zsprfs()

subroutine zsprfs	(	character	uplo,
		integer	n,
		integer	nrhs,
		complex*16, dimension( * )	ap,
		complex*16, dimension( * )	afp,
		integer, dimension( * )	ipiv,
		complex*16, dimension( ldb, * )	b,
		integer	ldb,
		complex*16, dimension( ldx, * )	x,
		integer	ldx,
		double precision, dimension( * )	ferr,
		double precision, dimension( * )	berr,
		complex*16, dimension( * )	work,
		double precision, dimension( * )	rwork,
		integer	info )

ZSPRFS

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Purpose:

!>
!> ZSPRFS improves the computed solution to a system of linear
!> equations when the coefficient matrix is symmetric indefinite
!> and packed, and provides error bounds and backward error estimates
!> for the solution.
!> 

Parameters

[in]	UPLO	!> UPLO is CHARACTER*1 !> = 'U': Upper triangle of A is stored; !> = 'L': Lower triangle of A is stored. !>
[in]	N	!> N is INTEGER !> The order of the matrix A. N >= 0. !>
[in]	NRHS	!> NRHS is INTEGER !> The number of right hand sides, i.e., the number of columns !> of the matrices B and X. NRHS >= 0. !>
[in]	AP	!> AP is COMPLEX*16 array, dimension (N*(N+1)/2) !> The upper or lower triangle of the symmetric matrix A, packed !> columnwise in a linear array. The j-th column of A is stored !> in the array AP as follows: !> if UPLO = 'U', AP(i + (j-1)*j/2) = A(i,j) for 1<=i<=j; !> if UPLO = 'L', AP(i + (j-1)*(2*n-j)/2) = A(i,j) for j<=i<=n. !>
[in]	AFP	!> AFP is COMPLEX*16 array, dimension (N*(N+1)/2) !> The factored form of the matrix A. AFP contains the block !> diagonal matrix D and the multipliers used to obtain the !> factor U or L from the factorization A = U*D*U**T or !> A = L*D*L**T as computed by ZSPTRF, stored as a packed !> triangular matrix. !>
[in]	IPIV	!> IPIV is INTEGER array, dimension (N) !> Details of the interchanges and the block structure of D !> as determined by ZSPTRF. !>
[in]	B	!> B is COMPLEX*16 array, dimension (LDB,NRHS) !> The right hand side matrix B. !>
[in]	LDB	!> LDB is INTEGER !> The leading dimension of the array B. LDB >= max(1,N). !>
[in,out]	X	!> X is COMPLEX*16 array, dimension (LDX,NRHS) !> On entry, the solution matrix X, as computed by ZSPTRS. !> On exit, the improved solution matrix X. !>
[in]	LDX	!> LDX is INTEGER !> The leading dimension of the array X. LDX >= max(1,N). !>
[out]	FERR	!> FERR is DOUBLE PRECISION 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. !>
[out]	BERR	!> BERR is DOUBLE PRECISION 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). !>
[out]	WORK	!> WORK is COMPLEX*16 array, dimension (2*N) !>
[out]	RWORK	!> RWORK is DOUBLE PRECISION array, dimension (N) !>
[out]	INFO	!> INFO is 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.
!> 

Author

Univ. of Tennessee

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

Definition at line 176 of file zsprfs.f.

