LAPACK  3.4.1
LAPACK: Linear Algebra PACKage
sporfs.f
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00001 *> \brief \b SPORFS
00002 *
00003 *  =========== DOCUMENTATION ===========
00004 *
00005 * Online html documentation available at 
00006 *            http://www.netlib.org/lapack/explore-html/ 
00007 *
00008 *> \htmlonly
00009 *> Download SPORFS + 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 SPORFS( UPLO, N, NRHS, A, LDA, AF, LDAF, B, LDB, X,
00022 *                          LDX, FERR, BERR, WORK, IWORK, INFO )
00023 * 
00024 *       .. Scalar Arguments ..
00025 *       CHARACTER          UPLO
00026 *       INTEGER            INFO, LDA, LDAF, LDB, LDX, N, NRHS
00027 *       ..
00028 *       .. Array Arguments ..
00029 *       INTEGER            IWORK( * )
00030 *       REAL               A( LDA, * ), AF( LDAF, * ), B( LDB, * ),
00031 *      $                   BERR( * ), FERR( * ), WORK( * ), X( LDX, * )
00032 *       ..
00033 *  
00034 *
00035 *> \par Purpose:
00036 *  =============
00037 *>
00038 *> \verbatim
00039 *>
00040 *> SPORFS improves the computed solution to a system of linear
00041 *> equations when the coefficient matrix is symmetric positive definite,
00042 *> and provides error bounds and backward error estimates for the
00043 *> 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 REAL array, dimension (LDA,N)
00072 *>          The symmetric 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 REAL array, dimension (LDAF,N)
00090 *>          The triangular factor U or L from the Cholesky factorization
00091 *>          A = U**T*U or A = L*L**T, as computed by SPOTRF.
00092 *> \endverbatim
00093 *>
00094 *> \param[in] LDAF
00095 *> \verbatim
00096 *>          LDAF is INTEGER
00097 *>          The leading dimension of the array AF.  LDAF >= max(1,N).
00098 *> \endverbatim
00099 *>
00100 *> \param[in] B
00101 *> \verbatim
00102 *>          B is REAL array, dimension (LDB,NRHS)
00103 *>          The right hand side matrix B.
00104 *> \endverbatim
00105 *>
00106 *> \param[in] LDB
00107 *> \verbatim
00108 *>          LDB is INTEGER
00109 *>          The leading dimension of the array B.  LDB >= max(1,N).
00110 *> \endverbatim
00111 *>
00112 *> \param[in,out] X
00113 *> \verbatim
00114 *>          X is REAL array, dimension (LDX,NRHS)
00115 *>          On entry, the solution matrix X, as computed by SPOTRS.
00116 *>          On exit, the improved solution matrix X.
00117 *> \endverbatim
00118 *>
00119 *> \param[in] LDX
00120 *> \verbatim
00121 *>          LDX is INTEGER
00122 *>          The leading dimension of the array X.  LDX >= max(1,N).
00123 *> \endverbatim
00124 *>
00125 *> \param[out] FERR
00126 *> \verbatim
00127 *>          FERR is REAL array, dimension (NRHS)
00128 *>          The estimated forward error bound for each solution vector
00129 *>          X(j) (the j-th column of the solution matrix X).
00130 *>          If XTRUE is the true solution corresponding to X(j), FERR(j)
00131 *>          is an estimated upper bound for the magnitude of the largest
00132 *>          element in (X(j) - XTRUE) divided by the magnitude of the
00133 *>          largest element in X(j).  The estimate is as reliable as
00134 *>          the estimate for RCOND, and is almost always a slight
00135 *>          overestimate of the true error.
00136 *> \endverbatim
00137 *>
00138 *> \param[out] BERR
00139 *> \verbatim
00140 *>          BERR is REAL array, dimension (NRHS)
00141 *>          The componentwise relative backward error of each solution
00142 *>          vector X(j) (i.e., the smallest relative change in
00143 *>          any element of A or B that makes X(j) an exact solution).
