LAPACK  3.4.1
LAPACK: Linear Algebra PACKage
zgrqts.f
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00001 *> \brief \b ZGRQTS
00002 *
00003 *  =========== DOCUMENTATION ===========
00004 *
00005 * Online html documentation available at 
00006 *            http://www.netlib.org/lapack/explore-html/ 
00007 *
00008 *  Definition:
00009 *  ===========
00010 *
00011 *       SUBROUTINE ZGRQTS( M, P, N, A, AF, Q, R, LDA, TAUA, B, BF, Z, T,
00012 *                          BWK, LDB, TAUB, WORK, LWORK, RWORK, RESULT )
00013 * 
00014 *       .. Scalar Arguments ..
00015 *       INTEGER            LDA, LDB, LWORK, M, N, P
00016 *       ..
00017 *       .. Array Arguments ..
00018 *       DOUBLE PRECISION   RESULT( 4 ), RWORK( * )
00019 *       COMPLEX*16         A( LDA, * ), AF( LDA, * ), B( LDB, * ),
00020 *      $                   BF( LDB, * ), BWK( LDB, * ), Q( LDA, * ),
00021 *      $                   R( LDA, * ), T( LDB, * ), TAUA( * ), TAUB( * ),
00022 *      $                   WORK( LWORK ), Z( LDB, * )
00023 *       ..
00024 *  
00025 *
00026 *> \par Purpose:
00027 *  =============
00028 *>
00029 *> \verbatim
00030 *>
00031 *> ZGRQTS tests ZGGRQF, which computes the GRQ factorization of an
00032 *> M-by-N matrix A and a P-by-N matrix B: A = R*Q and B = Z*T*Q.
00033 *> \endverbatim
00034 *
00035 *  Arguments:
00036 *  ==========
00037 *
00038 *> \param[in] M
00039 *> \verbatim
00040 *>          M is INTEGER
00041 *>          The number of rows of the matrix A.  M >= 0.
00042 *> \endverbatim
00043 *>
00044 *> \param[in] P
00045 *> \verbatim
00046 *>          P is INTEGER
00047 *>          The number of rows of the matrix B.  P >= 0.
00048 *> \endverbatim
00049 *>
00050 *> \param[in] N
00051 *> \verbatim
00052 *>          N is INTEGER
00053 *>          The number of columns of the matrices A and B.  N >= 0.
00054 *> \endverbatim
00055 *>
00056 *> \param[in] A
00057 *> \verbatim
00058 *>          A is COMPLEX*16 array, dimension (LDA,N)
00059 *>          The M-by-N matrix A.
00060 *> \endverbatim
00061 *>
00062 *> \param[out] AF
00063 *> \verbatim
00064 *>          AF is COMPLEX*16 array, dimension (LDA,N)
00065 *>          Details of the GRQ factorization of A and B, as returned
00066 *>          by ZGGRQF, see CGGRQF for further details.
00067 *> \endverbatim
00068 *>
00069 *> \param[out] Q
00070 *> \verbatim
00071 *>          Q is COMPLEX*16 array, dimension (LDA,N)
00072 *>          The N-by-N unitary matrix Q.
00073 *> \endverbatim
00074 *>
00075 *> \param[out] R
00076 *> \verbatim
00077 *>          R is COMPLEX*16 array, dimension (LDA,MAX(M,N))
00078 *> \endverbatim
00079 *>
00080 *> \param[in] LDA
00081 *> \verbatim
00082 *>          LDA is INTEGER
00083 *>          The leading dimension of the arrays A, AF, R and Q.
00084 *>          LDA >= max(M,N).
00085 *> \endverbatim
00086 *>
00087 *> \param[out] TAUA
00088 *> \verbatim
00089 *>          TAUA is COMPLEX*16 array, dimension (min(M,N))
00090 *>          The scalar factors of the elementary reflectors, as returned
00091 *>          by DGGQRC.
00092 *> \endverbatim
00093 *>
00094 *> \param[in] B
00095 *> \verbatim
00096 *>          B is COMPLEX*16 array, dimension (LDB,N)
00097 *>          On entry, the P-by-N matrix A.
