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LAPACK
3.4.1
LAPACK: Linear Algebra PACKage
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00001 *> \brief <b> SPPSVX computes the solution to system of linear equations A * X = B for OTHER matrices</b> 00002 * 00003 * =========== DOCUMENTATION =========== 00004 * 00005 * Online html documentation available at 00006 * http://www.netlib.org/lapack/explore-html/ 00007 * 00008 *> \htmlonly 00009 *> Download SPPSVX + dependencies 00010 *> <a href="http://www.netlib.org/cgi-bin/netlibfiles.tgz?format=tgz&filename=/lapack/lapack_routine/sppsvx.f"> 00011 *> [TGZ]</a> 00012 *> <a href="http://www.netlib.org/cgi-bin/netlibfiles.zip?format=zip&filename=/lapack/lapack_routine/sppsvx.f"> 00013 *> [ZIP]</a> 00014 *> <a href="http://www.netlib.org/cgi-bin/netlibfiles.txt?format=txt&filename=/lapack/lapack_routine/sppsvx.f"> 00015 *> [TXT]</a> 00016 *> \endhtmlonly 00017 * 00018 * Definition: 00019 * =========== 00020 * 00021 * SUBROUTINE SPPSVX( FACT, UPLO, N, NRHS, AP, AFP, EQUED, S, B, LDB, 00022 * X, LDX, RCOND, FERR, BERR, WORK, IWORK, INFO ) 00023 * 00024 * .. Scalar Arguments .. 00025 * CHARACTER EQUED, FACT, UPLO 00026 * INTEGER INFO, LDB, LDX, N, NRHS 00027 * REAL RCOND 00028 * .. 00029 * .. Array Arguments .. 00030 * INTEGER IWORK( * ) 00031 * REAL AFP( * ), AP( * ), B( LDB, * ), BERR( * ), 00032 * $ FERR( * ), S( * ), WORK( * ), X( LDX, * ) 00033 * .. 00034 * 00035 * 00036 *> \par Purpose: 00037 * ============= 00038 *> 00039 *> \verbatim 00040 *> 00041 *> SPPSVX uses the Cholesky factorization A = U**T*U or A = L*L**T to 00042 *> compute the solution to a real system of linear equations 00043 *> A * X = B, 00044 *> where A is an N-by-N symmetric positive definite matrix stored in 00045 *> packed format and X and B are N-by-NRHS matrices. 00046 *> 00047 *> Error bounds on the solution and a condition estimate are also 00048 *> provided. 00049 *> \endverbatim 00050 * 00051 *> \par Description: 00052 * ================= 00053 *> 00054 *> \verbatim 00055 *> 00056 *> The following steps are performed: 00057 *> 00058 *> 1. If FACT = 'E', real scaling factors are computed to equilibrate 00059 *> the system: 00060 *> diag(S) * A * diag(S) * inv(diag(S)) * X = diag(S) * B 00061 *> Whether or not the system will be equilibrated depends on the 00062 *> scaling of the matrix A, but if equilibration is used, A is 00063 *> overwritten by diag(S)*A*diag(S) and B by diag(S)*B. 00064 *> 00065 *> 2. If FACT = 'N' or 'E', the Cholesky decomposition is used to 00066 *> factor the matrix A (after equilibration if FACT = 'E') as 00067 *> A = U**T* U, if UPLO = 'U', or 00068 *> A = L * L**T, if UPLO = 'L', 00069 *> where U is an upper triangular matrix and L is a lower triangular 00070 *> matrix. 00071 *> 00072 *> 3. If the leading i-by-i principal minor is not positive definite, 00073 *> then the routine returns with INFO = i. Otherwise, the factored 00074 *> form of A is used to estimate the condition number of the matrix 00075 *> A. If the reciprocal of the condition number is less than machine 00076 *> precision, INFO = N+1 is returned as a warning, but the routine 00077 *> still goes on to solve for X and compute error bounds as 00078 *> described below. 