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LAPACK
3.4.1
LAPACK: Linear Algebra PACKage
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00001 *> \brief \b ZGERFS 00002 * 00003 * =========== DOCUMENTATION =========== 00004 * 00005 * Online html documentation available at 00006 * http://www.netlib.org/lapack/explore-html/ 00007 * 00008 *> \htmlonly 00009 *> Download ZGERFS + dependencies 00010 *> <a href="http://www.netlib.org/cgi-bin/netlibfiles.tgz?format=tgz&filename=/lapack/lapack_routine/zgerfs.f"> 00011 *> [TGZ]</a> 00012 *> <a href="http://www.netlib.org/cgi-bin/netlibfiles.zip?format=zip&filename=/lapack/lapack_routine/zgerfs.f"> 00013 *> [ZIP]</a> 00014 *> <a href="http://www.netlib.org/cgi-bin/netlibfiles.txt?format=txt&filename=/lapack/lapack_routine/zgerfs.f"> 00015 *> [TXT]</a> 00016 *> \endhtmlonly 00017 * 00018 * Definition: 00019 * =========== 00020 * 00021 * SUBROUTINE ZGERFS( TRANS, N, NRHS, A, LDA, AF, LDAF, IPIV, B, LDB, 00022 * X, LDX, FERR, BERR, WORK, RWORK, INFO ) 00023 * 00024 * .. Scalar Arguments .. 00025 * CHARACTER TRANS 00026 * INTEGER INFO, LDA, LDAF, LDB, LDX, N, NRHS 00027 * .. 00028 * .. Array Arguments .. 00029 * INTEGER IPIV( * ) 00030 * DOUBLE PRECISION BERR( * ), FERR( * ), RWORK( * ) 00031 * COMPLEX*16 A( LDA, * ), AF( LDAF, * ), B( LDB, * ), 00032 * $ WORK( * ), X( LDX, * ) 00033 * .. 00034 * 00035 * 00036 *> \par Purpose: 00037 * ============= 00038 *> 00039 *> \verbatim 00040 *> 00041 *> ZGERFS improves the computed solution to a system of linear 00042 *> equations and provides error bounds and backward error estimates for 00043 *> the solution. 00044 *> \endverbatim 00045 * 00046 * Arguments: 00047 * ========== 00048 * 00049 *> \param[in] TRANS 00050 *> \verbatim 00051 *> TRANS is CHARACTER*1 00052 *> Specifies the form of the system of equations: 00053 *> = 'N': A * X = B (No transpose) 00054 *> = 'T': A**T * X = B (Transpose) 00055 *> = 'C': A**H * X = B (Conjugate transpose) 00056 *> \endverbatim 00057 *> 00058 *> \param[in] N 00059 *> \verbatim 00060 *> N is INTEGER 00061 *> The order of the matrix A. N >= 0. 00062 *> \endverbatim 00063 *> 00064 *> \param[in] NRHS 00065 *> \verbatim 00066 *> NRHS is INTEGER 00067 *> The number of right hand sides, i.e., the number of columns 00068 *> of the matrices B and X. NRHS >= 0. 00069 *> \endverbatim 00070 *> 00071 *> \param[in] A 00072 *> \verbatim 00073 *> A is COMPLEX*16 array, dimension (LDA,N) 00074 *> The original N-by-N matrix A. 00075 *> \endverbatim 00076 *> 00077 *> \param[in] LDA 00078 *> \verbatim 00079 *> LDA is INTEGER 00080 *> The leading dimension of the array A. LDA >= max(1,N). 