LAPACK  3.10.0 LAPACK: Linear Algebra PACKage

## ◆ zrqt02()

 subroutine zrqt02 ( integer M, integer N, integer K, complex*16, dimension( lda, * ) A, complex*16, dimension( lda, * ) AF, complex*16, dimension( lda, * ) Q, complex*16, dimension( lda, * ) R, integer LDA, complex*16, dimension( * ) TAU, complex*16, dimension( lwork ) WORK, integer LWORK, double precision, dimension( * ) RWORK, double precision, dimension( * ) RESULT )

ZRQT02

Purpose:
``` ZRQT02 tests ZUNGRQ, which generates an m-by-n matrix Q with
orthonornmal rows that is defined as the product of k elementary
reflectors.

Given the RQ factorization of an m-by-n matrix A, ZRQT02 generates
the orthogonal matrix Q defined by the factorization of the last k
rows of A; it compares R(m-k+1:m,n-m+1:n) with
A(m-k+1:m,1:n)*Q(n-m+1:n,1:n)', and checks that the rows of Q are
orthonormal.```
Parameters
 [in] M ``` M is INTEGER The number of rows of the matrix Q to be generated. M >= 0.``` [in] N ``` N is INTEGER The number of columns of the matrix Q to be generated. N >= M >= 0.``` [in] K ``` K is INTEGER The number of elementary reflectors whose product defines the matrix Q. M >= K >= 0.``` [in] A ``` A is COMPLEX*16 array, dimension (LDA,N) The m-by-n matrix A which was factorized by ZRQT01.``` [in] AF ``` AF is COMPLEX*16 array, dimension (LDA,N) Details of the RQ factorization of A, as returned by ZGERQF. See ZGERQF for further details.``` [out] Q ` Q is COMPLEX*16 array, dimension (LDA,N)` [out] R ` R is COMPLEX*16 array, dimension (LDA,M)` [in] LDA ``` LDA is INTEGER The leading dimension of the arrays A, AF, Q and L. LDA >= N.``` [in] TAU ``` TAU is COMPLEX*16 array, dimension (M) The scalar factors of the elementary reflectors corresponding to the RQ factorization in AF.``` [out] WORK ` WORK is COMPLEX*16 array, dimension (LWORK)` [in] LWORK ``` LWORK is INTEGER The dimension of the array WORK.``` [out] RWORK ` RWORK is DOUBLE PRECISION array, dimension (M)` [out] RESULT ``` RESULT is DOUBLE PRECISION array, dimension (2) The test ratios: RESULT(1) = norm( R - A*Q' ) / ( N * norm(A) * EPS ) RESULT(2) = norm( I - Q*Q' ) / ( N * EPS )```

Definition at line 134 of file zrqt02.f.

136 *
137 * -- LAPACK test routine --
138 * -- LAPACK is a software package provided by Univ. of Tennessee, --
139 * -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
140 *
141 * .. Scalar Arguments ..
142  INTEGER K, LDA, LWORK, M, N
143 * ..
144 * .. Array Arguments ..
145  DOUBLE PRECISION RESULT( * ), RWORK( * )
146  COMPLEX*16 A( LDA, * ), AF( LDA, * ), Q( LDA, * ),
147  \$ R( LDA, * ), TAU( * ), WORK( LWORK )
148 * ..
149 *
150 * =====================================================================
151 *
152 * .. Parameters ..
153  DOUBLE PRECISION ZERO, ONE
154  parameter( zero = 0.0d+0, one = 1.0d+0 )
155  COMPLEX*16 ROGUE
156  parameter( rogue = ( -1.0d+10, -1.0d+10 ) )
157 * ..
158 * .. Local Scalars ..
159  INTEGER INFO
160  DOUBLE PRECISION ANORM, EPS, RESID
161 * ..
162 * .. External Functions ..
163  DOUBLE PRECISION DLAMCH, ZLANGE, ZLANSY
164  EXTERNAL dlamch, zlange, zlansy
165 * ..
166 * .. External Subroutines ..
167  EXTERNAL zgemm, zherk, zlacpy, zlaset, zungrq
