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# sgtrfs

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

SYNOPSIS
SUBROUTINE SGTRFS( TRANS, N, NRHS, DL, D, DU, DLF, DF, DUF,
DU2, IPIV, B, LDB, X, LDX, FERR, BERR,
WORK, IWORK, INFO )

CHARACTER      TRANS

INTEGER        INFO, LDB, LDX, N, NRHS

INTEGER        IPIV( * ), IWORK( * )

REAL           B( LDB, * ), BERR( * ), D( * ), DF( * ),
DL( * ), DLF( * ), DU( * ), DU2( * ),
DUF( * ), FERR( * ), WORK( * ), X( LDX, *
)

PURPOSE
SGTRFS improves the computed solution to a system of linear
equations when the coefficient matrix is tridiagonal, 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 = Tran-
spose)

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.

DL      (input) REAL array, dimension (N-1)
The (n-1) subdiagonal elements of A.

D       (input) REAL array, dimension (N)
The diagonal elements of A.

DU      (input) REAL array, dimension (N-1)
The (n-1) superdiagonal elements of A.

DLF     (input) REAL array, dimension (N-1)
The (n-1) multipliers that define the matrix L from
the LU factorization of A as computed by SGTTRF.

DF      (input) REAL array, dimension (N)
The n diagonal elements of the upper triangular
matrix U from the LU factorization of A.

DUF     (input) REAL array, dimension (N-1)
The (n-1) elements of the first superdiagonal of U.

DU2     (input) REAL array, dimension (N-2)
The (n-2) elements of the second superdiagonal of U.

IPIV    (input) INTEGER array, dimension (N)
The pivot indices; for 1 <= i <= n, row i of the
matrix was interchanged with row IPIV(i).  IPIV(i)
will always be either i or i+1; IPIV(i) = i indi-
cates a row interchange was not required.

B       (input) REAL 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) REAL array, dimension (LDX,NRHS)
On entry, the solution matrix X, as computed by
SGTTRS.  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) REAL array, dimension (3*N)

IWORK   (workspace) INTEGER 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 refine-
ment.
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