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# cptrfs

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

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

CHARACTER      UPLO

INTEGER        INFO, LDB, LDX, N, NRHS

REAL           BERR( * ), D( * ), DF( * ), FERR( * ),
RWORK( * )

COMPLEX        B( LDB, * ), E( * ), EF( * ), WORK( * ),
X( LDX, * )

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

ARGUMENTS
UPLO    (input) CHARACTER*1
Specifies whether the superdiagonal or the subdiago-
nal of the tridiagonal matrix A is stored and the
form of the factorization:
= 'U':  E is the superdiagonal of A, and A =
U**H*D*U;
= 'L':  E is the subdiagonal of A, and A = L*D*L**H.
(The two forms are equivalent if A is real.)

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) REAL array, dimension (N)
The n real diagonal elements of the tridiagonal
matrix A.

E       (input) COMPLEX array, dimension (N-1)
The (n-1) off-diagonal elements of the tridiagonal
matrix A (see UPLO).

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

EF      (input) COMPLEX array, dimension (N-1)
The (n-1) off-diagonal elements of the unit bidiago-
nal factor U or L from the factorization computed by
CPTTRF (see UPLO).

B       (input) COMPLEX 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 array, dimension (LDX,NRHS)
On entry, the solution matrix X, as computed by
CPTTRS.  On exit, the improved solution matrix X.

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

FERR    (output) REAL 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) REAL 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) COMPLEX array, dimension (N)

RWORK   (workspace) REAL 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.
```