NAG Library Function Document

nag_inteq_fredholm2_split (d05aac)

+− Contents

1 Purpose

2 Specification

3 Description

4 References

5 Arguments

6 Error Indicators and Warnings

7 Accuracy

8 Parallelism and Performance

9 Further Comments

+− 10 Example

10.1 Program Text

10.2 Program Data

10.3 Program Results

1 Purpose

nag_inteq_fredholm2_split (d05aac) solves a linear, nonsingular Fredholm equation of the second kind with a split kernel.

2 Specification

#include <nag.h>

#include <nagd05.h>

void

nag_inteq_fredholm2_split (double lambda, double a, double b, Integer n,

double

(*k1)(double x, double s, Nag_Comm *comm),

double

(*k2)(double x, double s, Nag_Comm *comm),

double

(*g)(double x, Nag_Comm *comm),

Nag_KernelForm kform, double f[], double c[], Nag_Comm *comm, NagError *fail)

3 Description

nag_inteq_fredholm2_split (d05aac) solves an integral equation of the form

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

for

a \leq x \leq b

, when the kernel

k

is defined in two parts:

k = k_{1}

for

a \leq s \leq x

and

k = k_{2}

for

x < s \leq b

. The method used is that of El–Gendi (1969) for which, it is important to note, each of the functions

k_{1}

and

k_{2}

must be defined, smooth and nonsingular, for all

x

and

s

in the interval

[a, b]

An approximation to the solution

f (x)

is found in the form of an

n

term Chebyshev series

{\sum^{'}}_{i = 1}^{n} c_{i} T_{i} (x)

, where

^{'}

indicates that the first term is halved in the sum. The coefficients

c_{i}

, for

i = 1, 2, \dots, n

, of this series are determined directly from approximate values

f_{i}

, for

i = 1, 2, \dots, n

, of the function

f (x)

at the first

n

of a set of

m + 1

Chebyshev points:

x_{i} = \frac{1}{2} (a + b + (b - a) \cos [(i - 1) π / m]), i = 1, 2, \dots, m + 1 .

The values

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 = n - 1

. However, if the kernel

k

is centro-symmetric in the interval

[a, b]

, i.e., if

k (x, s) = k (a + b - x, a + b - s)

, 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)

implies symmetry in the function

f (x)

. In particular, if

g (x)

is even about the mid-point of the range of integration, then so also is

f (x)

, which may be approximated by an even Chebyshev series with

m = 2 n - 1

. Similarly, if

g (x)

is odd about the mid-point then

f (x)

may be approximated by an odd series with

m = 2 n

4 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

5 Arguments

1: lambda – doubleInput

On entry: the value of the parameter

λ

of the integral equation.

2: a – doubleInput

On entry:

a

, the lower limit of integration.

3: b – doubleInput

On entry:

b

, the upper limit of integration.

Constraint:

b > a

4: n – IntegerInput

On entry: the number of terms in the Chebyshev series required to approximate

f (x)

Constraint:

n \geq 1

5: k1 – function, supplied by the userExternal Function

k1 must evaluate the kernel

k (x, s) = k_{1} (x, s)

of the integral equation for

a \leq s \leq x

The specification of k1 is:

double

k1 (double x, double s, Nag_Comm *comm)

1: x – doubleInput

2: s – doubleInput

On entry: the values of

x

and

s

at which

k_{1} (x, s)

is to be evaluated.

3: comm – Nag_Comm *

Pointer to structure of type Nag_Comm; the following members are relevant to k1.

user – double *
iuser – Integer *
p – Pointer: The type Pointer will be void *. Before calling nag_inteq_fredholm2_split (d05aac) you may allocate memory and initialize these pointers with various quantities for use by k1 when called from nag_inteq_fredholm2_split (d05aac) (see Section 3.2.1.1 in the Essential Introduction).

6: k2 – function, supplied by the userExternal Function

k2 must evaluate the kernel

k (x, s) = k_{2} (x, s)

of the integral equation for

x < s \leq b

The specification of k2 is:

double

k2 (double x, double s, Nag_Comm *comm)

1: x – doubleInput

2: s – doubleInput

On entry: the values of

x

and

s

at which

k_{2} (x, s)

is to be evaluated.

