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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 SUBROUTINE SBDSDC( UPLO, COMPQ, N, D, E, U, LDU, VT, LDVT, Q, IQ, 00002 $ WORK, IWORK, INFO ) 00003 * 00004 * -- LAPACK routine (version 3.2.2) -- 00005 * -- LAPACK is a software package provided by Univ. of Tennessee, -- 00006 * -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..-- 00007 * June 2010 00008 * 00009 * .. Scalar Arguments .. 00010 CHARACTER COMPQ, UPLO 00011 INTEGER INFO, LDU, LDVT, N 00012 * .. 00013 * .. Array Arguments .. 00014 INTEGER IQ( * ), IWORK( * ) 00015 REAL D( * ), E( * ), Q( * ), U( LDU, * ), 00016 $ VT( LDVT, * ), WORK( * ) 00017 * .. 00018 * 00019 * Purpose 00020 * ======= 00021 * 00022 * SBDSDC computes the singular value decomposition (SVD) of a real 00023 * N-by-N (upper or lower) bidiagonal matrix B: B = U * S * VT, 00024 * using a divide and conquer method, where S is a diagonal matrix 00025 * with non-negative diagonal elements (the singular values of B), and 00026 * U and VT are orthogonal matrices of left and right singular vectors, 00027 * respectively. SBDSDC can be used to compute all singular values, 00028 * and optionally, singular vectors or singular vectors in compact form. 00029 * 00030 * This code makes very mild assumptions about floating point 00031 * arithmetic. It will work on machines with a guard digit in 00032 * add/subtract, or on those binary machines without guard digits 00033 * which subtract like the Cray X-MP, Cray Y-MP, Cray C-90, or Cray-2. 00034 * It could conceivably fail on hexadecimal or decimal machines 00035 * without guard digits, but we know of none. See SLASD3 for details. 00036 * 00037 * The code currently calls SLASDQ if singular values only are desired. 00038 * However, it can be slightly modified to compute singular values 00039 * using the divide and conquer method. 00040 * 00041 * Arguments 00042 * ========= 00043 * 00044 * UPLO (input) CHARACTER*1 00045 * = 'U': B is upper bidiagonal. 00046 * = 'L': B is lower bidiagonal. 00047 * 00048 * COMPQ (input) CHARACTER*1 00049 * Specifies whether singular vectors are to be computed 00050 * as follows: 00051 * = 'N': Compute singular values only; 00052 * = 'P': Compute singular values and compute singular 00053 * vectors in compact form; 00054 * = 'I': Compute singular values and singular vectors. 00055 * 00056 * N (input) INTEGER 00057 * The order of the matrix B. N >= 0. 00058 * 00059 * D (input/output) REAL array, dimension (N) 00060 * On entry, the n diagonal elements of the bidiagonal matrix B. 00061 * On exit, if INFO=0, the singular values of B. 00062 * 00063 * E (input/output) REAL array, dimension (N-1) 00064 * On entry, the elements of E contain the offdiagonal 00065 * elements of the bidiagonal matrix whose SVD is desired. 