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Chapter Contents
Chapter Introduction
NAG Toolbox

# NAG Toolbox: nag_inteq_fredholm2_split (d05aa)

## Purpose

nag_inteq_fredholm2_split (d05aa) solves a linear, nonsingular Fredholm equation of the second kind with a split kernel.

## Syntax

[f, c, ifail] = d05aa(lambda, a, b, k1, k2, g, n, ind)
[f, c, ifail] = nag_inteq_fredholm2_split(lambda, a, b, k1, k2, g, n, ind)

## Description

nag_inteq_fredholm2_split (d05aa) solves an integral equation of the form
 b f(x) − λ ∫ k(x,s)f(s)ds = g(x) a
$f(x)-λ∫abk(x,s)f(s)ds=g(x)$
for axb$a\le x\le b$, when the kernel k$k$ is defined in two parts: k = k1$k={k}_{1}$ for asx$a\le s\le x$ and k = k2$k={k}_{2}$ for x < sb$x. The method used is that of El–Gendi (1969) for which, it is important to note, each of the functions k1${k}_{1}$ and k2${k}_{2}$ must be defined, smooth and nonsingular, for all x$x$ and s$s$ in the interval [a,b]$\left[a,b\right]$.
An approximation to the solution f(x)$f\left(x\right)$ is found in the form of an n$n$ term Chebyshev series i = 1nciTi(x)$\underset{i=1}{\overset{n}{{\sum }^{\prime }}}{c}_{i}{T}_{i}\left(x\right)$, where ${}^{\prime }$ indicates that the first term is halved in the sum. The coefficients ci${c}_{\mathit{i}}$, for i = 1,2,,n$\mathit{i}=1,2,\dots ,n$, of this series are determined directly from approximate values fi${f}_{\mathit{i}}$, for i = 1,2,,n$\mathit{i}=1,2,\dots ,n$, of the function f(x)$f\left(x\right)$ at the first n$n$ of a set of m + 1$m+1$ Chebyshev points:
 xi = (1/2)(a + b + (b − a)cos[(i − 1)π / m]),  i = 1,2, … ,m + 1. $xi=12(a+b+(b-a)cos[(i-1)π/m]), i=1,2,…,m+1.$
The values fi${f}_{i}$ are obtained by solving simultaneous linear algebraic equations formed by applying a quadrature formula (equivalent to the scheme of Clenshaw and Curtis (1960)) to the integral equation at the above points.
In general m = n1$m=n-1$. However, if the kernel k$k$ is centro-symmetric in the interval [a,b]$\left[a,b\right]$, i.e., if k(x,s) = k(a + bx,a + bs)$k\left(x,s\right)=k\left(a+b-x,a+b-s\right)$, then the function is designed to take advantage of this fact in the formation and solution of the algebraic equations. In this case, symmetry in the function g(x)$g\left(x\right)$ implies symmetry in the function f(x)$f\left(x\right)$. In particular, if g(x)$g\left(x\right)$ is even about the mid-point of the range of integration, then so also is f(x)$f\left(x\right)$, which may be approximated by an even Chebyshev series with m = 2n1$m=2n-1$. Similarly, if g(x)$g\left(x\right)$ is odd about the mid-point then f(x)$f\left(x\right)$ may be approximated by an odd series with m = 2n$m=2n$.

## References

Clenshaw C W and Curtis A R (1960) A method for numerical integration on an automatic computer Numer. Math. 2 197–205
El–Gendi S E (1969) Chebyshev solution of differential, integral and integro-differential equations Comput. J. 12 282–287

## Parameters

### Compulsory Input Parameters

1:     lambda – double scalar
The value of the parameter λ$\lambda$ of the integral equation.
2:     a – double scalar
a$a$, the lower limit of integration.
3:     b – double scalar
b$b$, the upper limit of integration.
Constraint: b > a${\mathbf{b}}>{\mathbf{a}}$.
4:     k1 – function handle or string containing name of m-file
k1 must evaluate the kernel k(x,s) = k1(x,s)$k\left(x,s\right)={k}_{1}\left(x,s\right)$ of the integral equation for asx$a\le s\le x$.
[result] = k1(x, s)

Input Parameters

1:     x – double scalar
2:     s – double scalar
The values of x$x$ and s$s$ at which k1(x,s)${k}_{1}\left(x,s\right)$ is to be evaluated.

Output Parameters

1:     result – double scalar
The result of the function.
5:     k2 – function handle or string containing name of m-file
k2 must evaluate the kernel k(x,s) = k2(x,s)$k\left(x,s\right)={k}_{2}\left(x,s\right)$ of the integral equation for x < sb$x.
[result] = k2(x, s)

Input Parameters

1:     x – double scalar
2:     s – double scalar
The values of x$x$ and s$s$ at which k2(x,s)${k}_{2}\left(x,s\right)$ is to be evaluated.

Output Parameters

1:     result – double scalar
The result of the function.
Note that the functions k1${k}_{1}$ and k2${k}_{2}$ must be defined, smooth and nonsingular for all x$x$ and s$s$ in the interval [a,b$a,b$].
6:     g – function handle or string containing name of m-file
g must evaluate the function g(x)$g\left(x\right)$ for axb$a\le x\le b$.
[result] = g(x)

Input Parameters

1:     x – double scalar
The values of x$x$ at which g(x)$g\left(x\right)$ is to be evaluated.

