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LAPACK 3.3.1
Linear Algebra PACKage
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LAPACK 3.3.1
Linear Algebra PACKage
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00001 RECURSIVE SUBROUTINE DORCSD( JOBU1, JOBU2, JOBV1T, JOBV2T, TRANS, 00002 $ SIGNS, M, P, Q, X11, LDX11, X12, 00003 $ LDX12, X21, LDX21, X22, LDX22, THETA, 00004 $ U1, LDU1, U2, LDU2, V1T, LDV1T, V2T, 00005 $ LDV2T, WORK, LWORK, IWORK, INFO ) 00006 IMPLICIT NONE 00007 * 00008 * -- LAPACK routine (version 3.3.1) -- 00009 * 00010 * -- Contributed by Brian Sutton of the Randolph-Macon College -- 00011 * -- November 2010 00012 * 00013 * -- LAPACK is a software package provided by Univ. of Tennessee, -- 00014 * -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..-- 00015 * 00016 * @precisions normal d -> s 00017 * 00018 * .. Scalar Arguments .. 00019 CHARACTER JOBU1, JOBU2, JOBV1T, JOBV2T, SIGNS, TRANS 00020 INTEGER INFO, LDU1, LDU2, LDV1T, LDV2T, LDX11, LDX12, 00021 $ LDX21, LDX22, LWORK, M, P, Q 00022 * .. 00023 * .. Array Arguments .. 00024 INTEGER IWORK( * ) 00025 DOUBLE PRECISION THETA( * ) 00026 DOUBLE PRECISION U1( LDU1, * ), U2( LDU2, * ), V1T( LDV1T, * ), 00027 $ V2T( LDV2T, * ), WORK( * ), X11( LDX11, * ), 00028 $ X12( LDX12, * ), X21( LDX21, * ), X22( LDX22, 00029 $ * ) 00030 * .. 00031 * 00032 * Purpose 00033 * ======= 00034 * 00035 * DORCSD computes the CS decomposition of an M-by-M partitioned 00036 * orthogonal matrix X: 00037 * 00038 * [ I 0 0 | 0 0 0 ] 00039 * [ 0 C 0 | 0 -S 0 ] 00040 * [ X11 | X12 ] [ U1 | ] [ 0 0 0 | 0 0 -I ] [ V1 | ]**T 00041 * X = [-----------] = [---------] [---------------------] [---------] . 00042 * [ X21 | X22 ] [ | U2 ] [ 0 0 0 | I 0 0 ] [ | V2 ] 00043 * [ 0 S 0 | 0 C 0 ] 00044 * [ 0 0 I | 0 0 0 ] 00045 * 00046 * X11 is P-by-Q. The orthogonal matrices U1, U2, V1, and V2 are P-by-P, 00047 * (M-P)-by-(M-P), Q-by-Q, and (M-Q)-by-(M-Q), respectively. C and S are 00048 * R-by-R nonnegative diagonal matrices satisfying C^2 + S^2 = I, in 00049 * which R = MIN(P,M-P,Q,M-Q). 00050 * 00051 * Arguments 00052 * ========= 00053 * 00054 * JOBU1 (input) CHARACTER 00055 * = 'Y': U1 is computed; 00056 * otherwise: U1 is not computed. 00057 * 00058 * JOBU2 (input) CHARACTER 00059 * = 'Y': U2 is computed; 00060 * otherwise: U2 is not computed. 00061 * 00062 * JOBV1T (input) CHARACTER 00063 * = 'Y': V1T is computed; 00064 * otherwise: V1T is not computed. 00065 * 00066 * JOBV2T (input) CHARACTER 00067 * = 'Y': V2T is computed; 00068 * otherwise: V2T is not computed. 