sporfs.f

          SUBROUTINE SPORFS( UPLO, N, NRHS, A, LDA, AF, LDAF, B, LDB, X,
         $                   LDX, FERR, BERR, WORK, IWORK, INFO )
    *
    *  -- LAPACK routine (version 3.1) --
    *     Univ. of Tennessee, Univ. of California Berkeley and NAG Ltd..
    *     November 2006
    *
    *     Modified to call SLACN2 in place of SLACON, 7 Feb 03, SJH.
    *
    *     .. Scalar Arguments ..
          CHARACTER          UPLO
          INTEGER            INFO, LDA, LDAF, LDB, LDX, N, NRHS
    *     ..
    *     .. Array Arguments ..
          INTEGER            IWORK( * )
          REAL               A( LDA, * ), AF( LDAF, * ), B( LDB, * ),
         $                   BERR( * ), FERR( * ), WORK( * ), X( LDX, * )
    *     ..
    *
    *  Purpose
    *  =======
    *
    *  SPORFS improves the computed solution to a system of linear
    *  equations when the coefficient matrix is symmetric positive definite,
    *  and provides error bounds and backward error estimates for the
    *  solution.
    *
    *  Arguments
    *  =========
    *
    *  UPLO    (input) CHARACTER*1
    *          = 'U':  Upper triangle of A is stored;
    *          = 'L':  Lower triangle of A is stored.
    *
    *  N       (input) INTEGER
    *          The order of the matrix A.  N >= 0.
    *
    *  NRHS    (input) INTEGER
    *          The number of right hand sides, i.e., the number of columns
    *          of the matrices B and X.  NRHS >= 0.
    *
    *  A       (input) REAL array, dimension (LDA,N)
    *          The symmetric matrix A.  If UPLO = 'U', the leading N-by-N
    *          upper triangular part of A contains the upper triangular part
    *          of the matrix A, and the strictly lower triangular part of A
    *          is not referenced.  If UPLO = 'L', the leading N-by-N lower
    *          triangular part of A contains the lower triangular part of
    *          the matrix A, and the strictly upper triangular part of A is
    *          not referenced.
    *
    *  LDA     (input) INTEGER
    *          The leading dimension of the array A.  LDA >= max(1,N).
    *
    *  AF      (input) REAL array, dimension (LDAF,N)
    *          The triangular factor U or L from the Cholesky factorization
    *          A = U**T*U or A = L*L**T, as computed by SPOTRF.
    *
    *  LDAF    (input) INTEGER
    *          The leading dimension of the array AF.  LDAF >= max(1,N).
    *
    *  B       (input) REAL array, dimension (LDB,NRHS)
    *          The right hand side matrix B.
    *
    *  LDB     (input) INTEGER
    *          The leading dimension of the array B.  LDB >= max(1,N).
    *
    *  X       (input/output) REAL array, dimension (LDX,NRHS)
    *          On entry, the solution matrix X, as computed by SPOTRS.
    *          On exit, the improved solution matrix X.
    *
    *  LDX     (input) INTEGER
    *          The leading dimension of the array X.  LDX >= max(1,N).
    *
    *  FERR    (output) REAL array, dimension (NRHS)
    *          The estimated forward error bound for each solution vector
    *          X(j) (the j-th column of the solution matrix X).
    *          If XTRUE is the true solution corresponding to X(j), FERR(j)
    *          is an estimated upper bound for the magnitude of the largest
    *          element in (X(j) - XTRUE) divided by the magnitude of the
    *          largest element in X(j).  The estimate is as reliable as
    *          the estimate for RCOND, and is almost always a slight
    *          overestimate of the true error.
    *
    *  BERR    (output) REAL array, dimension (NRHS)
    *          The componentwise relative backward error of each solution
    *          vector X(j) (i.e., the smallest relative change in
    *          any element of A or B that makes X(j) an exact solution).
    *
    *  WORK    (workspace) REAL array, dimension (3*N)
    *
    *  IWORK   (workspace) INTEGER array, dimension (N)
    *
    *  INFO    (output) INTEGER
    *          = 0:  successful exit
    *          < 0:  if INFO = -i, the i-th argument had an illegal value
    *
    *  Internal Parameters
    *  ===================
    *
    *  ITMAX is the maximum number of steps of iterative refinement.
    *
    *  =====================================================================
    *
    *     .. Parameters ..
