NAG Library Function Document

1Purpose

nag_inteq_volterra_weights (d05bwc) 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.

2Specification

 #include #include
 void nag_inteq_volterra_weights (Nag_ODEMethod method, Integer iorder, Integer nomg, double omega[], double sw[], NagError *fail)

3Description

nag_inteq_volterra_weights (d05bwc) computes the weights ${W}_{i,j}$ and ${\omega }_{i}$ 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:
 $∫0t ϕs ds ≃h ∑ j=0 p-1 Wi,j ϕj×h + h ∑ j=p i ωi-j ϕj×h , 0≤t≤T ,$ (1)
with $t=\mathit{i}×h$, for $\mathit{i}=0,1,\dots ,\mathit{n}$, for some given constant $h$.
In (1), $h$ is a uniform mesh, $\mathit{p}$ is related to the order of the method being used and ${W}_{i,j}$, ${\omega }_{i}$ are the starting and the convolution weights respectively. The mesh size $h$ is determined as $h=\frac{T}{\mathit{n}}$, where $\mathit{n}={\mathit{n}}_{w}+\mathit{p}-1$ and ${\mathit{n}}_{w}$ is the chosen number of convolution weights ${w}_{j}$, for $\mathit{j}=1,2,\dots ,{\mathit{n}}_{w}-1$. 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.
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

5Arguments

1:    $\mathbf{method}$Nag_ODEMethodInput
On entry: the type of method to be used.
${\mathbf{method}}=\mathrm{Nag_Adams}$
${\mathbf{method}}=\mathrm{Nag_BDF}$
For Backward Differentiation Formulae.
Constraint: ${\mathbf{method}}=\mathrm{Nag_Adams}$ or $\mathrm{Nag_BDF}$.
2:    $\mathbf{iorder}$IntegerInput
On entry: the order of the method to be used. The number of starting weights, $\mathit{p}$ is determined by method and iorder.
If ${\mathbf{method}}=\mathrm{Nag_Adams}$, $\mathit{p}={\mathbf{iorder}}-1$.
If ${\mathbf{method}}=\mathrm{Nag_BDF}$, $\mathit{p}={\mathbf{iorder}}$.
Constraints:
• if ${\mathbf{method}}=\mathrm{Nag_Adams}$, $3\le {\mathbf{iorder}}\le 6$;
• if ${\mathbf{method}}=\mathrm{Nag_BDF}$, $2\le {\mathbf{iorder}}\le 5$.
3:    $\mathbf{nomg}$IntegerInput
On entry: the number of convolution weights, ${\mathit{n}}_{w}$.
Constraint: ${\mathbf{nomg}}\ge 1$.
4:    $\mathbf{omega}\left[{\mathbf{nomg}}\right]$doubleOutput
On exit: contains the first nomg convolution weights.
5:    $\mathbf{sw}\left[\mathit{n}×\mathit{p}\right]$doubleOutput
Note: the $\left(i,j\right)$th element of the matrix is stored in ${\mathbf{sw}}\left[\left(j-1\right)×\mathit{n}+i-1\right]$.
On exit: ${\mathbf{sw}}\left[\mathit{j}×\mathit{n}+\mathit{i}-1\right]$ contains the weights ${W}_{\mathit{i},\mathit{j}}$, for $\mathit{i}=1,2,\dots ,\mathit{n}$ and $\mathit{j}=0,1,\dots ,\mathit{p}-1$, where $\mathit{n}$ is as defined in Section 3.
6:    $\mathbf{fail}$NagError *Input/Output
The NAG error argument (see Section 3.7 in How to Use the NAG Library and its Documentation).