00069 * 00070 * TRANS (input) CHARACTER 00071 * = 'T': X, U1, U2, V1T, and V2T are stored in row-major 00072 * order; 00073 * otherwise: X, U1, U2, V1T, and V2T are stored in column- 00074 * major order. 00075 * 00076 * SIGNS (input) CHARACTER 00077 * = 'O': The lower-left block is made nonpositive (the 00078 * "other" convention); 00079 * otherwise: The upper-right block is made nonpositive (the 00080 * "default" convention). 00081 * 00082 * M (input) INTEGER 00083 * The number of rows and columns in X. 00084 * 00085 * P (input) INTEGER 00086 * The number of rows in X11 and X12. 0 <= P <= M. 00087 * 00088 * Q (input) INTEGER 00089 * The number of columns in X11 and X21. 0 <= Q <= M. 00090 * 00091 * X (input/workspace) DOUBLE PRECISION array, dimension (LDX,M) 00092 * On entry, the orthogonal matrix whose CSD is desired. 00093 * 00094 * LDX (input) INTEGER 00095 * The leading dimension of X. LDX >= MAX(1,M). 00096 * 00097 * THETA (output) DOUBLE PRECISION array, dimension (R), in which R = 00098 * MIN(P,M-P,Q,M-Q). 00099 * C = DIAG( COS(THETA(1)), ... , COS(THETA(R)) ) and 00100 * S = DIAG( SIN(THETA(1)), ... , SIN(THETA(R)) ). 00101 * 00102 * U1 (output) DOUBLE PRECISION array, dimension (P) 00103 * If JOBU1 = 'Y', U1 contains the P-by-P orthogonal matrix U1. 00104 * 00105 * LDU1 (input) INTEGER 00106 * The leading dimension of U1. If JOBU1 = 'Y', LDU1 >= 00107 * MAX(1,P). 00108 * 00109 * U2 (output) DOUBLE PRECISION array, dimension (M-P) 00110 * If JOBU2 = 'Y', U2 contains the (M-P)-by-(M-P) orthogonal 00111 * matrix U2. 00112 * 00113 * LDU2 (input) INTEGER 00114 * The leading dimension of U2. If JOBU2 = 'Y', LDU2 >= 00115 * MAX(1,M-P). 00116 * 00117 * V1T (output) DOUBLE PRECISION array, dimension (Q) 00118 * If JOBV1T = 'Y', V1T contains the Q-by-Q matrix orthogonal 00119 * matrix V1**T. 00120 * 00121 * LDV1T (input) INTEGER 00122 * The leading dimension of V1T. If JOBV1T = 'Y', LDV1T >= 00123 * MAX(1,Q). 00124 * 00125 * V2T (output) DOUBLE PRECISION array, dimension (M-Q) 00126 * If JOBV2T = 'Y', V2T contains the (M-Q)-by-(M-Q) orthogonal 00127 * matrix V2**T. 00128 * 00129 * LDV2T (input) INTEGER 00130 * The leading dimension of V2T. If JOBV2T = 'Y', LDV2T >= 00131 * MAX(1,M-Q). 00132 * 00133 * WORK (workspace) DOUBLE PRECISION array, dimension (MAX(1,LWORK)) 00134 * On exit, if INFO = 0, WORK(1) returns the optimal LWORK. 