LAPACK  3.4.1
LAPACK: Linear Algebra PACKage
dsprfs.f
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00001 *> \brief \b DSPRFS
00002 *
00003 *  =========== DOCUMENTATION ===========
00004 *
00005 * Online html documentation available at 
00006 *            http://www.netlib.org/lapack/explore-html/ 
00007 *
00008 *> \htmlonly
00009 *> Download DSPRFS + dependencies 
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00011 *> [TGZ]</a> 
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00013 *> [ZIP]</a> 
00014 *> <a href="http://www.netlib.org/cgi-bin/netlibfiles.txt?format=txt&filename=/lapack/lapack_routine/dsprfs.f"> 
00015 *> [TXT]</a>
00016 *> \endhtmlonly 
00017 *
00018 *  Definition:
00019 *  ===========
00020 *
00021 *       SUBROUTINE DSPRFS( UPLO, N, NRHS, AP, AFP, IPIV, B, LDB, X, LDX,
00022 *                          FERR, BERR, WORK, IWORK, INFO )
00023 * 
00024 *       .. Scalar Arguments ..
00025 *       CHARACTER          UPLO
00026 *       INTEGER            INFO, LDB, LDX, N, NRHS
00027 *       ..
00028 *       .. Array Arguments ..
00029 *       INTEGER            IPIV( * ), IWORK( * )
00030 *       DOUBLE PRECISION   AFP( * ), AP( * ), B( LDB, * ), BERR( * ),
00031 *      $                   FERR( * ), WORK( * ), X( LDX, * )
00032 *       ..
00033 *  
00034 *
00035 *> \par Purpose:
00036 *  =============
00037 *>
00038 *> \verbatim
00039 *>
00040 *> DSPRFS improves the computed solution to a system of linear
00041 *> equations when the coefficient matrix is symmetric indefinite
00042 *> and packed, and provides error bounds and backward error estimates
00043 *> 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] AP
00070 *> \verbatim
00071 *>          AP is DOUBLE PRECISION array, dimension (N*(N+1)/2)
00072 *>          The upper or lower triangle of the symmetric matrix A, packed
00073 *>          columnwise in a linear array.  The j-th column of A is stored
00074 *>          in the array AP as follows:
00075 *>          if UPLO = 'U', AP(i + (j-1)*j/2) = A(i,j) for 1<=i<=j;
00076 *>          if UPLO = 'L', AP(i + (j-1)*(2*n-j)/2) = A(i,j) for j<=i<=n.
00077 *> \endverbatim
00078 *>
00079 *> \param[in] AFP
00080 *> \verbatim
00081 *>          AFP is DOUBLE PRECISION array, dimension (N*(N+1)/2)
00082 *>          The factored form of the matrix A.  AFP contains the block
00083 *>          diagonal matrix D and the multipliers used to obtain the
00084 *>          factor U or L from the factorization A = U*D*U**T or
00085 *>          A = L*D*L**T as computed by DSPTRF, stored as a packed
00086 *>          triangular matrix.
00087 *> \endverbatim
00088 *>
00089 *> \param[in] IPIV
00090 *> \verbatim
00091 *>          IPIV is INTEGER array, dimension (N)
00092 *>          Details of the interchanges and the block structure of D
00093 *>          as determined by DSPTRF.
00094 *> \endverbatim
00095 *>
00096 *> \param[in] B
00097 *> \verbatim
00098 *>          B is DOUBLE PRECISION array, dimension (LDB,NRHS)
00099 *>          The right hand side matrix B.
00100 *> \endverbatim
00101 *>
00102 *> \param[in] LDB
00103 *> \verbatim
00104 *>          LDB is INTEGER
00105 *>          The leading dimension of the array B.  LDB >= max(1,N).
00106 *> \endverbatim
00107 *>
00108 *> \param[in,out] X
00109 *> \verbatim
00110 *>          X is DOUBLE PRECISION array, dimension (LDX,NRHS)
00111 *>          On entry, the solution matrix X, as computed by DSPTRS.
00112 *>          On exit, the improved solution matrix X.
00113 *> \endverbatim
00114 *>
00115 *> \param[in] LDX
00116 *> \verbatim
00117 *>          LDX is INTEGER
00118 *>          The leading dimension of the array X.  LDX >= max(1,N).
00119 *> \endverbatim
00120 *>
00121 *> \param[out] FERR
00122 *> \verbatim
00123 *>          FERR is DOUBLE PRECISION array, dimension (NRHS)
00124 *>          The estimated forward error bound for each solution vector
00125 *>          X(j) (the j-th column of the solution matrix X).