Output Parameters

1:     result – double scalar
The result of the function.
7:     n – int64int32nag_int scalar
The number of terms in the Chebyshev series required to approximate f(x)$f\left(x\right)$.
Constraint: n1${\mathbf{n}}\ge 1$.
8:     ind – int64int32nag_int scalar
Determines the forms of the kernel, k(x,s)$k\left(x,s\right)$, and the function g(x)$g\left(x\right)$.
ind = 0${\mathbf{ind}}=0$
k(x,s)$k\left(x,s\right)$ is not centro-symmetric (or no account is to be taken of centro-symmetry).
ind = 1${\mathbf{ind}}=1$
k(x,s)$k\left(x,s\right)$ is centro-symmetric and g(x)$g\left(x\right)$ is odd.
ind = 2${\mathbf{ind}}=2$
k(x,s)$k\left(x,s\right)$ is centro-symmetric and g(x)$g\left(x\right)$ is even.
ind = 3${\mathbf{ind}}=3$
k(x,s)$k\left(x,s\right)$ is centro-symmetric but g(x)$g\left(x\right)$ is neither odd nor even.
Constraint: ind = 0${\mathbf{ind}}=0$, 1$1$, 2$2$ or 3$3$.

None.

### Input Parameters Omitted from the MATLAB Interface

w1 w2 wd ldw1 ldw2

### Output Parameters

1:     f(n) – double array
The approximate values fi${f}_{\mathit{i}}$, for i = 1,2,,n$\mathit{i}=1,2,\dots ,{\mathbf{n}}$, of f(x)$f\left(x\right)$ evaluated at the first n of m + 1$m+1$ Chebyshev points xi${x}_{i}$, (see Section [Description]).
If ind = 0${\mathbf{ind}}=0$ or 3$3$, m = n1$m={\mathbf{n}}-1$.
If ind = 1${\mathbf{ind}}=1$, m = 2 × n$m=2×{\mathbf{n}}$.
If ind = 2${\mathbf{ind}}=2$, m = 2 × n1$m=2×{\mathbf{n}}-1$.
2:     c(n) – double array
The coefficients ci${c}_{\mathit{i}}$, for i = 1,2,,n$\mathit{i}=1,2,\dots ,{\mathbf{n}}$, of the Chebyshev series approximation to f(x)$f\left(x\right)$.
If ind = 1${\mathbf{ind}}=1$ this series contains polynomials of odd order only and if ind = 2${\mathbf{ind}}=2$ the series contains even order polynomials only.
3:     ifail – int64int32nag_int scalar
${\mathrm{ifail}}={\mathbf{0}}$ unless the function detects an error (see [Error Indicators and Warnings]).

## Error Indicators and Warnings

Errors or warnings detected by the function:
ifail = 1${\mathbf{ifail}}=1$
 On entry, a ≥ b${\mathbf{a}}\ge {\mathbf{b}}$ or n < 1${\mathbf{n}}<1$.
ifail = 2${\mathbf{ifail}}=2$
A failure has occurred due to proximity to an eigenvalue. In general, if lambda is near an eigenvalue of the integral equation, the corresponding matrix will be nearly singular. In the special case, m = 1$m=1$, the matrix reduces to a zero-valued number.

## Accuracy

No explicit error estimate is provided by the function but it is usually possible to obtain a good indication of the accuracy of the solution either
 (i) by examining the size of the later Chebyshev coefficients ci${c}_{i}$, or (ii) by comparing the coefficients ci${c}_{i}$ or the function values fi${f}_{i}$ for two or more values of n.

The time taken by nag_inteq_fredholm2_split (d05aa) increases with n.
This function may be used to solve an equation with a continuous kernel by defining k1 and k2 to be identical.
This function may also be used to solve a Volterra equation by defining k2 (or k1) to be identically zero.

## Example

```function nag_inteq_fredholm2_split_example
lambda = 1;
a = 0;
b = 1;
g = @(x) sin(pi*x)*(1-1/(pi*pi));
k1 = @(x, s) s*(1-x);
k2 = @(x, s) x*(1-s);
n = int64(5);
ind = int64(2);
[f, c, ifail] = nag_inteq_fredholm2_split(lambda, a, b, k1, k2, g, n, ind)
```
```

f =

0.0000
0.0946
0.3593
0.7071
0.9630

c =

0.9440
-0.4994
0.0280
-0.0006
0.0000

ifail =

0

```
```function d05aa_example
lambda = 1;
a = 0;
b = 1;
g = @(x) sin(pi*x)*(1-1/(pi*pi));
k1 = @(x, s) s*(1-x);
k2 = @(x, s) x*(1-s);
n = int64(5);
ind = int64(2);
[f, c, ifail] = d05aa(lambda, a, b, k1, k2, g, n, ind)
```
```

f =

0.0000
0.0946
0.3593
0.7071
0.9630

c =

0.9440
-0.4994
0.0280
-0.0006
0.0000

ifail =

0

```