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LAPACK
3.4.1
LAPACK: Linear Algebra PACKage
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00001 *> \brief \b SLASDA 00002 * 00003 * =========== DOCUMENTATION =========== 00004 * 00005 * Online html documentation available at 00006 * http://www.netlib.org/lapack/explore-html/ 00007 * 00008 *> \htmlonly 00009 *> Download SLASDA + dependencies 00010 *> <a href="http://www.netlib.org/cgi-bin/netlibfiles.tgz?format=tgz&filename=/lapack/lapack_routine/slasda.f"> 00011 *> [TGZ]</a> 00012 *> <a href="http://www.netlib.org/cgi-bin/netlibfiles.zip?format=zip&filename=/lapack/lapack_routine/slasda.f"> 00013 *> [ZIP]</a> 00014 *> <a href="http://www.netlib.org/cgi-bin/netlibfiles.txt?format=txt&filename=/lapack/lapack_routine/slasda.f"> 00015 *> [TXT]</a> 00016 *> \endhtmlonly 00017 * 00018 * Definition: 00019 * =========== 00020 * 00021 * SUBROUTINE SLASDA( ICOMPQ, SMLSIZ, N, SQRE, D, E, U, LDU, VT, K, 00022 * DIFL, DIFR, Z, POLES, GIVPTR, GIVCOL, LDGCOL, 00023 * PERM, GIVNUM, C, S, WORK, IWORK, INFO ) 00024 * 00025 * .. Scalar Arguments .. 00026 * INTEGER ICOMPQ, INFO, LDGCOL, LDU, N, SMLSIZ, SQRE 00027 * .. 00028 * .. Array Arguments .. 00029 * INTEGER GIVCOL( LDGCOL, * ), GIVPTR( * ), IWORK( * ), 00030 * $ K( * ), PERM( LDGCOL, * ) 00031 * REAL C( * ), D( * ), DIFL( LDU, * ), DIFR( LDU, * ), 00032 * $ E( * ), GIVNUM( LDU, * ), POLES( LDU, * ), 00033 * $ S( * ), U( LDU, * ), VT( LDU, * ), WORK( * ), 00034 * $ Z( LDU, * ) 00035 * .. 00036 * 00037 * 00038 *> \par Purpose: 00039 * ============= 00040 *> 00041 *> \verbatim 00042 *> 00043 *> Using a divide and conquer approach, SLASDA computes the singular 00044 *> value decomposition (SVD) of a real upper bidiagonal N-by-M matrix 00045 *> B with diagonal D and offdiagonal E, where M = N + SQRE. The 00046 *> algorithm computes the singular values in the SVD B = U * S * VT. 00047 *> The orthogonal matrices U and VT are optionally computed in 00048 *> compact form. 00049 *> 00050 *> A related subroutine, SLASD0, computes the singular values and 00051 *> the singular vectors in explicit form. 00052 *> \endverbatim 00053 * 00054 * Arguments: 00055 * ========== 00056 * 00057 *> \param[in] ICOMPQ 00058 *> \verbatim 00059 *> ICOMPQ is INTEGER 00060 *> Specifies whether singular vectors are to be computed 00061 *> in compact form, as follows 00062 *> = 0: Compute singular values only. 00063 *> = 1: Compute singular vectors of upper bidiagonal 00064 *> matrix in compact form. 00065 *> \endverbatim 00066 *> 00067 *> \param[in] SMLSIZ 00068 *> \verbatim 00069 *> SMLSIZ is INTEGER 00070 *> The maximum size of the subproblems at the bottom of the 00071 *> computation tree. 00072 *> \endverbatim 00073 *> 00074 *> \param[in] N 00075 *> \verbatim 00076 *> N is INTEGER 00077 *> The row dimension of the upper bidiagonal matrix. This is 00078 *> also the dimension of the main diagonal array D. 00079 *> \endverbatim 00080 *> 00081 *> \param[in] SQRE 00082 *> \verbatim 00083 *> SQRE is INTEGER 00084 *> Specifies the column dimension of the bidiagonal matrix. 