DLAGTF - factorize the matrix (T - lambda*I), where T is an n by n tridiagonal matrix and lambda is a scalar, as T - lambda*I = PLU,

SUBROUTINE DLAGTF(

N, A, LAMBDA, B, C, TOL, D, IN, INFO )

INTEGER INFO, N

DOUBLE PRECISION LAMBDA, TOL

INTEGER IN( * )

DOUBLE PRECISION A( * ), B( * ), C( * ), D( * )

DLAGTF factorizes the matrix (T - lambda*I), where T is an n by n tridiagonal matrix and lambda is a scalar, as T - lambda*I = PLU, where P is a permutation matrix, L is a unit lower tridiagonal matrix with at most one non-zero sub-diagonal elements per column and U is an upper triangular matrix with at most two non-zero super-diagonal elements per column.

The factorization is obtained by Gaussian elimination with partial pivoting and implicit row scaling.

The parameter LAMBDA is included in the routine so that DLAGTF may be used, in conjunction with DLAGTS, to obtain eigenvectors of T by inverse iteration.

N (input) INTEGER

The order of the matrix T.

A (input/output) DOUBLE PRECISION array, dimension (N)

On entry, A must contain the diagonal elements of T.

On exit, A is overwritten by the n diagonal elements of the upper triangular matrix U of the factorization of T.

LAMBDA (input) DOUBLE PRECISION

On entry, the scalar lambda.

B (input/output) DOUBLE PRECISION array, dimension (N-1)

On entry, B must contain the (n-1) super-diagonal elements of T.

On exit, B is overwritten by the (n-1) super-diagonal elements of the matrix U of the factorization of T.

C (input/output) DOUBLE PRECISION array, dimension (N-1)

On entry, C must contain the (n-1) sub-diagonal elements of T.

On exit, C is overwritten by the (n-1) sub-diagonal elements of the matrix L of the factorization of T.

TOL (input) DOUBLE PRECISION

On entry, a relative tolerance used to indicate whether or not the matrix (T - lambda*I) is nearly singular. TOL should normally be chose as approximately the largest relative error in the elements of T. For example, if the elements of T are correct to about 4 significant figures, then TOL should be set to about 5*10**(-4). If TOL is supplied as less than eps, where eps is the relative machine precision, then the value eps is used in place of TOL.

D (output) DOUBLE PRECISION array, dimension (N-2)

On exit, D is overwritten by the (n-2) second super-diagonal elements of the matrix U of the factorization of T.

IN (output) INTEGER array, dimension (N)

On exit, IN contains details of the permutation matrix P. If an interchange occurred at the kth step of the elimination, then IN(k) = 1, otherwise IN(k) = 0. The element IN(n) returns the smallest positive integer j such that

abs( u(j,j) ).le. norm( (T - lambda*I)(j) )*TOL,

where norm( A(j) ) denotes the sum of the absolute values of the jth row of the matrix A. If no such j exists then IN(n) is returned as zero. If IN(n) is returned as positive, then a diagonal element of U is small, indicating that (T - lambda*I) is singular or nearly singular,

INFO (output) INTEGER

= 0 : successful exit