![]() |
LAPACK
3.4.1
LAPACK: Linear Algebra PACKage
|
00001 *> \brief \b CGGSVP 00002 * 00003 * =========== DOCUMENTATION =========== 00004 * 00005 * Online html documentation available at 00006 * http://www.netlib.org/lapack/explore-html/ 00007 * 00008 *> \htmlonly 00009 *> Download CGGSVP + dependencies 00010 *> <a href="http://www.netlib.org/cgi-bin/netlibfiles.tgz?format=tgz&filename=/lapack/lapack_routine/cggsvp.f"> 00011 *> [TGZ]</a> 00012 *> <a href="http://www.netlib.org/cgi-bin/netlibfiles.zip?format=zip&filename=/lapack/lapack_routine/cggsvp.f"> 00013 *> [ZIP]</a> 00014 *> <a href="http://www.netlib.org/cgi-bin/netlibfiles.txt?format=txt&filename=/lapack/lapack_routine/cggsvp.f"> 00015 *> [TXT]</a> 00016 *> \endhtmlonly 00017 * 00018 * Definition: 00019 * =========== 00020 * 00021 * SUBROUTINE CGGSVP( JOBU, JOBV, JOBQ, M, P, N, A, LDA, B, LDB, 00022 * TOLA, TOLB, K, L, U, LDU, V, LDV, Q, LDQ, 00023 * IWORK, RWORK, TAU, WORK, INFO ) 00024 * 00025 * .. Scalar Arguments .. 00026 * CHARACTER JOBQ, JOBU, JOBV 00027 * INTEGER INFO, K, L, LDA, LDB, LDQ, LDU, LDV, M, N, P 00028 * REAL TOLA, TOLB 00029 * .. 00030 * .. Array Arguments .. 00031 * INTEGER IWORK( * ) 00032 * REAL RWORK( * ) 00033 * COMPLEX A( LDA, * ), B( LDB, * ), Q( LDQ, * ), 00034 * $ TAU( * ), U( LDU, * ), V( LDV, * ), WORK( * ) 00035 * .. 00036 * 00037 * 00038 *> \par Purpose: 00039 * ============= 00040 *> 00041 *> \verbatim 00042 *> 00043 *> CGGSVP computes unitary matrices U, V and Q such that 00044 *> 00045 *> N-K-L K L 00046 *> U**H*A*Q = K ( 0 A12 A13 ) if M-K-L >= 0; 00047 *> L ( 0 0 A23 ) 00048 *> M-K-L ( 0 0 0 ) 00049 *> 00050 *> N-K-L K L 00051 *> = K ( 0 A12 A13 ) if M-K-L < 0; 00052 *> M-K ( 0 0 A23 ) 00053 *> 00054 *> N-K-L K L 00055 *> V**H*B*Q = L ( 0 0 B13 ) 00056 *> P-L ( 0 0 0 ) 00057 *> 00058 *> where the K-by-K matrix A12 and L-by-L matrix B13 are nonsingular 00059 *> upper triangular; A23 is L-by-L upper triangular if M-K-L >= 0, 00060 *> otherwise A23 is (M-K)-by-L upper trapezoidal. K+L = the effective 00061 *> numerical rank of the (M+P)-by-N matrix (A**H,B**H)**H. 00062 *> 00063 *> This decomposition is the preprocessing step for computing the 00064 *> Generalized Singular Value Decomposition (GSVD), see subroutine 00065 *> CGGSVD. 00066 *> \endverbatim 00067 * 00068 * Arguments: 00069 * ========== 00070 * 00071 *> \param[in] JOBU 00072 *> \verbatim 00073 *> JOBU is CHARACTER*1 00074 *> = 'U': Unitary matrix U is computed; 00075 *> = 'N': U is not computed. 00076 *> \endverbatim 00077 *> 00078 *> \param[in] JOBV 00079 *> \verbatim 00080 *> JOBV is CHARACTER*1 00081 *> = 'V': Unitary matrix V is computed; 00082 *> = 'N': V is not computed. 00083 *> \endverbatim 00084 *> 00085 *> \param[in] JOBQ 00086 *> \verbatim 00087 *> JOBQ is CHARACTER*1 00088 *> = 'Q': Unitary matrix Q is computed; 00089 *> = 'N': Q is not computed. 00090 *> \endverbatim 00091 *> 00092 *> \param[in] M 00093 *> \verbatim 00094 *> M is INTEGER 00095 *> The number of rows of the matrix A. M >= 0. 