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# dptrfs

```
NAME
DPTRFS - improve the computed solution to a system of linear
equations when the coefficient matrix is symmetric positive
definite and tridiagonal, and provides error bounds and
backward error estimates for the solution

SYNOPSIS
SUBROUTINE DPTRFS( N, NRHS, D, E, DF, EF, B, LDB, X, LDX,
FERR, BERR, WORK, INFO )

INTEGER        INFO, LDB, LDX, N, NRHS

DOUBLE         PRECISION B( LDB, * ), BERR( * ), D( * ),
DF( * ), E( * ), EF( * ), FERR( * ),
WORK( * ), X( LDX, * )

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

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

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

D       (input) DOUBLE PRECISION array, dimension (N)
The n diagonal elements of the tridiagonal matrix A.

E       (input) DOUBLE PRECISION array, dimension (N-1)
The (n-1) subdiagonal elements of the tridiagonal
matrix A.

DF      (input) DOUBLE PRECISION array, dimension (N)
The n diagonal elements of the diagonal matrix D
from the factorization computed by DPTTRF.

EF      (input) DOUBLE PRECISION array, dimension (N-1)
The (n-1) subdiagonal elements of the unit bidiago-
nal factor L from the factorization computed by
DPTTRF.

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

LDB     (input) INTEGER
The leading dimension of the array B.  LDB >=

max(1,N).

(LDX,NRHS)
X       (input/output) DOUBLE PRECISION array, dimension
On entry, the solution matrix X, as computed by
DPTTRS.  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 bounds for each solution
vector X(j) (the j-th column of the solution matrix
X).  If XTRUE is the true solution, FERR(j) bounds
the magnitude of the largest entry in (X(j) - XTRUE)
divided by the magnitude of the largest entry in
X(j).  The quality of the error bound depends on the
quality of the estimate of norm(inv(A)) computed in
the code; if the estimate of norm(inv(A)) is accu-
rate, the error bound is guaranteed.

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 entry of A or B that makes X(j) an
exact solution).

WORK    (workspace) DOUBLE PRECISION array, dimension (2*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 refine-
ment.
```