# dlagtf.f

Section: LAPACK (3)
Updated: Tue Nov 14 2017
Page Index

dlagtf.f

## SYNOPSIS

### Functions/Subroutines

subroutine dlagtf (N, A, LAMBDA, B, C, TOL, D, IN, INFO)
DLAGTF computes an LU factorization of a matrix T-λI, where T is a general tridiagonal matrix, and λ a scalar, using partial pivoting with row interchanges.

## Function/Subroutine Documentation

### subroutine dlagtf (integer N, double precision, dimension( * ) A, double precision LAMBDA, double precision, dimension( * ) B, double precision, dimension( * ) C, double precision TOL, double precision, dimension( * ) D, integer, dimension( * ) IN, integer INFO)

DLAGTF computes an LU factorization of a matrix T-λI, where T is a general tridiagonal matrix, and λ a scalar, using partial pivoting with row interchanges.

Purpose:

``` 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.
```

Parameters:

N

```          N is INTEGER
The order of the matrix T.
```

A

```          A is 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

```          LAMBDA is DOUBLE PRECISION
On entry, the scalar lambda.
```

B

```          B is 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

```          C is 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

```          TOL is 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

```          D is 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

```          IN is 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

```          INFO is INTEGER
= 0   : successful exit
.lt. 0: if INFO = -k, the kth argument had an illegal value
```

Author:

Univ. of Tennessee

Univ. of California Berkeley