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,
SYNOPSIS
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( *
)
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 trian
gular matrix with at most two non-zero super-diagonal ele
ments 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.
ARGUMENTS
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 ele
ments of the upper triangular matrix U of the fac
torization 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.
onal 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-diago
nal 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 approxi
mately the largest relative error in the elements
of T. For example, if the elements of T are cor
rect to about 4 significant figures, then TOL
should be set to about 5*10**(-4). If TOL is sup
plied 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
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