00079 *> 00080 *> 4. The system of equations is solved for X using the factored form 00081 *> of A. 00082 *> 00083 *> 5. Iterative refinement is applied to improve the computed solution 00084 *> matrix and calculate error bounds and backward error estimates 00085 *> for it. 00086 *> 00087 *> 6. If equilibration was used, the matrix X is premultiplied by 00088 *> diag(S) so that it solves the original system before 00089 *> equilibration. 00090 *> \endverbatim 00091 * 00092 * Arguments: 00093 * ========== 00094 * 00095 *> \param[in] FACT 00096 *> \verbatim 00097 *> FACT is CHARACTER*1 00098 *> Specifies whether or not the factored form of the matrix A is 00099 *> supplied on entry, and if not, whether the matrix A should be 00100 *> equilibrated before it is factored. 00101 *> = 'F': On entry, AFP contains the factored form of A. 00102 *> If EQUED = 'Y', the matrix A has been equilibrated 00103 *> with scaling factors given by S. AP and AFP will not 00104 *> be modified. 00105 *> = 'N': The matrix A will be copied to AFP and factored. 00106 *> = 'E': The matrix A will be equilibrated if necessary, then 00107 *> copied to AFP and factored. 00108 *> \endverbatim 00109 *> 00110 *> \param[in] UPLO 00111 *> \verbatim 00112 *> UPLO is CHARACTER*1 00113 *> = 'U': Upper triangle of A is stored; 00114 *> = 'L': Lower triangle of A is stored. 00115 *> \endverbatim 00116 *> 00117 *> \param[in] N 00118 *> \verbatim 00119 *> N is INTEGER 00120 *> The number of linear equations, i.e., the order of the 00121 *> matrix A. N >= 0. 00122 *> \endverbatim 00123 *> 00124 *> \param[in] NRHS 00125 *> \verbatim 00126 *> NRHS is INTEGER 00127 *> The number of right hand sides, i.e., the number of columns 00128 *> of the matrices B and X. NRHS >= 0. 00129 *> \endverbatim 00130 *> 00131 *> \param[in,out] AP 00132 *> \verbatim 00133 *> AP is REAL array, dimension (N*(N+1)/2) 00134 *> On entry, the upper or lower triangle of the symmetric matrix 00135 *> A, packed columnwise in a linear array, except if FACT = 'F' 00136 *> and EQUED = 'Y', then A must contain the equilibrated matrix 00137 *> diag(S)*A*diag(S). The j-th column of A is stored in the 00138 *> array AP as follows: 00139 *> if UPLO = 'U', AP(i + (j-1)*j/2) = A(i,j) for 1<=i<=j; 00140 *> if UPLO = 'L', AP(i + (j-1)*(2n-j)/2) = A(i,j) for j<=i<=n. 00141 *> See below for further details. A is not modified if 00142 *> FACT = 'F' or 'N', or if FACT = 'E' and EQUED = 'N' on exit. 00143 *> 00144 *> On exit, if FACT = 'E' and EQUED = 'Y', A is overwritten by 00145 *> diag(S)*A*diag(S). 00146 *> \endverbatim 00147 *> 00148 *> \param[in,out] AFP 00149 *> \verbatim 00150 *> AFP is REAL array, dimension 00151 *> (N*(N+1)/2) 00152 *> If FACT = 'F', then AFP is an input argument and on entry 00153 *> contains the triangular factor U or L from the Cholesky 00154 *> factorization A = U**T*U or A = L*L**T, in the same storage 00155 *> format as A. If EQUED .ne. 'N', then AFP is the factored 00156 *> form of the equilibrated matrix A. 00157 *> 00158 *> If FACT = 'N', then AFP is an output argument and on exit 00159 *> returns the triangular factor U or L from the Cholesky 00160 *> factorization A = U**T * U or A = L * L**T of the original 00161 *> matrix A. 00162 *> 00163 *> If FACT = 'E', then AFP is an output argument and on exit 00164 *> returns the triangular factor U or L from the Cholesky 00165 *> factorization A = U**T * U or A = L * L**T of the equilibrated 00166 *> matrix A (see the description of AP for the form of the 00167 *> equilibrated matrix). 