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

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, * )

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.

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 = 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) 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 indicates 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 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) REAL 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) 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

ITMAX is the maximum number of steps of iterative refinement.