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