00168 *> \endverbatim 00169 *> 00170 *> \param[in,out] EQUED 00171 *> \verbatim 00172 *> EQUED is CHARACTER*1 00173 *> Specifies the form of equilibration that was done. 00174 *> = 'N': No equilibration (always true if FACT = 'N'). 00175 *> = 'Y': Equilibration was done, i.e., A has been replaced by 00176 *> diag(S) * A * diag(S). 00177 *> EQUED is an input argument if FACT = 'F'; otherwise, it is an 00178 *> output argument. 00179 *> \endverbatim 00180 *> 00181 *> \param[in,out] S 00182 *> \verbatim 00183 *> S is REAL array, dimension (N) 00184 *> The scale factors for A; not accessed if EQUED = 'N'. S is 00185 *> an input argument if FACT = 'F'; otherwise, S is an output 00186 *> argument. If FACT = 'F' and EQUED = 'Y', each element of S 00187 *> must be positive. 00188 *> \endverbatim 00189 *> 00190 *> \param[in,out] B 00191 *> \verbatim 00192 *> B is REAL array, dimension (LDB,NRHS) 00193 *> On entry, the N-by-NRHS right hand side matrix B. 00194 *> On exit, if EQUED = 'N', B is not modified; if EQUED = 'Y', 00195 *> B is overwritten by diag(S) * B. 00196 *> \endverbatim 00197 *> 00198 *> \param[in] LDB 00199 *> \verbatim 00200 *> LDB is INTEGER 00201 *> The leading dimension of the array B. LDB >= max(1,N). 00202 *> \endverbatim 00203 *> 00204 *> \param[out] X 00205 *> \verbatim 00206 *> X is REAL array, dimension (LDX,NRHS) 00207 *> If INFO = 0 or INFO = N+1, the N-by-NRHS solution matrix X to 00208 *> the original system of equations. Note that if EQUED = 'Y', 00209 *> A and B are modified on exit, and the solution to the 00210 *> equilibrated system is inv(diag(S))*X. 00211 *> \endverbatim 00212 *> 00213 *> \param[in] LDX 00214 *> \verbatim 00215 *> LDX is INTEGER 00216 *> The leading dimension of the array X. LDX >= max(1,N). 00217 *> \endverbatim 00218 *> 00219 *> \param[out] RCOND 00220 *> \verbatim 00221 *> RCOND is REAL 00222 *> The estimate of the reciprocal condition number of the matrix 00223 *> A after equilibration (if done). If RCOND is less than the 00224 *> machine precision (in particular, if RCOND = 0), the matrix 00225 *> is singular to working precision. This condition is 00226 *> indicated by a return code of INFO > 0. 00227 *> \endverbatim 00228 *> 00229 *> \param[out] FERR 00230 *> \verbatim 00231 *> FERR is REAL array, dimension (NRHS) 00232 *> The estimated forward error bound for each solution vector 00233 *> X(j) (the j-th column of the solution matrix X). 00234 *> If XTRUE is the true solution corresponding to X(j), FERR(j) 00235 *> is an estimated upper bound for the magnitude of the largest 00236 *> element in (X(j) - XTRUE) divided by the magnitude of the 00237 *> largest element in X(j). The estimate is as reliable as 00238 *> the estimate for RCOND, and is almost always a slight 00239 *> overestimate of the true error. 00240 *> \endverbatim 00241 *> 00242 *> \param[out] BERR 00243 *> \verbatim 00244 *> BERR is REAL array, dimension (NRHS) 00245 *> The componentwise relative backward error of each solution 00246 *> vector X(j) (i.e., the smallest relative change in 00247 *> any element of A or B that makes X(j) an exact solution). 