00096 *> \endverbatim 00097 *> 00098 *> \param[in] P 00099 *> \verbatim 00100 *> P is INTEGER 00101 *> The number of rows of the matrix B. P >= 0. 00102 *> \endverbatim 00103 *> 00104 *> \param[in] N 00105 *> \verbatim 00106 *> N is INTEGER 00107 *> The number of columns of the matrices A and B. N >= 0. 00108 *> \endverbatim 00109 *> 00110 *> \param[in,out] A 00111 *> \verbatim 00112 *> A is COMPLEX array, dimension (LDA,N) 00113 *> On entry, the M-by-N matrix A. 00114 *> On exit, A contains the triangular (or trapezoidal) matrix 00115 *> described in the Purpose section. 00116 *> \endverbatim 00117 *> 00118 *> \param[in] LDA 00119 *> \verbatim 00120 *> LDA is INTEGER 00121 *> The leading dimension of the array A. LDA >= max(1,M). 00122 *> \endverbatim 00123 *> 00124 *> \param[in,out] B 00125 *> \verbatim 00126 *> B is COMPLEX array, dimension (LDB,N) 00127 *> On entry, the P-by-N matrix B. 00128 *> On exit, B contains the triangular matrix described in 00129 *> the Purpose section. 00130 *> \endverbatim 00131 *> 00132 *> \param[in] LDB 00133 *> \verbatim 00134 *> LDB is INTEGER 00135 *> The leading dimension of the array B. LDB >= max(1,P). 00136 *> \endverbatim 00137 *> 00138 *> \param[in] TOLA 00139 *> \verbatim 00140 *> TOLA is REAL 00141 *> \endverbatim 00142 *> 00143 *> \param[in] TOLB 00144 *> \verbatim 00145 *> TOLB is REAL 00146 *> 00147 *> TOLA and TOLB are the thresholds to determine the effective 00148 *> numerical rank of matrix B and a subblock of A. Generally, 00149 *> they are set to 00150 *> TOLA = MAX(M,N)*norm(A)*MACHEPS, 00151 *> TOLB = MAX(P,N)*norm(B)*MACHEPS. 00152 *> The size of TOLA and TOLB may affect the size of backward 00153 *> errors of the decomposition. 00154 *> \endverbatim 00155 *> 00156 *> \param[out] K 00157 *> \verbatim 00158 *> K is INTEGER 00159 *> \endverbatim 00160 *> 00161 *> \param[out] L 00162 *> \verbatim 00163 *> L is INTEGER 00164 *> 00165 *> On exit, K and L specify the dimension of the subblocks 00166 *> described in Purpose section. 00167 *> K + L = effective numerical rank of (A**H,B**H)**H. 00168 *> \endverbatim 00169 *> 00170 *> \param[out] U 00171 *> \verbatim 00172 *> U is COMPLEX array, dimension (LDU,M) 00173 *> If JOBU = 'U', U contains the unitary matrix U. 00174 *> If JOBU = 'N', U is not referenced. 00175 *> \endverbatim 00176 *> 00177 *> \param[in] LDU 00178 *> \verbatim 00179 *> LDU is INTEGER 00180 *> The leading dimension of the array U. LDU >= max(1,M) if 00181 *> JOBU = 'U'; LDU >= 1 otherwise. 00182 *> \endverbatim 00183 *> 00184 *> \param[out] V 00185 *> \verbatim 00186 *> V is COMPLEX array, dimension (LDV,P) 00187 *> If JOBV = 'V', V contains the unitary matrix V. 00188 *> If JOBV = 'N', V is not referenced. 00189 *> \endverbatim 00190 *> 00191 *> \param[in] LDV 00192 *> \verbatim 00193 *> LDV is INTEGER 00194 *> The leading dimension of the array V. LDV >= max(1,P) if 00195 *> JOBV = 'V'; LDV >= 1 otherwise. 00196 *> \endverbatim 00197 *> 00198 *> \param[out] Q 00199 *> \verbatim 00200 *> Q is COMPLEX array, dimension (LDQ,N) 00201 *> If JOBQ = 'Q', Q contains the unitary matrix Q. 00202 *> If JOBQ = 'N', Q is not referenced. 