f07jef computes the solution to a real system of linear equations
, where
is an
symmetric positive definite tridiagonal matrix and
and
are
matrices, using the
factorization returned by
f07jdf.
f07jef should be preceded by a call to
f07jdf, which computes a modified Cholesky factorization of the matrix
as
where
is a unit lower bidiagonal matrix and
is a diagonal matrix, with positive diagonal elements.
f07jef then utilizes the factorization to solve the required equations. Note that the factorization may also be regarded as having the form
, where
is a unit upper bidiagonal matrix.
Anderson E, Bai Z, Bischof C, Blackford S, Demmel J, Dongarra J J, Du Croz J J, Greenbaum A, Hammarling S, McKenney A and Sorensen D (1999)
LAPACK Users' Guide (3rd Edition) SIAM, Philadelphia
https://www.netlib.org/lapack/lugThe computed solution for a single right-hand side,
, satisfies an equation of the form
where
and
is the
machine precision. An approximate error bound for the computed solution is given by
where
, the condition number of
with respect to the solution of the linear equations. See Section 4.4 of
Anderson et al. (1999) for further details.
Following the use of this routine
f07jgf can be used to estimate the condition number of
and
f07jhf can be used to obtain approximate error bounds.
Please consult the
X06 Chapter Introduction for information on how to control and interrogate the OpenMP environment used within this routine. Please also consult the
Users' Note for your implementation for any additional implementation-specific information.
The complex analogue of this routine is
f07jsf.
This example solves the equations
where
is the symmetric positive definite tridiagonal matrix