00144 *> \endverbatim
00145 *>
00146 *> \param[out] WORK
00147 *> \verbatim
00148 *>          WORK is REAL array, dimension (3*N)
00149 *> \endverbatim
00150 *>
00151 *> \param[out] IWORK
00152 *> \verbatim
00153 *>          IWORK is INTEGER array, dimension (N)
00154 *> \endverbatim
00155 *>
00156 *> \param[out] INFO
00157 *> \verbatim
00158 *>          INFO is INTEGER
00159 *>          = 0:  successful exit
00160 *>          < 0:  if INFO = -i, the i-th argument had an illegal value
00161 *> \endverbatim
00162 *
00163 *> \par Internal Parameters:
00164 *  =========================
00165 *>
00166 *> \verbatim
00167 *>  ITMAX is the maximum number of steps of iterative refinement.
00168 *> \endverbatim
00169 *
00170 *  Authors:
00171 *  ========
00172 *
00173 *> \author Univ. of Tennessee 
00174 *> \author Univ. of California Berkeley 
00175 *> \author Univ. of Colorado Denver 
00176 *> \author NAG Ltd. 
00177 *
00178 *> \date November 2011
00179 *
00180 *> \ingroup realPOcomputational
00181 *
00182 *  =====================================================================
00183       SUBROUTINE SPORFS( UPLO, N, NRHS, A, LDA, AF, LDAF, B, LDB, X,
00184      $                   LDX, FERR, BERR, WORK, IWORK, INFO )
00185 *
00186 *  -- LAPACK computational routine (version 3.4.0) --
00187 *  -- LAPACK is a software package provided by Univ. of Tennessee,    --
00188 *  -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
00189 *     November 2011
00190 *
00191 *     .. Scalar Arguments ..
00192       CHARACTER          UPLO
00193       INTEGER            INFO, LDA, LDAF, LDB, LDX, N, NRHS
00194 *     ..
00195 *     .. Array Arguments ..
00196       INTEGER            IWORK( * )
00197       REAL               A( LDA, * ), AF( LDAF, * ), B( LDB, * ),
00198      $                   BERR( * ), FERR( * ), WORK( * ), X( LDX, * )
00199 *     ..
00200 *
00201 *  =====================================================================
00202 *
00203 *     .. Parameters ..
00204       INTEGER            ITMAX
00205       PARAMETER          ( ITMAX = 5 )
00206       REAL               ZERO
00207       PARAMETER          ( ZERO = 0.0E+0 )
00208       REAL               ONE
00209       PARAMETER          ( ONE = 1.0E+0 )
00210       REAL               TWO
00211       PARAMETER          ( TWO = 2.0E+0 )
00212       REAL               THREE
00213       PARAMETER          ( THREE = 3.0E+0 )
00214 *     ..
00215 *     .. Local Scalars ..
00216       LOGICAL            UPPER
00217       INTEGER            COUNT, I, J, K, KASE, NZ
00218       REAL               EPS, LSTRES, S, SAFE1, SAFE2, SAFMIN, XK
00219 *     ..
00220 *     .. Local Arrays ..
00221       INTEGER            ISAVE( 3 )
00222 *     ..
00223 *     .. External Subroutines ..
00224       EXTERNAL           SAXPY, SCOPY, SLACN2, SPOTRS, SSYMV, XERBLA
00225 *     ..
00226 *     .. Intrinsic Functions ..
00227       INTRINSIC          ABS, MAX
00228 *     ..
00229 *     .. External Functions ..
00230       LOGICAL            LSAME
00231       REAL               SLAMCH
00232       EXTERNAL           LSAME, SLAMCH
00233 *     ..
00234 *     .. Executable Statements ..
00235 *
00236 *     Test the input parameters.
