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