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

# zpbrfs(3lapack)

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

## NAME

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

## 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.