ZGBRFS (3lapack)

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

SYNOPSIS

  SUBROUTINE ZGBRFS( TRANS, N, KL, KU, NRHS, AB, LDAB, AFB, LDAFB, IPIV, B,
                     LDB, X, LDX, FERR, BERR, WORK, RWORK, INFO )

      CHARACTER      TRANS

      INTEGER        INFO, KL, KU, LDAB, LDAFB, LDB, LDX, N, NRHS

      INTEGER        IPIV( * )

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

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

PURPOSE

  ZGBRFS improves the computed solution to a system of linear equations when
  the coefficient matrix is banded, and provides error bounds and backward
  error estimates for the solution.

ARGUMENTS

  TRANS   (input) CHARACTER*1
          Specifies the form of the system of equations:
          = 'N':  A * X = B     (No transpose)
          = 'T':  A**T * X = B  (Transpose)
          = 'C':  A**H * X = B  (Conjugate transpose)

  N       (input) INTEGER
          The order of the matrix A.  N >= 0.

  KL      (input) INTEGER
          The number of subdiagonals within the band of A.  KL >= 0.

  KU      (input) INTEGER
          The number of superdiagonals within the band of A.  KU >= 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) COMPLEX*16 array, dimension (LDAB,N)
          The original band matrix A, stored in rows 1 to KL+KU+1.  The j-th
          column of A is stored in the j-th column of the array AB as
          follows: AB(ku+1+i-j,j) = A(i,j) for max(1,j-ku)<=i<=min(n,j+kl).

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

  AFB     (input) COMPLEX*16 array, dimension (LDAFB,N)
          Details of the LU factorization of the band matrix A, as computed
          by ZGBTRF.  U is stored as an upper triangular band matrix with
          KL+KU superdiagonals in rows 1 to KL+KU+1, and the multipliers used
          during the factorization are stored in rows KL+KU+2 to 2*KL+KU+1.

  LDAFB   (input) INTEGER
          The leading dimension of the array AFB.  LDAFB >= 2*KL*KU+1.

  IPIV    (input) INTEGER array, dimension (N)
          The pivot indices from ZGBTRF; for 1<=i<=N, row i of the matrix was
          interchanged with row IPIV(i).

  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 ZGBTRS.  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.