00203 *> \endverbatim 00204 *> 00205 *> \param[in] LDQ 00206 *> \verbatim 00207 *> LDQ is INTEGER 00208 *> The leading dimension of the array Q. LDQ >= max(1,N) if 00209 *> JOBQ = 'Q'; LDQ >= 1 otherwise. 00210 *> \endverbatim 00211 *> 00212 *> \param[out] IWORK 00213 *> \verbatim 00214 *> IWORK is INTEGER array, dimension (N) 00215 *> \endverbatim 00216 *> 00217 *> \param[out] RWORK 00218 *> \verbatim 00219 *> RWORK is REAL array, dimension (2*N) 00220 *> \endverbatim 00221 *> 00222 *> \param[out] TAU 00223 *> \verbatim 00224 *> TAU is COMPLEX array, dimension (N) 00225 *> \endverbatim 00226 *> 00227 *> \param[out] WORK 00228 *> \verbatim 00229 *> WORK is COMPLEX array, dimension (max(3*N,M,P)) 00230 *> \endverbatim 00231 *> 00232 *> \param[out] INFO 00233 *> \verbatim 00234 *> INFO is INTEGER 00235 *> = 0: successful exit 00236 *> < 0: if INFO = -i, the i-th argument had an illegal value. 00237 *> \endverbatim 00238 * 00239 * Authors: 00240 * ======== 00241 * 00242 *> \author Univ. of Tennessee 00243 *> \author Univ. of California Berkeley 00244 *> \author Univ. of Colorado Denver 00245 *> \author NAG Ltd. 00246 * 00247 *> \date November 2011 00248 * 00249 *> \ingroup complexOTHERcomputational 00250 * 00251 *> \par Further Details: 00252 * ===================== 00253 *> 00254 *> The subroutine uses LAPACK subroutine CGEQPF for the QR factorization 00255 *> with column pivoting to detect the effective numerical rank of the 00256 *> a matrix. It may be replaced by a better rank determination strategy. 00257 *> 00258 * ===================================================================== 00259 SUBROUTINE CGGSVP( JOBU, JOBV, JOBQ, M, P, N, A, LDA, B, LDB, 00260 $ TOLA, TOLB, K, L, U, LDU, V, LDV, Q, LDQ, 00261 $ IWORK, RWORK, TAU, WORK, INFO ) 00262 * 00263 * -- LAPACK computational routine (version 3.4.0) -- 00264 * -- LAPACK is a software package provided by Univ. of Tennessee, -- 00265 * -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..-- 00266 * November 2011 00267 * 00268 * .. Scalar Arguments .. 00269 CHARACTER JOBQ, JOBU, JOBV 00270 INTEGER INFO, K, L, LDA, LDB, LDQ, LDU, LDV, M, N, P 00271 REAL TOLA, TOLB 00272 * .. 00273 * .. Array Arguments .. 00274 INTEGER IWORK( * ) 00275 REAL RWORK( * ) 00276 COMPLEX A( LDA, * ), B( LDB, * ), Q( LDQ, * ), 00277 $ TAU( * ), U( LDU, * ), V( LDV, * ), WORK( * ) 00278 * .. 00279 * 00280 * ===================================================================== 00281 * 00282 * .. Parameters .. 00283 COMPLEX CZERO, CONE 00284 PARAMETER ( CZERO = ( 0.0E+0, 0.0E+0 ), 00285 $ CONE = ( 1.0E+0, 0.0E+0 ) ) 00286 * .. 00287 * .. Local Scalars .. 00288 LOGICAL FORWRD, WANTQ, WANTU, WANTV 00289 INTEGER I, J 00290 COMPLEX T 00291 * .. 00292 * .. External Functions .. 00293 LOGICAL LSAME 00294 EXTERNAL LSAME 00295 * .. 00296 * .. External Subroutines .. 00297 EXTERNAL CGEQPF, CGEQR2, CGERQ2, CLACPY, CLAPMT, CLASET, 00298 $ CUNG2R, CUNM2R, CUNMR2, XERBLA 00299 * .. 