00126 *>          If XTRUE is the true solution corresponding to X(j), FERR(j)
00127 *>          is an estimated upper bound for the magnitude of the largest
00128 *>          element in (X(j) - XTRUE) divided by the magnitude of the
00129 *>          largest element in X(j).  The estimate is as reliable as
00130 *>          the estimate for RCOND, and is almost always a slight
00131 *>          overestimate of the true error.
00132 *> \endverbatim
00133 *>
00134 *> \param[out] BERR
00135 *> \verbatim
00136 *>          BERR is DOUBLE PRECISION array, dimension (NRHS)
00137 *>          The componentwise relative backward error of each solution
00138 *>          vector X(j) (i.e., the smallest relative change in
00139 *>          any element of A or B that makes X(j) an exact solution).
00140 *> \endverbatim
00141 *>
00142 *> \param[out] WORK
00143 *> \verbatim
00144 *>          WORK is DOUBLE PRECISION array, dimension (3*N)
00145 *> \endverbatim
00146 *>
00147 *> \param[out] IWORK
00148 *> \verbatim
00149 *>          IWORK is INTEGER array, dimension (N)
00150 *> \endverbatim
00151 *>
00152 *> \param[out] INFO
00153 *> \verbatim
00154 *>          INFO is INTEGER
00155 *>          = 0:  successful exit
00156 *>          < 0:  if INFO = -i, the i-th argument had an illegal value
00157 *> \endverbatim
00158 *
00159 *> \par Internal Parameters:
00160 *  =========================
00161 *>
00162 *> \verbatim
00163 *>  ITMAX is the maximum number of steps of iterative refinement.
00164 *> \endverbatim
00165 *
00166 *  Authors:
00167 *  ========
00168 *
00169 *> \author Univ. of Tennessee 
00170 *> \author Univ. of California Berkeley 
00171 *> \author Univ. of Colorado Denver 
00172 *> \author NAG Ltd. 
00173 *
00174 *> \date November 2011
00175 *
00176 *> \ingroup doubleOTHERcomputational
00177 *
00178 *  =====================================================================
00179       SUBROUTINE DSPRFS( UPLO, N, NRHS, AP, AFP, IPIV, B, LDB, X, LDX,
00180      $                   FERR, BERR, WORK, IWORK, INFO )
00181 *
00182 *  -- LAPACK computational routine (version 3.4.0) --
00183 *  -- LAPACK is a software package provided by Univ. of Tennessee,    --
00184 *  -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
00185 *     November 2011
00186 *
00187 *     .. Scalar Arguments ..
00188       CHARACTER          UPLO
00189       INTEGER            INFO, LDB, LDX, N, NRHS
00190 *     ..
00191 *     .. Array Arguments ..
00192       INTEGER            IPIV( * ), IWORK( * )
00193       DOUBLE PRECISION   AFP( * ), AP( * ), B( LDB, * ), BERR( * ),
00194      $                   FERR( * ), WORK( * ), X( LDX, * )
00195 *     ..
00196 *
00197 *  =====================================================================
00198 *
00199 *     .. Parameters ..
00200       INTEGER            ITMAX
00201       PARAMETER          ( ITMAX = 5 )
00202       DOUBLE PRECISION   ZERO
00203       PARAMETER          ( ZERO = 0.0D+0 )
00204       DOUBLE PRECISION   ONE
00205       PARAMETER          ( ONE = 1.0D+0 )
00206       DOUBLE PRECISION   TWO
00207       PARAMETER          ( TWO = 2.0D+0 )
00208       DOUBLE PRECISION   THREE
00209       PARAMETER          ( THREE = 3.0D+0 )
00210 *     ..
00211 *     .. Local Scalars ..
00212       LOGICAL            UPPER
00213       INTEGER            COUNT, I, IK, J, K, KASE, KK, NZ
00214       DOUBLE PRECISION   EPS, LSTRES, S, SAFE1, SAFE2, SAFMIN, XK
00215 *     ..
00216 *     .. Local Arrays ..
00217       INTEGER            ISAVE( 3 )
00218 *     ..
00219 *     .. External Subroutines ..
00220       EXTERNAL           DAXPY, DCOPY, DLACN2, DSPMV, DSPTRS, XERBLA
00221 *     ..
00222 *     .. Intrinsic Functions ..
00223       INTRINSIC          ABS, MAX
00224 *     ..
00225 *     .. External Functions ..
