LAPACK  3.8.0
LAPACK: Linear Algebra PACKage
zrqt02.f
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1 *> \brief \b ZRQT02
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 ZRQT02( 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 * DOUBLE PRECISION RESULT( * ), RWORK( * )
19 * COMPLEX*16 A( LDA, * ), AF( LDA, * ), Q( LDA, * ),
20 * $ R( LDA, * ), TAU( * ), WORK( LWORK )
21 * ..
22 *
23 *
24 *> \par Purpose:
25 * =============
26 *>
27 *> \verbatim
28 *>
29 *> ZRQT02 tests ZUNGRQ, which generates an m-by-n matrix Q with
30 *> orthonornmal rows that is defined as the product of k elementary
31 *> reflectors.
32 *>
33 *> Given the RQ factorization of an m-by-n matrix A, ZRQT02 generates
34 *> the orthogonal matrix Q defined by the factorization of the last k
35 *> rows of A; it compares R(m-k+1:m,n-m+1:n) with
36 *> A(m-k+1:m,1:n)*Q(n-m+1:n,1:n)', and checks that the rows of Q are
37 *> orthonormal.
38 *> \endverbatim
39 *
40 * Arguments:
41 * ==========
42 *
43 *> \param[in] M
44 *> \verbatim
45 *> M is INTEGER
46 *> The number of rows of the matrix Q to be generated. M >= 0.
47 *> \endverbatim
48 *>
49 *> \param[in] N
50 *> \verbatim
51 *> N is INTEGER
52 *> The number of columns of the matrix Q to be generated.
53 *> N >= M >= 0.
54 *> \endverbatim
55 *>
56 *> \param[in] K
57 *> \verbatim
58 *> K is INTEGER
59 *> The number of elementary reflectors whose product defines the
60 *> matrix Q. M >= K >= 0.
61 *> \endverbatim
62 *>
63 *> \param[in] A
64 *> \verbatim
65 *> A is COMPLEX*16 array, dimension (LDA,N)
66 *> The m-by-n matrix A which was factorized by ZRQT01.
67 *> \endverbatim
68 *>
69 *> \param[in] AF
70 *> \verbatim
71 *> AF is COMPLEX*16 array, dimension (LDA,N)
72 *> Details of the RQ factorization of A, as returned by ZGERQF.
73 *> See ZGERQF for further details.
74 *> \endverbatim
75 *>
76 *> \param[out] Q
77 *> \verbatim
78 *> Q is COMPLEX*16 array, dimension (LDA,N)
79 *> \endverbatim
80 *>
81 *> \param[out] R
82 *> \verbatim
83 *> R is COMPLEX*16 array, dimension (LDA,M)
84 *> \endverbatim
85 *>
86 *> \param[in] LDA
87 *> \verbatim
88 *> LDA is INTEGER
89 *> The leading dimension of the arrays A, AF, Q and L. LDA >= N.
90 *> \endverbatim
91 *>
92 *> \param[in] TAU
93 *> \verbatim
94 *> TAU is COMPLEX*16 array, dimension (M)
95 *> The scalar factors of the elementary reflectors corresponding
96 *> to the RQ factorization in AF.
97 *> \endverbatim
98 *>
99 *> \param[out] WORK
100 *> \verbatim
101 *> WORK is COMPLEX*16 array, dimension (LWORK)
102 *> \endverbatim
103 *>
104 *> \param[in] LWORK
105 *> \verbatim
106 *> LWORK is INTEGER
107 *> The dimension of the array WORK.
108 *> \endverbatim
109 *>
110 *> \param[out] RWORK
111 *> \verbatim
112 *> RWORK is DOUBLE PRECISION array, dimension (M)
113 *> \endverbatim
114 *>
115 *> \param[out] RESULT
116 *> \verbatim
117 *> RESULT is DOUBLE PRECISION array, dimension (2)
118 *> The test ratios:
119 *> RESULT(1) = norm( R - A*Q' ) / ( N * norm(A) * EPS )
120 *> RESULT(2) = norm( I - Q*Q' ) / ( N * EPS )
121 *> \endverbatim
122 *
123 * Authors:
124 * ========
125 *
126 *> \author Univ. of Tennessee
127 *> \author Univ. of California Berkeley
128 *> \author Univ. of Colorado Denver
129 *> \author NAG Ltd.
130 *
131 *> \date December 2016
132 *
133 *> \ingroup complex16_lin
134 *
135 * =====================================================================
136  SUBROUTINE zrqt02( M, N, K, A, AF, Q, R, LDA, TAU, WORK, LWORK,
137  $ RWORK, RESULT )
138 *
139 * -- LAPACK test routine (version 3.7.0) --
140 * -- LAPACK is a software package provided by Univ. of Tennessee, --
141 * -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
142 * December 2016
143 *
144 * .. Scalar Arguments ..
145  INTEGER K, LDA, LWORK, M, N
146 * ..
147 * .. Array Arguments ..
148  DOUBLE PRECISION RESULT( * ), RWORK( * )
149  COMPLEX*16 A( lda, * ), AF( lda, * ), Q( lda, * ),
150  $ r( lda, * ), tau( * ), work( lwork )
151 * ..
152 *
153 * =====================================================================
154 *
155 * .. Parameters ..
