NAG FL Interface
d05bwf (volterra_​weights)

1 Purpose

d05bwf computes the quadrature weights associated with the Adams' methods of orders three to six and the Backward Differentiation Formulae (BDF) methods of orders two to five. These rules, which are referred to as reducible quadrature rules, can then be used in the solution of Volterra integral and integro-differential equations.

2 Specification

Fortran Interface
Subroutine d05bwf ( method, iorder, omega, nomg, lensw, sw, ldsw, nwt, ifail)
Integer, Intent (In) :: iorder, nomg, ldsw, nwt
Integer, Intent (Inout) :: ifail
Integer, Intent (Out) :: lensw
Real (Kind=nag_wp), Intent (Inout) :: sw(ldsw,nwt)
Real (Kind=nag_wp), Intent (Out) :: omega(nomg)
Character (1), Intent (In) :: method
C Header Interface
#include <nag.h>
void  d05bwf_ (const char *method, const Integer *iorder, double omega[], const Integer *nomg, Integer *lensw, double sw[], const Integer *ldsw, const Integer *nwt, Integer *ifail, const Charlen length_method)
The routine may be called by the names d05bwf or nagf_inteq_volterra_weights.

3 Description

d05bwf computes the weights Wi,j and ωi 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:
0t ϕs ds h j=0 p-1 Wi,j ϕj×h + h j=p i ωi-j ϕj×h ,   0tT , (1)
with t=i×h, for i=0,1,,n, for some given constant h.
In (1), h is a uniform mesh, p is related to the order of the method being used and Wi,j, ωi are the starting and the convolution weights respectively. The mesh size h is determined as h=Tn, where n=nw+p-1 and nw is the chosen number of convolution weights wj, for j=1,2,,nw-1. 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.

4 References

Lambert J D (1973) Computational Methods in Ordinary Differential Equations John Wiley
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

5 Arguments

1: method Character(1) Input
On entry: the type of method to be used.
For Adams' type formulae.
For Backward Differentiation Formulae.
Constraint: method='A' or 'B'.
2: iorder Integer Input
On entry: the order of the method to be used. The number of starting weights, p is determined by method and iorder.
If method='A', p=iorder-1.
If method='B', p=iorder.
  • if method='A', 3iorder6;
  • if method='B', 2iorder5.
3: omeganomg Real (Kind=nag_wp) array Output
On exit: contains the first nomg convolution weights.
4: nomg Integer Input
On entry: the number of convolution weights, nw.
Constraint: nomg1.
5: lensw Integer Output
On exit: the number of rows in the weights Wi,j.
6: swldswnwt Real (Kind=nag_wp) array Output
On exit: swij+1 contains the weights Wi,j, for i=1,2,,lensw and j=0,1,,nwt-1, where n is as defined in Section 3.
7: ldsw Integer Input
On entry: the first dimension of the array sw as declared in the (sub)program from which d05bwf is called.
  • if method='A', ldswnomg+iorder-2;
  • if method='B', ldswnomg+iorder-1.
8: nwt Integer Input
On entry: p, the number of columns in the starting weights.
  • if method='A', nwt=iorder-1;
  • if method='B', nwt=iorder.
9: ifail Integer Input/Output
On entry: ifail must be set to 0, -1 or 1 to set behaviour on detection of an error; these values have no effect when no error is detected.
A value of 0 causes the printing of an error message and program execution will be halted; otherwise program execution continues. A value of -1 means that an error message is printed while a value of 1 means that it is not.
If halting is not appropriate, the value -1 or 1 is recommended. If message printing is undesirable, then the value 1 is recommended. Otherwise, the value 0 is recommended. When the value -1 or 1 is used it is essential to test the value of ifail on exit.
On exit: ifail=0 unless the routine detects an error or a warning has been flagged (see Section 6).

6 Error Indicators and Warnings

If on entry ifail=0 or -1, explanatory error messages are output on the current error message unit (as defined by x04aaf).
Errors or warnings detected by the routine:
On entry, method=value.
Constraint: method='A' or 'B'.
On entry, iorder=value.
Constraint: 2iorder6.
On entry, nomg=value.
Constraint: nomg1.
On entry, method='A' and iorder=2.
Constraint: if method='A', 3iorder6.
On entry, method='B' and iorder=6.
Constraint: if method='B', 2iorder5.
On entry, method='A', iorder=value and nwt=value.
Constraint: if method='A', nwt=iorder-1.
On entry, method='B', iorder=value and nwt=value.
Constraint: if method='B', nwt=iorder.
On entry, method='A', iorder=value and nomg=value, ldsw=value.
Constraint: if method='A', ldswnomg+iorder-2.
On entry, method='B', iorder=value and nomg=value, ldsw=value.
Constraint: if method='B', ldswnomg+iorder-1.
An unexpected error has been triggered by this routine. Please contact NAG.
See Section 7 in the Introduction to the NAG Library FL Interface for further information.
Your licence key may have expired or may not have been installed correctly.
See Section 8 in the Introduction to the NAG Library FL Interface for further information.
Dynamic memory allocation failed.
See Section 9 in the Introduction to the NAG Library FL Interface for further information.

7 Accuracy

Not applicable.

8 Parallelism and Performance

d05bwf is not threaded in any implementation.

9 Further Comments

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
yt=ft+0tKt,sysds,  0tT, (2)
using d05bwf. In (2), Kt,s and ft are given and the solution yt is sought on a uniform mesh of size h such that T=nh. Discretization of (2) yields
yi=fi×h+hj=0 p-1Wi,jKi,h,j,hyj+hj=piωi-jKi,h,j,hyj, (3)
where yiyi×h. We propose the following algorithm for computing yi from (3) after a call to d05bwf:
  1. (a)Equation (3) requires starting values, yj, for j=1,2,,nwt-1, with y0=f0. These starting values can be computed by solving the linear system
    yi = fi×h + h j=0 nwt-1 swij+1 K i,h,j,h yj ,   i=1,2,,nwt-1 .  
  2. (b)Compute the inhomogeneous terms
    σi = fi×h + h j= 0 nwt-1 swij+1 Ki,h,j,h yj ,   i=nwt,nwt+ 1,,n .  
  3. (c)Start the iteration for i=nwt,nwt+1,,n to compute yi from:
    1 - h × omega1 K i,h,i,h y i = σ i + h j = nwt i - 1 omega i - j + 1 K i,h,j,h y j .  
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).

10 Example

The following example generates the first ten convolution and thirteen starting weights generated by the fourth-order BDF method.

10.1 Program Text

Program Text (d05bwfe.f90)

10.2 Program Data


10.3 Program Results

Program Results (d05bwfe.r)