00081 *> \endverbatim 00082 *> 00083 *> \param[in] AF 00084 *> \verbatim 00085 *> AF is COMPLEX*16 array, dimension (LDAF,N) 00086 *> The factors L and U from the factorization A = P*L*U 00087 *> as computed by ZGETRF. 00088 *> \endverbatim 00089 *> 00090 *> \param[in] LDAF 00091 *> \verbatim 00092 *> LDAF is INTEGER 00093 *> The leading dimension of the array AF. LDAF >= max(1,N). 00094 *> \endverbatim 00095 *> 00096 *> \param[in] IPIV 00097 *> \verbatim 00098 *> IPIV is INTEGER array, dimension (N) 00099 *> The pivot indices from ZGETRF; for 1<=i<=N, row i of the 00100 *> matrix was interchanged with row IPIV(i). 00101 *> \endverbatim 00102 *> 00103 *> \param[in] B 00104 *> \verbatim 00105 *> B is COMPLEX*16 array, dimension (LDB,NRHS) 00106 *> The right hand side matrix B. 00107 *> \endverbatim 00108 *> 00109 *> \param[in] LDB 00110 *> \verbatim 00111 *> LDB is INTEGER 00112 *> The leading dimension of the array B. LDB >= max(1,N). 00113 *> \endverbatim 00114 *> 00115 *> \param[in,out] X 00116 *> \verbatim 00117 *> X is COMPLEX*16 array, dimension (LDX,NRHS) 00118 *> On entry, the solution matrix X, as computed by ZGETRS. 00119 *> On exit, the improved solution matrix X. 00120 *> \endverbatim 00121 *> 00122 *> \param[in] LDX 00123 *> \verbatim 00124 *> LDX is INTEGER 00125 *> The leading dimension of the array X. LDX >= max(1,N). 00126 *> \endverbatim 00127 *> 00128 *> \param[out] FERR 00129 *> \verbatim 00130 *> FERR is DOUBLE PRECISION array, dimension (NRHS) 00131 *> The estimated forward error bound for each solution vector 00132 *> X(j) (the j-th column of the solution matrix X). 00133 *> If XTRUE is the true solution corresponding to X(j), FERR(j) 00134 *> is an estimated upper bound for the magnitude of the largest 00135 *> element in (X(j) - XTRUE) divided by the magnitude of the 00136 *> largest element in X(j). The estimate is as reliable as 00137 *> the estimate for RCOND, and is almost always a slight 00138 *> overestimate of the true error. 00139 *> \endverbatim 00140 *> 00141 *> \param[out] BERR 00142 *> \verbatim 00143 *> BERR is DOUBLE PRECISION array, dimension (NRHS) 00144 *> The componentwise relative backward error of each solution 00145 *> vector X(j) (i.e., the smallest relative change in 00146 *> any element of A or B that makes X(j) an exact solution). 00147 *> \endverbatim 00148 *> 00149 *> \param[out] WORK 00150 *> \verbatim 00151 *> WORK is COMPLEX*16 array, dimension (2*N) 00152 *> \endverbatim 00153 *> 00154 *> \param[out] RWORK 00155 *> \verbatim 00156 *> RWORK is DOUBLE PRECISION array, dimension (N) 00157 *> \endverbatim 00158 *> 00159 *> \param[out] INFO 00160 *> \verbatim 00161 *> INFO is INTEGER 00162 *> = 0: successful exit 00163 *> < 0: if INFO = -i, the i-th argument had an illegal value 00164 *> \endverbatim 00165 * 00166 *> \par Internal Parameters: 00167 * ========================= 00168 *> 00169 *> \verbatim 00170 *> ITMAX is the maximum number of steps of iterative refinement. 