*
*  -- LAPACK computational routine --
*  -- LAPACK is a software package provided by Univ. of Tennessee,    --
*  -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
*
*     .. Scalar Arguments ..
      CHARACTER          UPLO
      INTEGER            INFO, LDB, LDX, N, NRHS
*     ..
*     .. Array Arguments ..
      INTEGER            IPIV( * )
      DOUBLE PRECISION   BERR( * ), FERR( * ), RWORK( * )
      COMPLEX*16         AFP( * ), AP( * ), B( LDB, * ), WORK( * ),
     $                   X( LDX, * )
*     ..
*
*  =====================================================================
*
*     .. Parameters ..
      INTEGER            ITMAX
      parameter( itmax = 5 )
      DOUBLE PRECISION   ZERO
      parameter( zero = 0.0d+0 )
      COMPLEX*16         ONE
      parameter( one = ( 1.0d+0, 0.0d+0 ) )
      DOUBLE PRECISION   TWO
      parameter( two = 2.0d+0 )
      DOUBLE PRECISION   THREE
      parameter( three = 3.0d+0 )
*     ..
*     .. Local Scalars ..
      LOGICAL            UPPER
      INTEGER            COUNT, I, IK, J, K, KASE, KK, NZ
      DOUBLE PRECISION   EPS, LSTRES, S, SAFE1, SAFE2, SAFMIN, XK
      COMPLEX*16         ZDUM
*     ..
*     .. Local Arrays ..
      INTEGER            ISAVE( 3 )
*     ..
*     .. External Subroutines ..
      EXTERNAL           xerbla, zaxpy, zcopy, zlacn2, zspmv,
     $                   zsptrs
*     ..
*     .. Intrinsic Functions ..
      INTRINSIC          abs, dble, dimag, max
*     ..
*     .. External Functions ..
      LOGICAL            LSAME
      DOUBLE PRECISION   DLAMCH
      EXTERNAL           lsame, dlamch
*     ..
*     .. Statement Functions ..
      DOUBLE PRECISION   CABS1
*     ..
*     .. Statement Function definitions ..
      cabs1( zdum ) = abs( dble( zdum ) ) + abs( dimag( zdum ) )
*     ..
*     .. 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( 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( 'ZSPRFS', -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 = dlamch( 'Epsilon' )
      safmin = dlamch( '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 zcopy( n, b( 1, j ), 1, work, 1 )
         CALL zspmv( uplo, n, -one, ap, x( 1, j ), 1, one, work, 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
            rwork( i ) = cabs1( b( i, j ) )
   30    CONTINUE
*
*        Compute abs(A)*abs(X) + abs(B).
*
         kk = 1
         IF( upper ) THEN
            DO 50 k = 1, n
               s = zero
               xk = cabs1( x( k, j ) )
               ik = kk
               DO 40 i = 1, k - 1
                  rwork( i ) = rwork( i ) + cabs1( ap( ik ) )*xk
                  s = s + cabs1( ap( ik ) )*cabs1( x( i, j ) )
                  ik = ik + 1
   40          CONTINUE
               rwork( k ) = rwork( k ) + cabs1( ap( kk+k-1 ) )*xk + s
               kk = kk + k
   50       CONTINUE
         ELSE
            DO 70 k = 1, n
               s = zero
               xk = cabs1( x( k, j ) )
               rwork( k ) = rwork( k ) + cabs1( ap( kk ) )*xk
               ik = kk + 1
               DO 60 i = k + 1, n
                  rwork( i ) = rwork( i ) + cabs1( ap( ik ) )*xk
                  s = s + cabs1( ap( ik ) )*cabs1( x( i, j ) )
                  ik = ik + 1
   60          CONTINUE
               rwork( k ) = rwork( k ) + s
               kk = kk + ( n-k+1 )
   70       CONTINUE
         END IF
         s = zero
         DO 80 i = 1, n
            IF( rwork( i ).GT.safe2 ) THEN
               s = max( s, cabs1( work( i ) ) / rwork( i ) )
            ELSE
               s = max( s, ( cabs1( work( i ) )+safe1 ) /
     $             ( rwork( 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 zsptrs( uplo, n, 1, afp, ipiv, work, n, info )
            CALL zaxpy( n, one, work, 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 ZLACN2 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( rwork( i ).GT.safe2 ) THEN
               rwork( i ) = cabs1( work( i ) ) + nz*eps*rwork( i )
            ELSE
               rwork( i ) = cabs1( work( i ) ) + nz*eps*rwork( i ) +
     $                      safe1
            END IF
   90    CONTINUE
*
         kase = 0
  100    CONTINUE
         CALL zlacn2( n, work( n+1 ), work, ferr( j ), kase, isave )
         IF( kase.NE.0 ) THEN
            IF( kase.EQ.1 ) THEN
*
*              Multiply by diag(W)*inv(A**T).
*
               CALL zsptrs( uplo, n, 1, afp, ipiv, work, n, info )
               DO 110 i = 1, n
                  work( i ) = rwork( i )*work( i )
  110          CONTINUE
            ELSE IF( kase.EQ.2 ) THEN
*
*              Multiply by inv(A)*diag(W).
*
               DO 120 i = 1, n
                  work( i ) = rwork( i )*work( i )
  120          CONTINUE
               CALL zsptrs( uplo, n, 1, afp, ipiv, work, n, info )
            END IF
            GO TO 100
         END IF
*
*        Normalize error.
*
         lstres = zero
         DO 130 i = 1, n
            lstres = max( lstres, cabs1( x( i, j ) ) )
  130    CONTINUE
         IF( lstres.NE.zero )
     $      ferr( j ) = ferr( j ) / lstres
*
  140 CONTINUE
*
      RETURN
*
*     End of ZSPRFS
*

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