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LAPACK
3.4.1
LAPACK: Linear Algebra PACKage
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00001 *> \brief \b DSYRFS 00002 * 00003 * =========== DOCUMENTATION =========== 00004 * 00005 * Online html documentation available at 00006 * http://www.netlib.org/lapack/explore-html/ 00007 * 00008 *> \htmlonly 00009 *> Download DSYRFS + dependencies 00010 *> <a href="http://www.netlib.org/cgi-bin/netlibfiles.tgz?format=tgz&filename=/lapack/lapack_routine/dsyrfs.f"> 00011 *> [TGZ]</a> 00012 *> <a href="http://www.netlib.org/cgi-bin/netlibfiles.zip?format=zip&filename=/lapack/lapack_routine/dsyrfs.f"> 00013 *> [ZIP]</a> 00014 *> <a href="http://www.netlib.org/cgi-bin/netlibfiles.txt?format=txt&filename=/lapack/lapack_routine/dsyrfs.f"> 00015 *> [TXT]</a> 00016 *> \endhtmlonly 00017 * 00018 * Definition: 00019 * =========== 00020 * 00021 * SUBROUTINE DSYRFS( UPLO, N, NRHS, A, LDA, AF, LDAF, IPIV, B, LDB, 00022 * X, LDX, FERR, BERR, WORK, IWORK, INFO ) 00023 * 00024 * .. Scalar Arguments .. 00025 * CHARACTER UPLO 00026 * INTEGER INFO, LDA, LDAF, LDB, LDX, N, NRHS 00027 * .. 00028 * .. Array Arguments .. 00029 * INTEGER IPIV( * ), IWORK( * ) 00030 * DOUBLE PRECISION A( LDA, * ), AF( LDAF, * ), B( LDB, * ), 00031 * $ BERR( * ), FERR( * ), WORK( * ), X( LDX, * ) 00032 * .. 00033 * 00034 * 00035 *> \par Purpose: 00036 * ============= 00037 *> 00038 *> \verbatim 00039 *> 00040 *> DSYRFS improves the computed solution to a system of linear 00041 *> equations when the coefficient matrix is symmetric indefinite, and 00042 *> provides error bounds and backward error estimates for the solution. 00043 *> \endverbatim 00044 * 00045 * Arguments: 00046 * ========== 00047 * 00048 *> \param[in] UPLO 00049 *> \verbatim 00050 *> UPLO is CHARACTER*1 00051 *> = 'U': Upper triangle of A is stored; 00052 *> = 'L': Lower triangle of A is stored. 00053 *> \endverbatim 00054 *> 00055 *> \param[in] N 00056 *> \verbatim 00057 *> N is INTEGER 00058 *> The order of the matrix A. N >= 0. 00059 *> \endverbatim 00060 *> 00061 *> \param[in] NRHS 00062 *> \verbatim 00063 *> NRHS is INTEGER 00064 *> The number of right hand sides, i.e., the number of columns 00065 *> of the matrices B and X. NRHS >= 0. 00066 *> \endverbatim 00067 *> 00068 *> \param[in] A 00069 *> \verbatim 00070 *> A is DOUBLE PRECISION array, dimension (LDA,N) 00071 *> The symmetric matrix A. If UPLO = 'U', the leading N-by-N 00072 *> upper triangular part of A contains the upper triangular part 00073 *> of the matrix A, and the strictly lower triangular part of A 00074 *> is not referenced. If UPLO = 'L', the leading N-by-N lower 00075 *> triangular part of A contains the lower triangular part of 00076 *> the matrix A, and the strictly upper triangular part of A is 00077 *> not referenced. 