LAPACK 3.11.0 LAPACK: Linear Algebra PACKage
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sqrt02.f
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1*> \brief \b SQRT02
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 SQRT02( M, N, K, A, AF, Q, R, LDA, TAU, WORK, LWORK,
12* RWORK, RESULT )
13*
14* .. Scalar Arguments ..
15* INTEGER K, 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*> SQRT02 tests SORGQR, which generates an m-by-n matrix Q with
30*> orthonornmal columns that is defined as the product of k elementary
31*> reflectors.
32*>
33*> Given the QR factorization of an m-by-n matrix A, SQRT02 generates
34*> the orthogonal matrix Q defined by the factorization of the first k
35*> columns of A; it compares R(1:n,1:k) with Q(1:m,1:n)'*A(1:m,1:k),
36*> and checks that the columns of Q are orthonormal.
37*> \endverbatim
38*
39* Arguments:
40* ==========
41*
42*> \param[in] M
43*> \verbatim
44*> M is INTEGER
45*> The number of rows of the matrix Q to be generated. M >= 0.
46*> \endverbatim
47*>
48*> \param[in] N
49*> \verbatim
50*> N is INTEGER
51*> The number of columns of the matrix Q to be generated.
52*> M >= N >= 0.
53*> \endverbatim
54*>
55*> \param[in] K
56*> \verbatim
57*> K is INTEGER
58*> The number of elementary reflectors whose product defines the
59*> matrix Q. N >= K >= 0.
60*> \endverbatim
61*>
62*> \param[in] A
63*> \verbatim
64*> A is REAL array, dimension (LDA,N)
65*> The m-by-n matrix A which was factorized by SQRT01.
66*> \endverbatim
67*>
68*> \param[in] AF
69*> \verbatim
70*> AF is REAL array, dimension (LDA,N)
71*> Details of the QR factorization of A, as returned by SGEQRF.
72*> See SGEQRF for further details.
73*> \endverbatim
74*>
75*> \param[out] Q
76*> \verbatim
77*> Q is REAL array, dimension (LDA,N)
78*> \endverbatim
79*>
80*> \param[out] R
81*> \verbatim
82*> R is REAL array, dimension (LDA,N)
83*> \endverbatim
84*>
85*> \param[in] LDA
86*> \verbatim
87*> LDA is INTEGER
88*> The leading dimension of the arrays A, AF, Q and R. LDA >= M.
89*> \endverbatim
90*>
91*> \param[in] TAU
92*> \verbatim
93*> TAU is REAL array, dimension (N)
94*> The scalar factors of the elementary reflectors corresponding
95*> to the QR factorization in AF.
96*> \endverbatim
97*>
98*> \param[out] WORK
99*> \verbatim
100*> WORK is REAL array, dimension (LWORK)
101*> \endverbatim
102*>
103*> \param[in] LWORK
104*> \verbatim
105*> LWORK is INTEGER
106*> The dimension of the array WORK.
107*> \endverbatim
108*>
109*> \param[out] RWORK
110*> \verbatim
111*> RWORK is REAL array, dimension (M)
112*> \endverbatim
113*>
114*> \param[out] RESULT
115*> \verbatim
116*> RESULT is REAL array, dimension (2)
117*> The test ratios:
118*> RESULT(1) = norm( R - Q'*A ) / ( M * norm(A) * EPS )
119*> RESULT(2) = norm( I - Q'*Q ) / ( M * EPS )
120*> \endverbatim
121*
122* Authors:
123* ========
124*
125*> \author Univ. of Tennessee
126*> \author Univ. of California Berkeley
127*> \author Univ. of Colorado Denver
128*> \author NAG Ltd.
129*
130*> \ingroup single_lin
131*
132* =====================================================================
133 SUBROUTINE sqrt02( M, N, K, A, AF, Q, R, LDA, TAU, WORK, LWORK,
134 \$ RWORK, RESULT )
135*
136* -- LAPACK test routine --
137* -- LAPACK is a software package provided by Univ. of Tennessee, --
138* -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
139*
140* .. Scalar Arguments ..
141 INTEGER K, LDA, LWORK, M, N
142* ..
143* .. Array Arguments ..
144 REAL A( LDA, * ), AF( LDA, * ), Q( LDA, * ),
145 \$ r( lda, * ), result( * ), rwork( * ), tau( * ),
146 \$ work( lwork )
147* ..
148*
149* =====================================================================
150*
151* .. Parameters ..
152 REAL ZERO, ONE
153 parameter( zero = 0.0e+0, one = 1.0e+0 )
154 REAL ROGUE
155 parameter( rogue = -1.0e+10 )
156* ..
157* .. Local Scalars ..
158 INTEGER INFO
159 REAL ANORM, EPS, RESID
160* ..
161* .. External Functions ..
162 REAL SLAMCH, SLANGE, SLANSY
163 EXTERNAL slamch, slange, slansy
164* ..
165* .. External Subroutines ..
166 EXTERNAL sgemm, slacpy, slaset, sorgqr, ssyrk
167* ..
168* .. Intrinsic Functions ..
169 INTRINSIC max, real
170* ..
171* .. Scalars in Common ..
172 CHARACTER*32 SRNAMT
173* ..
174* .. Common blocks ..
175 COMMON / srnamc / srnamt
176* ..
177* .. Executable Statements ..
178*
179 eps = slamch( 'Epsilon' )
180*
181* Copy the first k columns of the factorization to the array Q
182*
183 CALL slaset( 'Full', m, n, rogue, rogue, q, lda )
184 CALL slacpy( 'Lower', m-1, k, af( 2, 1 ), lda, q( 2, 1 ), lda )
185*
186* Generate the first n columns of the matrix Q
187*
188 srnamt = 'SORGQR'
189 CALL sorgqr( m, n, k, q, lda, tau, work, lwork, info )
190*
191* Copy R(1:n,1:k)
192*
193 CALL slaset( 'Full', n, k, zero, zero, r, lda )
194 CALL slacpy( 'Upper', n, k, af, lda, r, lda )
195*
196* Compute R(1:n,1:k) - Q(1:m,1:n)' * A(1:m,1:k)
197*
198 CALL sgemm( 'Transpose', 'No transpose', n, k, m, -one, q, lda, a,
199 \$ lda, one, r, lda )
200*
201* Compute norm( R - Q'*A ) / ( M * norm(A) * EPS ) .
202*
203 anorm = slange( '1', m, k, a, lda, rwork )
204 resid = slange( '1', n, k, r, lda, rwork )
205 IF( anorm.GT.zero ) THEN
206 result( 1 ) = ( ( resid / real( max( 1, m ) ) ) / anorm ) / eps
207 ELSE
208 result( 1 ) = zero
209 END IF
210*
211* Compute I - Q'*Q
212*
213 CALL slaset( 'Full', n, n, zero, one, r, lda )
214 CALL ssyrk( 'Upper', 'Transpose', n, m, -one, q, lda, one, r,
215 \$ lda )
216*
217* Compute norm( I - Q'*Q ) / ( M * EPS ) .
218*
219 resid = slansy( '1', 'Upper', n, r, lda, rwork )
220*
221 result( 2 ) = ( resid / real( max( 1, m ) ) ) / eps
222*
223 RETURN
224*
225* End of SQRT02
226*
227 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 sorgqr(M, N, K, A, LDA, TAU, WORK, LWORK, INFO)
SORGQR
Definition: sorgqr.f:128
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 sqrt02(M, N, K, A, AF, Q, R, LDA, TAU, WORK, LWORK, RWORK, RESULT)
SQRT02
Definition: sqrt02.f:135