# ?lagtf

Computes an LU factorization of a matrix T-λ*I, where T is a general tridiagonal matrix, and λ is a scalar, using partial pivoting with row interchanges.

## Syntax

call slagtf( n, a, lambda, b, c, tol, d, in, info )

call dlagtf( n, a, lambda, b, c, tol, d, in, info )

## Description

This routine factorizes the matrix (`T - lambda*I`), where T is an n-by-n tridiagonal matrix and lambda is a scalar, as

`T - lambda*I = P*L*U`,

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 ?lagtf may be used, in conjunction with ?lagts, to obtain eigenvectors of T by inverse iteration.

## Input Parameters

n

  INTEGER. The order of the matrix T (`n ≥ 0`).

a, b, c

  REAL for slagtf

  DOUBLE PRECISION for dlagtf

  Arrays, dimension a(n), b(n-1), c(n-1):

  On entry, a(*) must contain the diagonal elements of the matrix T.

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

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

tol

  REAL for slagtf

  DOUBLE PRECISION for dlagtf

  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.

## Output Parameters

a

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

b

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

c

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

d

  REAL for slagtf

  DOUBLE PRECISION for dlagtf

  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

  INTEGER.

  Array, dimension (n).

  On exit, in contains details of the permutation matrix p. If an interchange occurred at the k-th 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)) ≤ norm((T - lambda*I)(j))*tol`,

  where norm( A(j)) denotes the sum of the absolute values of the j-th 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

  INTEGER.

  If `info = 0`, the execution is successful.

  If `info = -k`, the k-th parameter had an illegal value.

Parent topic: Auxiliary Routines