sptrfs.f

          SUBROUTINE SPTRFS( N, NRHS, D, E, DF, EF, B, LDB, X, LDX, FERR,
         $                   BERR, WORK, INFO )
    *
    *  -- LAPACK routine (version 3.1) --
    *     Univ. of Tennessee, Univ. of California Berkeley and NAG Ltd..
    *     November 2006
    *
    *     .. Scalar Arguments ..
          INTEGER            INFO, LDB, LDX, N, NRHS
    *     ..
    *     .. Array Arguments ..
          REAL               B( LDB, * ), BERR( * ), D( * ), DF( * ),
         $                   E( * ), EF( * ), FERR( * ), WORK( * ),
         $                   X( LDX, * )
    *     ..
    *
    *  Purpose
    *  =======
    *
    *  SPTRFS improves the computed solution to a system of linear
    *  equations when the coefficient matrix is symmetric positive definite
    *  and tridiagonal, and provides error bounds and backward error
    *  estimates for the solution.
    *
    *  Arguments
    *  =========
    *
    *  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 matrix B.  NRHS >= 0.
    *
    *  D       (input) REAL array, dimension (N)
    *          The n diagonal elements of the tridiagonal matrix A.
    *
    *  E       (input) REAL array, dimension (N-1)
    *          The (n-1) subdiagonal elements of the tridiagonal matrix A.
    *
    *  DF      (input) REAL array, dimension (N)
    *          The n diagonal elements of the diagonal matrix D from the
    *          factorization computed by SPTTRF.
    *
    *  EF      (input) REAL array, dimension (N-1)
    *          The (n-1) subdiagonal elements of the unit bidiagonal factor
    *          L from the factorization computed by SPTTRF.
    *
    *  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 SPTTRS.
    *          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 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).
    *
    *  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 (2*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 ..
          INTEGER            COUNT, I, IX, J, NZ
          REAL               BI, CX, DX, EPS, EX, LSTRES, S, SAFE1, SAFE2,
         $                   SAFMIN
    *     ..
    *     .. External Subroutines ..
          EXTERNAL           SAXPY, SPTTRS, XERBLA
    *     ..
    *     .. Intrinsic Functions ..
          INTRINSIC          ABS, MAX
    *     ..
    *     .. External Functions ..
          INTEGER            ISAMAX
          REAL               SLAMCH
          EXTERNAL           ISAMAX, SLAMCH
    *     ..
    *     .. Executable Statements ..
    *
    *     Test the input parameters.
    *
          INFO = 0
          IF( N.LT.0 ) THEN
             INFO = -1
          ELSE IF( NRHS.LT.0 ) THEN
             INFO = -2
          ELSE IF( LDB.LT.MAX( 1, N ) ) THEN
             INFO = -8
          ELSE IF( LDX.LT.MAX( 1, N ) ) THEN
             INFO = -10
          END IF
          IF( INFO.NE.0 ) THEN
             CALL XERBLA( 'SPTRFS', -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 = 4
          EPS = SLAMCH( 'Epsilon' )
          SAFMIN = SLAMCH( 'Safe minimum' )
          SAFE1 = NZ*SAFMIN
          SAFE2 = SAFE1 / EPS
    *
    *     Do for each right hand side
    *
          DO 90 J = 1, NRHS
    *
             COUNT = 1
             LSTRES = THREE
       20    CONTINUE
    *
    *        Loop until stopping criterion is satisfied.
    *
    *        Compute residual R = B - A * X.  Also compute
    *        abs(A)*abs(x) + abs(b) for use in the backward error bound.
    *
             IF( N.EQ.1 ) THEN
                BI = B( 1, J )
                DX = D( 1 )*X( 1, J )
                WORK( N+1 ) = BI - DX
                WORK( 1 ) = ABS( BI ) + ABS( DX )
             ELSE
                BI = B( 1, J )
                DX = D( 1 )*X( 1, J )
                EX = E( 1 )*X( 2, J )
                WORK( N+1 ) = BI - DX - EX
                WORK( 1 ) = ABS( BI ) + ABS( DX ) + ABS( EX )
                DO 30 I = 2, N - 1
                   BI = B( I, J )
                   CX = E( I-1 )*X( I-1, J )
                   DX = D( I )*X( I, J )
                   EX = E( I )*X( I+1, J )
                   WORK( N+I ) = BI - CX - DX - EX
                   WORK( I ) = ABS( BI ) + ABS( CX ) + ABS( DX ) + ABS( EX )
       30       CONTINUE
                BI = B( N, J )
                CX = E( N-1 )*X( N-1, J )
                DX = D( N )*X( N, J )
                WORK( N+N ) = BI - CX - DX
                WORK( N ) = ABS( BI ) + ABS( CX ) + ABS( DX )
             END IF
    *
    *        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.
    *
             S = ZERO
             DO 40 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
       40    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 SPTTRS( N, 1, DF, EF, 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.
    *
             DO 50 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
       50    CONTINUE
             IX = ISAMAX( N, WORK, 1 )
             FERR( J ) = WORK( IX )
    *
    *        Estimate the norm of inv(A).
    *
    *        Solve M(A) * x = e, where M(A) = (m(i,j)) is given by
    *
    *           m(i,j) =  abs(A(i,j)), i = j,
    *           m(i,j) = -abs(A(i,j)), i .ne. j,
    *
    *        and e = [ 1, 1, ..., 1 ]'.  Note M(A) = M(L)*D*M(L)'.
    *
    *        Solve M(L) * x = e.
    *
             WORK( 1 ) = ONE
             DO 60 I = 2, N
                WORK( I ) = ONE + WORK( I-1 )*ABS( EF( I-1 ) )
       60    CONTINUE
    *
    *        Solve D * M(L)' * x = b.
    *
             WORK( N ) = WORK( N ) / DF( N )
             DO 70 I = N - 1, 1, -1
                WORK( I ) = WORK( I ) / DF( I ) + WORK( I+1 )*ABS( EF( I ) )
       70    CONTINUE
    *
    *        Compute norm(inv(A)) = max(x(i)), 1<=i<=n.
    *
             IX = ISAMAX( N, WORK, 1 )
             FERR( J ) = FERR( J )*ABS( WORK( IX ) )
    *
    *        Normalize error.
    *
             LSTRES = ZERO
             DO 80 I = 1, N
                LSTRES = MAX( LSTRES, ABS( X( I, J ) ) )
       80    CONTINUE
             IF( LSTRES.NE.ZERO )
         $      FERR( J ) = FERR( J ) / LSTRES
    *
       90 CONTINUE
    *
          RETURN
    *
    *     End of SPTRFS
    *
          END