DGTRFS - 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 DGTRFS(

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( * )

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

DGTRFS 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) DOUBLE PRECISION array, dimension (N-1)

The (n-1) subdiagonal elements of A.

D (input) DOUBLE PRECISION array, dimension (N)

The diagonal elements of A.

DU (input) DOUBLE PRECISION array, dimension (N-1)

The (n-1) superdiagonal elements of A.

DLF (input) DOUBLE PRECISION array, dimension (N-1)

The (n-1) multipliers that define the matrix L from the LU factorization of A as computed by DGTTRF.

DF (input) DOUBLE PRECISION array, dimension (N)

The n diagonal elements of the upper triangular matrix U from the LU factorization of A.

DUF (input) DOUBLE PRECISION array, dimension (N-1)

The (n-1) elements of the first superdiagonal of U.

DU2 (input) DOUBLE PRECISION 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) 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).

X (input/output) DOUBLE PRECISION array, dimension (LDX,NRHS)

On entry, the solution matrix X, as computed by DGTTRS. 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) DOUBLE PRECISION 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.