00248 *> \endverbatim 00249 *> 00250 *> \param[out] WORK 00251 *> \verbatim 00252 *> WORK is REAL array, dimension (3*N) 00253 *> \endverbatim 00254 *> 00255 *> \param[out] IWORK 00256 *> \verbatim 00257 *> IWORK is INTEGER array, dimension (N) 00258 *> \endverbatim 00259 *> 00260 *> \param[out] INFO 00261 *> \verbatim 00262 *> INFO is INTEGER 00263 *> = 0: successful exit 00264 *> < 0: if INFO = -i, the i-th argument had an illegal value 00265 *> > 0: if INFO = i, and i is 00266 *> <= N: the leading minor of order i of A is 00267 *> not positive definite, so the factorization 00268 *> could not be completed, and the solution has not 00269 *> been computed. RCOND = 0 is returned. 00270 *> = N+1: U is nonsingular, but RCOND is less than machine 00271 *> precision, meaning that the matrix is singular 00272 *> to working precision. Nevertheless, the 00273 *> solution and error bounds are computed because 00274 *> there are a number of situations where the 00275 *> computed solution can be more accurate than the 00276 *> value of RCOND would suggest. 00277 *> \endverbatim 00278 * 00279 * Authors: 00280 * ======== 00281 * 00282 *> \author Univ. of Tennessee 00283 *> \author Univ. of California Berkeley 00284 *> \author Univ. of Colorado Denver 00285 *> \author NAG Ltd. 00286 * 00287 *> \date April 2012 00288 * 00289 *> \ingroup realOTHERsolve 00290 * 00291 *> \par Further Details: 00292 * ===================== 00293 *> 00294 *> \verbatim 00295 *> 00296 *> The packed storage scheme is illustrated by the following example 00297 *> when N = 4, UPLO = 'U': 00298 *> 00299 *> Two-dimensional storage of the symmetric matrix A: 00300 *> 00301 *> a11 a12 a13 a14 00302 *> a22 a23 a24 00303 *> a33 a34 (aij = conjg(aji)) 00304 *> a44 00305 *> 00306 *> Packed storage of the upper triangle of A: 00307 *> 00308 *> AP = [ a11, a12, a22, a13, a23, a33, a14, a24, a34, a44 ] 00309 *> \endverbatim 00310 *> 00311 * ===================================================================== 00312 SUBROUTINE SPPSVX( FACT, UPLO, N, NRHS, AP, AFP, EQUED, S, B, LDB, 00313 $ X, LDX, RCOND, FERR, BERR, WORK, IWORK, INFO ) 00314 * 00315 * -- LAPACK driver routine (version 3.4.1) -- 00316 * -- LAPACK is a software package provided by Univ. of Tennessee, -- 00317 * -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..-- 00318 * April 2012 00319 * 00320 * .. Scalar Arguments .. 00321 CHARACTER EQUED, FACT, UPLO 00322 INTEGER INFO, LDB, LDX, N, NRHS 00323 REAL RCOND 00324 * .. 00325 * .. Array Arguments .. 00326 INTEGER IWORK( * ) 00327 REAL AFP( * ), AP( * ), B( LDB, * ), BERR( * ), 00328 $ FERR( * ), S( * ), WORK( * ), X( LDX, * ) 00329 * .. 00330 * 00331 * ===================================================================== 00332 * 00333 * .. Parameters .. 00334 REAL ZERO, ONE 00335 PARAMETER ( ZERO = 0.0E+0, ONE = 1.0E+0 ) 00336 * .. 00337 * .. Local Scalars .. 00338 LOGICAL EQUIL, NOFACT, RCEQU 00339 INTEGER I, INFEQU, J 00340 REAL AMAX, ANORM, BIGNUM, SCOND, SMAX, SMIN, SMLNUM 00341 * .. 00342 * .. External Functions .. 00343 LOGICAL LSAME 00344 REAL SLAMCH, SLANSP 00345 EXTERNAL LSAME, SLAMCH, SLANSP 00346 * .. 00347 * .. External Subroutines .. 00348 EXTERNAL SCOPY, SLACPY, SLAQSP, SPPCON, SPPEQU, SPPRFS, 00349 $ SPPTRF, SPPTRS, XERBLA 00350 * .. 00351 * .. Intrinsic Functions .. 00352 INTRINSIC MAX, MIN 00353 * .. 00354 * .. Executable Statements .. 00355 * 00356 INFO = 0 00357 NOFACT = LSAME( FACT, 'N' ) 00358 EQUIL = LSAME( FACT, 'E' ) 00359 IF( NOFACT .OR. EQUIL ) THEN 00360 EQUED = 'N' 00361 RCEQU = .FALSE. 