00237 *
00238       INFO = 0
00239       UPPER = LSAME( UPLO, 'U' )
00240       IF( .NOT.UPPER .AND. .NOT.LSAME( UPLO, 'L' ) ) THEN
00241          INFO = -1
00242       ELSE IF( N.LT.0 ) THEN
00243          INFO = -2
00244       ELSE IF( NRHS.LT.0 ) THEN
00245          INFO = -3
00246       ELSE IF( LDA.LT.MAX( 1, N ) ) THEN
00247          INFO = -5
00248       ELSE IF( LDAF.LT.MAX( 1, N ) ) THEN
00249          INFO = -7
00250       ELSE IF( LDB.LT.MAX( 1, N ) ) THEN
00251          INFO = -9
00252       ELSE IF( LDX.LT.MAX( 1, N ) ) THEN
00253          INFO = -11
00254       END IF
00255       IF( INFO.NE.0 ) THEN
00256          CALL XERBLA( 'SPORFS', -INFO )
00257          RETURN
00258       END IF
00259 *
00260 *     Quick return if possible
00261 *
00262       IF( N.EQ.0 .OR. NRHS.EQ.0 ) THEN
00263          DO 10 J = 1, NRHS
00264             FERR( J ) = ZERO
00265             BERR( J ) = ZERO
00266    10    CONTINUE
00267          RETURN
00268       END IF
00269 *
00270 *     NZ = maximum number of nonzero elements in each row of A, plus 1
00271 *
00272       NZ = N + 1
00273       EPS = SLAMCH( 'Epsilon' )
00274       SAFMIN = SLAMCH( 'Safe minimum' )
00275       SAFE1 = NZ*SAFMIN
00276       SAFE2 = SAFE1 / EPS
00277 *
00278 *     Do for each right hand side
00279 *
00280       DO 140 J = 1, NRHS
00281 *
00282          COUNT = 1
00283          LSTRES = THREE
00284    20    CONTINUE
00285 *
00286 *        Loop until stopping criterion is satisfied.
00287 *
00288 *        Compute residual R = B - A * X
00289 *
00290          CALL SCOPY( N, B( 1, J ), 1, WORK( N+1 ), 1 )
00291          CALL SSYMV( UPLO, N, -ONE, A, LDA, X( 1, J ), 1, ONE,
00292      $               WORK( N+1 ), 1 )
00293 *
00294 *        Compute componentwise relative backward error from formula
00295 *
00296 *        max(i) ( abs(R(i)) / ( abs(A)*abs(X) + abs(B) )(i) )
00297 *
00298 *        where abs(Z) is the componentwise absolute value of the matrix
00299 *        or vector Z.  If the i-th component of the denominator is less
00300 *        than SAFE2, then SAFE1 is added to the i-th components of the
00301 *        numerator and denominator before dividing.
00302 *
00303          DO 30 I = 1, N
00304             WORK( I ) = ABS( B( I, J ) )
00305    30    CONTINUE
00306 *
00307 *        Compute abs(A)*abs(X) + abs(B).
00308 *
00309          IF( UPPER ) THEN
00310             DO 50 K = 1, N
00311                S = ZERO
00312                XK = ABS( X( K, J ) )
00313                DO 40 I = 1, K - 1
00314                   WORK( I ) = WORK( I ) + ABS( A( I, K ) )*XK
00315                   S = S + ABS( A( I, K ) )*ABS( X( I, J ) )
00316    40          CONTINUE
00317                WORK( K ) = WORK( K ) + ABS( A( K, K ) )*XK + S
00318    50       CONTINUE
00319          ELSE
00320             DO 70 K = 1, N
00321                S = ZERO
00322                XK = ABS( X( K, J ) )
00323                WORK( K ) = WORK( K ) + ABS( A( K, K ) )*XK
00324                DO 60 I = K + 1, N
00325                   WORK( I ) = WORK( I ) + ABS( A( I, K ) )*XK
00326                   S = S + ABS( A( I, K ) )*ABS( X( I, J ) )
00327    60          CONTINUE
00328                WORK( K ) = WORK( K ) + S
00329    70       CONTINUE
00330          END IF
00331          S = ZERO
00332          DO 80 I = 1, N
00333             IF( WORK( I ).GT.SAFE2 ) THEN
00334                S = MAX( S, ABS( WORK( N+I ) ) / WORK( I ) )
00335             ELSE
00336                S = MAX( S, ( ABS( WORK( N+I ) )+SAFE1 ) /
00337      $             ( WORK( I )+SAFE1 ) )
00338             END IF
00339    80    CONTINUE
00340          BERR( J ) = S
00341 *
00342 *        Test stopping criterion. Continue iterating if
00343 *           1) The residual BERR(J) is larger than machine epsilon, and
00344 *           2) BERR(J) decreased by at least a factor of 2 during the
00345 *              last iteration, and
00346 *           3) At most ITMAX iterations tried.