00098 *> \endverbatim
00099 *>
00100 *> \param[out] BF
00101 *> \verbatim
00102 *>          BF is COMPLEX*16 array, dimension (LDB,N)
00103 *>          Details of the GQR factorization of A and B, as returned
00104 *>          by ZGGRQF, see CGGRQF for further details.
00105 *> \endverbatim
00106 *>
00107 *> \param[out] Z
00108 *> \verbatim
00109 *>          Z is DOUBLE PRECISION array, dimension (LDB,P)
00110 *>          The P-by-P unitary matrix Z.
00111 *> \endverbatim
00112 *>
00113 *> \param[out] T
00114 *> \verbatim
00115 *>          T is COMPLEX*16 array, dimension (LDB,max(P,N))
00116 *> \endverbatim
00117 *>
00118 *> \param[out] BWK
00119 *> \verbatim
00120 *>          BWK is COMPLEX*16 array, dimension (LDB,N)
00121 *> \endverbatim
00122 *>
00123 *> \param[in] LDB
00124 *> \verbatim
00125 *>          LDB is INTEGER
00126 *>          The leading dimension of the arrays B, BF, Z and T.
00127 *>          LDB >= max(P,N).
00128 *> \endverbatim
00129 *>
00130 *> \param[out] TAUB
00131 *> \verbatim
00132 *>          TAUB is COMPLEX*16 array, dimension (min(P,N))
00133 *>          The scalar factors of the elementary reflectors, as returned
00134 *>          by DGGRQF.
00135 *> \endverbatim
00136 *>
00137 *> \param[out] WORK
00138 *> \verbatim
00139 *>          WORK is COMPLEX*16 array, dimension (LWORK)
00140 *> \endverbatim
00141 *>
00142 *> \param[in] LWORK
00143 *> \verbatim
00144 *>          LWORK is INTEGER
00145 *>          The dimension of the array WORK, LWORK >= max(M,P,N)**2.
00146 *> \endverbatim
00147 *>
00148 *> \param[out] RWORK
00149 *> \verbatim
00150 *>          RWORK is DOUBLE PRECISION array, dimension (M)
00151 *> \endverbatim
00152 *>
00153 *> \param[out] RESULT
00154 *> \verbatim
00155 *>          RESULT is DOUBLE PRECISION array, dimension (4)
00156 *>          The test ratios:
00157 *>            RESULT(1) = norm( R - A*Q' ) / ( MAX(M,N)*norm(A)*ULP)
00158 *>            RESULT(2) = norm( T*Q - Z'*B ) / (MAX(P,N)*norm(B)*ULP)
00159 *>            RESULT(3) = norm( I - Q'*Q ) / ( N*ULP )
00160 *>            RESULT(4) = norm( I - Z'*Z ) / ( P*ULP )
00161 *> \endverbatim
00162 *
00163 *  Authors:
00164 *  ========
00165 *
00166 *> \author Univ. of Tennessee 
00167 *> \author Univ. of California Berkeley 
00168 *> \author Univ. of Colorado Denver 
00169 *> \author NAG Ltd. 
00170 *
00171 *> \date November 2011
00172 *
00173 *> \ingroup complex16_eig
00174 *
00175 *  =====================================================================
00176       SUBROUTINE ZGRQTS( M, P, N, A, AF, Q, R, LDA, TAUA, B, BF, Z, T,
00177      $                   BWK, LDB, TAUB, WORK, LWORK, RWORK, RESULT )
00178 *
00179 *  -- LAPACK test routine (version 3.4.0) --
00180 *  -- LAPACK is a software package provided by Univ. of Tennessee,    --
00181 *  -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
00182 *     November 2011
00183 *
00184 *     .. Scalar Arguments ..
00185       INTEGER            LDA, LDB, LWORK, M, N, P
00186 *     ..
00187 *     .. Array Arguments ..