00300 * .. Intrinsic Functions .. 00301 INTRINSIC ABS, AIMAG, MAX, MIN, REAL 00302 * .. 00303 * .. Statement Functions .. 00304 REAL CABS1 00305 * .. 00306 * .. Statement Function definitions .. 00307 CABS1( T ) = ABS( REAL( T ) ) + ABS( AIMAG( T ) ) 00308 * .. 00309 * .. Executable Statements .. 00310 * 00311 * Test the input parameters 00312 * 00313 WANTU = LSAME( JOBU, 'U' ) 00314 WANTV = LSAME( JOBV, 'V' ) 00315 WANTQ = LSAME( JOBQ, 'Q' ) 00316 FORWRD = .TRUE. 00317 * 00318 INFO = 0 00319 IF( .NOT.( WANTU .OR. LSAME( JOBU, 'N' ) ) ) THEN 00320 INFO = -1 00321 ELSE IF( .NOT.( WANTV .OR. LSAME( JOBV, 'N' ) ) ) THEN 00322 INFO = -2 00323 ELSE IF( .NOT.( WANTQ .OR. LSAME( JOBQ, 'N' ) ) ) THEN 00324 INFO = -3 00325 ELSE IF( M.LT.0 ) THEN 00326 INFO = -4 00327 ELSE IF( P.LT.0 ) THEN 00328 INFO = -5 00329 ELSE IF( N.LT.0 ) THEN 00330 INFO = -6 00331 ELSE IF( LDA.LT.MAX( 1, M ) ) THEN 00332 INFO = -8 00333 ELSE IF( LDB.LT.MAX( 1, P ) ) THEN 00334 INFO = -10 00335 ELSE IF( LDU.LT.1 .OR. ( WANTU .AND. LDU.LT.M ) ) THEN 00336 INFO = -16 00337 ELSE IF( LDV.LT.1 .OR. ( WANTV .AND. LDV.LT.P ) ) THEN 00338 INFO = -18 00339 ELSE IF( LDQ.LT.1 .OR. ( WANTQ .AND. LDQ.LT.N ) ) THEN 00340 INFO = -20 00341 END IF 00342 IF( INFO.NE.0 ) THEN 00343 CALL XERBLA( 'CGGSVP', -INFO ) 00344 RETURN 00345 END IF 00346 * 00347 * QR with column pivoting of B: B*P = V*( S11 S12 ) 00348 * ( 0 0 ) 00349 * 00350 DO 10 I = 1, N 00351 IWORK( I ) = 0 00352 10 CONTINUE 00353 CALL CGEQPF( P, N, B, LDB, IWORK, TAU, WORK, RWORK, INFO ) 00354 * 00355 * Update A := A*P 00356 * 00357 CALL CLAPMT( FORWRD, M, N, A, LDA, IWORK ) 00358 * 00359 * Determine the effective rank of matrix B. 00360 * 00361 L = 0 00362 DO 20 I = 1, MIN( P, N ) 00363 IF( CABS1( B( I, I ) ).GT.TOLB ) 00364 $ L = L + 1 00365 20 CONTINUE 00366 * 00367 IF( WANTV ) THEN 00368 * 00369 * Copy the details of V, and form V. 00370 * 00371 CALL CLASET( 'Full', P, P, CZERO, CZERO, V, LDV ) 00372 IF( P.GT.1 ) 00373 $ CALL CLACPY( 'Lower', P-1, N, B( 2, 1 ), LDB, V( 2, 1 ), 00374 $ LDV ) 00375 CALL CUNG2R( P, P, MIN( P, N ), V, LDV, TAU, WORK, INFO ) 00376 END IF 00377 * 00378 * Clean up B 00379 * 00380 DO 40 J = 1, L - 1 00381 DO 30 I = J + 1, L 00382 B( I, J ) = CZERO 00383 30 CONTINUE 00384 40 CONTINUE 00385 IF( P.GT.L ) 00386 $ CALL CLASET( 'Full', P-L, N, CZERO, CZERO, B( L+1, 1 ), LDB ) 00387 * 00388 IF( WANTQ ) THEN 00389 * 00390 * Set Q = I and Update Q := Q*P 00391 * 00392 CALL CLASET( 'Full', N, N, CZERO, CONE, Q, LDQ ) 00393 CALL CLAPMT( FORWRD, N, N, Q, LDQ, IWORK ) 00394 END IF 00395 * 00396 IF( P.GE.L .AND. N.NE.L ) THEN 00397 * 00398 * RQ factorization of ( S11 S12 ) = ( 0 S12 )*Z 00399 * 00400 CALL CGERQ2( L, N, B, LDB, TAU, WORK, INFO ) 00401 * 00402 * Update A := A*Z**H 00403 * 00404 CALL CUNMR2( 'Right', 'Conjugate transpose', M, N, L, B, LDB, 00405 $ TAU, A, LDA, WORK, INFO ) 00406 IF( WANTQ ) THEN 00407 * 00408 * Update Q := Q*Z**H 00409 * 00410 CALL CUNMR2( 'Right', 'Conjugate transpose', N, N, L, B, 00411 $ LDB, TAU, Q, LDQ, WORK, INFO ) 00412 END IF 00413 * 