LAPACK  3.4.1
LAPACK: Linear Algebra PACKage
cherfs.f
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00001 *> \brief \b CHERFS
00002 *
00003 *  =========== DOCUMENTATION ===========
00004 *
00005 * Online html documentation available at 
00006 *            http://www.netlib.org/lapack/explore-html/ 
00007 *
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00015 *> [TXT]</a>
00016 *> \endhtmlonly 
00017 *
00018 *  Definition:
00019 *  ===========
00020 *
00021 *       SUBROUTINE CHERFS( UPLO, N, NRHS, A, LDA, AF, LDAF, IPIV, B, LDB,
00022 *                          X, LDX, FERR, BERR, WORK, RWORK, INFO )
00023 * 
00024 *       .. Scalar Arguments ..
00025 *       CHARACTER          UPLO
00026 *       INTEGER            INFO, LDA, LDAF, LDB, LDX, N, NRHS
00027 *       ..
00028 *       .. Array Arguments ..
00029 *       INTEGER            IPIV( * )
00030 *       REAL               BERR( * ), FERR( * ), RWORK( * )
00031 *       COMPLEX            A( LDA, * ), AF( LDAF, * ), B( LDB, * ),
00032 *      $                   WORK( * ), X( LDX, * )
00033 *       ..
00034 *  
00035 *
00036 *> \par Purpose:
00037 *  =============
00038 *>
00039 *> \verbatim
00040 *>
00041 *> CHERFS improves the computed solution to a system of linear
00042 *> equations when the coefficient matrix is Hermitian indefinite, and
00043 *> provides error bounds and backward error estimates for the solution.
00044 *> \endverbatim
00045 *
00046 *  Arguments:
00047 *  ==========
00048 *
00049 *> \param[in] UPLO
00050 *> \verbatim
00051 *>          UPLO is CHARACTER*1
00052 *>          = 'U':  Upper triangle of A is stored;
00053 *>          = 'L':  Lower triangle of A is stored.
00054 *> \endverbatim
00055 *>
00056 *> \param[in] N
00057 *> \verbatim
00058 *>          N is INTEGER
00059 *>          The order of the matrix A.  N >= 0.
00060 *> \endverbatim
00061 *>
00062 *> \param[in] NRHS
00063 *> \verbatim
00064 *>          NRHS is INTEGER
00065 *>          The number of right hand sides, i.e., the number of columns
00066 *>          of the matrices B and X.  NRHS >= 0.
00067 *> \endverbatim
00068 *>
00069 *> \param[in] A
00070 *> \verbatim
00071 *>          A is COMPLEX array, dimension (LDA,N)
00072 *>          The Hermitian matrix A.  If UPLO = 'U', the leading N-by-N
00073 *>          upper triangular part of A contains the upper triangular part
00074 *>          of the matrix A, and the strictly lower triangular part of A
00075 *>          is not referenced.  If UPLO = 'L', the leading N-by-N lower
00076 *>          triangular part of A contains the lower triangular part of
00077 *>          the matrix A, and the strictly upper triangular part of A is
00078 *>          not referenced.
00079 *> \endverbatim
00080 *>
00081 *> \param[in] LDA
00082 *> \verbatim
00083 *>          LDA is INTEGER
00084 *>          The leading dimension of the array A.  LDA >= max(1,N).
00085 *> \endverbatim
00086 *>
00087 *> \param[in] AF
00088 *> \verbatim
00089 *>          AF is COMPLEX array, dimension (LDAF,N)
00090 *>          The factored form of the matrix A.  AF contains the block
00091 *>          diagonal matrix D and the multipliers used to obtain the
00092 *>          factor U or L from the factorization A = U*D*U**H or
00093 *>          A = L*D*L**H as computed by CHETRF.
00094 *> \endverbatim
00095 *>
00096 *> \param[in] LDAF
00097 *> \verbatim
00098 *>          LDAF is INTEGER
00099 *>          The leading dimension of the array AF.  LDAF >= max(1,N).
00100 *> \endverbatim
00101 *>
00102 *> \param[in] IPIV
00103 *> \verbatim
00104 *>          IPIV is INTEGER array, dimension (N)
00105 *>          Details of the interchanges and the block structure of D
00106 *>          as determined by CHETRF.