00085 *> = 0: The bidiagonal matrix has column dimension M = N; 00086 *> = 1: The bidiagonal matrix has column dimension M = N + 1. 00087 *> \endverbatim 00088 *> 00089 *> \param[in,out] D 00090 *> \verbatim 00091 *> D is REAL array, dimension ( N ) 00092 *> On entry D contains the main diagonal of the bidiagonal 00093 *> matrix. On exit D, if INFO = 0, contains its singular values. 00094 *> \endverbatim 00095 *> 00096 *> \param[in] E 00097 *> \verbatim 00098 *> E is REAL array, dimension ( M-1 ) 00099 *> Contains the subdiagonal entries of the bidiagonal matrix. 00100 *> On exit, E has been destroyed. 00101 *> \endverbatim 00102 *> 00103 *> \param[out] U 00104 *> \verbatim 00105 *> U is REAL array, 00106 *> dimension ( LDU, SMLSIZ ) if ICOMPQ = 1, and not referenced 00107 *> if ICOMPQ = 0. If ICOMPQ = 1, on exit, U contains the left 00108 *> singular vector matrices of all subproblems at the bottom 00109 *> level. 00110 *> \endverbatim 00111 *> 00112 *> \param[in] LDU 00113 *> \verbatim 00114 *> LDU is INTEGER, LDU = > N. 00115 *> The leading dimension of arrays U, VT, DIFL, DIFR, POLES, 00116 *> GIVNUM, and Z. 00117 *> \endverbatim 00118 *> 00119 *> \param[out] VT 00120 *> \verbatim 00121 *> VT is REAL array, 00122 *> dimension ( LDU, SMLSIZ+1 ) if ICOMPQ = 1, and not referenced 00123 *> if ICOMPQ = 0. If ICOMPQ = 1, on exit, VT**T contains the right 00124 *> singular vector matrices of all subproblems at the bottom 00125 *> level. 00126 *> \endverbatim 00127 *> 00128 *> \param[out] K 00129 *> \verbatim 00130 *> K is INTEGER array, dimension ( N ) 00131 *> if ICOMPQ = 1 and dimension 1 if ICOMPQ = 0. 00132 *> If ICOMPQ = 1, on exit, K(I) is the dimension of the I-th 00133 *> secular equation on the computation tree. 00134 *> \endverbatim 00135 *> 00136 *> \param[out] DIFL 00137 *> \verbatim 00138 *> DIFL is REAL array, dimension ( LDU, NLVL ), 00139 *> where NLVL = floor(log_2 (N/SMLSIZ))). 00140 *> \endverbatim 00141 *> 00142 *> \param[out] DIFR 00143 *> \verbatim 00144 *> DIFR is REAL array, 00145 *> dimension ( LDU, 2 * NLVL ) if ICOMPQ = 1 and 00146 *> dimension ( N ) if ICOMPQ = 0. 00147 *> If ICOMPQ = 1, on exit, DIFL(1:N, I) and DIFR(1:N, 2 * I - 1) 00148 *> record distances between singular values on the I-th 00149 *> level and singular values on the (I -1)-th level, and 00150 *> DIFR(1:N, 2 * I ) contains the normalizing factors for 00151 *> the right singular vector matrix. See SLASD8 for details. 00152 *> \endverbatim 00153 *> 00154 *> \param[out] Z 00155 *> \verbatim 00156 *> Z is REAL array, 00157 *> dimension ( LDU, NLVL ) if ICOMPQ = 1 and 00158 *> dimension ( N ) if ICOMPQ = 0. 