156  DOUBLE PRECISION ZERO, ONE
157  parameter( zero = 0.0d+0, one = 1.0d+0 )
158  COMPLEX*16 ROGUE
159  parameter( rogue = ( -1.0d+10, -1.0d+10 ) )
160 * ..
161 * .. Local Scalars ..
162  INTEGER INFO
163  DOUBLE PRECISION ANORM, EPS, RESID
164 * ..
165 * .. External Functions ..
166  DOUBLE PRECISION DLAMCH, ZLANGE, ZLANSY
167  EXTERNAL dlamch, zlange, zlansy
168 * ..
169 * .. External Subroutines ..
170  EXTERNAL zgemm, zherk, zlacpy, zlaset, zungrq
171 * ..
172 * .. Intrinsic Functions ..
173  INTRINSIC dble, dcmplx, max
174 * ..
175 * .. Scalars in Common ..
176  CHARACTER*32 SRNAMT
177 * ..
178 * .. Common blocks ..
179  COMMON / srnamc / srnamt
180 * ..
181 * .. Executable Statements ..
182 *
183 * Quick return if possible
184 *
185  IF( m.EQ.0 .OR. n.EQ.0 .OR. k.EQ.0 ) THEN
186  result( 1 ) = zero
187  result( 2 ) = zero
188  RETURN
189  END IF
190 *
191  eps = dlamch( 'Epsilon' )
192 *
193 * Copy the last k rows of the factorization to the array Q
194 *
195  CALL zlaset( 'Full', m, n, rogue, rogue, q, lda )
196  IF( k.LT.n )
197  $ CALL zlacpy( 'Full', k, n-k, af( m-k+1, 1 ), lda,
198  $ q( m-k+1, 1 ), lda )
199  IF( k.GT.1 )
200  $ CALL zlacpy( 'Lower', k-1, k-1, af( m-k+2, n-k+1 ), lda,
201  $ q( m-k+2, n-k+1 ), lda )
202 *
203 * Generate the last n rows of the matrix Q
204 *
205  srnamt = 'ZUNGRQ'
206  CALL zungrq( m, n, k, q, lda, tau( m-k+1 ), work, lwork, info )
207 *
208 * Copy R(m-k+1:m,n-m+1:n)
209 *
210  CALL zlaset( 'Full', k, m, dcmplx( zero ), dcmplx( zero ),
211  $ r( m-k+1, n-m+1 ), lda )
212  CALL zlacpy( 'Upper', k, k, af( m-k+1, n-k+1 ), lda,
213  $ r( m-k+1, n-k+1 ), lda )
214 *
215 * Compute R(m-k+1:m,n-m+1:n) - A(m-k+1:m,1:n) * Q(n-m+1:n,1:n)'
216 *
217  CALL zgemm( 'No transpose', 'Conjugate transpose', k, m, n,
218  $ dcmplx( -one ), a( m-k+1, 1 ), lda, q, lda,
219  $ dcmplx( one ), r( m-k+1, n-m+1 ), lda )
220 *
221 * Compute norm( R - A*Q' ) / ( N * norm(A) * EPS ) .
222 *
223  anorm = zlange( '1', k, n, a( m-k+1, 1 ), lda, rwork )
224  resid = zlange( '1', k, m, r( m-k+1, n-m+1 ), lda, rwork )
225  IF( anorm.GT.zero ) THEN
226  result( 1 ) = ( ( resid / dble( max( 1, n ) ) ) / anorm ) / eps
227  ELSE
228  result( 1 ) = zero
229  END IF
230 *
231 * Compute I - Q*Q'
232 *
233  CALL zlaset( 'Full', m, m, dcmplx( zero ), dcmplx( one ), r, lda )
234  CALL zherk( 'Upper', 'No transpose', m, n, -one, q, lda, one, r,
235  $ lda )
236 *
237 * Compute norm( I - Q*Q' ) / ( N * EPS ) .
238 *
239  resid = zlansy( '1', 'Upper', m, r, lda, rwork )
240 *
241  result( 2 ) = ( resid / dble( max( 1, n ) ) ) / eps
242 *
243  RETURN
244 *
245 * End of ZRQT02
246 *
247  END
subroutine zlaset(UPLO, M, N, ALPHA, BETA, A, LDA)
ZLASET initializes the off-diagonal elements and the diagonal elements of a matrix to given values...
Definition: zlaset.f:108
subroutine zgemm(TRANSA, TRANSB, M, N, K, ALPHA, A, LDA, B, LDB, BETA, C, LDC)
ZGEMM
Definition: zgemm.f:189
subroutine zungrq(M, N, K, A, LDA, TAU, WORK, LWORK, INFO)
ZUNGRQ
Definition: zungrq.f:130
subroutine zrqt02(M, N, K, A, AF, Q, R, LDA, TAU, WORK, LWORK, RWORK, RESULT)
ZRQT02
Definition: zrqt02.f:138
subroutine zherk(UPLO, TRANS, N, K, ALPHA, A, LDA, BETA, C, LDC)
ZHERK
Definition: zherk.f:175
subroutine zlacpy(UPLO, M, N, A, LDA, B, LDB)
ZLACPY copies all or part of one two-dimensional array to another.
Definition: zlacpy.f:105