00171 *> \endverbatim 00172 * 00173 * Authors: 00174 * ======== 00175 * 00176 *> \author Univ. of Tennessee 00177 *> \author Univ. of California Berkeley 00178 *> \author Univ. of Colorado Denver 00179 *> \author NAG Ltd. 00180 * 00181 *> \date November 2011 00182 * 00183 *> \ingroup complex16GEcomputational 00184 * 00185 * ===================================================================== 00186 SUBROUTINE ZGERFS( TRANS, N, NRHS, A, LDA, AF, LDAF, IPIV, B, LDB, 00187 $ X, LDX, FERR, BERR, WORK, RWORK, INFO ) 00188 * 00189 * -- LAPACK computational routine (version 3.4.0) -- 00190 * -- LAPACK is a software package provided by Univ. of Tennessee, -- 00191 * -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..-- 00192 * November 2011 00193 * 00194 * .. Scalar Arguments .. 00195 CHARACTER TRANS 00196 INTEGER INFO, LDA, LDAF, LDB, LDX, N, NRHS 00197 * .. 00198 * .. Array Arguments .. 00199 INTEGER IPIV( * ) 00200 DOUBLE PRECISION BERR( * ), FERR( * ), RWORK( * ) 00201 COMPLEX*16 A( LDA, * ), AF( LDAF, * ), B( LDB, * ), 00202 $ WORK( * ), X( LDX, * ) 00203 * .. 00204 * 00205 * ===================================================================== 00206 * 00207 * .. Parameters .. 00208 INTEGER ITMAX 00209 PARAMETER ( ITMAX = 5 ) 00210 DOUBLE PRECISION ZERO 00211 PARAMETER ( ZERO = 0.0D+0 ) 00212 COMPLEX*16 ONE 00213 PARAMETER ( ONE = ( 1.0D+0, 0.0D+0 ) ) 00214 DOUBLE PRECISION TWO 00215 PARAMETER ( TWO = 2.0D+0 ) 00216 DOUBLE PRECISION THREE 00217 PARAMETER ( THREE = 3.0D+0 ) 00218 * .. 00219 * .. Local Scalars .. 00220 LOGICAL NOTRAN 00221 CHARACTER TRANSN, TRANST 00222 INTEGER COUNT, I, J, K, KASE, NZ 00223 DOUBLE PRECISION EPS, LSTRES, S, SAFE1, SAFE2, SAFMIN, XK 00224 COMPLEX*16 ZDUM 00225 * .. 00226 * .. Local Arrays .. 00227 INTEGER ISAVE( 3 ) 00228 * .. 00229 * .. External Functions .. 00230 LOGICAL LSAME 00231 DOUBLE PRECISION DLAMCH 00232 EXTERNAL LSAME, DLAMCH 00233 * .. 00234 * .. External Subroutines .. 00235 EXTERNAL XERBLA, ZAXPY, ZCOPY, ZGEMV, ZGETRS, ZLACN2 00236 * .. 00237 * .. Intrinsic Functions .. 00238 INTRINSIC ABS, DBLE, DIMAG, MAX 00239 * .. 00240 * .. Statement Functions .. 00241 DOUBLE PRECISION CABS1 00242 * .. 00243 * .. Statement Function definitions .. 00244 CABS1( ZDUM ) = ABS( DBLE( ZDUM ) ) + ABS( DIMAG( ZDUM ) ) 00245 * .. 00246 * .. Executable Statements .. 00247 * 00248 * Test the input parameters. 