LAPACK  3.4.1
LAPACK: Linear Algebra PACKage
zgerfs.f
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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 
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00011 *> [TGZ]</a> 
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00013 *> [ZIP]</a> 
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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
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