00159 *> The first K elements of Z(1, I) contain the components of 00160 *> the deflation-adjusted updating row vector for subproblems 00161 *> on the I-th level. 00162 *> \endverbatim 00163 *> 00164 *> \param[out] POLES 00165 *> \verbatim 00166 *> POLES is REAL array, 00167 *> dimension ( LDU, 2 * NLVL ) if ICOMPQ = 1, and not referenced 00168 *> if ICOMPQ = 0. If ICOMPQ = 1, on exit, POLES(1, 2*I - 1) and 00169 *> POLES(1, 2*I) contain the new and old singular values 00170 *> involved in the secular equations on the I-th level. 00171 *> \endverbatim 00172 *> 00173 *> \param[out] GIVPTR 00174 *> \verbatim 00175 *> GIVPTR is INTEGER array, 00176 *> dimension ( N ) if ICOMPQ = 1, and not referenced if 00177 *> ICOMPQ = 0. If ICOMPQ = 1, on exit, GIVPTR( I ) records 00178 *> the number of Givens rotations performed on the I-th 00179 *> problem on the computation tree. 00180 *> \endverbatim 00181 *> 00182 *> \param[out] GIVCOL 00183 *> \verbatim 00184 *> GIVCOL is INTEGER array, 00185 *> dimension ( LDGCOL, 2 * NLVL ) if ICOMPQ = 1, and not 00186 *> referenced if ICOMPQ = 0. If ICOMPQ = 1, on exit, for each I, 00187 *> GIVCOL(1, 2 *I - 1) and GIVCOL(1, 2 *I) record the locations 00188 *> of Givens rotations performed on the I-th level on the 00189 *> computation tree. 00190 *> \endverbatim 00191 *> 00192 *> \param[in] LDGCOL 00193 *> \verbatim 00194 *> LDGCOL is INTEGER, LDGCOL = > N. 00195 *> The leading dimension of arrays GIVCOL and PERM. 00196 *> \endverbatim 00197 *> 00198 *> \param[out] PERM 00199 *> \verbatim 00200 *> PERM is INTEGER array, dimension ( LDGCOL, NLVL ) 00201 *> if ICOMPQ = 1, and not referenced 00202 *> if ICOMPQ = 0. If ICOMPQ = 1, on exit, PERM(1, I) records 00203 *> permutations done on the I-th level of the computation tree. 00204 *> \endverbatim 00205 *> 00206 *> \param[out] GIVNUM 00207 *> \verbatim 00208 *> GIVNUM is REAL array, 00209 *> dimension ( LDU, 2 * NLVL ) if ICOMPQ = 1, and not 00210 *> referenced if ICOMPQ = 0. If ICOMPQ = 1, on exit, for each I, 00211 *> GIVNUM(1, 2 *I - 1) and GIVNUM(1, 2 *I) record the C- and S- 00212 *> values of Givens rotations performed on the I-th level on 00213 *> the computation tree. 00214 *> \endverbatim 00215 *> 00216 *> \param[out] C 00217 *> \verbatim 00218 *> C is REAL array, 00219 *> dimension ( N ) if ICOMPQ = 1, and dimension 1 if ICOMPQ = 0. 00220 *> If ICOMPQ = 1 and the I-th subproblem is not square, on exit, 00221 *> C( I ) contains the C-value of a Givens rotation related to 00222 *> the right null space of the I-th subproblem. 00223 *> \endverbatim 00224 *> 00225 *> \param[out] S 00226 *> \verbatim 00227 *> S is REAL array, dimension ( N ) if 00228 *> ICOMPQ = 1, and dimension 1 if ICOMPQ = 0. If ICOMPQ = 1 00229 *> and the I-th subproblem is not square, on exit, S( I ) 00230 *> contains the S-value of a Givens rotation related to 00231 *> the right null space of the I-th subproblem. 