LAPACK  3.6.1 LAPACK: Linear Algebra PACKage
sqrt01.f
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1 *> \brief \b SQRT01
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 SQRT01( M, N, A, AF, Q, R, LDA, TAU, WORK, LWORK,
12 * RWORK, RESULT )
13 *
14 * .. Scalar Arguments ..
15 * INTEGER LDA, LWORK, M, N
16 * ..
17 * .. Array Arguments ..
18 * REAL A( LDA, * ), AF( LDA, * ), Q( LDA, * ),
19 * \$ R( LDA, * ), RESULT( * ), RWORK( * ), TAU( * ),
20 * \$ WORK( LWORK )
21 * ..
22 *
23 *
24 *> \par Purpose:
25 * =============
26 *>
27 *> \verbatim
28 *>
29 *> SQRT01 tests SGEQRF, which computes the QR factorization of an m-by-n
30 *> matrix A, and partially tests SORGQR which forms the m-by-m
31 *> orthogonal matrix Q.
32 *>
33 *> SQRT01 compares R with Q'*A, and checks that Q is orthogonal.
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] N
46 *> \verbatim
47 *> N is INTEGER
48 *> The number of columns of the matrix A. N >= 0.
49 *> \endverbatim
50 *>
51 *> \param[in] A
52 *> \verbatim
53 *> A is REAL array, dimension (LDA,N)
54 *> The m-by-n matrix A.
55 *> \endverbatim
56 *>
57 *> \param[out] AF
58 *> \verbatim
59 *> AF is REAL array, dimension (LDA,N)
60 *> Details of the QR factorization of A, as returned by SGEQRF.
61 *> See SGEQRF for further details.
62 *> \endverbatim
63 *>
64 *> \param[out] Q
65 *> \verbatim
66 *> Q is REAL array, dimension (LDA,M)
67 *> The m-by-m orthogonal matrix Q.
68 *> \endverbatim
69 *>
70 *> \param[out] R
71 *> \verbatim
72 *> R is REAL array, dimension (LDA,max(M,N))
73 *> \endverbatim
74 *>
75 *> \param[in] LDA
76 *> \verbatim
77 *> LDA is INTEGER
78 *> The leading dimension of the arrays A, AF, Q and R.
79 *> LDA >= max(M,N).
80 *> \endverbatim
81 *>
82 *> \param[out] TAU
83 *> \verbatim
84 *> TAU is REAL array, dimension (min(M,N))
85 *> The scalar factors of the elementary reflectors, as returned
86 *> by SGEQRF.
87 *> \endverbatim
88 *>
89 *> \param[out] WORK
90 *> \verbatim
91 *> WORK is REAL array, dimension (LWORK)
92 *> \endverbatim
93 *>
94 *> \param[in] LWORK
95 *> \verbatim
96 *> LWORK is INTEGER
97 *> The dimension of the array WORK.
98 *> \endverbatim
99 *>
100 *> \param[out] RWORK
101 *> \verbatim
102 *> RWORK is REAL array, dimension (M)
103 *> \endverbatim
104 *>
105 *> \param[out] RESULT
106 *> \verbatim
107 *> RESULT is REAL array, dimension (2)
108 *> The test ratios:
109 *> RESULT(1) = norm( R - Q'*A ) / ( M * norm(A) * EPS )
110 *> RESULT(2) = norm( I - Q'*Q ) / ( M * EPS )
111 *> \endverbatim
112 *
113 * Authors:
114 * ========
115 *
116 *> \author Univ. of Tennessee
117 *> \author Univ. of California Berkeley
118 *> \author Univ. of Colorado Denver
119 *> \author NAG Ltd.
120 *
121 *> \date November 2011
122 *
123 *> \ingroup single_lin
124 *
125 * =====================================================================
126  SUBROUTINE sqrt01( M, N, A, AF, Q, R, LDA, TAU, WORK, LWORK,
127  \$ rwork, result )
128 *
129 * -- LAPACK test routine (version 3.4.0) --
130 * -- LAPACK is a software package provided by Univ. of Tennessee, --
131 * -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
132 * November 2011
133 *
134 * .. Scalar Arguments ..
135  INTEGER LDA, LWORK, M, N
136 * ..
137 * .. Array Arguments ..
138  REAL A( lda, * ), AF( lda, * ), Q( lda, * ),
139  \$ r( lda, * ), result( * ), rwork( * ), tau( * ),
140  \$ work( lwork )
141 * ..
