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# LAPACK-3

## man page of LAPACK-3

### LAPACK-3: improves the computed solution to a system of linear equations when the coefficient matrix is tridiagonal, and provides

```NAME
LAPACK-3  -  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

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

CHARACTER      TRANS

INTEGER        INFO, LDB, LDX, N, NRHS

INTEGER        IPIV( * )

DOUBLE         PRECISION BERR( * ), FERR( * ), RWORK( * )

COMPLEX*16     B( LDB, * ), D( * ), DF( * ), DL( * ), DLF( * ), DU(
* ), DU2( * ), DUF( * ), WORK( * ), X( LDX, * )

PURPOSE
ZGTRFS  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)

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) COMPLEX*16 array, dimension (N-1)
The (n-1) subdiagonal elements of A.

D       (input) COMPLEX*16 array, dimension (N)
The diagonal elements of A.

DU      (input) COMPLEX*16 array, dimension (N-1)
The (n-1) superdiagonal elements of A.

DLF     (input) COMPLEX*16 array, dimension (N-1)
The (n-1) multipliers that define the matrix L from the
LU factorization of A as computed by ZGTTRF.

DF      (input) COMPLEX*16 array, dimension (N)
The n diagonal elements of the upper triangular matrix U from
the LU factorization of A.

DUF     (input) COMPLEX*16 array, dimension (N-1)
The (n-1) elements of the first superdiagonal of U.

DU2     (input) COMPLEX*16 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 indicates a row interchange was not
required.

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 ZGTTRS.
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.

LAPACK routine (version 3.2)     March 2011                       LAPACK-3(3)
```

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