00135 * If INFO > 0 on exit, WORK(2:R) contains the values PHI(1), 00136 * ..., PHI(R-1) that, together with THETA(1), ..., THETA(R), 00137 * define the matrix in intermediate bidiagonal-block form 00138 * remaining after nonconvergence. INFO specifies the number 00139 * of nonzero PHI's. 00140 * 00141 * LWORK (input) INTEGER 00142 * The dimension of the array WORK. 00143 * 00144 * If LWORK = -1, then a workspace query is assumed; the routine 00145 * only calculates the optimal size of the WORK array, returns 00146 * this value as the first entry of the work array, and no error 00147 * message related to LWORK is issued by XERBLA. 00148 * 00149 * IWORK (workspace) INTEGER array, dimension (M-MIN(P, M-P, Q, M-Q)) 00150 * 00151 * INFO (output) INTEGER 00152 * = 0: successful exit. 00153 * < 0: if INFO = -i, the i-th argument had an illegal value. 00154 * > 0: DBBCSD did not converge. See the description of WORK 00155 * above for details. 00156 * 00157 * Reference 00158 * ========= 00159 * 00160 * [1] Brian D. Sutton. Computing the complete CS decomposition. Numer. 00161 * Algorithms, 50(1):33-65, 2009. 00162 * 00163 * =================================================================== 00164 * 00165 * .. Parameters .. 00166 DOUBLE PRECISION REALONE 00167 PARAMETER ( REALONE = 1.0D0 ) 00168 DOUBLE PRECISION NEGONE, ONE, PIOVER2, ZERO 00169 PARAMETER ( NEGONE = -1.0D0, ONE = 1.0D0, 00170 $ PIOVER2 = 1.57079632679489662D0, 00171 $ ZERO = 0.0D0 ) 00172 * .. 00173 * .. Local Scalars .. 00174 CHARACTER TRANST, SIGNST 00175 INTEGER CHILDINFO, I, IB11D, IB11E, IB12D, IB12E, 00176 $ IB21D, IB21E, IB22D, IB22E, IBBCSD, IORBDB, 00177 $ IORGLQ, IORGQR, IPHI, ITAUP1, ITAUP2, ITAUQ1, 00178 $ ITAUQ2, J, LBBCSDWORK, LBBCSDWORKMIN, 00179 $ LBBCSDWORKOPT, LORBDBWORK, LORBDBWORKMIN, 00180 $ LORBDBWORKOPT, LORGLQWORK, LORGLQWORKMIN, 00181 $ LORGLQWORKOPT, LORGQRWORK, LORGQRWORKMIN, 00182 $ LORGQRWORKOPT, LWORKMIN, LWORKOPT 00183 LOGICAL COLMAJOR, DEFAULTSIGNS, LQUERY, WANTU1, WANTU2, 00184 $ WANTV1T, WANTV2T 00185 * .. 00186 * .. External Subroutines .. 00187 EXTERNAL DBBCSD, DLACPY, DLAPMR, DLAPMT, DLASCL, DLASET, 00188 $ DORBDB, DORGLQ, DORGQR, XERBLA 00189 * .. 00190 * .. External Functions .. 00191 LOGICAL LSAME 00192 EXTERNAL LSAME 00193 * .. 00194 * .. Intrinsic Functions 00195 INTRINSIC COS, INT, MAX, MIN, SIN 00196 * .. 00197 * .. Executable Statements .. 