00107 *> \endverbatim
00108 *>
00109 *> \param[in] B
00110 *> \verbatim
00111 *>          B is COMPLEX array, dimension (LDB,NRHS)
00112 *>          The right hand side matrix B.
00113 *> \endverbatim
00114 *>
00115 *> \param[in] LDB
00116 *> \verbatim
00117 *>          LDB is INTEGER
00118 *>          The leading dimension of the array B.  LDB >= max(1,N).
00119 *> \endverbatim
00120 *>
00121 *> \param[in,out] X
00122 *> \verbatim
00123 *>          X is COMPLEX array, dimension (LDX,NRHS)
00124 *>          On entry, the solution matrix X, as computed by CHETRS.
00125 *>          On exit, the improved solution matrix X.
00126 *> \endverbatim
00127 *>
00128 *> \param[in] LDX
00129 *> \verbatim
00130 *>          LDX is INTEGER
00131 *>          The leading dimension of the array X.  LDX >= max(1,N).
00132 *> \endverbatim
00133 *>
00134 *> \param[out] FERR
00135 *> \verbatim
00136 *>          FERR is REAL array, dimension (NRHS)
00137 *>          The estimated forward error bound for each solution vector
00138 *>          X(j) (the j-th column of the solution matrix X).
00139 *>          If XTRUE is the true solution corresponding to X(j), FERR(j)
00140 *>          is an estimated upper bound for the magnitude of the largest
00141 *>          element in (X(j) - XTRUE) divided by the magnitude of the
00142 *>          largest element in X(j).  The estimate is as reliable as
00143 *>          the estimate for RCOND, and is almost always a slight
00144 *>          overestimate of the true error.
00145 *> \endverbatim
00146 *>
00147 *> \param[out] BERR
00148 *> \verbatim
00149 *>          BERR is REAL array, dimension (NRHS)
00150 *>          The componentwise relative backward error of each solution
00151 *>          vector X(j) (i.e., the smallest relative change in
00152 *>          any element of A or B that makes X(j) an exact solution).
00153 *> \endverbatim
00154 *>
00155 *> \param[out] WORK
00156 *> \verbatim
00157 *>          WORK is COMPLEX array, dimension (2*N)
00158 *> \endverbatim
00159 *>
00160 *> \param[out] RWORK
00161 *> \verbatim
00162 *>          RWORK is REAL array, dimension (N)
00163 *> \endverbatim
00164 *>
00165 *> \param[out] INFO
00166 *> \verbatim
00167 *>          INFO is INTEGER
00168 *>          = 0:  successful exit
00169 *>          < 0:  if INFO = -i, the i-th argument had an illegal value
00170 *> \endverbatim
00171 *
00172 *> \par Internal Parameters:
00173 *  =========================
00174 *>
00175 *> \verbatim
00176 *>  ITMAX is the maximum number of steps of iterative refinement.
00177 *> \endverbatim
00178 *
00179 *  Authors:
00180 *  ========
00181 *
00182 *> \author Univ. of Tennessee 
00183 *> \author Univ. of California Berkeley 
00184 *> \author Univ. of Colorado Denver 
00185 *> \author NAG Ltd. 
00186 *
00187 *> \date November 2011
00188 *
00189 *> \ingroup complexHEcomputational
00190 *
00191 *  =====================================================================
00192       SUBROUTINE CHERFS( UPLO, N, NRHS, A, LDA, AF, LDAF, IPIV, B, LDB,
00193      $                   X, LDX, FERR, BERR, WORK, RWORK, INFO )
00194 *
00195 *  -- LAPACK computational routine (version 3.4.0) --
00196 *  -- LAPACK is a software package provided by Univ. of Tennessee,    --
00197 *  -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
00198 *     November 2011
00199 *
00200 *     .. Scalar Arguments ..
00201       CHARACTER          UPLO
00202       INTEGER            INFO, LDA, LDAF, LDB, LDX, N, NRHS
00203 *     ..
00204 *     .. Array Arguments ..
