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# slagtf

```
NAME
SLAGTF - 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,

SYNOPSIS
SUBROUTINE SLAGTF( N, A, LAMBDA, B, C, TOL, D, IN, INFO )

INTEGER        INFO, N

REAL           LAMBDA, TOL

INTEGER        IN( * )

REAL           A( * ), B( * ), C( * ), D( * )

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

where P is a permutation matrix, L is a unit lower tridiago-
nal 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
SLAGTF may be used, in conjunction with SLAGTS, to obtain
eigenvectors of T by inverse iteration.

ARGUMENTS
N       (input) INTEGER
The order of the matrix T.

A       (input/output) REAL 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 factoriza-
tion of T.

LAMBDA  (input) REAL
On entry, the scalar lambda.

B       (input/output) REAL 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 factoriza-
tion of T.

C       (input/output) REAL array, dimension (N-1)
On entry, C must contain the (n-1) sub-diagonal ele-
ments 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) REAL
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 approxi-
mately 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) REAL array, dimension (N-2)
On exit, D is overwritten by the (n-2) second
super-diagonal elements of the matrix U of the fac-
torization 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)
= 0   : successful exit
< 0: if INFO = -k, the kth argument had an illegal
value
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