          INTEGER            ITMAX
          PARAMETER          ( ITMAX = 5 )
          REAL               ZERO
          PARAMETER          ( ZERO = 0.0E+0 )
          REAL               ONE
          PARAMETER          ( ONE = 1.0E+0 )
          REAL               TWO
          PARAMETER          ( TWO = 2.0E+0 )
          REAL               THREE
          PARAMETER          ( THREE = 3.0E+0 )
    *     ..
    *     .. Local Scalars ..
          LOGICAL            UPPER
          INTEGER            COUNT, I, J, K, KASE, NZ
          REAL               EPS, LSTRES, S, SAFE1, SAFE2, SAFMIN, XK
    *     ..
    *     .. Local Arrays ..
          INTEGER            ISAVE( 3 )
    *     ..
    *     .. External Subroutines ..
          EXTERNAL           SAXPY, SCOPY, SLACN2, SPOTRS, SSYMV, XERBLA
    *     ..
    *     .. Intrinsic Functions ..
          INTRINSIC          ABS, MAX
    *     ..
    *     .. External Functions ..
          LOGICAL            LSAME
          REAL               SLAMCH
          EXTERNAL           LSAME, SLAMCH
    *     ..
    *     .. Executable Statements ..
    *
    *     Test the input parameters.
    *
          INFO = 0
          UPPER = LSAME( UPLO, 'U' )
          IF( .NOT.UPPER .AND. .NOT.LSAME( UPLO, 'L' ) ) THEN
             INFO = -1
          ELSE IF( N.LT.0 ) THEN
             INFO = -2
          ELSE IF( NRHS.LT.0 ) THEN
             INFO = -3
          ELSE IF( LDA.LT.MAX( 1, N ) ) THEN
             INFO = -5
          ELSE IF( LDAF.LT.MAX( 1, N ) ) THEN
             INFO = -7
          ELSE IF( LDB.LT.MAX( 1, N ) ) THEN
             INFO = -9
          ELSE IF( LDX.LT.MAX( 1, N ) ) THEN
             INFO = -11
          END IF
          IF( INFO.NE.0 ) THEN
             CALL XERBLA( 'SPORFS', -INFO )
             RETURN
          END IF
    *
    *     Quick return if possible
    *
          IF( N.EQ.0 .OR. NRHS.EQ.0 ) THEN
             DO 10 J = 1, NRHS
                FERR( J ) = ZERO
                BERR( J ) = ZERO
       10    CONTINUE
             RETURN
          END IF
    *
    *     NZ = maximum number of nonzero elements in each row of A, plus 1
    *
          NZ = N + 1
          EPS = SLAMCH( 'Epsilon' )
          SAFMIN = SLAMCH( 'Safe minimum' )
          SAFE1 = NZ*SAFMIN
          SAFE2 = SAFE1 / EPS
    *
    *     Do for each right hand side
    *
          DO 140 J = 1, NRHS
    *
             COUNT = 1
             LSTRES = THREE
       20    CONTINUE
    *
    *        Loop until stopping criterion is satisfied.
    *
    *        Compute residual R = B - A * X
    *
             CALL SCOPY( N, B( 1, J ), 1, WORK( N+1 ), 1 )
             CALL SSYMV( UPLO, N, -ONE, A, LDA, X( 1, J ), 1, ONE,
         $               WORK( N+1 ), 1 )
    *
    *        Compute componentwise relative backward error from formula
    *
    *        max(i) ( abs(R(i)) / ( abs(A)*abs(X) + abs(B) )(i) )
    *
    *        where abs(Z) is the componentwise absolute value of the matrix
    *        or vector Z.  If the i-th component of the denominator is less
    *        than SAFE2, then SAFE1 is added to the i-th components of the
    *        numerator and denominator before dividing.
    *
             DO 30 I = 1, N
                WORK( I ) = ABS( B( I, J ) )
       30    CONTINUE
    *
    *        Compute abs(A)*abs(X) + abs(B).