00347 *
00348          IF( BERR( J ).GT.EPS .AND. TWO*BERR( J ).LE.LSTRES .AND.
00349      $       COUNT.LE.ITMAX ) THEN
00350 *
00351 *           Update solution and try again.
00352 *
00353             CALL SPOTRS( UPLO, N, 1, AF, LDAF, WORK( N+1 ), N, INFO )
00354             CALL SAXPY( N, ONE, WORK( N+1 ), 1, X( 1, J ), 1 )
00355             LSTRES = BERR( J )
00356             COUNT = COUNT + 1
00357             GO TO 20
00358          END IF
00359 *
00360 *        Bound error from formula
00361 *
00362 *        norm(X - XTRUE) / norm(X) .le. FERR =
00363 *        norm( abs(inv(A))*
00364 *           ( abs(R) + NZ*EPS*( abs(A)*abs(X)+abs(B) ))) / norm(X)
00365 *
00366 *        where
00367 *          norm(Z) is the magnitude of the largest component of Z
00368 *          inv(A) is the inverse of A
00369 *          abs(Z) is the componentwise absolute value of the matrix or
00370 *             vector Z
00371 *          NZ is the maximum number of nonzeros in any row of A, plus 1
00372 *          EPS is machine epsilon
00373 *
00374 *        The i-th component of abs(R)+NZ*EPS*(abs(A)*abs(X)+abs(B))
00375 *        is incremented by SAFE1 if the i-th component of
00376 *        abs(A)*abs(X) + abs(B) is less than SAFE2.
00377 *
00378 *        Use SLACN2 to estimate the infinity-norm of the matrix
00379 *           inv(A) * diag(W),
00380 *        where W = abs(R) + NZ*EPS*( abs(A)*abs(X)+abs(B) )))
00381 *
00382          DO 90 I = 1, N
00383             IF( WORK( I ).GT.SAFE2 ) THEN
00384                WORK( I ) = ABS( WORK( N+I ) ) + NZ*EPS*WORK( I )
00385             ELSE
00386                WORK( I ) = ABS( WORK( N+I ) ) + NZ*EPS*WORK( I ) + SAFE1
00387             END IF
00388    90    CONTINUE
00389 *
00390          KASE = 0
00391   100    CONTINUE
00392          CALL SLACN2( N, WORK( 2*N+1 ), WORK( N+1 ), IWORK, FERR( J ),
00393      $                KASE, ISAVE )
00394          IF( KASE.NE.0 ) THEN
00395             IF( KASE.EQ.1 ) THEN
00396 *
00397 *              Multiply by diag(W)*inv(A**T).
00398 *
00399                CALL SPOTRS( UPLO, N, 1, AF, LDAF, WORK( N+1 ), N, 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 SPOTRS( UPLO, N, 1, AF, LDAF, WORK( N+1 ), N, INFO )
00411             END IF
00412             GO TO 100
00413          END IF
00414 *
00415 *        Normalize error.
00416 *
00417          LSTRES = ZERO
00418          DO 130 I = 1, N
00419             LSTRES = MAX( LSTRES, ABS( X( I, J ) ) )
00420   130    CONTINUE
00421          IF( LSTRES.NE.ZERO )
00422      $      FERR( J ) = FERR( J ) / LSTRES
00423 *
00424   140 CONTINUE
00425 *
00426       RETURN
00427 *
00428 *     End of SPORFS
00429 *
00430       END
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