00232 *> \endverbatim 00233 *> 00234 *> \param[out] WORK 00235 *> \verbatim 00236 *> WORK is REAL array, dimension 00237 *> (6 * N + (SMLSIZ + 1)*(SMLSIZ + 1)). 00238 *> \endverbatim 00239 *> 00240 *> \param[out] IWORK 00241 *> \verbatim 00242 *> IWORK is INTEGER array, dimension (7*N). 00243 *> \endverbatim 00244 *> 00245 *> \param[out] INFO 00246 *> \verbatim 00247 *> INFO is INTEGER 00248 *> = 0: successful exit. 00249 *> < 0: if INFO = -i, the i-th argument had an illegal value. 00250 *> > 0: if INFO = 1, a singular value did not converge 00251 *> \endverbatim 00252 * 00253 * Authors: 00254 * ======== 00255 * 00256 *> \author Univ. of Tennessee 00257 *> \author Univ. of California Berkeley 00258 *> \author Univ. of Colorado Denver 00259 *> \author NAG Ltd. 00260 * 00261 *> \date November 2011 00262 * 00263 *> \ingroup auxOTHERauxiliary 00264 * 00265 *> \par Contributors: 00266 * ================== 00267 *> 00268 *> Ming Gu and Huan Ren, Computer Science Division, University of 00269 *> California at Berkeley, USA 00270 *> 00271 * ===================================================================== 00272 SUBROUTINE SLASDA( ICOMPQ, SMLSIZ, N, SQRE, D, E, U, LDU, VT, K, 00273 $ DIFL, DIFR, Z, POLES, GIVPTR, GIVCOL, LDGCOL, 00274 $ PERM, GIVNUM, C, S, WORK, IWORK, INFO ) 00275 * 00276 * -- LAPACK auxiliary routine (version 3.4.0) -- 00277 * -- LAPACK is a software package provided by Univ. of Tennessee, -- 00278 * -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..-- 00279 * November 2011 00280 * 00281 * .. Scalar Arguments .. 00282 INTEGER ICOMPQ, INFO, LDGCOL, LDU, N, SMLSIZ, SQRE 00283 * .. 00284 * .. Array Arguments .. 00285 INTEGER GIVCOL( LDGCOL, * ), GIVPTR( * ), IWORK( * ), 00286 $ K( * ), PERM( LDGCOL, * ) 00287 REAL C( * ), D( * ), DIFL( LDU, * ), DIFR( LDU, * ), 00288 $ E( * ), GIVNUM( LDU, * ), POLES( LDU, * ), 00289 $ S( * ), U( LDU, * ), VT( LDU, * ), WORK( * ), 00290 $ Z( LDU, * ) 00291 * .. 00292 * 00293 * ===================================================================== 00294 * 00295 * .. Parameters .. 00296 REAL ZERO, ONE 00297 PARAMETER ( ZERO = 0.0E+0, ONE = 1.0E+0 ) 00298 * .. 00299 * .. Local Scalars .. 00300 INTEGER I, I1, IC, IDXQ, IDXQI, IM1, INODE, ITEMP, IWK, 00301 $ J, LF, LL, LVL, LVL2, M, NCC, ND, NDB1, NDIML, 00302 $ NDIMR, NL, NLF, NLP1, NLVL, NR, NRF, NRP1, NRU, 00303 $ NWORK1, NWORK2, SMLSZP, SQREI, VF, VFI, VL, VLI 00304 REAL ALPHA, BETA 00305 * .. 00306 * .. External Subroutines .. 00307 EXTERNAL SCOPY, SLASD6, SLASDQ, SLASDT, SLASET, XERBLA 00308 * .. 00309 * .. Executable Statements .. 00310 * 00311 * Test the input parameters. 