00078 *> \endverbatim 00079 *> 00080 *> \param[in] LDA 00081 *> \verbatim 00082 *> LDA is INTEGER 00083 *> The leading dimension of the array A. LDA >= max(1,N). 00084 *> \endverbatim 00085 *> 00086 *> \param[in] AF 00087 *> \verbatim 00088 *> AF is DOUBLE PRECISION array, dimension (LDAF,N) 00089 *> The factored form of the matrix A. AF contains the block 00090 *> diagonal matrix D and the multipliers used to obtain the 00091 *> factor U or L from the factorization A = U*D*U**T or 00092 *> A = L*D*L**T as computed by DSYTRF. 00093 *> \endverbatim 00094 *> 00095 *> \param[in] LDAF 00096 *> \verbatim 00097 *> LDAF is INTEGER 00098 *> The leading dimension of the array AF. LDAF >= max(1,N). 00099 *> \endverbatim 00100 *> 00101 *> \param[in] IPIV 00102 *> \verbatim 00103 *> IPIV is INTEGER array, dimension (N) 00104 *> Details of the interchanges and the block structure of D 00105 *> as determined by DSYTRF. 00106 *> \endverbatim 00107 *> 00108 *> \param[in] B 00109 *> \verbatim 00110 *> B is DOUBLE PRECISION array, dimension (LDB,NRHS) 00111 *> The right hand side matrix B. 00112 *> \endverbatim 00113 *> 00114 *> \param[in] LDB 00115 *> \verbatim 00116 *> LDB is INTEGER 00117 *> The leading dimension of the array B. LDB >= max(1,N). 00118 *> \endverbatim 00119 *> 00120 *> \param[in,out] X 00121 *> \verbatim 00122 *> X is DOUBLE PRECISION array, dimension (LDX,NRHS) 00123 *> On entry, the solution matrix X, as computed by DSYTRS. 00124 *> On exit, the improved solution matrix X. 00125 *> \endverbatim 00126 *> 00127 *> \param[in] LDX 00128 *> \verbatim 00129 *> LDX is INTEGER 00130 *> The leading dimension of the array X. LDX >= max(1,N). 00131 *> \endverbatim 00132 *> 00133 *> \param[out] FERR 00134 *> \verbatim 00135 *> FERR is DOUBLE PRECISION array, dimension (NRHS) 00136 *> The estimated forward error bound for each solution vector 00137 *> X(j) (the j-th column of the solution matrix X). 00138 *> If XTRUE is the true solution corresponding to X(j), FERR(j) 00139 *> is an estimated upper bound for the magnitude of the largest 00140 *> element in (X(j) - XTRUE) divided by the magnitude of the 00141 *> largest element in X(j). The estimate is as reliable as 00142 *> the estimate for RCOND, and is almost always a slight 00143 *> overestimate of the true error. 00144 *> \endverbatim 00145 *> 00146 *> \param[out] BERR 00147 *> \verbatim 00148 *> BERR is DOUBLE PRECISION array, dimension (NRHS) 00149 *> The componentwise relative backward error of each solution 00150 *> vector X(j) (i.e., the smallest relative change in 00151 *> any element of A or B that makes X(j) an exact solution). 