00249 * 00250 INFO = 0 00251 NOTRAN = LSAME( TRANS, 'N' ) 00252 IF( .NOT.NOTRAN .AND. .NOT.LSAME( TRANS, 'T' ) .AND. .NOT. 00253 $ LSAME( TRANS, 'C' ) ) THEN 00254 INFO = -1 00255 ELSE IF( N.LT.0 ) THEN 00256 INFO = -2 00257 ELSE IF( NRHS.LT.0 ) THEN 00258 INFO = -3 00259 ELSE IF( LDA.LT.MAX( 1, N ) ) THEN 00260 INFO = -5 00261 ELSE IF( LDAF.LT.MAX( 1, N ) ) THEN 00262 INFO = -7 00263 ELSE IF( LDB.LT.MAX( 1, N ) ) THEN 00264 INFO = -10 00265 ELSE IF( LDX.LT.MAX( 1, N ) ) THEN 00266 INFO = -12 00267 END IF 00268 IF( INFO.NE.0 ) THEN 00269 CALL XERBLA( 'ZGERFS', -INFO ) 00270 RETURN 00271 END IF 00272 * 00273 * Quick return if possible 00274 * 00275 IF( N.EQ.0 .OR. NRHS.EQ.0 ) THEN 00276 DO 10 J = 1, NRHS 00277 FERR( J ) = ZERO 00278 BERR( J ) = ZERO 00279 10 CONTINUE 00280 RETURN 00281 END IF 00282 * 00283 IF( NOTRAN ) THEN 00284 TRANSN = 'N' 00285 TRANST = 'C' 00286 ELSE 00287 TRANSN = 'C' 00288 TRANST = 'N' 00289 END IF 00290 * 00291 * NZ = maximum number of nonzero elements in each row of A, plus 1 00292 * 00293 NZ = N + 1 00294 EPS = DLAMCH( 'Epsilon' ) 00295 SAFMIN = DLAMCH( 'Safe minimum' ) 00296 SAFE1 = NZ*SAFMIN 00297 SAFE2 = SAFE1 / EPS 00298 * 00299 * Do for each right hand side 00300 * 00301 DO 140 J = 1, NRHS 00302 * 00303 COUNT = 1 00304 LSTRES = THREE 00305 20 CONTINUE 00306 * 00307 * Loop until stopping criterion is satisfied. 00308 * 00309 * Compute residual R = B - op(A) * X, 00310 * where op(A) = A, A**T, or A**H, depending on TRANS. 00311 * 00312 CALL ZCOPY( N, B( 1, J ), 1, WORK, 1 ) 00313 CALL ZGEMV( TRANS, N, N, -ONE, A, LDA, X( 1, J ), 1, ONE, WORK, 00314 $ 1 ) 00315 * 00316 * Compute componentwise relative backward error from formula 00317 * 00318 * max(i) ( abs(R(i)) / ( abs(op(A))*abs(X) + abs(B) )(i) ) 00319 * 00320 * where abs(Z) is the componentwise absolute value of the matrix 00321 * or vector Z. If the i-th component of the denominator is less 00322 * than SAFE2, then SAFE1 is added to the i-th components of the 00323 * numerator and denominator before dividing. 00324 * 00325 DO 30 I = 1, N 00326 RWORK( I ) = CABS1( B( I, J ) ) 00327 30 CONTINUE 00328 * 00329 * Compute abs(op(A))*abs(X) + abs(B). 00330 * 00331 IF( NOTRAN ) THEN 00332 DO 50 K = 1, N 00333 XK = CABS1( X( K, J ) ) 00334 DO 40 I = 1, N 00335 RWORK( I ) = RWORK( I ) + CABS1( A( I, K ) )*XK 00336 40 CONTINUE 00337 50 CONTINUE 00338 ELSE 00339 DO 70 K = 1, N 00340 S = ZERO 00341 DO 60 I = 1, N 00342 S = S + CABS1( A( I, K ) )*CABS1( X( I, J ) ) 00343 60 CONTINUE 00344 RWORK( K ) = RWORK( K ) + S 00345 70 CONTINUE 00346 END IF 00347 S = ZERO 00348 DO 80 I = 1, N 00349 IF( RWORK( I ).GT.SAFE2 ) THEN 00350 S = MAX( S, CABS1( WORK( I ) ) / RWORK( I ) ) 00351 ELSE 00352 S = MAX( S, ( CABS1( WORK( I ) )+SAFE1 ) / 00353 $ ( RWORK( I )+SAFE1 ) ) 00354 END IF 00355 80 CONTINUE 00356 BERR( J ) = S 00357 * 00358 * Test stopping criterion. Continue iterating if 00359 * 1) The residual BERR(J) is larger than machine epsilon, and 00360 * 2) BERR(J) decreased by at least a factor of 2 during the 00361 * last iteration, and 00362 * 3) At most ITMAX iterations tried. 