00188       DOUBLE PRECISION   RESULT( 4 ), RWORK( * )
00189       COMPLEX*16         A( LDA, * ), AF( LDA, * ), B( LDB, * ),
00190      $                   BF( LDB, * ), BWK( LDB, * ), Q( LDA, * ),
00191      $                   R( LDA, * ), T( LDB, * ), TAUA( * ), TAUB( * ),
00192      $                   WORK( LWORK ), Z( LDB, * )
00193 *     ..
00194 *
00195 *  =====================================================================
00196 *
00197 *     .. Parameters ..
00198       DOUBLE PRECISION   ZERO, ONE
00199       PARAMETER          ( ZERO = 0.0D+0, ONE = 1.0D+0 )
00200       COMPLEX*16         CZERO, CONE
00201       PARAMETER          ( CZERO = ( 0.0D+0, 0.0D+0 ),
00202      $                   CONE = ( 1.0D+0, 0.0D+0 ) )
00203       COMPLEX*16         CROGUE
00204       PARAMETER          ( CROGUE = ( -1.0D+10, 0.0D+0 ) )
00205 *     ..
00206 *     .. Local Scalars ..
00207       INTEGER            INFO
00208       DOUBLE PRECISION   ANORM, BNORM, RESID, ULP, UNFL
00209 *     ..
00210 *     .. External Functions ..
00211       DOUBLE PRECISION   DLAMCH, ZLANGE, ZLANHE
00212       EXTERNAL           DLAMCH, ZLANGE, ZLANHE
00213 *     ..
00214 *     .. External Subroutines ..
00215       EXTERNAL           ZGEMM, ZGGRQF, ZHERK, ZLACPY, ZLASET, ZUNGQR,
00216      $                   ZUNGRQ
00217 *     ..
00218 *     .. Intrinsic Functions ..
00219       INTRINSIC          DBLE, MAX, MIN
00220 *     ..
00221 *     .. Executable Statements ..
00222 *
00223       ULP = DLAMCH( 'Precision' )
00224       UNFL = DLAMCH( 'Safe minimum' )
00225 *
00226 *     Copy the matrix A to the array AF.
00227 *
00228       CALL ZLACPY( 'Full', M, N, A, LDA, AF, LDA )
00229       CALL ZLACPY( 'Full', P, N, B, LDB, BF, LDB )
00230 *
00231       ANORM = MAX( ZLANGE( '1', M, N, A, LDA, RWORK ), UNFL )
00232       BNORM = MAX( ZLANGE( '1', P, N, B, LDB, RWORK ), UNFL )
00233 *
00234 *     Factorize the matrices A and B in the arrays AF and BF.
00235 *
00236       CALL ZGGRQF( M, P, N, AF, LDA, TAUA, BF, LDB, TAUB, WORK, LWORK,
00237      $             INFO )
00238 *
00239 *     Generate the N-by-N matrix Q
00240 *
00241       CALL ZLASET( 'Full', N, N, CROGUE, CROGUE, Q, LDA )
00242       IF( M.LE.N ) THEN
00243          IF( M.GT.0 .AND. M.LT.N )
00244      $      CALL ZLACPY( 'Full', M, N-M, AF, LDA, Q( N-M+1, 1 ), LDA )
00245          IF( M.GT.1 )
00246      $      CALL ZLACPY( 'Lower', M-1, M-1, AF( 2, N-M+1 ), LDA,
00247      $                   Q( N-M+2, N-M+1 ), LDA )
00248       ELSE
00249          IF( N.GT.1 )
00250      $      CALL ZLACPY( 'Lower', N-1, N-1, AF( M-N+2, 1 ), LDA,
00251      $                   Q( 2, 1 ), LDA )
00252       END IF
00253       CALL ZUNGRQ( N, N, MIN( M, N ), Q, LDA, TAUA, WORK, LWORK, INFO )
00254 *
00255 *     Generate the P-by-P matrix Z
00256 *
00257       CALL ZLASET( 'Full', P, P, CROGUE, CROGUE, Z, LDB )
00258       IF( P.GT.1 )