00152 *> \endverbatim 00153 *> 00154 *> \param[out] WORK 00155 *> \verbatim 00156 *> WORK is DOUBLE PRECISION array, dimension (3*N) 00157 *> \endverbatim 00158 *> 00159 *> \param[out] IWORK 00160 *> \verbatim 00161 *> IWORK is INTEGER array, dimension (N) 00162 *> \endverbatim 00163 *> 00164 *> \param[out] INFO 00165 *> \verbatim 00166 *> INFO is INTEGER 00167 *> = 0: successful exit 00168 *> < 0: if INFO = -i, the i-th argument had an illegal value 00169 *> \endverbatim 00170 * 00171 *> \par Internal Parameters: 00172 * ========================= 00173 *> 00174 *> \verbatim 00175 *> ITMAX is the maximum number of steps of iterative refinement. 00176 *> \endverbatim 00177 * 00178 * Authors: 00179 * ======== 00180 * 00181 *> \author Univ. of Tennessee 00182 *> \author Univ. of California Berkeley 00183 *> \author Univ. of Colorado Denver 00184 *> \author NAG Ltd. 00185 * 00186 *> \date November 2011 00187 * 00188 *> \ingroup doubleSYcomputational 00189 * 00190 * ===================================================================== 00191 SUBROUTINE DSYRFS( UPLO, N, NRHS, A, LDA, AF, LDAF, IPIV, B, LDB, 00192 $ X, LDX, FERR, BERR, WORK, IWORK, INFO ) 00193 * 00194 * -- LAPACK computational routine (version 3.4.0) -- 00195 * -- LAPACK is a software package provided by Univ. of Tennessee, -- 00196 * -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..-- 00197 * November 2011 00198 * 00199 * .. Scalar Arguments .. 00200 CHARACTER UPLO 00201 INTEGER INFO, LDA, LDAF, LDB, LDX, N, NRHS 00202 * .. 00203 * .. Array Arguments .. 00204 INTEGER IPIV( * ), IWORK( * ) 00205 DOUBLE PRECISION A( LDA, * ), AF( LDAF, * ), B( LDB, * ), 00206 $ BERR( * ), FERR( * ), WORK( * ), X( LDX, * ) 00207 * .. 00208 * 00209 * ===================================================================== 00210 * 00211 * .. Parameters .. 00212 INTEGER ITMAX 00213 PARAMETER ( ITMAX = 5 ) 00214 DOUBLE PRECISION ZERO 00215 PARAMETER ( ZERO = 0.0D+0 ) 00216 DOUBLE PRECISION ONE 00217 PARAMETER ( ONE = 1.0D+0 ) 00218 DOUBLE PRECISION TWO 00219 PARAMETER ( TWO = 2.0D+0 ) 00220 DOUBLE PRECISION THREE 00221 PARAMETER ( THREE = 3.0D+0 ) 00222 * .. 00223 * .. Local Scalars .. 00224 LOGICAL UPPER 00225 INTEGER COUNT, I, J, K, KASE, NZ 00226 DOUBLE PRECISION EPS, LSTRES, S, SAFE1, SAFE2, SAFMIN, XK 00227 * .. 00228 * .. Local Arrays .. 00229 INTEGER ISAVE( 3 ) 00230 * .. 00231 * .. External Subroutines .. 00232 EXTERNAL DAXPY, DCOPY, DLACN2, DSYMV, DSYTRS, XERBLA 00233 * .. 00234 * .. Intrinsic Functions .. 00235 INTRINSIC ABS, MAX 00236 * .. 00237 * .. External Functions .. 00238 LOGICAL LSAME 00239 DOUBLE PRECISION DLAMCH 00240 EXTERNAL LSAME, DLAMCH 00241 * .. 00242 * .. Executable Statements .. 00243 * 00244 * Test the input parameters. 