    *
             IF( UPPER ) THEN
                DO 50 K = 1, N
                   S = ZERO
                   XK = ABS( X( K, J ) )
                   DO 40 I = 1, K - 1
                      WORK( I ) = WORK( I ) + ABS( A( I, K ) )*XK
                      S = S + ABS( A( I, K ) )*ABS( X( I, J ) )
       40          CONTINUE
                   WORK( K ) = WORK( K ) + ABS( A( K, K ) )*XK + S
       50       CONTINUE
             ELSE
                DO 70 K = 1, N
                   S = ZERO
                   XK = ABS( X( K, J ) )
                   WORK( K ) = WORK( K ) + ABS( A( K, K ) )*XK
                   DO 60 I = K + 1, N
                      WORK( I ) = WORK( I ) + ABS( A( I, K ) )*XK
                      S = S + ABS( A( I, K ) )*ABS( X( I, J ) )
       60          CONTINUE
                   WORK( K ) = WORK( K ) + S
       70       CONTINUE
             END IF
             S = ZERO
             DO 80 I = 1, N
                IF( WORK( I ).GT.SAFE2 ) THEN
                   S = MAX( S, ABS( WORK( N+I ) ) / WORK( I ) )
                ELSE
                   S = MAX( S, ( ABS( WORK( N+I ) )+SAFE1 ) /
         $             ( WORK( I )+SAFE1 ) )
                END IF
       80    CONTINUE
             BERR( J ) = S
    *
    *        Test stopping criterion. Continue iterating if
    *           1) The residual BERR(J) is larger than machine epsilon, and
    *           2) BERR(J) decreased by at least a factor of 2 during the
    *              last iteration, and
    *           3) At most ITMAX iterations tried.
    *
             IF( BERR( J ).GT.EPS .AND. TWO*BERR( J ).LE.LSTRES .AND.
         $       COUNT.LE.ITMAX ) THEN
    *
    *           Update solution and try again.
    *
                CALL SPOTRS( UPLO, N, 1, AF, LDAF, WORK( N+1 ), N, INFO )
                CALL SAXPY( N, ONE, WORK( N+1 ), 1, X( 1, J ), 1 )
                LSTRES = BERR( J )
                COUNT = COUNT + 1
                GO TO 20
             END IF
    *
    *        Bound error from formula
    *
    *        norm(X - XTRUE) / norm(X) .le. FERR =
    *        norm( abs(inv(A))*
    *           ( abs(R) + NZ*EPS*( abs(A)*abs(X)+abs(B) ))) / norm(X)
    *
    *        where
    *          norm(Z) is the magnitude of the largest component of Z
    *          inv(A) is the inverse of A
    *          abs(Z) is the componentwise absolute value of the matrix or
    *             vector Z
    *          NZ is the maximum number of nonzeros in any row of A, plus 1
    *          EPS is machine epsilon
    *
    *        The i-th component of abs(R)+NZ*EPS*(abs(A)*abs(X)+abs(B))
    *        is incremented by SAFE1 if the i-th component of
    *        abs(A)*abs(X) + abs(B) is less than SAFE2.
    *
    *        Use SLACN2 to estimate the infinity-norm of the matrix
    *           inv(A) * diag(W),
    *        where W = abs(R) + NZ*EPS*( abs(A)*abs(X)+abs(B) )))
    *
             DO 90 I = 1, N
                IF( WORK( I ).GT.SAFE2 ) THEN
                   WORK( I ) = ABS( WORK( N+I ) ) + NZ*EPS*WORK( I )
                ELSE
                   WORK( I ) = ABS( WORK( N+I ) ) + NZ*EPS*WORK( I ) + SAFE1
                END IF
       90    CONTINUE
    *
             KASE = 0
      100    CONTINUE
             CALL SLACN2( N, WORK( 2*N+1 ), WORK( N+1 ), IWORK, FERR( J ),
         $                KASE, ISAVE )
             IF( KASE.NE.0 ) THEN
                IF( KASE.EQ.1 ) THEN
    *
    *              Multiply by diag(W)*inv(A').
    *
                   CALL SPOTRS( UPLO, N, 1, AF, LDAF, WORK( N+1 ), N, INFO )
                   DO 110 I = 1, N
                      WORK( N+I ) = WORK( I )*WORK( N+I )
      110          CONTINUE
                ELSE IF( KASE.EQ.2 ) THEN
    *
    *              Multiply by inv(A)*diag(W).
    *
                   DO 120 I = 1, N
                      WORK( N+I ) = WORK( I )*WORK( N+I )
      120          CONTINUE
                   CALL SPOTRS( UPLO, N, 1, AF, LDAF, WORK( N+1 ), N, INFO )
                END IF
                GO TO 100
             END IF
    *
    *        Normalize error.
    *
             LSTRES = ZERO
             DO 130 I = 1, N
                LSTRES = MAX( LSTRES, ABS( X( I, J ) ) )
      130    CONTINUE
             IF( LSTRES.NE.ZERO )
         $      FERR( J ) = FERR( J ) / LSTRES
    *
      140 CONTINUE
    *
          RETURN
    *
    *     End of SPORFS
    *
          END