00066 * On exit, E has been destroyed. 00067 * 00068 * U (output) REAL array, dimension (LDU,N) 00069 * If COMPQ = 'I', then: 00070 * On exit, if INFO = 0, U contains the left singular vectors 00071 * of the bidiagonal matrix. 00072 * For other values of COMPQ, U is not referenced. 00073 * 00074 * LDU (input) INTEGER 00075 * The leading dimension of the array U. LDU >= 1. 00076 * If singular vectors are desired, then LDU >= max( 1, N ). 00077 * 00078 * VT (output) REAL array, dimension (LDVT,N) 00079 * If COMPQ = 'I', then: 00080 * On exit, if INFO = 0, VT**T contains the right singular 00081 * vectors of the bidiagonal matrix. 00082 * For other values of COMPQ, VT is not referenced. 00083 * 00084 * LDVT (input) INTEGER 00085 * The leading dimension of the array VT. LDVT >= 1. 00086 * If singular vectors are desired, then LDVT >= max( 1, N ). 00087 * 00088 * Q (output) REAL array, dimension (LDQ) 00089 * If COMPQ = 'P', then: 00090 * On exit, if INFO = 0, Q and IQ contain the left 00091 * and right singular vectors in a compact form, 00092 * requiring O(N log N) space instead of 2*N**2. 00093 * In particular, Q contains all the REAL data in 00094 * LDQ >= N*(11 + 2*SMLSIZ + 8*INT(LOG_2(N/(SMLSIZ+1)))) 00095 * words of memory, where SMLSIZ is returned by ILAENV and 00096 * is equal to the maximum size of the subproblems at the 00097 * bottom of the computation tree (usually about 25). 00098 * For other values of COMPQ, Q is not referenced. 00099 * 00100 * IQ (output) INTEGER array, dimension (LDIQ) 00101 * If COMPQ = 'P', then: 00102 * On exit, if INFO = 0, Q and IQ contain the left 00103 * and right singular vectors in a compact form, 00104 * requiring O(N log N) space instead of 2*N**2. 00105 * In particular, IQ contains all INTEGER data in 00106 * LDIQ >= N*(3 + 3*INT(LOG_2(N/(SMLSIZ+1)))) 00107 * words of memory, where SMLSIZ is returned by ILAENV and 00108 * is equal to the maximum size of the subproblems at the 00109 * bottom of the computation tree (usually about 25). 00110 * For other values of COMPQ, IQ is not referenced. 00111 * 00112 * WORK (workspace) REAL array, dimension (MAX(1,LWORK)) 00113 * If COMPQ = 'N' then LWORK >= (4 * N). 00114 * If COMPQ = 'P' then LWORK >= (6 * N). 00115 * If COMPQ = 'I' then LWORK >= (3 * N**2 + 4 * N). 00116 * 00117 * IWORK (workspace) INTEGER array, dimension (8*N) 00118 * 00119 * INFO (output) INTEGER 00120 * = 0: successful exit. 00121 * < 0: if INFO = -i, the i-th argument had an illegal value. 00122 * > 0: The algorithm failed to compute a singular value. 00123 * The update process of divide and conquer failed. 