00259      $   CALL ZLACPY( 'Lower', P-1, N, BF( 2, 1 ), LDB, Z( 2, 1 ), LDB )
00260       CALL ZUNGQR( P, P, MIN( P, N ), Z, LDB, TAUB, WORK, LWORK, INFO )
00261 *
00262 *     Copy R
00263 *
00264       CALL ZLASET( 'Full', M, N, CZERO, CZERO, R, LDA )
00265       IF( M.LE.N ) THEN
00266          CALL ZLACPY( 'Upper', M, M, AF( 1, N-M+1 ), LDA, R( 1, N-M+1 ),
00267      $                LDA )
00268       ELSE
00269          CALL ZLACPY( 'Full', M-N, N, AF, LDA, R, LDA )
00270          CALL ZLACPY( 'Upper', N, N, AF( M-N+1, 1 ), LDA, R( M-N+1, 1 ),
00271      $                LDA )
00272       END IF
00273 *
00274 *     Copy T
00275 *
00276       CALL ZLASET( 'Full', P, N, CZERO, CZERO, T, LDB )
00277       CALL ZLACPY( 'Upper', P, N, BF, LDB, T, LDB )
00278 *
00279 *     Compute R - A*Q'
00280 *
00281       CALL ZGEMM( 'No transpose', 'Conjugate transpose', M, N, N, -CONE,
00282      $            A, LDA, Q, LDA, CONE, R, LDA )
00283 *
00284 *     Compute norm( R - A*Q' ) / ( MAX(M,N)*norm(A)*ULP ) .
00285 *
00286       RESID = ZLANGE( '1', M, N, R, LDA, RWORK )
00287       IF( ANORM.GT.ZERO ) THEN
00288          RESULT( 1 ) = ( ( RESID / DBLE( MAX( 1, M, N ) ) ) / ANORM ) /
00289      $                 ULP
00290       ELSE
00291          RESULT( 1 ) = ZERO
00292       END IF
00293 *
00294 *     Compute T*Q - Z'*B
00295 *
00296       CALL ZGEMM( 'Conjugate transpose', 'No transpose', P, N, P, CONE,
00297      $            Z, LDB, B, LDB, CZERO, BWK, LDB )
00298       CALL ZGEMM( 'No transpose', 'No transpose', P, N, N, CONE, T, LDB,
00299      $            Q, LDA, -CONE, BWK, LDB )
00300 *
00301 *     Compute norm( T*Q - Z'*B ) / ( MAX(P,N)*norm(A)*ULP ) .
00302 *
00303       RESID = ZLANGE( '1', P, N, BWK, LDB, RWORK )
00304       IF( BNORM.GT.ZERO ) THEN
00305          RESULT( 2 ) = ( ( RESID / DBLE( MAX( 1, P, M ) ) ) / BNORM ) /
00306      $                 ULP
00307       ELSE
00308          RESULT( 2 ) = ZERO
00309       END IF
00310 *
00311 *     Compute I - Q*Q'
00312 *
00313       CALL ZLASET( 'Full', N, N, CZERO, CONE, R, LDA )
00314       CALL ZHERK( 'Upper', 'No Transpose', N, N, -ONE, Q, LDA, ONE, R,
00315      $            LDA )
00316 *
00317 *     Compute norm( I - Q'*Q ) / ( N * ULP ) .
00318 *
00319       RESID = ZLANHE( '1', 'Upper', N, R, LDA, RWORK )
00320       RESULT( 3 ) = ( RESID / DBLE( MAX( 1, N ) ) ) / ULP
00321 *
00322 *     Compute I - Z'*Z
00323 *
00324       CALL ZLASET( 'Full', P, P, CZERO, CONE, T, LDB )
00325       CALL ZHERK( 'Upper', 'Conjugate transpose', P, P, -ONE, Z, LDB,
00326      $            ONE, T, LDB )
00327 *
00328 *     Compute norm( I - Z'*Z ) / ( P*ULP ) .
00329 *
00330       RESID = ZLANHE( '1', 'Upper', P, T, LDB, RWORK )
00331       RESULT( 4 ) = ( RESID / DBLE( MAX( 1, P ) ) ) / ULP
00332 *
00333       RETURN
00334 *
00335 *     End of ZGRQTS
00336 *
00337       END
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