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SUBROUTINE ZRQT02( M, N, K, A, AF, Q, R, LDA, TAU, WORK, LWORK,
$ RWORK, RESULT )
*
* -- LAPACK test routine (version 3.0) --
* Univ. of Tennessee, Univ. of California Berkeley, NAG Ltd.,
* Courant Institute, Argonne National Lab, and Rice University
* September 30, 1994
*
* .. Scalar Arguments ..
INTEGER K, LDA, LWORK, M, N
* ..
* .. Array Arguments ..
DOUBLE PRECISION RESULT( * ), RWORK( * )
COMPLEX*16 A( LDA, * ), AF( LDA, * ), Q( LDA, * ),
$ R( LDA, * ), TAU( * ), WORK( LWORK )
* ..
*
* 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.
*
* Arguments
* =========
*
* M (input) INTEGER
* The number of rows of the matrix Q to be generated. M >= 0.
*
* N (input) INTEGER
* The number of columns of the matrix Q to be generated.
* N >= M >= 0.
*
* K (input) INTEGER
* The number of elementary reflectors whose product defines the
* matrix Q. M >= K >= 0.
*
* A (input) COMPLEX*16 array, dimension (LDA,N)
* The m-by-n matrix A which was factorized by ZRQT01.
*
* AF (input) COMPLEX*16 array, dimension (LDA,N)
* Details of the RQ factorization of A, as returned by ZGERQF.
* See ZGERQF for further details.
*
* Q (workspace) COMPLEX*16 array, dimension (LDA,N)
*
* R (workspace) COMPLEX*16 array, dimension (LDA,M)
*
* LDA (input) INTEGER
* The leading dimension of the arrays A, AF, Q and L. LDA >= N.
*
* TAU (input) COMPLEX*16 array, dimension (M)
* The scalar factors of the elementary reflectors corresponding
* to the RQ factorization in AF.
*
* WORK (workspace) COMPLEX*16 array, dimension (LWORK)
*
* LWORK (input) INTEGER
* The dimension of the array WORK.
*
* RWORK (workspace) DOUBLE PRECISION array, dimension (M)
*
* RESULT (output) 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 )
*
* =====================================================================
*
* .. Parameters ..
DOUBLE PRECISION ZERO, ONE
PARAMETER ( ZERO = 0.0D+0, ONE = 1.0D+0 )
COMPLEX*16 ROGUE
PARAMETER ( ROGUE = ( -1.0D+10, -1.0D+10 ) )
* ..
* .. Local Scalars ..
INTEGER INFO
DOUBLE PRECISION ANORM, EPS, RESID
* ..
* .. External Functions ..
DOUBLE PRECISION DLAMCH, ZLANGE, ZLANSY
EXTERNAL DLAMCH, ZLANGE, ZLANSY
* ..
* .. External Subroutines ..
EXTERNAL ZGEMM, ZHERK, ZLACPY, ZLASET, ZUNGRQ
* ..
* .. Intrinsic Functions ..
INTRINSIC DBLE, DCMPLX, MAX
* ..
* .. Scalars in Common ..
CHARACTER*6 SRNAMT
* ..
* .. Common blocks ..
COMMON / SRNAMC / SRNAMT
* ..
* .. Executable Statements ..
*
* Quick return if possible
*
IF( M.EQ.0 .OR. N.EQ.0 .OR. K.EQ.0 ) THEN
RESULT( 1 ) = ZERO
RESULT( 2 ) = ZERO
RETURN
END IF
*
EPS = DLAMCH( 'Epsilon' )
*
* Copy the last k rows of the factorization to the array Q
*
CALL ZLASET( 'Full', M, N, ROGUE, ROGUE, Q, LDA )
IF( K.LT.N )
$ CALL ZLACPY( 'Full', K, N-K, AF( M-K+1, 1 ), LDA,
$ Q( M-K+1, 1 ), LDA )
IF( K.GT.1 )
$ CALL ZLACPY( 'Lower', K-1, K-1, AF( M-K+2, N-K+1 ), LDA,
$ Q( M-K+2, N-K+1 ), LDA )
*
* Generate the last n rows of the matrix Q
*
SRNAMT = 'ZUNGRQ'
CALL ZUNGRQ( M, N, K, Q, LDA, TAU( M-K+1 ), WORK, LWORK, INFO )
*
* Copy R(m-k+1:m,n-m+1:n)
*
CALL ZLASET( 'Full', K, M, DCMPLX( ZERO ), DCMPLX( ZERO ),
$ R( M-K+1, N-M+1 ), LDA )
CALL ZLACPY( 'Upper', K, K, AF( M-K+1, N-K+1 ), LDA,
$ R( M-K+1, N-K+1 ), LDA )
*
* Compute R(m-k+1:m,n-m+1:n) - A(m-k+1:m,1:n) * Q(n-m+1:n,1:n)'
*
CALL ZGEMM( 'No transpose', 'Conjugate transpose', K, M, N,
$ DCMPLX( -ONE ), A( M-K+1, 1 ), LDA, Q, LDA,
$ DCMPLX( ONE ), R( M-K+1, N-M+1 ), LDA )
*
* Compute norm( R - A*Q' ) / ( N * norm(A) * EPS ) .
*
ANORM = ZLANGE( '1', K, N, A( M-K+1, 1 ), LDA, RWORK )
RESID = ZLANGE( '1', K, M, R( M-K+1, N-M+1 ), LDA, RWORK )
IF( ANORM.GT.ZERO ) THEN
RESULT( 1 ) = ( ( RESID / DBLE( MAX( 1, N ) ) ) / ANORM ) / EPS
ELSE
RESULT( 1 ) = ZERO
END IF
*
* Compute I - Q*Q'
*
CALL ZLASET( 'Full', M, M, DCMPLX( ZERO ), DCMPLX( ONE ), R, LDA )
CALL ZHERK( 'Upper', 'No transpose', M, N, -ONE, Q, LDA, ONE, R,
$ LDA )
*
* Compute norm( I - Q*Q' ) / ( N * EPS ) .
*
RESID = ZLANSY( '1', 'Upper', M, R, LDA, RWORK )
*
RESULT( 2 ) = ( RESID / DBLE( MAX( 1, N ) ) ) / EPS
*
RETURN
*
* End of ZRQT02
*
END
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