00226       LOGICAL            LSAME
00227       DOUBLE PRECISION   DLAMCH
00228       EXTERNAL           LSAME, DLAMCH
00229 *     ..
00230 *     .. Executable Statements ..
00231 *
00232 *     Test the input parameters.
00233 *
00234       INFO = 0
00235       UPPER = LSAME( UPLO, 'U' )
00236       IF( .NOT.UPPER .AND. .NOT.LSAME( UPLO, 'L' ) ) THEN
00237          INFO = -1
00238       ELSE IF( N.LT.0 ) THEN
00239          INFO = -2
00240       ELSE IF( NRHS.LT.0 ) THEN
00241          INFO = -3
00242       ELSE IF( LDB.LT.MAX( 1, N ) ) THEN
00243          INFO = -8
00244       ELSE IF( LDX.LT.MAX( 1, N ) ) THEN
00245          INFO = -10
00246       END IF
00247       IF( INFO.NE.0 ) THEN
00248          CALL XERBLA( 'DSPRFS', -INFO )
00249          RETURN
00250       END IF
00251 *
00252 *     Quick return if possible
00253 *
00254       IF( N.EQ.0 .OR. NRHS.EQ.0 ) THEN
00255          DO 10 J = 1, NRHS
00256             FERR( J ) = ZERO
00257             BERR( J ) = ZERO
00258    10    CONTINUE
00259          RETURN
00260       END IF
00261 *
00262 *     NZ = maximum number of nonzero elements in each row of A, plus 1
00263 *
00264       NZ = N + 1
00265       EPS = DLAMCH( 'Epsilon' )
00266       SAFMIN = DLAMCH( 'Safe minimum' )
00267       SAFE1 = NZ*SAFMIN
00268       SAFE2 = SAFE1 / EPS
00269 *
00270 *     Do for each right hand side
00271 *
00272       DO 140 J = 1, NRHS
00273 *
00274          COUNT = 1
00275          LSTRES = THREE
00276    20    CONTINUE
00277 *
00278 *        Loop until stopping criterion is satisfied.
00279 *
00280 *        Compute residual R = B - A * X
00281 *
00282          CALL DCOPY( N, B( 1, J ), 1, WORK( N+1 ), 1 )
00283          CALL DSPMV( UPLO, N, -ONE, AP, X( 1, J ), 1, ONE, WORK( N+1 ),
00284      $               1 )
00285 *
00286 *        Compute componentwise relative backward error from formula
00287 *
00288 *        max(i) ( abs(R(i)) / ( abs(A)*abs(X) + abs(B) )(i) )
00289 *
00290 *        where abs(Z) is the componentwise absolute value of the matrix
00291 *        or vector Z.  If the i-th component of the denominator is less
00292 *        than SAFE2, then SAFE1 is added to the i-th components of the
00293 *        numerator and denominator before dividing.
00294 *
00295          DO 30 I = 1, N
00296             WORK( I ) = ABS( B( I, J ) )
00297    30    CONTINUE
00298 *
00299 *        Compute abs(A)*abs(X) + abs(B).
00300 *
00301          KK = 1
00302          IF( UPPER ) THEN
00303             DO 50 K = 1, N
00304                S = ZERO
00305                XK = ABS( X( K, J ) )
00306                IK = KK
00307                DO 40 I = 1, K - 1
00308                   WORK( I ) = WORK( I ) + ABS( AP( IK ) )*XK
00309                   S = S + ABS( AP( IK ) )*ABS( X( I, J ) )
00310                   IK = IK + 1
00311    40          CONTINUE
00312                WORK( K ) = WORK( K ) + ABS( AP( KK+K-1 ) )*XK + S
00313                KK = KK + K
00314    50       CONTINUE
00315          ELSE
00316             DO 70 K = 1, N
00317                S = ZERO
00318                XK = ABS( X( K, J ) )
00319                WORK( K ) = WORK( K ) + ABS( AP( KK ) )*XK
00320                IK = KK + 1
00321                DO 60 I = K + 1, N
00322                   WORK( I ) = WORK( I ) + ABS( AP( IK ) )*XK
00323                   S = S + ABS( AP( IK ) )*ABS( X( I, J ) )
00324                   IK = IK + 1
00325    60          CONTINUE
00326                WORK( K ) = WORK( K ) + S
00327                KK = KK + ( N-K+1 )
00328    70       CONTINUE
00329          END IF
00330          S = ZERO
00331          DO 80 I = 1, N
00332             IF( WORK( I ).GT.SAFE2 ) THEN
00333                S = MAX( S, ABS( WORK( N+I ) ) / WORK( I ) )
00334             ELSE
00335                S = MAX( S, ( ABS( WORK( N+I ) )+SAFE1 ) /
00336      $             ( WORK( I )+SAFE1 ) )
00337             END IF
00338    80    CONTINUE
00339          BERR( J ) = S
00340 *
00341 *        Test stopping criterion. Continue iterating if
00342 *           1) The residual BERR(J) is larger than machine epsilon, and
00343 *           2) BERR(J) decreased by at least a factor of 2 during the
00344 *              last iteration, and
00345 *           3) At most ITMAX iterations tried.
