# dpbrfs.f

Section: LAPACK (3)
Updated: Tue Nov 14 2017
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dpbrfs.f

## SYNOPSIS

### Functions/Subroutines

subroutine dpbrfs (UPLO, N, KD, NRHS, AB, LDAB, AFB, LDAFB, B, LDB, X, LDX, FERR, BERR, WORK, IWORK, INFO)
DPBRFS

## Function/Subroutine Documentation

### subroutine dpbrfs (character UPLO, integer N, integer KD, integer NRHS, double precision, dimension( ldab, * ) AB, integer LDAB, double precision, dimension( ldafb, * ) AFB, integer LDAFB, double precision, dimension( ldb, * ) B, integer LDB, double precision, dimension( ldx, * ) X, integer LDX, double precision, dimension( * ) FERR, double precision, dimension( * ) BERR, double precision, dimension( * ) WORK, integer, dimension( * ) IWORK, integer INFO)

DPBRFS

Purpose:

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

Parameters:

UPLO

```          UPLO is CHARACTER*1
= 'U':  Upper triangle of A is stored;
= 'L':  Lower triangle of A is stored.
```

N

```          N is INTEGER
The order of the matrix A.  N >= 0.
```

KD

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

NRHS

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

AB

```          AB is DOUBLE PRECISION array, dimension (LDAB,N)
The upper or lower triangle of the symmetric 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

```          LDAB is INTEGER
The leading dimension of the array AB.  LDAB >= KD+1.
```

AFB

```          AFB is DOUBLE PRECISION array, dimension (LDAFB,N)
The triangular factor U or L from the Cholesky factorization
A = U**T*U or A = L*L**T of the band matrix A as computed by
DPBTRF, in the same storage format as A (see AB).
```

LDAFB

```          LDAFB is INTEGER
The leading dimension of the array AFB.  LDAFB >= KD+1.
```

B

```          B is DOUBLE PRECISION array, dimension (LDB,NRHS)
The right hand side matrix B.
```

LDB

```          LDB is INTEGER
The leading dimension of the array B.  LDB >= max(1,N).
```

X

```          X is DOUBLE PRECISION array, dimension (LDX,NRHS)
On entry, the solution matrix X, as computed by DPBTRS.
On exit, the improved solution matrix X.
```

LDX

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

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.
```

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).
```

WORK

```          WORK is DOUBLE PRECISION array, dimension (3*N)
```

IWORK

```          IWORK is INTEGER array, dimension (N)
```

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