Full', n, n, zero, one, r, lda )
305  CALL ssyrk( 'Upper', 'Transpose', n, n, -one, q, lda, one, r,
306  \$ lda )
307 *
308 * Compute norm( I - Q'*Q ) / ( N * ULP ) .
309 *
310  resid = slansy( '1', 'Upper', n, r, lda, rwork )
311  result( 3 ) = ( resid / real( max( 1, n ) ) ) / ulp
312 *
313 * Compute I - Z'*Z
314 *
315  CALL slaset( 'Full', p, p, zero, one, t, ldb )
316  CALL ssyrk( 'Upper', 'Transpose', p, p, -one, z, ldb, one, t,
317  \$ ldb )
318 *
319 * Compute norm( I - Z'*Z ) / ( P*ULP ) .
320 *
321  resid = slansy( '1', 'Upper', p, t, ldb, rwork )
322  result( 4 ) = ( resid / real( max( 1, p ) ) ) / ulp
323 *
324  RETURN
325 *
326 * End of SGQRTS
327 *
328  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 sggqrf(N, M, P, A, LDA, TAUA, B, LDB, TAUB, WORK, LWORK, INFO)
SGGQRF
Definition: sggqrf.f:215
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 sgqrts(N, M, P, A, AF, Q, R, LDA, TAUA, B, BF, Z, T, BWK, LDB, TAUB, WORK, LWORK, RWORK, RESULT)
SGQRTS
Definition: sgqrts.f:176