00312 * 00313 INFO = 0 00314 * 00315 IF( ( ICOMPQ.LT.0 ) .OR. ( ICOMPQ.GT.1 ) ) THEN 00316 INFO = -1 00317 ELSE IF( SMLSIZ.LT.3 ) THEN 00318 INFO = -2 00319 ELSE IF( N.LT.0 ) THEN 00320 INFO = -3 00321 ELSE IF( ( SQRE.LT.0 ) .OR. ( SQRE.GT.1 ) ) THEN 00322 INFO = -4 00323 ELSE IF( LDU.LT.( N+SQRE ) ) THEN 00324 INFO = -8 00325 ELSE IF( LDGCOL.LT.N ) THEN 00326 INFO = -17 00327 END IF 00328 IF( INFO.NE.0 ) THEN 00329 CALL XERBLA( 'SLASDA', -INFO ) 00330 RETURN 00331 END IF 00332 * 00333 M = N + SQRE 00334 * 00335 * If the input matrix is too small, call SLASDQ to find the SVD. 00336 * 00337 IF( N.LE.SMLSIZ ) THEN 00338 IF( ICOMPQ.EQ.0 ) THEN 00339 CALL SLASDQ( 'U', SQRE, N, 0, 0, 0, D, E, VT, LDU, U, LDU, 00340 $ U, LDU, WORK, INFO ) 00341 ELSE 00342 CALL SLASDQ( 'U', SQRE, N, M, N, 0, D, E, VT, LDU, U, LDU, 00343 $ U, LDU, WORK, INFO ) 00344 END IF 00345 RETURN 00346 END IF 00347 * 00348 * Book-keeping and set up the computation tree. 00349 * 00350 INODE = 1 00351 NDIML = INODE + N 00352 NDIMR = NDIML + N 00353 IDXQ = NDIMR + N 00354 IWK = IDXQ + N 00355 * 00356 NCC = 0 00357 NRU = 0 00358 * 00359 SMLSZP = SMLSIZ + 1 00360 VF = 1 00361 VL = VF + M 00362 NWORK1 = VL + M 00363 NWORK2 = NWORK1 + SMLSZP*SMLSZP 00364 * 00365 CALL SLASDT( N, NLVL, ND, IWORK( INODE ), IWORK( NDIML ), 00366 $ IWORK( NDIMR ), SMLSIZ ) 00367 * 00368 * for the nodes on bottom level of the tree, solve 00369 * their subproblems by SLASDQ. 00370 * 00371 NDB1 = ( ND+1 ) / 2 00372 DO 30 I = NDB1, ND 00373 * 00374 * IC : center row of each node 00375 * NL : number of rows of left subproblem 00376 * NR : number of rows of right subproblem 00377 * NLF: starting row of the left subproblem 00378 * NRF: starting row of the right subproblem 00379 * 00380 I1 = I - 1 00381 IC = IWORK( INODE+I1 ) 00382 NL = IWORK( NDIML+I1 ) 00383 NLP1 = NL + 1 00384 NR = IWORK( NDIMR+I1 ) 00385 NLF = IC - NL 00386 NRF = IC + 1 00387 IDXQI = IDXQ + NLF - 2 00388 VFI = VF + NLF - 1 00389 VLI = VL + NLF - 1 00390 SQREI = 1 00391 IF( ICOMPQ.EQ.0 ) THEN 00392 CALL SLASET( 'A', NLP1, NLP1, ZERO, ONE, WORK( NWORK1 ), 00393 $ SMLSZP ) 00394 CALL SLASDQ( 'U', SQREI, NL, NLP1, NRU, NCC, D( NLF ), 00395 $ E( NLF ), WORK( NWORK1 ), SMLSZP, 00396 $ WORK( NWORK2 ), NL, WORK( NWORK2 ), NL, 00397 $ WORK( NWORK2 ), INFO ) 00398 ITEMP = NWORK1 + NL*SMLSZP 00399 CALL SCOPY( NLP1, WORK( NWORK1 ), 1, WORK( VFI ), 1 ) 00400 CALL SCOPY( NLP1, WORK( ITEMP ), 1, WORK( VLI ), 1 ) 00401 ELSE 00402 CALL SLASET( 'A', NL, NL, ZERO, ONE, U( NLF, 1 ), LDU ) 00403 CALL SLASET( 'A', NLP1, NLP1, ZERO, ONE, VT( NLF, 1 ), LDU ) 00404 CALL SLASDQ( 'U', SQREI, NL, NLP1, NL, NCC, D( NLF ), 00405 $ E( NLF ), VT( NLF, 1 ), LDU, U( NLF, 1 ), LDU, 00406 $ U( NLF, 1 ), LDU, WORK( NWORK1 ), INFO ) 00407 CALL SCOPY( NLP1, VT( NLF, 1 ), 1, WORK( VFI ), 1 ) 00408 CALL SCOPY( NLP1, VT( NLF, NLP1 ), 1, WORK( VLI ), 1 ) 00409 END IF 00410 IF( INFO.NE.0 ) THEN 00411 RETURN 00412 END IF 00413 DO 10 J = 1, NL 00414 IWORK( IDXQI+J ) = J 00415 10 CONTINUE 