00245 * 00246 INFO = 0 00247 UPPER = LSAME( UPLO, 'U' ) 00248 IF( .NOT.UPPER .AND. .NOT.LSAME( UPLO, 'L' ) ) THEN 00249 INFO = -1 00250 ELSE IF( N.LT.0 ) THEN 00251 INFO = -2 00252 ELSE IF( NRHS.LT.0 ) THEN 00253 INFO = -3 00254 ELSE IF( LDA.LT.MAX( 1, N ) ) THEN 00255 INFO = -5 00256 ELSE IF( LDAF.LT.MAX( 1, N ) ) THEN 00257 INFO = -7 00258 ELSE IF( LDB.LT.MAX( 1, N ) ) THEN 00259 INFO = -10 00260 ELSE IF( LDX.LT.MAX( 1, N ) ) THEN 00261 INFO = -12 00262 END IF 00263 IF( INFO.NE.0 ) THEN 00264 CALL XERBLA( 'DSYRFS', -INFO ) 00265 RETURN 00266 END IF 00267 * 00268 * Quick return if possible 00269 * 00270 IF( N.EQ.0 .OR. NRHS.EQ.0 ) THEN 00271 DO 10 J = 1, NRHS 00272 FERR( J ) = ZERO 00273 BERR( J ) = ZERO 00274 10 CONTINUE 00275 RETURN 00276 END IF 00277 * 00278 * NZ = maximum number of nonzero elements in each row of A, plus 1 00279 * 00280 NZ = N + 1 00281 EPS = DLAMCH( 'Epsilon' ) 00282 SAFMIN = DLAMCH( 'Safe minimum' ) 00283 SAFE1 = NZ*SAFMIN 00284 SAFE2 = SAFE1 / EPS 00285 * 00286 * Do for each right hand side 00287 * 00288 DO 140 J = 1, NRHS 00289 * 00290 COUNT = 1 00291 LSTRES = THREE 00292 20 CONTINUE 00293 * 00294 * Loop until stopping criterion is satisfied. 00295 * 00296 * Compute residual R = B - A * X 00297 * 00298 CALL DCOPY( N, B( 1, J ), 1, WORK( N+1 ), 1 ) 00299 CALL DSYMV( UPLO, N, -ONE, A, LDA, X( 1, J ), 1, ONE, 00300 $ WORK( N+1 ), 1 ) 00301 * 00302 * Compute componentwise relative backward error from formula 00303 * 00304 * max(i) ( abs(R(i)) / ( abs(A)*abs(X) + abs(B) )(i) ) 00305 * 00306 * where abs(Z) is the componentwise absolute value of the matrix 00307 * or vector Z. If the i-th component of the denominator is less 00308 * than SAFE2, then SAFE1 is added to the i-th components of the 00309 * numerator and denominator before dividing. 00310 * 00311 DO 30 I = 1, N 00312 WORK( I ) = ABS( B( I, J ) ) 00313 30 CONTINUE 00314 * 00315 * Compute abs(A)*abs(X) + abs(B). 00316 * 00317 IF( UPPER ) THEN 00318 DO 50 K = 1, N 00319 S = ZERO 00320 XK = ABS( X( K, J ) ) 00321 DO 40 I = 1, K - 1 00322 WORK( I ) = WORK( I ) + ABS( A( I, K ) )*XK 00323 S = S + ABS( A( I, K ) )*ABS( X( I, J ) ) 00324 40 CONTINUE 00325 WORK( K ) = WORK( K ) + ABS( A( K, K ) )*XK + S 00326 50 CONTINUE 00327 ELSE 00328 DO 70 K = 1, N 00329 S = ZERO 00330 XK = ABS( X( K, J ) ) 00331 WORK( K ) = WORK( K ) + ABS( A( K, K ) )*XK 00332 DO 60 I = K + 1, N 00333 WORK( I ) = WORK( I ) + ABS( A( I, K ) )*XK 00334 S = S + ABS( A( I, K ) )*ABS( X( I, J ) ) 00335 60 CONTINUE 00336 WORK( K ) = WORK( K ) + S 00337 70 CONTINUE 00338 END IF 00339 S = ZERO 00340 DO 80 I = 1, N 00341 IF( WORK( I ).GT.SAFE2 ) THEN 00342 S = MAX( S, ABS( WORK( N+I ) ) / WORK( I ) ) 00343 ELSE 00344 S = MAX( S, ( ABS( WORK( N+I ) )+SAFE1 ) / 00345 $ ( WORK( I )+SAFE1 ) ) 00346 END IF 00347 80 CONTINUE 00348 BERR( J ) = S 00349 * 00350 * Test stopping criterion. Continue iterating if 00351 * 1) The residual BERR(J) is larger than machine epsilon, and 00352 * 2) BERR(J) decreased by at least a factor of 2 during the 00353 * last iteration, and 00354 * 3) At most ITMAX iterations tried. 