00205       INTEGER            IPIV( * )
00206       REAL               BERR( * ), FERR( * ), RWORK( * )
00207       COMPLEX            A( LDA, * ), AF( LDAF, * ), B( LDB, * ),
00208      $                   WORK( * ), X( LDX, * )
00209 *     ..
00210 *
00211 *  =====================================================================
00212 *
00213 *     .. Parameters ..
00214       INTEGER            ITMAX
00215       PARAMETER          ( ITMAX = 5 )
00216       REAL               ZERO
00217       PARAMETER          ( ZERO = 0.0E+0 )
00218       COMPLEX            ONE
00219       PARAMETER          ( ONE = ( 1.0E+0, 0.0E+0 ) )
00220       REAL               TWO
00221       PARAMETER          ( TWO = 2.0E+0 )
00222       REAL               THREE
00223       PARAMETER          ( THREE = 3.0E+0 )
00224 *     ..
00225 *     .. Local Scalars ..
00226       LOGICAL            UPPER
00227       INTEGER            COUNT, I, J, K, KASE, NZ
00228       REAL               EPS, LSTRES, S, SAFE1, SAFE2, SAFMIN, XK
00229       COMPLEX            ZDUM
00230 *     ..
00231 *     .. Local Arrays ..
00232       INTEGER            ISAVE( 3 )
00233 *     ..
00234 *     .. External Subroutines ..
00235       EXTERNAL           CAXPY, CCOPY, CHEMV, CHETRS, CLACN2, XERBLA
00236 *     ..
00237 *     .. Intrinsic Functions ..
00238       INTRINSIC          ABS, AIMAG, MAX, REAL
00239 *     ..
00240 *     .. External Functions ..
00241       LOGICAL            LSAME
00242       REAL               SLAMCH
00243       EXTERNAL           LSAME, SLAMCH
00244 *     ..
00245 *     .. Statement Functions ..
00246       REAL               CABS1
00247 *     ..
00248 *     .. Statement Function definitions ..
00249       CABS1( ZDUM ) = ABS( REAL( ZDUM ) ) + ABS( AIMAG( ZDUM ) )
00250 *     ..
00251 *     .. Executable Statements ..
00252 *
00253 *     Test the input parameters.
00254 *
00255       INFO = 0
00256       UPPER = LSAME( UPLO, 'U' )
00257       IF( .NOT.UPPER .AND. .NOT.LSAME( UPLO, 'L' ) ) THEN
00258          INFO = -1
00259       ELSE IF( N.LT.0 ) THEN
00260          INFO = -2
00261       ELSE IF( NRHS.LT.0 ) THEN
00262          INFO = -3
00263       ELSE IF( LDA.LT.MAX( 1, N ) ) THEN
00264          INFO = -5
00265       ELSE IF( LDAF.LT.MAX( 1, N ) ) THEN
00266          INFO = -7
00267       ELSE IF( LDB.LT.MAX( 1, N ) ) THEN
00268          INFO = -10
00269       ELSE IF( LDX.LT.MAX( 1, N ) ) THEN
00270          INFO = -12
00271       END IF
00272       IF( INFO.NE.0 ) THEN
00273          CALL XERBLA( 'CHERFS', -INFO )
00274          RETURN
00275       END IF
00276 *
00277 *     Quick return if possible
00278 *
00279       IF( N.EQ.0 .OR. NRHS.EQ.0 ) THEN
00280          DO 10 J = 1, NRHS
00281             FERR( J ) = ZERO
00282             BERR( J ) = ZERO
00283    10    CONTINUE
00284          RETURN
00285       END IF
00286 *
00287 *     NZ = maximum number of nonzero elements in each row of A, plus 1
00288 *
00289       NZ = N + 1
00290       EPS = SLAMCH( 'Epsilon' )
00291       SAFMIN = SLAMCH( 'Safe minimum' )
00292       SAFE1 = NZ*SAFMIN
00293       SAFE2 = SAFE1 / EPS
00294 *
00295 *     Do for each right hand side
00296 *
00297       DO 140 J = 1, NRHS
00298 *
00299          COUNT = 1
00300          LSTRES = THREE
00301    20    CONTINUE
00302 *
00303 *        Loop until stopping criterion is satisfied.
