naginterfaces.library.inteq.abel_​weak_​weights

naginterfaces.library.inteq.abel_weak_weights(iorder, iq, lenfw)

abel_weak_weights computes the fractional quadrature weights associated with the Backward Differentiation Formulae (BDF) of orders $4$, $5$ and $6$. These weights can then be used in the solution of weakly singular equations of Abel type.

For full information please refer to the NAG Library document for d05by

https://support.nag.com/numeric/nl/nagdoc_30.3/flhtml/d05/d05byf.html

Parameters

iorderint

$p$, the order of the BDF method to be used.

iqint

Determines the number of weights to be computed. By setting $iq$ to a value, $2^{iq+1}$ fractional convolution weights are computed.

lenfwint

The dimension of the array $wt$.

Returns

wtfloat, ndarray, shape $\left(lenfw\right)$

The first $2^{iq+1}$ elements of $wt$ contains the fractional convolution weights ${\omega }_i$, for $\text{i}=0,1,\dots ,2^{iq+1}-1$. The remainder of the array is used as workspace.

swfloat, ndarray, shape $(2^{iq+1}+2\times iorder-1,2\times iorder-1)$

$sw[\text{i}-1,\text{j}]$ contains the fractional starting weights $W_{\text{i}-1,\text{j}}$, for $\text{j}=0,1,\dots ,2\times iorder-2$, for $\text{i}=1,2,\dots ,\text{N}$, where $\text{N}=(2^{iq+1}+2\times iorder-1)$.

Raises

NagValueError

(errno $1$)

On entry, $lenfw=⟨value⟩$ and $iq=⟨value⟩$.

Constraint: $lenfw\ge 2^{iq+2}$.

(errno $1$)

On entry, $\text{ldsw}=⟨value⟩$, $iq=⟨value⟩$ and $iorder=⟨value⟩$.

Constraint: $\text{ldsw}\ge 2^{iq+1}+2\times iorder-1$.

(errno $1$)

On entry, $iq=⟨value⟩$.

Constraint: $iq\ge 0$.

(errno $1$)

On entry, $iorder=⟨value⟩$.

Constraint: $4\le iorder\le 6$.

Notes

abel_weak_weights computes the weights $W_{i,j}$ and ${\omega }_i$ for a family of quadrature rules related to a BDF method for approximating the integral:

$$\frac{1}{\sqrt{\pi }}{\int }_0^t\frac{\varphi \left(s\right)}{\sqrt{t-s}}ds\simeq \sqrt{h}\sum _{j=0}^{2p-2}{W}_{i,j}\varphi (j\times h)+\sqrt{h}\sum _{j=2p-1}^i{\omega }_{i-j}\varphi (j\times h)\text{,}0\le t\le T\text{,}$$

with $t=i\times h(i\ge 0)$, for some given $h$. In (1), $p$ is the order of the BDF method used and $W_{i,j}$, ${\omega }_i$ are the fractional starting and the fractional convolution weights respectively. The algorithm for the generation of ${\omega }_i$ is based on Newton’s iteration. Fast Fourier transform (FFT) techniques are used for computing these weights and subsequently $W_{i,j}$ (see Baker and Derakhshan (1987) and Henrici (1979) for practical details and Lubich (1986) for theoretical details). Some special functions can be represented as the fractional integrals of simpler functions and fractional quadratures can be employed for their computation (see Lubich (1986)). A description of how these weights can be used in the solution of weakly singular equations of Abel type is given in Further Comments.

References

Baker, C T H and Derakhshan, M S, 1987, Computational approximations to some power series, Approximation Theory, (eds L Collatz, G Meinardus and G Nürnberger) (81), 11–20

Henrici, P, 1979, Fast Fourier methods in computational complex analysis, SIAM Rev. (21), 481–529

Lubich, Ch, 1986, Discretized fractional calculus, SIAM J. Math. Anal. (17), 704–719