00355 * 00356 IF( BERR( J ).GT.EPS .AND. TWO*BERR( J ).LE.LSTRES .AND. 00357 $ COUNT.LE.ITMAX ) THEN 00358 * 00359 * Update solution and try again. 00360 * 00361 CALL DSYTRS( UPLO, N, 1, AF, LDAF, IPIV, WORK( N+1 ), N, 00362 $ INFO ) 00363 CALL DAXPY( N, ONE, WORK( N+1 ), 1, X( 1, J ), 1 ) 00364 LSTRES = BERR( J ) 00365 COUNT = COUNT + 1 00366 GO TO 20 00367 END IF 00368 * 00369 * Bound error from formula 00370 * 00371 * norm(X - XTRUE) / norm(X) .le. FERR = 00372 * norm( abs(inv(A))* 00373 * ( abs(R) + NZ*EPS*( abs(A)*abs(X)+abs(B) ))) / norm(X) 00374 * 00375 * where 00376 * norm(Z) is the magnitude of the largest component of Z 00377 * inv(A) is the inverse of A 00378 * abs(Z) is the componentwise absolute value of the matrix or 00379 * vector Z 00380 * NZ is the maximum number of nonzeros in any row of A, plus 1 00381 * EPS is machine epsilon 00382 * 00383 * The i-th component of abs(R)+NZ*EPS*(abs(A)*abs(X)+abs(B)) 00384 * is incremented by SAFE1 if the i-th component of 00385 * abs(A)*abs(X) + abs(B) is less than SAFE2. 00386 * 00387 * Use DLACN2 to estimate the infinity-norm of the matrix 00388 * inv(A) * diag(W), 00389 * where W = abs(R) + NZ*EPS*( abs(A)*abs(X)+abs(B) ))) 00390 * 00391 DO 90 I = 1, N 00392 IF( WORK( I ).GT.SAFE2 ) THEN 00393 WORK( I ) = ABS( WORK( N+I ) ) + NZ*EPS*WORK( I ) 00394 ELSE 00395 WORK( I ) = ABS( WORK( N+I ) ) + NZ*EPS*WORK( I ) + SAFE1 00396 END IF 00397 90 CONTINUE 00398 * 00399 KASE = 0 00400 100 CONTINUE 00401 CALL DLACN2( N, WORK( 2*N+1 ), WORK( N+1 ), IWORK, FERR( J ), 00402 $ KASE, ISAVE ) 00403 IF( KASE.NE.0 ) THEN 00404 IF( KASE.EQ.1 ) THEN 00405 * 00406 * Multiply by diag(W)*inv(A**T). 00407 * 00408 CALL DSYTRS( UPLO, N, 1, AF, LDAF, IPIV, WORK( N+1 ), N, 00409 $ INFO ) 00410 DO 110 I = 1, N 00411 WORK( N+I ) = WORK( I )*WORK( N+I ) 00412 110 CONTINUE 00413 ELSE IF( KASE.EQ.2 ) THEN 00414 * 00415 * Multiply by inv(A)*diag(W). 00416 * 00417 DO 120 I = 1, N 00418 WORK( N+I ) = WORK( I )*WORK( N+I ) 00419 120 CONTINUE 00420 CALL DSYTRS( UPLO, N, 1, AF, LDAF, IPIV, WORK( N+1 ), N, 00421 $ INFO ) 00422 END IF 00423 GO TO 100 00424 END IF 00425 * 00426 * Normalize error. 00427 * 00428 LSTRES = ZERO 00429 DO 130 I = 1, N 00430 LSTRES = MAX( LSTRES, ABS( X( I, J ) ) ) 00431 130 CONTINUE 00432 IF( LSTRES.NE.ZERO ) 00433 $ FERR( J ) = FERR( J ) / LSTRES 00434 * 00435 140 CONTINUE 00436 * 00437 RETURN 00438 * 00439 * End of DSYRFS 00440 * 00441 END