00198 * 00199 * Test input arguments 00200 * 00201 INFO = 0 00202 WANTU1 = LSAME( JOBU1, 'Y' ) 00203 WANTU2 = LSAME( JOBU2, 'Y' ) 00204 WANTV1T = LSAME( JOBV1T, 'Y' ) 00205 WANTV2T = LSAME( JOBV2T, 'Y' ) 00206 COLMAJOR = .NOT. LSAME( TRANS, 'T' ) 00207 DEFAULTSIGNS = .NOT. LSAME( SIGNS, 'O' ) 00208 LQUERY = LWORK .EQ. -1 00209 IF( M .LT. 0 ) THEN 00210 INFO = -7 00211 ELSE IF( P .LT. 0 .OR. P .GT. M ) THEN 00212 INFO = -8 00213 ELSE IF( Q .LT. 0 .OR. Q .GT. M ) THEN 00214 INFO = -9 00215 ELSE IF( ( COLMAJOR .AND. LDX11 .LT. MAX(1,P) ) .OR. 00216 $ ( .NOT.COLMAJOR .AND. LDX11 .LT. MAX(1,Q) ) ) THEN 00217 INFO = -11 00218 ELSE IF( WANTU1 .AND. LDU1 .LT. P ) THEN 00219 INFO = -14 00220 ELSE IF( WANTU2 .AND. LDU2 .LT. M-P ) THEN 00221 INFO = -16 00222 ELSE IF( WANTV1T .AND. LDV1T .LT. Q ) THEN 00223 INFO = -18 00224 ELSE IF( WANTV2T .AND. LDV2T .LT. M-Q ) THEN 00225 INFO = -20 00226 END IF 00227 * 00228 * Work with transpose if convenient 00229 * 00230 IF( INFO .EQ. 0 .AND. MIN( P, M-P ) .LT. MIN( Q, M-Q ) ) THEN 00231 IF( COLMAJOR ) THEN 00232 TRANST = 'T' 00233 ELSE 00234 TRANST = 'N' 00235 END IF 00236 IF( DEFAULTSIGNS ) THEN 00237 SIGNST = 'O' 00238 ELSE 00239 SIGNST = 'D' 00240 END IF 00241 CALL DORCSD( JOBV1T, JOBV2T, JOBU1, JOBU2, TRANST, SIGNST, M, 00242 $ Q, P, X11, LDX11, X21, LDX21, X12, LDX12, X22, 00243 $ LDX22, THETA, V1T, LDV1T, V2T, LDV2T, U1, LDU1, 00244 $ U2, LDU2, WORK, LWORK, IWORK, INFO ) 00245 RETURN 00246 END IF 00247 * 00248 * Work with permutation [ 0 I; I 0 ] * X * [ 0 I; I 0 ] if 00249 * convenient 00250 * 00251 IF( INFO .EQ. 0 .AND. M-Q .LT. Q ) THEN 00252 IF( DEFAULTSIGNS ) THEN 00253 SIGNST = 'O' 00254 ELSE 00255 SIGNST = 'D' 00256 END IF 00257 CALL DORCSD( JOBU2, JOBU1, JOBV2T, JOBV1T, TRANS, SIGNST, M, 00258 $ M-P, M-Q, X22, LDX22, X21, LDX21, X12, LDX12, X11, 00259 $ LDX11, THETA, U2, LDU2, U1, LDU1, V2T, LDV2T, V1T, 00260 $ LDV1T, WORK, LWORK, IWORK, INFO ) 00261 RETURN 00262 END IF 00263 * 00264 * Compute workspace 00265 * 00266 IF( INFO .EQ. 0 ) THEN 00267 * 00268 IPHI = 2 00269 ITAUP1 = IPHI + MAX( 1, Q - 1 ) 00270 ITAUP2 = ITAUP1 + MAX( 1, P ) 00271 ITAUQ1 = ITAUP2 + MAX( 1, M - P ) 00272 ITAUQ2 = ITAUQ1 + MAX( 1, Q ) 00273 IORGQR = ITAUQ2 + MAX( 1, M - Q ) 00274 CALL DORGQR( M-Q, M-Q, M-Q, 0, MAX(1,M-Q), 0, WORK, -1, 00275 $ CHILDINFO ) 00276 LORGQRWORKOPT = INT( WORK(1) ) 00277 LORGQRWORKMIN = MAX( 1, M - Q ) 00278 IORGLQ = ITAUQ2 + MAX( 1, M - Q ) 00279 CALL DORGLQ( M-Q, M-Q, M-Q, 0, MAX(1,M-Q), 0, WORK, -1, 00280 $ CHILDINFO ) 00281 LORGLQWORKOPT = INT( WORK(1) ) 00282 LORGLQWORKMIN = MAX( 1, M - Q ) 00283 IORBDB = ITAUQ2 + MAX( 1, M - Q ) 00284 CALL DORBDB( TRANS, SIGNS, M, P, Q, X11, LDX11, X12, LDX12, 00285 $ X21, LDX21, X22, LDX22, 0, 0, 0, 0, 0, 0, WORK, 00286 $ -1, CHILDINFO ) 00287 LORBDBWORKOPT = INT( WORK(1) ) 00288 LORBDBWORKMIN = LORBDBWORKOPT 00289 IB11D = ITAUQ2 + MAX( 1, M - Q ) 00290 IB11E = IB11D + MAX( 1, Q ) 00291 IB12D = IB11E + MAX( 1, Q - 1 ) 00292 IB12E = IB12D + MAX( 1, Q ) 00293 IB21D = IB12E + MAX( 1, Q - 1 ) 00294 IB21E = IB21D + MAX( 1, Q ) 00295 IB22D = IB21E + MAX( 1, Q - 1 ) 00296 IB22E = IB22D + MAX( 1, Q ) 00297 IBBCSD = IB22E + MAX( 1, Q - 1 ) 00298 CALL DBBCSD( JOBU1, JOBU2, JOBV1T, JOBV2T, TRANS, M, P, Q, 0, 00299 $ 0, U1, LDU1, U2, LDU2, V1T, LDV1T, V2T, LDV2T, 0, 00300 $ 0, 0, 0, 0, 0, 0, 0, WORK, -1, CHILDINFO ) 00301 LBBCSDWORKOPT = INT( WORK(1) ) 00302 LBBCSDWORKMIN = LBBCSDWORKOPT 00303 LWORKOPT = MAX( IORGQR + LORGQRWORKOPT, IORGLQ + LORGLQWORKOPT, 00304 $ IORBDB + LORBDBWORKOPT, IBBCSD + LBBCSDWORKOPT ) - 1 00305 LWORKMIN = MAX( IORGQR + LORGQRWORKMIN, IORGLQ + LORGLQWORKMIN, 00306 $ IORBDB + LORBDBWORKOPT, IBBCSD + LBBCSDWORKMIN ) - 1 00307 WORK(1) = MAX(LWORKOPT,LWORKMIN) 00308 * 00309 IF( LWORK .LT. LWORKMIN .AND. .NOT. LQUERY ) THEN 00310 INFO = -22 00311 ELSE 00312 LORGQRWORK = LWORK - IORGQR + 1 00313 LORGLQWORK = LWORK - IORGLQ + 1 00314 LORBDBWORK = LWORK - IORBDB + 1 00315 LBBCSDWORK = LWORK - IBBCSD + 1 00316 END IF 00317 END IF 00318 * 00319 * Abort if any illegal arguments 00320 * 00321 IF( INFO .NE. 0 ) THEN 00322 CALL XERBLA( 'DORCSD', -INFO ) 00323 RETURN 00324 ELSE IF( LQUERY ) THEN 00325 RETURN 00326 END IF 00327 * 00328 * Transform to bidiagonal block form 00329 * 00330 CALL DORBDB( TRANS, SIGNS, M, P, Q, X11, LDX11, X12, LDX12, X21, 00331 $ LDX21, X22, LDX22, THETA, WORK(IPHI), WORK(ITAUP1), 00332 $ WORK(ITAUP2), WORK(ITAUQ1), WORK(ITAUQ2), 00333 $ WORK(IORBDB), LORBDBWORK, CHILDINFO ) 00334 * 00335 * Accumulate Householder reflectors 00336 * 00337 IF( COLMAJOR ) THEN 00338 IF( WANTU1 .AND. P .GT. 0 ) THEN 00339 CALL DLACPY( 'L', P, Q, X11, LDX11, U1, LDU1 ) 00340 CALL DORGQR( P, P, Q, U1, LDU1, WORK(ITAUP1), WORK(IORGQR), 00341 $ LORGQRWORK, INFO) 00342 END IF 00343 IF( WANTU2 .AND. M-P .GT. 0 ) THEN 00344 CALL DLACPY( 'L', M-P, Q, X21, LDX21, U2, LDU2 ) 00345 CALL DORGQR( M-P, M-P, Q, U2, LDU2, WORK(ITAUP2), 00346 $ WORK(IORGQR), LORGQRWORK, INFO ) 00347 END IF 00348 IF( WANTV1T .AND. Q .GT. 0 ) THEN 00349 CALL DLACPY( 'U', Q-1, Q-1, X11(1,2), LDX11, V1T(2,2), 00350 $ LDV1T ) 00351 V1T(1, 1) = ONE 00352 DO J = 2, Q 00353 V1T(1,J) = ZERO 00354 V1T(J,1) = ZERO 00355 END DO 00356 CALL DORGLQ( Q-1, Q-1, Q-1, V1T(2,2), LDV1T, WORK(ITAUQ1), 00357 $ WORK(IORGLQ), LORGLQWORK, INFO ) 00358 END IF 00359 IF( WANTV2T .AND. M-Q .GT. 0 ) THEN 00360 CALL DLACPY( 'U', P, M-Q, X12, LDX12, V2T, LDV2T ) 00361 CALL DLACPY( 'U', M-P-Q, M-P-Q, X22(Q+1,P+1), LDX22, 00362 $ V2T(P+1,P+1), LDV2T ) 00363 CALL