00124 * 00125 * Further Details 00126 * =============== 00127 * 00128 * Based on contributions by 00129 * Ming Gu and Huan Ren, Computer Science Division, University of 00130 * California at Berkeley, USA 00131 * ===================================================================== 00132 * Changed dimension statement in comment describing E from (N) to 00133 * (N-1). Sven, 17 Feb 05. 00134 * ===================================================================== 00135 * 00136 * .. Parameters .. 00137 REAL ZERO, ONE, TWO 00138 PARAMETER ( ZERO = 0.0E+0, ONE = 1.0E+0, TWO = 2.0E+0 ) 00139 * .. 00140 * .. Local Scalars .. 00141 INTEGER DIFL, DIFR, GIVCOL, GIVNUM, GIVPTR, I, IC, 00142 $ ICOMPQ, IERR, II, IS, IU, IUPLO, IVT, J, K, KK, 00143 $ MLVL, NM1, NSIZE, PERM, POLES, QSTART, SMLSIZ, 00144 $ SMLSZP, SQRE, START, WSTART, Z 00145 REAL CS, EPS, ORGNRM, P, R, SN 00146 * .. 00147 * .. External Functions .. 00148 LOGICAL LSAME 00149 INTEGER ILAENV 00150 REAL SLAMCH, SLANST 00151 EXTERNAL SLAMCH, SLANST, ILAENV, LSAME 00152 * .. 00153 * .. External Subroutines .. 00154 EXTERNAL SCOPY, SLARTG, SLASCL, SLASD0, SLASDA, SLASDQ, 00155 $ SLASET, SLASR, SSWAP, XERBLA 00156 * .. 00157 * .. Intrinsic Functions .. 00158 INTRINSIC REAL, ABS, INT, LOG, SIGN 00159 * .. 00160 * .. Executable Statements .. 00161 * 00162 * Test the input parameters. 00163 * 00164 INFO = 0 00165 * 00166 IUPLO = 0 00167 IF( LSAME( UPLO, 'U' ) ) 00168 $ IUPLO = 1 00169 IF( LSAME( UPLO, 'L' ) ) 00170 $ IUPLO = 2 00171 IF( LSAME( COMPQ, 'N' ) ) THEN 00172 ICOMPQ = 0 00173 ELSE IF( LSAME( COMPQ, 'P' ) ) THEN 00174 ICOMPQ = 1 00175 ELSE IF( LSAME( COMPQ, 'I' ) ) THEN 00176 ICOMPQ = 2 00177 ELSE 00178 ICOMPQ = -1 00179 END IF 00180 IF( IUPLO.EQ.0 ) THEN 00181 INFO = -1 00182 ELSE IF( ICOMPQ.LT.0 ) THEN 00183 INFO = -2 00184 ELSE IF( N.LT.0 ) THEN 00185 INFO = -3 00186 ELSE IF( ( LDU.LT.1 ) .OR. ( ( ICOMPQ.EQ.2 ) .AND. ( LDU.LT. 00187 $ N ) ) ) THEN 00188 INFO = -7 00189 ELSE IF( ( LDVT.LT.1 ) .OR. ( ( ICOMPQ.EQ.2 ) .AND. ( LDVT.LT. 00190 $ N ) ) ) THEN 00191 INFO = -9 00192 END IF 00193 IF( INFO.NE.0 ) THEN 00194 CALL XERBLA( 'SBDSDC', -INFO ) 00195 RETURN 00196 END IF 00197 * 00198 * Quick return if possible 00199 * 00200 IF( N.EQ.0 ) 00201 $ RETURN 00202 SMLSIZ = ILAENV( 9, 'SBDSDC', ' ', 0, 0, 0, 0 ) 00203 IF( N.EQ.1 ) THEN 00204 IF( ICOMPQ.EQ.1 ) THEN 00205 Q( 1 ) = SIGN( ONE, D( 1 ) ) 00206 Q( 1+SMLSIZ*N ) = ONE 00207 ELSE IF( ICOMPQ.EQ.2 ) THEN 00208 U( 1, 1 ) = SIGN( ONE, D( 1 ) ) 00209 VT( 1, 1 ) = ONE 00210 END IF 00211 D( 1 ) = ABS( D( 1 ) ) 00212 RETURN 00213 END IF 00214 NM1 = N - 1 00215 * 00216 * If matrix lower bidiagonal, rotate to be upper bidiagonal 00217 * by applying Givens rotations on the left 00218 * 00219 WSTART = 1 00220 QSTART = 3 00221 IF( ICOMPQ.EQ.1 ) THEN 00222 CALL SCOPY( N, D, 1, Q( 1 ), 1 ) 00223 CALL SCOPY( N-1, E, 1, Q( N+1 ), 1 ) 00224 END IF 00225 IF( IUPLO.EQ.2 ) THEN 00226 QSTART = 