---
name: "zgbrfs"
section: "3lapack"
platform: "linux"
package: "liblapack-doc"
version: "3.2.1-8"
release: "Debian 6.0 (Squeeze)"
platforms: ["linux"]
url: "https://new.carta.tech/zgbrfs/3lapack"
---

# zgbrfs(3lapack)

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

## NAME

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

## 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.