naginterfaces.library.inteq.volterra_​weights

naginterfaces.library.inteq.volterra_weights(method, iorder, nomg, nwt)

volterra_weights 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.

For full information please refer to the NAG Library document for d05bw

https://support.nag.com/numeric/nl/nagdoc_30.2/flhtml/d05/d05bwf.html

Parameters

methodstr, length 1

The type of method to be used.

$method=\text{'A'}$

For Adams’ type formulae.

$method=\text{'B'}$

For Backward Differentiation Formulae.

iorderint

The order of the method to be used. The number of starting weights, $\text{p}$ is determined by $method$ and $iorder$.

If $method=\text{'A'}$, $\text{p}=iorder-1$.

If $method=\text{'B'}$, $\text{p}=iorder$.

nomgint

The number of convolution weights, ${\text{n}}_w$.

nwtint

$\text{p}$, the number of columns in the starting weights.

Returns

omegafloat, ndarray, shape $\left(nomg\right)$

Contains the first $nomg$ convolution weights.

lenswint

The number of rows in the weights $W_{i,j}$.

swfloat, ndarray, shape $(\text{n},nwt)$

$sw[\text{i}-1,\text{j}]$ contains the weights $W_{\text{i},\text{j}}$, for $\text{j}=0,1,\dots ,nwt-1$, for $\text{i}=1,2,\dots ,lensw$, where $\text{n}$ is as defined in Notes.

Raises

NagValueError

(errno $1$)

On entry, $method=⟨value⟩$.

Constraint: $method=\text{'A'}$ or $\text{'B'}$.

(errno $2$)

On entry, $nomg=⟨value⟩$.

Constraint: $nomg\ge 1$.

(errno $2$)

On entry, $iorder=⟨value⟩$.

Constraint: $2\le iorder\le 6$.

(errno $3$)

On entry, $method=\text{'B'}$ and $iorder=6$.

Constraint: if $method=\text{'B'}$, $2\le iorder\le 5$.

(errno $3$)

On entry, $method=\text{'A'}$ and $iorder=2$.

Constraint: if $method=\text{'A'}$, $3\le iorder\le 6$.

(errno $4$)

On entry, $method=\text{'B'}$, $iorder=⟨value⟩$ and $nwt=⟨value⟩$.

Constraint: if $method=\text{'B'}$, $nwt=iorder$.

(errno $4$)

On entry, $method=\text{'A'}$, $iorder=⟨value⟩$ and $nwt=⟨value⟩$.

Constraint: if $method=\text{'A'}$, $nwt=iorder-1$.

(errno $5$)

On entry, $method=\text{'B'}$, $iorder=⟨value⟩$ and $nomg=⟨value⟩$, $\text{ldsw}=⟨value⟩$.

Constraint: if $method=\text{'B'}$, $\text{ldsw}\ge nomg+iorder-1$.

(errno $5$)

On entry, $method=\text{'A'}$, $iorder=⟨value⟩$ and $nomg=⟨value⟩$, $\text{ldsw}=⟨value⟩$.

Constraint: if $method=\text{'A'}$, $\text{ldsw}\ge nomg+iorder-2$.

Notes

volterra_weights 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:

$${\int }_0^t\varphi \left(s\right)ds\simeq h\sum _{j=0}^{\text{p}-1}{W}_{i,j}\varphi (j\times h)+h\sum _{j=\text{p}}^i{\omega }_{i-j}\varphi (j\times h)\text{,}0\le t\le T\text{,}$$

with $t=\text{i}\times h$, for $\text{i}=0,1,\dots ,\text{n}$, for some given constant $h$.

In (1), $h$ is a uniform mesh, $\text{p}$ 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 $h=\frac{T}{\text{n}}$, where $\text{n}={\text{n}}_w+\text{p}-1$ and ${\text{n}}_w$ is the chosen number of convolution weights $w_j$, for $\text{j}=1,2,\dots ,{\text{n}}_w-1$. A description of how these weights can be used in the solution of a Volterra integral equation of the second kind is given in Further Comments. For a general discussion of these methods, see Wolkenfelt (1982) for more details.

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