ZPBRFS (3lapack)

improve the computed solution to a system of linear equations when the coefficient matrix is Hermitian positive definite and banded, and provides error bounds and backward error estimates for the solution

SYNOPSIS

  SUBROUTINE ZPBRFS( UPLO, N, KD, NRHS, AB, LDAB, AFB, LDAFB, B, LDB, X, LDX,
                     FERR, BERR, WORK, RWORK, INFO )

      CHARACTER      UPLO

      INTEGER        INFO, KD, LDAB, LDAFB, LDB, LDX, N, NRHS

      DOUBLE         PRECISION BERR( * ), FERR( * ), RWORK( * )

      COMPLEX*16     AB( LDAB, * ), AFB( LDAFB, * ), B( LDB, * ), WORK( * ),
                     X( LDX, * )

PURPOSE

  ZPBRFS improves the computed solution to a system of linear equations when
  the coefficient matrix is Hermitian positive definite and banded, 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.

  KD      (input) INTEGER
          The number of superdiagonals of the matrix A if UPLO = 'U', or the
          number of subdiagonals if UPLO = 'L'.  KD >= 0.

  NRHS    (input) INTEGER
          The number of right hand sides, i.e., the number of columns of the
          matrices B and X.  NRHS >= 0.

  AB      (input) DOUBLE PRECISION array, dimension (LDAB,N)
          The upper or lower triangle of the Hermitian band matrix A, stored
          in the first KD+1 rows of the array.  The j-th column of A is
          stored in the j-th column of the array AB as follows: if UPLO =
          'U', AB(kd+1+i-j,j) = A(i,j) for max(1,j-kd)<=i<=j; if UPLO = 'L',
          AB(1+i-j,j)    = A(i,j) for j<=i<=min(n,j+kd).

  LDAB    (input) INTEGER
          The leading dimension of the array AB.  LDAB >= KD+1.

  AFB     (input) COMPLEX*16 array, dimension (LDAFB,N)
          The triangular factor U or L from the Cholesky factorization A =
          U**H*U or A = L*L**H of the band matrix A as computed by ZPBTRF, in
          the same storage format as A (see AB).

  LDAFB   (input) INTEGER
          The leading dimension of the array AFB.  LDAFB >= KD+1.

  B       (input) COMPLEX*16 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) COMPLEX*16 array, dimension (LDX,NRHS)
          On entry, the solution matrix X, as computed by ZPBTRS.  On exit,
          the improved solution matrix X.

  LDX     (input) INTEGER
          The leading dimension of the array X.  LDX >= max(1,N).

  FERR    (output) 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.

  BERR    (output) 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).

  WORK    (workspace) COMPLEX*16 array, dimension (2*N)

  RWORK   (workspace) DOUBLE PRECISION array, dimension (N)

  INFO    (output) INTEGER
          = 0:  successful exit
          < 0:  if INFO = -i, the i-th argument had an illegal value

PARAMETERS

  ITMAX is the maximum number of steps of iterative refinement.