00416 IF( ( I.EQ.ND ) .AND. ( SQRE.EQ.0 ) ) THEN 00417 SQREI = 0 00418 ELSE 00419 SQREI = 1 00420 END IF 00421 IDXQI = IDXQI + NLP1 00422 VFI = VFI + NLP1 00423 VLI = VLI + NLP1 00424 NRP1 = NR + SQREI 00425 IF( ICOMPQ.EQ.0 ) THEN 00426 CALL SLASET( 'A', NRP1, NRP1, ZERO, ONE, WORK( NWORK1 ), 00427 $ SMLSZP ) 00428 CALL SLASDQ( 'U', SQREI, NR, NRP1, NRU, NCC, D( NRF ), 00429 $ E( NRF ), WORK( NWORK1 ), SMLSZP, 00430 $ WORK( NWORK2 ), NR, WORK( NWORK2 ), NR, 00431 $ WORK( NWORK2 ), INFO ) 00432 ITEMP = NWORK1 + ( NRP1-1 )*SMLSZP 00433 CALL SCOPY( NRP1, WORK( NWORK1 ), 1, WORK( VFI ), 1 ) 00434 CALL SCOPY( NRP1, WORK( ITEMP ), 1, WORK( VLI ), 1 ) 00435 ELSE 00436 CALL SLASET( 'A', NR, NR, ZERO, ONE, U( NRF, 1 ), LDU ) 00437 CALL SLASET( 'A', NRP1, NRP1, ZERO, ONE, VT( NRF, 1 ), LDU ) 00438 CALL SLASDQ( 'U', SQREI, NR, NRP1, NR, NCC, D( NRF ), 00439 $ E( NRF ), VT( NRF, 1 ), LDU, U( NRF, 1 ), LDU, 00440 $ U( NRF, 1 ), LDU, WORK( NWORK1 ), INFO ) 00441 CALL SCOPY( NRP1, VT( NRF, 1 ), 1, WORK( VFI ), 1 ) 00442 CALL SCOPY( NRP1, VT( NRF, NRP1 ), 1, WORK( VLI ), 1 ) 00443 END IF 00444 IF( INFO.NE.0 ) THEN 00445 RETURN 00446 END IF 00447 DO 20 J = 1, NR 00448 IWORK( IDXQI+J ) = J 00449 20 CONTINUE 00450 30 CONTINUE 00451 * 00452 * Now conquer each subproblem bottom-up. 00453 * 00454 J = 2**NLVL 00455 DO 50 LVL = NLVL, 1, -1 00456 LVL2 = LVL*2 - 1 00457 * 00458 * Find the first node LF and last node LL on 00459 * the current level LVL. 00460 * 00461 IF( LVL.EQ.1 ) THEN 00462 LF = 1 00463 LL = 1 00464 ELSE 00465 LF = 2**( LVL-1 ) 00466 LL = 2*LF - 1 00467 END IF 00468 DO 40 I = LF, LL 00469 IM1 = I - 1 00470 IC = IWORK( INODE+IM1 ) 00471 NL = IWORK( NDIML+IM1 ) 00472 NR = IWORK( NDIMR+IM1 ) 00473 NLF = IC - NL 00474 NRF = IC + 1 00475 IF( I.EQ.LL ) THEN 00476 SQREI = SQRE 00477 ELSE 00478 SQREI = 1 00479 END IF 00480 VFI = VF + NLF - 1 00481 VLI = VL + NLF - 1 00482 IDXQI = IDXQ + NLF - 1 00483 ALPHA = D( IC ) 00484 BETA = E( IC ) 00485 IF( ICOMPQ.EQ.0 ) THEN 00486 CALL SLASD6( ICOMPQ, NL, NR, SQREI, D( NLF ), 00487 $ WORK( VFI ), WORK( VLI ), ALPHA, BETA, 00488 $ IWORK( IDXQI ), PERM, GIVPTR( 1 ), GIVCOL, 00489 $ LDGCOL, GIVNUM, LDU, POLES, DIFL, DIFR, Z, 00490 $ K( 1 ), C( 1 ), S( 1 ), WORK( NWORK1 ), 00491 $ IWORK( IWK ), INFO ) 00492 ELSE 00493 J = J - 1 00494 CALL SLASD6( ICOMPQ, NL, NR, SQREI, D( NLF ), 00495 $ WORK( VFI ), WORK( VLI ), ALPHA, BETA, 00496 $ IWORK( IDXQI ), PERM( NLF, LVL ), 00497 $ GIVPTR( J ), GIVCOL( NLF, LVL2 ), LDGCOL, 00498 $ GIVNUM( NLF, LVL2 ), LDU, 00499 $ POLES( NLF, LVL2 ), DIFL( NLF, LVL ), 00500 $ DIFR( NLF, LVL2 ), Z( NLF, LVL ), K( J ), 00501 $ C( J ), S( J ), WORK( NWORK1 ), 00502 $ IWORK( IWK ), INFO ) 00503 END IF 00504 IF( INFO.NE.0 ) THEN 00505 RETURN 00506 END IF 00507 40 CONTINUE 00508 50 CONTINUE 00509 * 00510 RETURN 00511 * 00512 * End of SLASDA 00513 * 00514 END