168 * ..
169 * .. Intrinsic Functions ..
170  INTRINSIC dble, dcmplx, max
171 * ..
172 * .. Scalars in Common ..
173  CHARACTER*32 SRNAMT
174 * ..
175 * .. Common blocks ..
176  COMMON / srnamc / srnamt
177 * ..
178 * .. Executable Statements ..
179 *
180 * Quick return if possible
181 *
182  IF( m.EQ.0 .OR. n.EQ.0 .OR. k.EQ.0 ) THEN
183  result( 1 ) = zero
184  result( 2 ) = zero
185  RETURN
186  END IF
187 *
188  eps = dlamch( 'Epsilon' )
189 *
190 * Copy the last k rows of the factorization to the array Q
191 *
192  CALL zlaset( 'Full', m, n, rogue, rogue, q, lda )
193  IF( k.LT.n )
194  \$ CALL zlacpy( 'Full', k, n-k, af( m-k+1, 1 ), lda,
195  \$ q( m-k+1, 1 ), lda )
196  IF( k.GT.1 )
197  \$ CALL zlacpy( 'Lower', k-1, k-1, af( m-k+2, n-k+1 ), lda,
198  \$ q( m-k+2, n-k+1 ), lda )
199 *
200 * Generate the last n rows of the matrix Q
201 *
202  srnamt = 'ZUNGRQ'
203  CALL zungrq( m, n, k, q, lda, tau( m-k+1 ), work, lwork, info )
204 *
205 * Copy R(m-k+1:m,n-m+1:n)
206 *
207  CALL zlaset( 'Full', k, m, dcmplx( zero ), dcmplx( zero ),
208  \$ r( m-k+1, n-m+1 ), lda )
209  CALL zlacpy( 'Upper', k, k, af( m-k+1, n-k+1 ), lda,
210  \$ r( m-k+1, n-k+1 ), lda )
211 *
212 * Compute R(m-k+1:m,n-m+1:n) - A(m-k+1:m,1:n) * Q(n-m+1:n,1:n)'
213 *
214  CALL zgemm( 'No transpose', 'Conjugate transpose', k, m, n,
215  \$ dcmplx( -one ), a( m-k+1, 1 ), lda, q, lda,
216  \$ dcmplx( one ), r( m-k+1, n-m+1 ), lda )
217 *
218 * Compute norm( R - A*Q' ) / ( N * norm(A) * EPS ) .
219 *
220  anorm = zlange( '1', k, n, a( m-k+1, 1 ), lda, rwork )
221  resid = zlange( '1', k, m, r( m-k+1, n-m+1 ), lda, rwork )
222  IF( anorm.GT.zero ) THEN
223  result( 1 ) = ( ( resid / dble( max( 1, n ) ) ) / anorm ) / eps
224  ELSE
225  result( 1 ) = zero
226  END IF
227 *
228 * Compute I - Q*Q'
229 *
230  CALL zlaset( 'Full', m, m, dcmplx( zero ), dcmplx( one ), r, lda )
231  CALL zherk( 'Upper', 'No transpose', m, n, -one, q, lda, one, r,
232  \$ lda )
233 *
234 * Compute norm( I - Q*Q' ) / ( N * EPS ) .
235 *
236  resid = zlansy( '1', 'Upper', m, r, lda, rwork )
237 *
238  result( 2 ) = ( resid / dble( max( 1, n ) ) ) / eps
239 *
240  RETURN
241 *
242 * End of ZRQT02
243 *
double precision function dlamch(CMACH)
DLAMCH
Definition: dlamch.f:69
subroutine zgemm(TRANSA, TRANSB, M, N, K, ALPHA, A, LDA, B, LDB, BETA, C, LDC)
ZGEMM
Definition: zgemm.f:187
subroutine zherk(UPLO, TRANS, N, K, ALPHA, A, LDA, BETA, C, LDC)
ZHERK
Definition: zherk.f:173
double precision function zlange(NORM, M, N, A, LDA, WORK)
ZLANGE returns the value of the 1-norm, Frobenius norm, infinity-norm, or the largest absolute value ...
Definition: zlange.f:115
subroutine zlacpy(UPLO, M, N, A, LDA, B, LDB)
ZLACPY copies all or part of one two-dimensional array to another.
Definition: zlacpy.f:103
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:106
subroutine zungrq(M, N, K, A, LDA, TAU, WORK, LWORK, INFO)
ZUNGRQ
Definition: zungrq.f:128
double precision function zlansy(NORM, UPLO, N, A, LDA, WORK)
ZLANSY returns the value of the 1-norm, or the Frobenius norm, or the infinity norm,...
Definition: zlansy.f:123
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