00362 ELSE 00363 RCEQU = LSAME( EQUED, 'Y' ) 00364 SMLNUM = SLAMCH( 'Safe minimum' ) 00365 BIGNUM = ONE / SMLNUM 00366 END IF 00367 * 00368 * Test the input parameters. 00369 * 00370 IF( .NOT.NOFACT .AND. .NOT.EQUIL .AND. .NOT.LSAME( FACT, 'F' ) ) 00371 $ THEN 00372 INFO = -1 00373 ELSE IF( .NOT.LSAME( UPLO, 'U' ) .AND. .NOT.LSAME( UPLO, 'L' ) ) 00374 $ THEN 00375 INFO = -2 00376 ELSE IF( N.LT.0 ) THEN 00377 INFO = -3 00378 ELSE IF( NRHS.LT.0 ) THEN 00379 INFO = -4 00380 ELSE IF( LSAME( FACT, 'F' ) .AND. .NOT. 00381 $ ( RCEQU .OR. LSAME( EQUED, 'N' ) ) ) THEN 00382 INFO = -7 00383 ELSE 00384 IF( RCEQU ) THEN 00385 SMIN = BIGNUM 00386 SMAX = ZERO 00387 DO 10 J = 1, N 00388 SMIN = MIN( SMIN, S( J ) ) 00389 SMAX = MAX( SMAX, S( J ) ) 00390 10 CONTINUE 00391 IF( SMIN.LE.ZERO ) THEN 00392 INFO = -8 00393 ELSE IF( N.GT.0 ) THEN 00394 SCOND = MAX( SMIN, SMLNUM ) / MIN( SMAX, BIGNUM ) 00395 ELSE 00396 SCOND = ONE 00397 END IF 00398 END IF 00399 IF( INFO.EQ.0 ) THEN 00400 IF( LDB.LT.MAX( 1, N ) ) THEN 00401 INFO = -10 00402 ELSE IF( LDX.LT.MAX( 1, N ) ) THEN 00403 INFO = -12 00404 END IF 00405 END IF 00406 END IF 00407 * 00408 IF( INFO.NE.0 ) THEN 00409 CALL XERBLA( 'SPPSVX', -INFO ) 00410 RETURN 00411 END IF 00412 * 00413 IF( EQUIL ) THEN 00414 * 00415 * Compute row and column scalings to equilibrate the matrix A. 00416 * 00417 CALL SPPEQU( UPLO, N, AP, S, SCOND, AMAX, INFEQU ) 00418 IF( INFEQU.EQ.0 ) THEN 00419 * 00420 * Equilibrate the matrix. 00421 * 00422 CALL SLAQSP( UPLO, N, AP, S, SCOND, AMAX, EQUED ) 00423 RCEQU = LSAME( EQUED, 'Y' ) 00424 END IF 00425 END IF 00426 * 00427 * Scale the right-hand side. 00428 * 00429 IF( RCEQU ) THEN 00430 DO 30 J = 1, NRHS 00431 DO 20 I = 1, N 00432 B( I, J ) = S( I )*B( I, J ) 00433 20 CONTINUE 00434 30 CONTINUE 00435 END IF 00436 * 00437 IF( NOFACT .OR. EQUIL ) THEN 00438 * 00439 * Compute the Cholesky factorization A = U**T * U or A = L * L**T. 00440 * 00441 CALL SCOPY( N*( N+1 ) / 2, AP, 1, AFP, 1 ) 00442 CALL SPPTRF( UPLO, N, AFP, INFO ) 00443 * 00444 * Return if INFO is non-zero. 00445 * 00446 IF( INFO.GT.0 )THEN 00447 RCOND = ZERO 00448 RETURN 00449 END IF 00450 END IF 00451 * 00452 * Compute the norm of the matrix A. 00453 * 00454 ANORM = SLANSP( 'I', UPLO, N, AP, WORK ) 00455 * 00456 * Compute the reciprocal of the condition number of A. 00457 * 00458 CALL SPPCON( UPLO, N, AFP, ANORM, RCOND, WORK, IWORK, INFO ) 00459 * 00460 * Compute the solution matrix X. 00461 * 00462 CALL SLACPY( 'Full', N, NRHS, B, LDB, X, LDX ) 00463 CALL SPPTRS( UPLO, N, NRHS, AFP, X, LDX, INFO ) 00464 * 00465 * Use iterative refinement to improve the computed solution and 00466 * compute error bounds and backward error estimates for it. 00467 * 00468 CALL SPPRFS( UPLO, N, NRHS, AP, AFP, B, LDB, X, LDX, FERR, BERR, 00469 $ WORK, IWORK, INFO ) 00470 * 00471 * Transform the solution matrix X to a solution of the original 00472 * system. 00473 * 00474 IF( RCEQU ) THEN 00475 DO 50 J = 1, NRHS 00476 DO 40 I = 1, N 00477 X( I, J ) = S( I )*X( I, J ) 00478 40 CONTINUE 00479 50 CONTINUE 00480 DO 60 J = 1, NRHS 00481 FERR( J ) = FERR( J ) / SCOND 00482 60 CONTINUE 00483 END IF 00484 * 00485 * Set INFO = N+1 if the matrix is singular to working precision. 00486 * 00487 IF( RCOND.LT.SLAMCH( 'Epsilon' ) ) 00488 $ INFO = N + 1 00489 * 00490 RETURN 00491 * 00492 * End of SPPSVX 00493 * 00494 END