5 00227 WSTART = 2*N - 1 00228 DO 10 I = 1, N - 1 00229 CALL SLARTG( D( I ), E( I ), CS, SN, R ) 00230 D( I ) = R 00231 E( I ) = SN*D( I+1 ) 00232 D( I+1 ) = CS*D( I+1 ) 00233 IF( ICOMPQ.EQ.1 ) THEN 00234 Q( I+2*N ) = CS 00235 Q( I+3*N ) = SN 00236 ELSE IF( ICOMPQ.EQ.2 ) THEN 00237 WORK( I ) = CS 00238 WORK( NM1+I ) = -SN 00239 END IF 00240 10 CONTINUE 00241 END IF 00242 * 00243 * If ICOMPQ = 0, use SLASDQ to compute the singular values. 00244 * 00245 IF( ICOMPQ.EQ.0 ) THEN 00246 CALL SLASDQ( 'U', 0, N, 0, 0, 0, D, E, VT, LDVT, U, LDU, U, 00247 $ LDU, WORK( WSTART ), INFO ) 00248 GO TO 40 00249 END IF 00250 * 00251 * If N is smaller than the minimum divide size SMLSIZ, then solve 00252 * the problem with another solver. 00253 * 00254 IF( N.LE.SMLSIZ ) THEN 00255 IF( ICOMPQ.EQ.2 ) THEN 00256 CALL SLASET( 'A', N, N, ZERO, ONE, U, LDU ) 00257 CALL SLASET( 'A', N, N, ZERO, ONE, VT, LDVT ) 00258 CALL SLASDQ( 'U', 0, N, N, N, 0, D, E, VT, LDVT, U, LDU, U, 00259 $ LDU, WORK( WSTART ), INFO ) 00260 ELSE IF( ICOMPQ.EQ.1 ) THEN 00261 IU = 1 00262 IVT = IU + N 00263 CALL SLASET( 'A', N, N, ZERO, ONE, Q( IU+( QSTART-1 )*N ), 00264 $ N ) 00265 CALL SLASET( 'A', N, N, ZERO, ONE, Q( IVT+( QSTART-1 )*N ), 00266 $ N ) 00267 CALL SLASDQ( 'U', 0, N, N, N, 0, D, E, 00268 $ Q( IVT+( QSTART-1 )*N ), N, 00269 $ Q( IU+( QSTART-1 )*N ), N, 00270 $ Q( IU+( QSTART-1 )*N ), N, WORK( WSTART ), 00271 $ INFO ) 00272 END IF 00273 GO TO 40 00274 END IF 00275 * 00276 IF( ICOMPQ.EQ.2 ) THEN 00277 CALL SLASET( 'A', N, N, ZERO, ONE, U, LDU ) 00278 CALL SLASET( 'A', N, N, ZERO, ONE, VT, LDVT ) 00279 END IF 00280 * 00281 * Scale. 00282 * 00283 ORGNRM = SLANST( 'M', N, D, E ) 00284 IF( ORGNRM.EQ.ZERO ) 00285 $ RETURN 00286 CALL SLASCL( 'G', 0, 0, ORGNRM, ONE, N, 1, D, N, IERR ) 00287 CALL SLASCL( 'G', 0, 0, ORGNRM, ONE, NM1, 1, E, NM1, IERR ) 00288 * 00289 EPS = SLAMCH( 'Epsilon' ) 00290 * 00291 MLVL = INT( LOG( REAL( N ) / REAL( SMLSIZ+1 ) ) / LOG( TWO ) ) + 1 00292 SMLSZP = SMLSIZ + 1 00293 * 00294 IF( ICOMPQ.EQ.1 ) THEN 00295 IU = 1 00296 IVT = 1 + SMLSIZ 00297 DIFL = IVT + SMLSZP 00298 DIFR = DIFL + MLVL 00299 Z = DIFR + MLVL*2 00300 IC = Z + MLVL 00301 IS = IC + 1 00302 POLES = IS + 1 00303 GIVNUM = POLES + 2*MLVL 00304 * 00305 K = 1 00306 GIVPTR = 2 00307 PERM = 3 00308 GIVCOL = PERM + MLVL 00309 END IF 00310 * 00311 DO 20 I = 1, N 00312 IF( ABS( D( I ) ).LT.EPS ) THEN 00313 D( I ) = SIGN( EPS, D( I ) ) 00314 END IF 00315 20 CONTINUE 00316 * 00317 START = 1 00318 SQRE = 0 00319 * 00320 DO 30 I = 1, NM1 00321 IF( ( ABS( E( I ) ).LT.EPS ) .OR. ( I.EQ.NM1 ) ) THEN 00322 * 00323 * Subproblem found. First determine its size and then 00324 * apply divide and conquer on it. 00325 * 00326 IF( I.LT.NM1 ) THEN 00327 * 00328 * A subproblem with E(I) small for I < NM1. 