Integer, Intent (In)	::	n, ind, ldw1, ldw2
Integer, Intent (Inout)	::	ifail
Real (Kind=nag_wp), External	::	k1, k2, g
Real (Kind=nag_wp), Intent (In)	::	lambda, a, b
Real (Kind=nag_wp), Intent (Inout)	::	w1(ldw1,ldw2), w2(ldw2,4)
Real (Kind=nag_wp), Intent (Out)	::	f(n), c(n), wd(ldw2)

C Header Interface

#include nagmk26.h

void

d05aaf_ ( const double *lambda, const double *a, const double *b,
double (NAG_CALL *k1)( const double *x, const double *s),
double (NAG_CALL *k2)( const double *x, const double *s),
double (NAG_CALL *g)( const double *x),
double f[], double c[], const Integer *n, const Integer *ind, double w1[], double w2[], double wd[], const Integer *ldw1, const Integer *ldw2, Integer *ifail)

3

Description

d05aaf 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 routine 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: $lambda$ – Real (Kind=nag_wp)Input

On entry: the value of the parameter

λ

of the integral equation.

2: $a$ – Real (Kind=nag_wp)Input

On entry:

a

, the lower limit of integration.

3: $b$ – Real (Kind=nag_wp)Input

On entry:

b

, the upper limit of integration.

Constraint:

b > a

4: $k1$ – real (Kind=nag_wp) Function, supplied by the user.External Procedure

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:

Fortran Interface

Function k1 (

x, s)

Real (Kind=nag_wp)	::	k1
Real (Kind=nag_wp), Intent (In)	::	x, s

C Header Interface

#include nagmk26.h

double

k1 ( const double *x, const double *s)

1: $x$ – Real (Kind=nag_wp)Input
2: $s$ – Real (Kind=nag_wp)Input: On entry: the values of $x$ and $s$ at which $k_{1} (x, s)$ is to be evaluated.

k1 must either be a module subprogram USEd by, or declared as EXTERNAL in, the (sub)program from which d05aaf is called. Arguments denoted as Input must not be changed by this procedure.

Note: k1 should not return floating-point NaN (Not a Number) or infinity values, since these are not handled by d05aaf. If your code inadvertently does return any NaNs or infinities, d05aaf is likely to produce unexpected results.

5: $k2$ – real (Kind=nag_wp) Function, supplied by the user.External Procedure

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:

Fortran Interface

Function k2 (

x, s)

Real (Kind=nag_wp)	::	k2
Real (Kind=nag_wp), Intent (In)	::	x, s

C Header Interface

#include nagmk26.h

double

k2 ( const double *x, const double *s)

1: $x$ – Real (Kind=nag_wp)Input
2: $s$ – Real (Kind=nag_wp)Input: On entry: the values of $x$ and $s$ at which $k_{2} (x, s)$ is to be evaluated.

k2 must either be a module subprogram USEd by, or declared as EXTERNAL in, the (sub)program from which d05aaf is called. Arguments denoted as Input must not be changed by this procedure.

Note: k2 should not return floating-point NaN (Not a Number) or infinity values, since these are not handled by d05aaf. If your code inadvertently does return any NaNs or infinities, d05aaf is likely to produce unexpected results.

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

6: $g$ – real (Kind=nag_wp) Function, supplied by the user.External Procedure

g must evaluate the function

g (x)

for

a \leq x \leq b

The specification of g is:

Fortran Interface

Function g (

Real (Kind=nag_wp)	::	g
Real (Kind=nag_wp), Intent (In)	::	x

C Header Interface

#include nagmk26.h

double

g ( const double *x)

1: $x$ – Real (Kind=nag_wp)Input: On entry: the values of $x$ at which $g (x)$ is to be evaluated.

g must either be a module subprogram USEd by, or declared as EXTERNAL in, the (sub)program from which d05aaf is called. Arguments denoted as Input must not be changed by this procedure.

Note: g should not return floating-point NaN (Not a Number) or infinity values, since these are not handled by d05aaf. If your code inadvertently does return any NaNs or infinities, d05aaf is likely to produce unexpected results.

7: $f (n)$ – Real (Kind=nag_wp) arrayOutput

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).

ind = 0

3

m = n - 1

ind = 1

m = 2 \times n

ind = 2

m = 2 \times n - 1

8: $c (n)$ – Real (Kind=nag_wp) arrayOutput

On exit: the coefficients

c_{i}

, for

i = 1, 2, \dots, n

, of the Chebyshev series approximation to

f (x)

ind = 1

this series contains polynomials of odd order only and if

ind = 2

the series contains even order polynomials only.