6Error Indicators and Warnings

NE_ALLOC_FAIL
Dynamic memory allocation failed.
See Section 2.3.1.2 in How to Use the NAG Library and its Documentation for further information.
On entry, argument $〈\mathit{\text{value}}〉$ had an illegal value.
NE_ENUM_INT
On entry, ${\mathbf{method}}=\mathrm{Nag_Adams}$ and ${\mathbf{iorder}}=2$.
Constraint: if ${\mathbf{method}}=\mathrm{Nag_Adams}$, $3\le {\mathbf{iorder}}\le 6$.
On entry, ${\mathbf{method}}=\mathrm{Nag_BDF}$ and ${\mathbf{iorder}}=6$.
Constraint: if ${\mathbf{method}}=\mathrm{Nag_BDF}$, $2\le {\mathbf{iorder}}\le 5$.
On entry, ${\mathbf{method}}=〈\mathit{\text{value}}〉$ and ${\mathbf{iorder}}=〈\mathit{\text{value}}〉$.
Constraint: if ${\mathbf{method}}=\mathrm{Nag_Adams}$, $3\le {\mathbf{iorder}}\le 6$.
On entry, ${\mathbf{method}}=〈\mathit{\text{value}}〉$ and ${\mathbf{iorder}}=〈\mathit{\text{value}}〉$.
Constraint: if ${\mathbf{method}}=\mathrm{Nag_BDF}$, $2\le {\mathbf{iorder}}\le 5$.
NE_INT
On entry, ${\mathbf{iorder}}=〈\mathit{\text{value}}〉$.
Constraint: $2\le {\mathbf{iorder}}\le 6$.
On entry, ${\mathbf{nomg}}=〈\mathit{\text{value}}〉$.
Constraint: ${\mathbf{nomg}}\ge 1$.
NE_INTERNAL_ERROR
An internal error has occurred in this function. Check the function call and any array sizes. If the call is correct then please contact NAG for assistance.
See Section 2.7.6 in How to Use the NAG Library and its Documentation for further information.
NE_NO_LICENCE
Your licence key may have expired or may not have been installed correctly.
See Section 2.7.5 in How to Use the NAG Library and its Documentation for further information.

Not applicable.

8Parallelism and Performance

nag_inteq_volterra_weights (d05bwc) is not threaded in any implementation.

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
 $yt=ft+∫0tKt,sysds, 0≤t≤T,$ (2)
using nag_inteq_volterra_weights (d05bwc). In (2), $K\left(t,s\right)$ and $f\left(t\right)$ are given and the solution $y\left(t\right)$ is sought on a uniform mesh of size $h$ such that $T=\mathit{n}h$. Discretization of (2) yields
 $yi=fi×h+h∑j=0 p-1Wi,jKi,h,j,hyj+h∑j=piωi-jKi,h,j,hyj,$ (3)
where ${y}_{i}\simeq y\left(i×h\right)$. We propose the following algorithm for computing ${y}_{i}$ from (3) after a call to nag_inteq_volterra_weights (d05bwc):
(a) Equation (3) requires starting values, ${y}_{\mathit{j}}$, for $\mathit{j}=1,2,\dots ,\mathit{p}-1$, with ${y}_{0}=f\left(0\right)$. These starting values can be computed by solving the linear system
 $yi = fi×h + h ∑ j=0 p-1 sw[j×n+i-1] K i,h,j,h yj , i=1,2,…,p-1 .$
(b) Compute the inhomogeneous terms
 $σi = fi×h + h ∑ j= 0 p-1 sw[j×n+i-1] Ki,h,j,h yj , i=p,p+ 1,…,n .$
(c) Start the iteration for $i=\mathit{p},\mathit{p}+1,\dots ,\mathit{n}$ to compute ${y}_{i}$ from:
 $1 - h × omega K i,h,i,h y i = σ i + h ∑ j = p i - 1 omega[i-j] K i,h,j,h y j .$
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).

10Example

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

10.1Program Text

Program Text (d05bwce.c)

10.2Program Data

Program Data (d05bwce.d)

10.3Program Results

Program Results (d05bwce.r)

© The Numerical Algorithms Group Ltd, Oxford, UK. 2017