DLAGTF (3lapack)

factorize the matrix (T - lambda*I), where T is an n by n tridiagonal matrix and lambda is a scalar, as T (3lapack)
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

  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.

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 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)
          = 0   : successful exit



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