00363 * 00364 IF( BERR( J ).GT.EPS .AND. TWO*BERR( J ).LE.LSTRES .AND. 00365 $ COUNT.LE.ITMAX ) THEN 00366 * 00367 * Update solution and try again. 00368 * 00369 CALL ZGETRS( TRANS, N, 1, AF, LDAF, IPIV, WORK, N, INFO ) 00370 CALL ZAXPY( N, ONE, WORK, 1, X( 1, J ), 1 ) 00371 LSTRES = BERR( J ) 00372 COUNT = COUNT + 1 00373 GO TO 20 00374 END IF 00375 * 00376 * Bound error from formula 00377 * 00378 * norm(X - XTRUE) / norm(X) .le. FERR = 00379 * norm( abs(inv(op(A)))* 00380 * ( abs(R) + NZ*EPS*( abs(op(A))*abs(X)+abs(B) ))) / norm(X) 00381 * 00382 * where 00383 * norm(Z) is the magnitude of the largest component of Z 00384 * inv(op(A)) is the inverse of op(A) 00385 * abs(Z) is the componentwise absolute value of the matrix or 00386 * vector Z 00387 * NZ is the maximum number of nonzeros in any row of A, plus 1 00388 * EPS is machine epsilon 00389 * 00390 * The i-th component of abs(R)+NZ*EPS*(abs(op(A))*abs(X)+abs(B)) 00391 * is incremented by SAFE1 if the i-th component of 00392 * abs(op(A))*abs(X) + abs(B) is less than SAFE2. 00393 * 00394 * Use ZLACN2 to estimate the infinity-norm of the matrix 00395 * inv(op(A)) * diag(W), 00396 * where W = abs(R) + NZ*EPS*( abs(op(A))*abs(X)+abs(B) ))) 00397 * 00398 DO 90 I = 1, N 00399 IF( RWORK( I ).GT.SAFE2 ) THEN 00400 RWORK( I ) = CABS1( WORK( I ) ) + NZ*EPS*RWORK( I ) 00401 ELSE 00402 RWORK( I ) = CABS1( WORK( I ) ) + NZ*EPS*RWORK( I ) + 00403 $ SAFE1 00404 END IF 00405 90 CONTINUE 00406 * 00407 KASE = 0 00408 100 CONTINUE 00409 CALL ZLACN2( N, WORK( N+1 ), WORK, FERR( J ), KASE, ISAVE ) 00410 IF( KASE.NE.0 ) THEN 00411 IF( KASE.EQ.1 ) THEN 00412 * 00413 * Multiply by diag(W)*inv(op(A)**H). 00414 * 00415 CALL ZGETRS( TRANST, N, 1, AF, LDAF, IPIV, WORK, N, 00416 $ INFO ) 00417 DO 110 I = 1, N 00418 WORK( I ) = RWORK( I )*WORK( I ) 00419 110 CONTINUE 00420 ELSE 00421 * 00422 * Multiply by inv(op(A))*diag(W). 00423 * 00424 DO 120 I = 1, N 00425 WORK( I ) = RWORK( I )*WORK( I ) 00426 120 CONTINUE 00427 CALL ZGETRS( TRANSN, N, 1, AF, LDAF, IPIV, WORK, N, 00428 $ INFO ) 00429 END IF 00430 GO TO 100 00431 END IF 00432 * 00433 * Normalize error. 00434 * 00435 LSTRES = ZERO 00436 DO 130 I = 1, N 00437 LSTRES = MAX( LSTRES, CABS1( X( I, J ) ) ) 00438 130 CONTINUE 00439 IF( LSTRES.NE.ZERO ) 00440 $ FERR( J ) = FERR( J ) / LSTRES 00441 * 00442 140 CONTINUE 00443 * 00444 RETURN 00445 * 00446 * End of ZGERFS 00447 * 00448 END