d05bwc computes the weights
and
for a family of quadrature rules related to the Adams' methods of orders three to six and the BDF methods of orders two to five, for approximating the integral:
with
, for
, for some given constant
.
In
(1),
is a uniform mesh,
is related to the order of the method being used and
,
are the starting and the convolution weights respectively. The mesh size
is determined as
, where
and
is the chosen number of convolution weights
, for
. A description of how these weights can be used in the solution of a Volterra integral equation of the second kind is given in
Section 9. For a general discussion of these methods, see
Wolkenfelt (1982) for more details.
Wolkenfelt P H M (1982) The construction of reducible quadrature rules for Volterra integral and integro-differential equations IMA J. Numer. Anal. 2 131–152
Not applicable.
Background information to multithreading can be found in the
Multithreading documentation.
Reducible quadrature rules are most appropriate for solving Volterra integral equations (and integro-differential equations). In this section, we propose the following algorithm which you may find useful in solving a linear Volterra integral equation of the form
using
d05bwc. In
(2),
and
are given and the solution
is sought on a uniform mesh of size
such that
. Discretization of
(2) yields
where
. We propose the following algorithm for computing
from
(3) after a call to
d05bwc:
-
(a)Equation (3) requires starting values, , for , with . These starting values can be computed by solving the linear system
-
(b)Compute the inhomogeneous terms
-
(c)Start the iteration for to compute from:
Note that for a nonlinear integral equation, the solution of a nonlinear algebraic system is required at step
(a) and a single nonlinear equation at step
(c).
The following example generates the first ten convolution and thirteen starting weights generated by the fourth-order BDF method.