3: comm – Nag_Comm *

Pointer to structure of type Nag_Comm; the following members are relevant to k2.

user – double *
iuser – Integer *
p – Pointer: The type Pointer will be void *. Before calling nag_inteq_fredholm2_split (d05aac) you may allocate memory and initialize these pointers with various quantities for use by k2 when called from nag_inteq_fredholm2_split (d05aac) (see Section 3.2.1.1 in the Essential Introduction).

Note that the functions

k_{1}

and

k_{2}

must be defined, smooth and nonsingular for all

x

and

s

in the interval [

a, b

7: g – function, supplied by the userExternal Function

g must evaluate the function

g (x)

for

a \leq x \leq b

The specification of g is:

double

g (double x, Nag_Comm *comm)

1: x – doubleInput

On entry: the values of

x

at which

g (x)

is to be evaluated.

2: comm – Nag_Comm *

Pointer to structure of type Nag_Comm; the following members are relevant to g.

user – double *
iuser – Integer *
p – Pointer: The type Pointer will be void *. Before calling nag_inteq_fredholm2_split (d05aac) you may allocate memory and initialize these pointers with various quantities for use by g when called from nag_inteq_fredholm2_split (d05aac) (see Section 3.2.1.1 in the Essential Introduction).

8: kform – Nag_KernelFormInput

On entry: determines the forms of the kernel,

k (x, s)

, and the function

g (x)

$kform = Nag_NoCentroSymm$: $k (x, s)$ is not centro-symmetric (or no account is to be taken of centro-symmetry).
$kform = Nag_CentroSymmOdd$: $k (x, s)$ is centro-symmetric and $g (x)$ is odd.
$kform = Nag_CentroSymmEven$: $k (x, s)$ is centro-symmetric and $g (x)$ is even.
$kform = Nag_CentroSymmNeither$: $k (x, s)$ is centro-symmetric but $g (x)$ is neither odd nor even.

Constraint:

kform = Nag_NoCentroSymm

Nag_CentroSymmOdd

Nag_CentroSymmEven

Nag_CentroSymmNeither

9: f[n] – doubleOutput

On exit: the approximate values

f_{i}

, for

i = 1, 2, \dots, n

, of

f (x)

evaluated at the first n of

m + 1

Chebyshev points

x_{i}

, (see Section 3).

kform = Nag_NoCentroSymm

Nag_CentroSymmNeither

m = n - 1

kform = Nag_CentroSymmOdd

m = 2 \times n

kform = Nag_CentroSymmEven

m = 2 \times n - 1

10: c[n] – doubleOutput

On exit: the coefficients

c_{i}

, for

i = 1, 2, \dots, n

, of the Chebyshev series approximation to

f (x)

kform = Nag_CentroSymmOdd

this series contains polynomials of odd order only and if

kform = Nag_CentroSymmEven

the series contains even order polynomials only.

11: comm – Nag_Comm *Communication Structure

The NAG communication argument (see Section 3.2.1.1 in the Essential Introduction).

12: fail – NagError *Input/Output

The NAG error argument (see Section 3.6 in the Essential Introduction).

6 Error Indicators and Warnings

NE_ALLOC_FAIL: Dynamic memory allocation failed.
NE_BAD_PARAM: On entry, argument $⟨value⟩$ had an illegal value.
NE_EIGENVALUES: A failure has occurred due to proximity of an eigenvalue.
NE_INT: On entry, $n = ⟨value⟩$ .
Constraint: $n \geq 1$ .
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.
NE_REAL_2: On entry, $a = ⟨value⟩$ and $b = ⟨value⟩$ .
Constraint: $b > a$ .

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

8 Parallelism and Performance

nag_inteq_fredholm2_split (d05aac) is threaded by NAG for parallel execution in multithreaded implementations of the NAG Library.

nag_inteq_fredholm2_split (d05aac) makes calls to BLAS and/or LAPACK routines, which may be threaded within the vendor library used by this implementation. Consult the documentation for the vendor library for further information.

Please consult the Users' Note for your implementation for any additional implementation-specific information.