00329 * 00330 NSIZE = I - START + 1 00331 ELSE IF( ABS( E( I ) ).GE.EPS ) THEN 00332 * 00333 * A subproblem with E(NM1) not too small but I = NM1. 00334 * 00335 NSIZE = N - START + 1 00336 ELSE 00337 * 00338 * A subproblem with E(NM1) small. This implies an 00339 * 1-by-1 subproblem at D(N). Solve this 1-by-1 problem 00340 * first. 00341 * 00342 NSIZE = I - START + 1 00343 IF( ICOMPQ.EQ.2 ) THEN 00344 U( N, N ) = SIGN( ONE, D( N ) ) 00345 VT( N, N ) = ONE 00346 ELSE IF( ICOMPQ.EQ.1 ) THEN 00347 Q( N+( QSTART-1 )*N ) = SIGN( ONE, D( N ) ) 00348 Q( N+( SMLSIZ+QSTART-1 )*N ) = ONE 00349 END IF 00350 D( N ) = ABS( D( N ) ) 00351 END IF 00352 IF( ICOMPQ.EQ.2 ) THEN 00353 CALL SLASD0( NSIZE, SQRE, D( START ), E( START ), 00354 $ U( START, START ), LDU, VT( START, START ), 00355 $ LDVT, SMLSIZ, IWORK, WORK( WSTART ), INFO ) 00356 ELSE 00357 CALL SLASDA( ICOMPQ, SMLSIZ, NSIZE, SQRE, D( START ), 00358 $ E( START ), Q( START+( IU+QSTART-2 )*N ), N, 00359 $ Q( START+( IVT+QSTART-2 )*N ), 00360 $ IQ( START+K*N ), Q( START+( DIFL+QSTART-2 )* 00361 $ N ), Q( START+( DIFR+QSTART-2 )*N ), 00362 $ Q( START+( Z+QSTART-2 )*N ), 00363 $ Q( START+( POLES+QSTART-2 )*N ), 00364 $ IQ( START+GIVPTR*N ), IQ( START+GIVCOL*N ), 00365 $ N, IQ( START+PERM*N ), 00366 $ Q( START+( GIVNUM+QSTART-2 )*N ), 00367 $ Q( START+( IC+QSTART-2 )*N ), 00368 $ Q( START+( IS+QSTART-2 )*N ), 00369 $ WORK( WSTART ), IWORK, INFO ) 00370 END IF 00371 IF( INFO.NE.0 ) THEN 00372 RETURN 00373 END IF 00374 START = I + 1 00375 END IF 00376 30 CONTINUE 00377 * 00378 * Unscale 00379 * 00380 CALL SLASCL( 'G', 0, 0, ONE, ORGNRM, N, 1, D, N, IERR ) 00381 40 CONTINUE 00382 * 00383 * Use Selection Sort to minimize swaps of singular vectors 00384 * 00385 DO 60 II = 2, N 00386 I = II - 1 00387 KK = I 00388 P = D( I ) 00389 DO 50 J = II, N 00390 IF( D( J ).GT.P ) THEN 00391 KK = J 00392 P = D( J ) 00393 END IF 00394 50 CONTINUE 00395 IF( KK.NE.I ) THEN 00396 D( KK ) = D( I ) 00397 D( I ) = P 00398 IF( ICOMPQ.EQ.1 ) THEN 00399 IQ( I ) = KK 00400 ELSE IF( ICOMPQ.EQ.2 ) THEN 00401 CALL SSWAP( N, U( 1, I ), 1, U( 1, KK ), 1 ) 00402 CALL SSWAP( N, VT( I, 1 ), LDVT, VT( KK, 1 ), LDVT ) 00403 END IF 00404 ELSE IF( ICOMPQ.EQ.1 ) THEN 00405 IQ( I ) = I 00406 END IF 00407 60 CONTINUE 00408 * 00409 * If ICOMPQ = 1, use IQ(N,1) as the indicator for UPLO 00410 * 00411 IF( ICOMPQ.EQ.1 ) THEN 00412 IF( IUPLO.EQ.1 ) THEN 00413 IQ( N ) = 1 00414 ELSE 00415 IQ( N ) = 0 00416 END IF 00417 END IF 00418 * 00419 * If B is lower bidiagonal, update U by those Givens rotations 00420 * which rotated B to be upper bidiagonal 00421 * 00422 IF( ( IUPLO.EQ.2 ) .AND. ( ICOMPQ.EQ.2 ) ) 00423 $ CALL SLASR( 'L', 'V', 'B', N, N, WORK( 1 ), WORK( N ), U, LDU ) 00424 * 00425 RETURN 00426 * 00427 * End of SBDSDC 00428 * 00429 END
1.7.3 