00414 * Clean up B 00415 * 00416 CALL CLASET( 'Full', L, N-L, CZERO, CZERO, B, LDB ) 00417 DO 60 J = N - L + 1, N 00418 DO 50 I = J - N + L + 1, L 00419 B( I, J ) = CZERO 00420 50 CONTINUE 00421 60 CONTINUE 00422 * 00423 END IF 00424 * 00425 * Let N-L L 00426 * A = ( A11 A12 ) M, 00427 * 00428 * then the following does the complete QR decomposition of A11: 00429 * 00430 * A11 = U*( 0 T12 )*P1**H 00431 * ( 0 0 ) 00432 * 00433 DO 70 I = 1, N - L 00434 IWORK( I ) = 0 00435 70 CONTINUE 00436 CALL CGEQPF( M, N-L, A, LDA, IWORK, TAU, WORK, RWORK, INFO ) 00437 * 00438 * Determine the effective rank of A11 00439 * 00440 K = 0 00441 DO 80 I = 1, MIN( M, N-L ) 00442 IF( CABS1( A( I, I ) ).GT.TOLA ) 00443 $ K = K + 1 00444 80 CONTINUE 00445 * 00446 * Update A12 := U**H*A12, where A12 = A( 1:M, N-L+1:N ) 00447 * 00448 CALL CUNM2R( 'Left', 'Conjugate transpose', M, L, MIN( M, N-L ), 00449 $ A, LDA, TAU, A( 1, N-L+1 ), LDA, WORK, INFO ) 00450 * 00451 IF( WANTU ) THEN 00452 * 00453 * Copy the details of U, and form U 00454 * 00455 CALL CLASET( 'Full', M, M, CZERO, CZERO, U, LDU ) 00456 IF( M.GT.1 ) 00457 $ CALL CLACPY( 'Lower', M-1, N-L, A( 2, 1 ), LDA, U( 2, 1 ), 00458 $ LDU ) 00459 CALL CUNG2R( M, M, MIN( M, N-L ), U, LDU, TAU, WORK, INFO ) 00460 END IF 00461 * 00462 IF( WANTQ ) THEN 00463 * 00464 * Update Q( 1:N, 1:N-L ) = Q( 1:N, 1:N-L )*P1 00465 * 00466 CALL CLAPMT( FORWRD, N, N-L, Q, LDQ, IWORK ) 00467 END IF 00468 * 00469 * Clean up A: set the strictly lower triangular part of 00470 * A(1:K, 1:K) = 0, and A( K+1:M, 1:N-L ) = 0. 00471 * 00472 DO 100 J = 1, K - 1 00473 DO 90 I = J + 1, K 00474 A( I, J ) = CZERO 00475 90 CONTINUE 00476 100 CONTINUE 00477 IF( M.GT.K ) 00478 $ CALL CLASET( 'Full', M-K, N-L, CZERO, CZERO, A( K+1, 1 ), LDA ) 00479 * 00480 IF( N-L.GT.K ) THEN 00481 * 00482 * RQ factorization of ( T11 T12 ) = ( 0 T12 )*Z1 00483 * 00484 CALL CGERQ2( K, N-L, A, LDA, TAU, WORK, INFO ) 00485 * 00486 IF( WANTQ ) THEN 00487 * 00488 * Update Q( 1:N,1:N-L ) = Q( 1:N,1:N-L )*Z1**H 00489 * 00490 CALL CUNMR2( 'Right', 'Conjugate transpose', N, N-L, K, A, 00491 $ LDA, TAU, Q, LDQ, WORK, INFO ) 00492 END IF 00493 * 00494 * Clean up A 00495 * 00496 CALL CLASET( 'Full', K, N-L-K, CZERO, CZERO, A, LDA ) 00497 DO 120 J = N - L - K + 1, N - L 00498 DO 110 I = J - N + L + K + 1, K 00499 A( I, J ) = CZERO 00500 110 CONTINUE 00501 120 CONTINUE 00502 * 00503 END IF 00504 * 00505 IF( M.GT.K ) THEN 00506 * 00507 * QR factorization of A( K+1:M,N-L+1:N ) 00508 * 00509 CALL CGEQR2( M-K, L, A( K+1, N-L+1 ), LDA, TAU, WORK, INFO ) 00510 * 00511 IF( WANTU ) THEN 00512 * 00513 * Update U(:,K+1:M) := U(:,K+1:M)*U1 00514 * 00515 CALL CUNM2R( 'Right', 'No transpose', M, M-K, MIN( M-K, L ), 00516 $ A( K+1, N-L+1 ), LDA, TAU, U( 1, K+1 ), LDU, 00517 $ WORK, INFO ) 00518 END IF 00519 * 00520 * Clean up 00521 * 00522 DO 140 J = N - L + 1, N 00523 DO 130 I = J - N + K + L + 1, M 00524 A( I, J ) = CZERO 00525 130 CONTINUE 00526 140 CONTINUE 00527 * 00528 END IF 00529 * 00530 RETURN 00531 * 00532 * End of CGGSVP 00533 * 00534 END