00304 *
00305 *        Compute residual R = B - A * X
00306 *
00307          CALL CCOPY( N, B( 1, J ), 1, WORK, 1 )
00308          CALL CHEMV( UPLO, N, -ONE, A, LDA, X( 1, J ), 1, ONE, WORK, 1 )
00309 *
00310 *        Compute componentwise relative backward error from formula
00311 *
00312 *        max(i) ( abs(R(i)) / ( abs(A)*abs(X) + abs(B) )(i) )
00313 *
00314 *        where abs(Z) is the componentwise absolute value of the matrix
00315 *        or vector Z.  If the i-th component of the denominator is less
00316 *        than SAFE2, then SAFE1 is added to the i-th components of the
00317 *        numerator and denominator before dividing.
00318 *
00319          DO 30 I = 1, N
00320             RWORK( I ) = CABS1( B( I, J ) )
00321    30    CONTINUE
00322 *
00323 *        Compute abs(A)*abs(X) + abs(B).
00324 *
00325          IF( UPPER ) THEN
00326             DO 50 K = 1, N
00327                S = ZERO
00328                XK = CABS1( X( K, J ) )
00329                DO 40 I = 1, K - 1
00330                   RWORK( I ) = RWORK( I ) + CABS1( A( I, K ) )*XK
00331                   S = S + CABS1( A( I, K ) )*CABS1( X( I, J ) )
00332    40          CONTINUE
00333                RWORK( K ) = RWORK( K ) + ABS( REAL( A( K, K ) ) )*XK + S
00334    50       CONTINUE
00335          ELSE
00336             DO 70 K = 1, N
00337                S = ZERO
00338                XK = CABS1( X( K, J ) )
00339                RWORK( K ) = RWORK( K ) + ABS( REAL( A( K, K ) ) )*XK
00340                DO 60 I = K + 1, N
00341                   RWORK( I ) = RWORK( I ) + CABS1( A( I, K ) )*XK
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 CHETRS( UPLO, N, 1, AF, LDAF, IPIV, WORK, N, INFO )
00370             CALL CAXPY( 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(A))*
00380 *           ( abs(R) + NZ*EPS*( abs(A)*abs(X)+abs(B) ))) / norm(X)
00381 *
00382 *        where
00383 *          norm(Z) is the magnitude of the largest component of Z
00384 *          inv(A) is the inverse of 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(A)*abs(X)+abs(B))
00391 *        is incremented by SAFE1 if the i-th component of
00392 *        abs(A)*abs(X) + abs(B) is less than SAFE2.
00393 *
00394 *        Use CLACN2 to estimate the infinity-norm of the matrix
00395 *           inv(A) * diag(W),
00396 *        where W = abs(R) + NZ*EPS*( abs(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 CLACN2( 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(A**H).
00414 *
00415                CALL CHETRS( UPLO, N, 1, AF, LDAF, IPIV, WORK, N, INFO )
00416                DO 110 I = 1, N
00417                   WORK( I ) = RWORK( I )*WORK( I )
00418   110          CONTINUE
00419             ELSE IF( KASE.EQ.2 ) THEN
00420 *
00421 *              Multiply by inv(A)*diag(W).
00422 *
00423                DO 120 I = 1, N
00424                   WORK( I ) = RWORK( I )*WORK( I )
00425   120          CONTINUE
00426                CALL CHETRS( UPLO, N, 1, AF, LDAF, IPIV, WORK, N, INFO )
00427             END IF
00428             GO TO 100
00429          END IF
00430 *
00431 *        Normalize error.
00432 *
00433          LSTRES = ZERO
00434          DO 130 I = 1, N
00435             LSTRES = MAX( LSTRES, CABS1( X( I, J ) ) )
00436   130    CONTINUE
00437          IF( LSTRES.NE.ZERO )
00438      $      FERR( J ) = FERR( J ) / LSTRES
00439 *
00440   140 CONTINUE
00441 *
00442       RETURN
00443 *
00444 *     End of CHERFS
00445 *
00446       END
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