DORGLQ( M-Q, M-Q, M-Q, V2T, LDV2T, WORK(ITAUQ2), 00364 $ WORK(IORGLQ), LORGLQWORK, INFO ) 00365 END IF 00366 ELSE 00367 IF( WANTU1 .AND. P .GT. 0 ) THEN 00368 CALL DLACPY( 'U', Q, P, X11, LDX11, U1, LDU1 ) 00369 CALL DORGLQ( P, P, Q, U1, LDU1, WORK(ITAUP1), WORK(IORGLQ), 00370 $ LORGLQWORK, INFO) 00371 END IF 00372 IF( WANTU2 .AND. M-P .GT. 0 ) THEN 00373 CALL DLACPY( 'U', Q, M-P, X21, LDX21, U2, LDU2 ) 00374 CALL DORGLQ( M-P, M-P, Q, U2, LDU2, WORK(ITAUP2), 00375 $ WORK(IORGLQ), LORGLQWORK, INFO ) 00376 END IF 00377 IF( WANTV1T .AND. Q .GT. 0 ) THEN 00378 CALL DLACPY( 'L', Q-1, Q-1, X11(2,1), LDX11, V1T(2,2), 00379 $ LDV1T ) 00380 V1T(1, 1) = ONE 00381 DO J = 2, Q 00382 V1T(1,J) = ZERO 00383 V1T(J,1) = ZERO 00384 END DO 00385 CALL DORGQR( Q-1, Q-1, Q-1, V1T(2,2), LDV1T, WORK(ITAUQ1), 00386 $ WORK(IORGQR), LORGQRWORK, INFO ) 00387 END IF 00388 IF( WANTV2T .AND. M-Q .GT. 0 ) THEN 00389 CALL DLACPY( 'L', M-Q, P, X12, LDX12, V2T, LDV2T ) 00390 CALL DLACPY( 'L', M-P-Q, M-P-Q, X22(P+1,Q+1), LDX22, 00391 $ V2T(P+1,P+1), LDV2T ) 00392 CALL DORGQR( M-Q, M-Q, M-Q, V2T, LDV2T, WORK(ITAUQ2), 00393 $ WORK(IORGQR), LORGQRWORK, INFO ) 00394 END IF 00395 END IF 00396 * 00397 * Compute the CSD of the matrix in bidiagonal-block form 00398 * 00399 CALL DBBCSD( JOBU1, JOBU2, JOBV1T, JOBV2T, TRANS, M, P, Q, THETA, 00400 $ WORK(IPHI), U1, LDU1, U2, LDU2, V1T, LDV1T, V2T, 00401 $ LDV2T, WORK(IB11D), WORK(IB11E), WORK(IB12D), 00402 $ WORK(IB12E), WORK(IB21D), WORK(IB21E), WORK(IB22D), 00403 $ WORK(IB22E), WORK(IBBCSD), LBBCSDWORK, INFO ) 00404 * 00405 * Permute rows and columns to place identity submatrices in top- 00406 * left corner of (1,1)-block and/or bottom-right corner of (1,2)- 00407 * block and/or bottom-right corner of (2,1)-block and/or top-left 00408 * corner of (2,2)-block 00409 * 00410 IF( Q .GT. 0 .AND. WANTU2 ) THEN 00411 DO I = 1, Q 00412 IWORK(I) = M - P - Q + I 00413 END DO 00414 DO I = Q + 1, M - P 00415 IWORK(I) = I - Q 00416 END DO 00417 IF( COLMAJOR ) THEN 00418 CALL DLAPMT( .FALSE., M-P, M-P, U2, LDU2, IWORK ) 00419 ELSE 00420 CALL DLAPMR( .FALSE., M-P, M-P, U2, LDU2, IWORK ) 00421 END IF 00422 END IF 00423 IF( M .GT. 0 .AND. WANTV2T ) THEN 00424 DO I = 1, P 00425 IWORK(I) = M - P - Q + I 00426 END DO 00427 DO I = P + 1, M - Q 00428 IWORK(I) = I - P 00429 END DO 00430 IF( .NOT. COLMAJOR ) THEN 00431 CALL DLAPMT( .FALSE., M-Q, M-Q, V2T, LDV2T, IWORK ) 00432 ELSE 00433 CALL DLAPMR( .FALSE., M-Q, M-Q, V2T, LDV2T, IWORK ) 00434 END IF 00435 END IF 00436 * 00437 RETURN 00438 * 00439 * End DORCSD 00440 * 00441 END 00442
1.7.3 
