dlagts(l) [redhat man page]
DLAGTS(l) ) DLAGTS(l) NAME
DLAGTS - may be used to solve one of the systems of equations (T - lambda*I)*x = y or (T - lambda*I)'*x = y, SYNOPSIS
SUBROUTINE DLAGTS( JOB, N, A, B, C, D, IN, Y, TOL, INFO ) INTEGER INFO, JOB, N DOUBLE PRECISION TOL INTEGER IN( * ) DOUBLE PRECISION A( * ), B( * ), C( * ), D( * ), Y( * ) PURPOSE
DLAGTS may be used to solve one of the systems of equations (T - lambda*I)*x = y or (T - lambda*I)'*x = y, where T is an n by n tridiagonal matrix, for x, following the factorization of (T - lambda*I) as (T - lambda*I) = P*L*U , by routine DLAGTF. The choice of equation to be solved is controlled by the argument JOB, and in each case there is an option to perturb zero or very small diagonal elements of U, this option being intended for use in applications such as inverse iteration. ARGUMENTS
JOB (input) INTEGER Specifies the job to be performed by DLAGTS as follows: = 1: The equations (T - lambda*I)x = y are to be solved, but diagonal elements of U are not to be perturbed. = -1: The equa- tions (T - lambda*I)x = y are to be solved and, if overflow would otherwise occur, the diagonal elements of U are to be per- turbed. See argument TOL below. = 2: The equations (T - lambda*I)'x = y are to be solved, but diagonal elements of U are not to be perturbed. = -2: The equations (T - lambda*I)'x = y are to be solved and, if overflow would otherwise occur, the diagonal elements of U are to be perturbed. See argument TOL below. N (input) INTEGER The order of the matrix T. A (input) DOUBLE PRECISION array, dimension (N) On entry, A must contain the diagonal elements of U as returned from DLAGTF. B (input) DOUBLE PRECISION array, dimension (N-1) On entry, B must contain the first super-diagonal elements of U as returned from DLAGTF. C (input) DOUBLE PRECISION array, dimension (N-1) On entry, C must contain the sub-diagonal elements of L as returned from DLAGTF. D (input) DOUBLE PRECISION array, dimension (N-2) On entry, D must contain the second super-diagonal elements of U as returned from DLAGTF. IN (input) INTEGER array, dimension (N) On entry, IN must contain details of the matrix P as returned from DLAGTF. Y (input/output) DOUBLE PRECISION array, dimension (N) On entry, the right hand side vector y. On exit, Y is overwritten by the solution vector x. TOL (input/output) DOUBLE PRECISION On entry, with JOB .lt. 0, TOL should be the minimum perturbation to be made to very small diagonal elements of U. TOL should normally be chosen as about eps*norm(U), where eps is the relative machine precision, but if TOL is supplied as non-positive, then it is reset to eps*max( abs( u(i,j) ) ). If JOB .gt. 0 then TOL is not referenced. On exit, TOL is changed as described above, only if TOL is non-positive on entry. Otherwise TOL is unchanged. INFO (output) INTEGER = 0 : successful exit element of the solution vector x. This can only occur when JOB is supplied as positive and either means that a diagonal element of U is very small, or that the elements of the right-hand side vector y are very large. LAPACK version 3.0 15 June 2000 DLAGTS(l)
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DLAGTF(l) ) DLAGTF(l) NAME
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 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 singu- lar or nearly singular, INFO (output) INTEGER = 0 : successful exit LAPACK version 3.0 15 June 2000 DLAGTF(l)