9: $n$ – IntegerInput

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

f (x)

Constraint:

n \geq 1

10: $ind$ – IntegerInput

On entry: determines the forms of the kernel,

k (x, s)

, and the function

g (x)

$ind = 0$: $k (x, s)$ is not centro-symmetric (or no account is to be taken of centro-symmetry).
$ind = 1$: $k (x, s)$ is centro-symmetric and $g (x)$ is odd.
$ind = 2$: $k (x, s)$ is centro-symmetric and $g (x)$ is even.
$ind = 3$: $k (x, s)$ is centro-symmetric but $g (x)$ is neither odd nor even.

Constraint:

ind = 0

1

2

3

11: $w1 (ldw1, ldw2)$ – Real (Kind=nag_wp) arrayWorkspace

12: $w2 (ldw2, 4)$ – Real (Kind=nag_wp) arrayWorkspace

13: $wd (ldw2)$ – Real (Kind=nag_wp) arrayWorkspace

14: $ldw1$ – IntegerInput

On entry: the first dimension of the array w1 as declared in the (sub)program from which d05aaf is called.

Constraint:

ldw1 \geq n

15: $ldw2$ – IntegerInput

On entry: the second dimension of the array w1 and the first dimension of the array w2 and the dimension of the array wd as declared in the (sub)program from which d05aaf is called.

Constraint:

ldw2 \geq 2 \times n + 2

16: $ifail$ – IntegerInput/Output

On entry: ifail must be set to

0

- 1 ​ or ​ 1

. If you are unfamiliar with this argument you should refer to Section 3.4 in How to Use the NAG Library and its Documentation for details.

For environments where it might be inappropriate to halt program execution when an error is detected, the value

- 1 ​ or ​ 1

is recommended. If the output of error messages is undesirable, then the value

1

is recommended. Otherwise, if you are not familiar with this argument, the recommended value is

0

. When the value $- 1 or 1$ is used it is essential to test the value of ifail on exit.

On exit:

ifail = 0

unless the routine detects an error or a warning has been flagged (see Section 6).

6

Error Indicators and Warnings

If on entry

ifail = 0

- 1

, explanatory error messages are output on the current error message unit (as defined by x04aaf).

Errors or warnings detected by the routine:

$ifail = 1$

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.
See Section 3.9 in How to Use the NAG Library and its Documentation for further information.

$ifail = - 399$: Your licence key may have expired or may not have been installed correctly.
See Section 3.8 in How to Use the NAG Library and its Documentation for further information.

$ifail = - 999$: Dynamic memory allocation failed.
See Section 3.7 in How to Use the NAG Library and its Documentation for further information.

7

Accuracy

No explicit error estimate is provided by the routine 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

d05aaf is threaded by NAG for parallel execution in multithreaded implementations of the NAG Library.

d05aaf 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 X06 Chapter Introduction for information on how to control and interrogate the OpenMP environment used within this routine. Please also consult the Users' Note for your implementation for any additional implementation-specific information.

9

Further Comments

The time taken by d05aaf increases with n.

This routine may be used to solve an equation with a continuous kernel by defining k1 and k2 to be identical.

This routine may also be used to solve a Volterra equation by defining k2 (or k1) to be identically zero.

10

Example

This example solves the equation

f (x) - \int_{0}^{1} k (x, s) f (s) d s = (1 - \frac{1}{π^{2}}) \sin (π x)

where

k (x, s) = \{\begin{matrix} s (1 - x) for ​ 0 \leq s \leq x, \\ x (1 - s) for ​ x < s \leq 1 . \end{matrix}

Five terms of the Chebyshev series are sought, taking advantage of the centro-symmetry of the

k (x, s)

and even nature of

g (x)

about the mid-point of the range

[0, 1]

The approximate solution at the point

x = 0.1

is calculated by calling c06dcf.

10.1

Program Text

Program Text (d05aafe.f90)

10.2

Program Data

None.

10.3

Program Results

Program Results (d05aafe.r)

d05aaf (PDF version)

D05 (inteq) Chapter Contents

D05 (inteq) Chapter Introduction

NAG Library Manual

NAG Library Routine Document

d05aaf (fredholm2_split)

▸▿ 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

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