142 *
143 * =====================================================================
144 *
145 * .. Parameters ..
146  REAL ZERO, ONE
147  parameter ( zero = 0.0e+0, one = 1.0e+0 )
148  REAL ROGUE
149  parameter ( rogue = -1.0e+10 )
150 * ..
151 * .. Local Scalars ..
152  INTEGER INFO, MINMN
153  REAL ANORM, EPS, RESID
154 * ..
155 * .. External Functions ..
156  REAL SLAMCH, SLANGE, SLANSY
157  EXTERNAL slamch, slange, slansy
158 * ..
159 * .. External Subroutines ..
160  EXTERNAL sgemm, sgeqrf, slacpy, slaset, sorgqr, ssyrk
161 * ..
162 * .. Intrinsic Functions ..
163  INTRINSIC max, min, real
164 * ..
165 * .. Scalars in Common ..
166  CHARACTER*32 SRNAMT
167 * ..
168 * .. Common blocks ..
169  COMMON / srnamc / srnamt
170 * ..
171 * .. Executable Statements ..
172 *
173  minmn = min( m, n )
174  eps = slamch( 'Epsilon' )
175 *
176 * Copy the matrix A to the array AF.
177 *
178  CALL slacpy( 'Full', m, n, a, lda, af, lda )
179 *
180 * Factorize the matrix A in the array AF.
181 *
182  srnamt = 'SGEQRF'
183  CALL sgeqrf( m, n, af, lda, tau, work, lwork, info )
184 *
185 * Copy details of Q
186 *
187  CALL slaset( 'Full', m, m, rogue, rogue, q, lda )
188  CALL slacpy( 'Lower', m-1, n, af( 2, 1 ), lda, q( 2, 1 ), lda )
189 *
190 * Generate the m-by-m matrix Q
191 *
192  srnamt = 'SORGQR'
193  CALL sorgqr( m, m, minmn, q, lda, tau, work, lwork, info )
194 *
195 * Copy R
196 *
197  CALL slaset( 'Full', m, n, zero, zero, r, lda )
198  CALL slacpy( 'Upper', m, n, af, lda, r, lda )
199 *
200 * Compute R - Q'*A
201 *
202  CALL sgemm( 'Transpose', 'No transpose', m, n, m, -one, q, lda, a,
203  \$ lda, one, r, lda )
204 *
205 * Compute norm( R - Q'*A ) / ( M * norm(A) * EPS ) .
206 *
207  anorm = slange( '1', m, n, a, lda, rwork )
208  resid = slange( '1', m, n, r, lda, rwork )
209  IF( anorm.GT.zero ) THEN
210  result( 1 ) = ( ( resid / REAL( MAX( 1, M ) ) ) / anorm ) / eps
211  ELSE
212  result( 1 ) = zero
213  END IF
214 *
215 * Compute I - Q'*Q
216 *
217  CALL slaset( 'Full', m, m, zero, one, r, lda )
218  CALL ssyrk( 'Upper', 'Transpose', m, m, -one, q, lda, one, r,
219  \$ lda )
220 *
221 * Compute norm( I - Q'*Q ) / ( M * EPS ) .
222 *
223  resid = slansy( '1', 'Upper', m, r, lda, rwork )
224 *
225  result( 2 ) = ( resid / REAL( MAX( 1, M ) ) ) / eps
226 *
227  RETURN
228 *
229 * End of SQRT01
230 *
231  END
subroutine ssyrk(UPLO, TRANS, N, K, ALPHA, A, LDA, BETA, C, LDC)
SSYRK
Definition: ssyrk.f:171
subroutine sqrt01(M, N, A, AF, Q, R, LDA, TAU, WORK, LWORK, RWORK, RESULT)
SQRT01
Definition: sqrt01.f:128
subroutine sgemm(TRANSA, TRANSB, M, N, K, ALPHA, A, LDA, B, LDB, BETA, C, LDC)
SGEMM
Definition: sgemm.f:189
subroutine sgeqrf(M, N, A, LDA, TAU, WORK, LWORK, INFO)
SGEQRF
Definition: sgeqrf.f:138
subroutine slacpy(UPLO, M, N, A, LDA, B, LDB)
SLACPY copies all or part of one two-dimensional array to another.
Definition: slacpy.f:105
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:112
subroutine sorgqr(M, N, K, A, LDA, TAU, WORK, LWORK, INFO)
SORGQR
Definition: sorgqr.f:130