ZPTRFS - 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

SUBROUTINE ZPTRFS(

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

CHARACTER UPLO

INTEGER INFO, LDB, LDX, N, NRHS

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

COMPLEX*16 B( LDB, * ), E( * ), EF( * ), WORK( * ), X( LDX, * )

ZPTRFS 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.

UPLO (input) CHARACTER*1

Specifies whether the superdiagonal or the subdiagonal 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) DOUBLE PRECISION array, dimension (N)

The n real diagonal elements of the tridiagonal matrix A.

E (input) COMPLEX*16 array, dimension (N-1)

The (n-1) off-diagonal elements of the tridiagonal matrix A (see UPLO).

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

The n diagonal elements of the diagonal matrix D from the factorization computed by ZPTTRF.

EF (input) COMPLEX*16 array, dimension (N-1)

The (n-1) off-diagonal elements of the unit bidiagonal factor U or L from the factorization computed by ZPTTRF (see UPLO).

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 ZPTTRS. 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 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).

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 (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

ITMAX is the maximum number of steps of iterative refinement.