00346 *
00347          IF( BERR( J ).GT.EPS .AND. TWO*BERR( J ).LE.LSTRES .AND.
00348      $       COUNT.LE.ITMAX ) THEN
00349 *
00350 *           Update solution and try again.
00351 *
00352             CALL DSPTRS( UPLO, N, 1, AFP, IPIV, WORK( N+1 ), N, INFO )
00353             CALL DAXPY( N, ONE, WORK( N+1 ), 1, X( 1, J ), 1 )
00354             LSTRES = BERR( J )
00355             COUNT = COUNT + 1
00356             GO TO 20
00357          END IF
00358 *
00359 *        Bound error from formula
00360 *
00361 *        norm(X - XTRUE) / norm(X) .le. FERR =
00362 *        norm( abs(inv(A))*
00363 *           ( abs(R) + NZ*EPS*( abs(A)*abs(X)+abs(B) ))) / norm(X)
00364 *
00365 *        where
00366 *          norm(Z) is the magnitude of the largest component of Z
00367 *          inv(A) is the inverse of A
00368 *          abs(Z) is the componentwise absolute value of the matrix or
00369 *             vector Z
00370 *          NZ is the maximum number of nonzeros in any row of A, plus 1
00371 *          EPS is machine epsilon
00372 *
00373 *        The i-th component of abs(R)+NZ*EPS*(abs(A)*abs(X)+abs(B))
00374 *        is incremented by SAFE1 if the i-th component of
00375 *        abs(A)*abs(X) + abs(B) is less than SAFE2.
00376 *
00377 *        Use DLACN2 to estimate the infinity-norm of the matrix
00378 *           inv(A) * diag(W),
00379 *        where W = abs(R) + NZ*EPS*( abs(A)*abs(X)+abs(B) )))
00380 *
00381          DO 90 I = 1, N
00382             IF( WORK( I ).GT.SAFE2 ) THEN
00383                WORK( I ) = ABS( WORK( N+I ) ) + NZ*EPS*WORK( I )
00384             ELSE
00385                WORK( I ) = ABS( WORK( N+I ) ) + NZ*EPS*WORK( I ) + SAFE1
00386             END IF
00387    90    CONTINUE
00388 *
00389          KASE = 0
00390   100    CONTINUE
00391          CALL DLACN2( N, WORK( 2*N+1 ), WORK( N+1 ), IWORK, FERR( J ),
00392      $                KASE, ISAVE )
00393          IF( KASE.NE.0 ) THEN
00394             IF( KASE.EQ.1 ) THEN
00395 *
00396 *              Multiply by diag(W)*inv(A**T).
00397 *
00398                CALL DSPTRS( UPLO, N, 1, AFP, IPIV, WORK( N+1 ), N,
00399      $                      INFO )
00400                DO 110 I = 1, N
00401                   WORK( N+I ) = WORK( I )*WORK( N+I )
00402   110          CONTINUE
00403             ELSE IF( KASE.EQ.2 ) THEN
00404 *
00405 *              Multiply by inv(A)*diag(W).
00406 *
00407                DO 120 I = 1, N
00408                   WORK( N+I ) = WORK( I )*WORK( N+I )
00409   120          CONTINUE
00410                CALL DSPTRS( UPLO, N, 1, AFP, IPIV, WORK( N+1 ), N,
00411      $                      INFO )
00412             END IF
00413             GO TO 100
00414          END IF
00415 *
00416 *        Normalize error.
00417 *
00418          LSTRES = ZERO
00419          DO 130 I = 1, N
00420             LSTRES = MAX( LSTRES, ABS( X( I, J ) ) )
00421   130    CONTINUE
00422          IF( LSTRES.NE.ZERO )
00423      $      FERR( J ) = FERR( J ) / LSTRES
00424 *
00425   140 CONTINUE
00426 *
00427       RETURN
00428 *
00429 *     End of DSPRFS
00430 *
00431       END
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