are obtained by solving a set of simultaneous linear algebraic equations formed by applying a quadrature formula (equivalent to the scheme of Clenshaw and Curtis (1960)) to the integral equation at each of the above points.

In general

m = n - 1

. However, advantage may be taken of any prior knowledge of the symmetry of

f (x)

. Thus if

f (x)

is symmetric (i.e., even) about the mid-point of the range

(a, b)

, it may be approximated by an even Chebyshev series with

m = 2 n - 1

. Similarly, if

f (x)

is anti-symmetric (i.e., odd) about the mid-point of the range of integration, it may be approximated by an odd Chebyshev series with

m = 2 n

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: $k$ – function handle or string containing name of m-file

k must compute the value of the kernel

k (x, s)

of the integral equation over the square

a \leq x \leq b

a \leq s \leq b

[result] = k(x, s)

Input Parameters

1: $x$ – double scalar
2: $s$ – double scalar: The values of $x$ and $s$ at which $k (x, s)$ is to be calculated.

Output Parameters

1: $result$ – double scalar: The value of the kernel $k (x, s)$ evaluated at x and s.

2: $g$ – function handle or string containing name of m-file

g must compute the value of the function

g (x)

of the integral equation in the interval

a \leq x \leq b

[result] = g(x)

Input Parameters

1: $x$ – double scalar: The value of $x$ at which $g (x)$ is to be calculated.

Output Parameters

1: $result$ – double scalar: The value of $g (x)$ evaluated at x.

3: $lambda$ – double scalar

The value of the parameter

λ

of the integral equation.

4: $a$ – double scalar

a

, the lower limit of integration.

5: $b$ – double scalar

b

, the upper limit of integration.

Constraint:

b > a

6: $odorev$ – logical scalar

Indicates whether it is known that the solution

f (x)

is odd or even about the mid-point of the range of integration. If odorev is true then an odd or even solution is sought depending upon the value of ev.

7: $ev$ – logical scalar

Is ignored if odorev is false. Otherwise, if ev is true, an even solution is sought, whilst if ev is false, an odd solution is sought.

8: $n$ – int64int32nag_int scalar

The number of terms in the Chebyshev series which approximates the solution

f (x)

Constraint:

n \geq 1

Optional Input Parameters

None.

Output Parameters

1: $f (n)$ – double array

The approximate values

f_{i}

, for

i = 1, 2, \dots, n

, of the function

f (x)

at the first n of

m + 1

Chebyshev points (see Description), where

$m = 2 n - 1$	if $odorev = true$ and $ev = true$ .
$m = 2 n$	if $odorev = true$ and $ev = false$ .
$m = n - 1$	if $odorev = false$ .

2: $c (n)$ – double array

The coefficients

c_{i}

, for

i = 1, 2, \dots, n

, of the Chebyshev series approximation to

f (x)

. When odorev is true, this series contains polynomials of even order only or of odd order only, according to ev being true or false respectively.

3: $ifail$ – int64int32nag_int scalar

ifail = 0

unless the function detects an error (see Error Indicators and Warnings).

Error Indicators and Warnings

Errors or warnings detected by the function:

$ifail = 1$

On entry,

a \geq b

n < 1

$ifail = 2$: 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, $m = 1$ , the matrix reduces to a zero-valued number.

$ifail = - 99$: An unexpected error has been triggered by this routine. Please contact NAG.

$ifail = - 399$: Your licence key may have expired or may not have been installed correctly.

$ifail = - 999$: Dynamic memory allocation failed.

Accuracy

No explicit error estimate is provided by the function but it is possible to obtain a good indication of the accuracy of the solution either

(i)	by examining the size of the later Chebyshev coefficients $c_{i}$ , or
(ii)	by comparing the coefficients $c_{i}$ or the function values $f_{i}$ for two or more values of n.

Further Comments

The time taken by nag_inteq_fredholm2_smooth (d05ab) depends upon the value of n and upon the complexity of the kernel function

k (x, s)

Example

This example solves Love's equation:

f (x) + \frac{1}{π} \int_{- 1}^{1} \frac{f (s)}{1 + {(x - s)}^{2}} d s = 1 .

It will solve the slightly more general equation:

f (x) - λ \int_{a}^{b} k (x, s) f (s) d s = 1

where

k (x, s) = α / (α^{2} + {(x - s)}^{2})

. The values

λ = - 1 / π, a = - 1, b = 1, α = 1

are used below.

It is evident from the symmetry of the given equation that

f (x)

is an even function. Advantage is taken of this fact both in the application of nag_inteq_fredholm2_smooth (d05ab), to obtain the

f_{i} ≃ f (x_{i})

and the

c_{i}

, and in subsequent applications of nag_sum_chebyshev (c06dc) to obtain

f (x)

at selected points.

The program runs for

n = 5

and

n = 10

Open in the MATLAB editor: d05ab_example

function d05ab_example


fprintf('d05ab example results\n\n');

k = @(x, s) 1/(1+(x-s)*(x-s));
g = @(x) 1;
lambda = -0.3183;
a = -1;
b = 1;
odorev = true;
ev     = true;
xval = [0:0.25:1];
ss = int64(2);

for n = 5:5:10;
  in = int64(n);
  [f, c, ifail] = d05ab(k, g, lambda, a, b, odorev, ev, in);
  fprintf('\nResults for N = %2d\n\n', n);
  fprintf('Solution and coefficients on first %2d Chebyshev points\n',n);
  fprintf('  i           x             f(i)         c(i)\n');
  cheb(1:n,1) = cos(pi*(1:n)/(2*n-1));
  fprintf('%3d%15.5f%15.5f%15.5e\n', [[1:n]' cheb f c]');

  [chebr, ifail] = c06dc(xval, a, b, c, ss);

  fprintf('\nSolution on evenly spaced grid\n');
  fprintf('     x           f(x)\n');
  fprintf('%8.4f%15.5f\n',[xval' chebr]')

end

d05ab example results


Results for N =  5

Solution and coefficients on first  5 Chebyshev points
  i           x             f(i)         c(i)
  1        0.93969        0.75572    1.41520e+00
  2        0.76604        0.74534    4.93840e-02
  3        0.50000        0.71729   -1.04758e-03
  4        0.17365        0.68319   -2.32817e-04
  5       -0.17365        0.66051    2.08903e-05

Solution on evenly spaced grid
     x           f(x)
  0.0000        0.65742
  0.2500        0.66383
  0.5000        0.68319
  0.7500        0.71489
  1.0000        0.75572

Results for N = 10

Solution and coefficients on first 10 Chebyshev points
  i           x             f(i)         c(i)
  1        0.98636        0.75572    1.41520e+00
  2        0.94582        0.75336    4.93840e-02
  3        0.87947        0.74639   -1.04751e-03
  4        0.78914        0.73525   -2.32749e-04
  5        0.67728        0.72081    1.99856e-05
  6        0.54695        0.70452    9.86754e-07
  7        0.40170        0.68825   -2.37956e-07
  8        0.24549        0.67404    1.85810e-09
  9        0.08258        0.66361    2.44829e-09
 10       -0.08258        0.65812   -1.65268e-10

Solution on evenly spaced grid
     x           f(x)
  0.0000        0.65742
  0.2500        0.66384
  0.5000        0.68319
  0.7500        0.71489
  1.0000        0.75572

PDF version (NAG web site, 64-bit version, 64-bit version)

Chapter Contents

Chapter Introduction

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