NAG Library Routine Document

D05ABF

+− Contents

1 Purpose

2 Specification

3 Description

4 References

5 Parameters

6 Error Indicators and Warnings

7 Accuracy

8 Further Comments

+− 9 Example

9.1 Program Text

9.2 Program Data

9.3 Program Results

1 Purpose

D05ABF solves any linear nonsingular Fredholm integral equation of the second kind with a smooth kernel.

2 Specification

SUBROUTINE D05ABF (

K, G, LAMBDA, A, B, ODOREV, EV, N, CM, F1, WK, LDCM, NT2P1, F, C, IFAIL)

INTEGER	N, LDCM, NT2P1, IFAIL
REAL (KIND=nag_wp)	K, G, LAMBDA, A, B, CM(LDCM,LDCM), F1(LDCM,1), WK(2,NT2P1), F(N), C(N)
LOGICAL	ODOREV, EV
EXTERNAL	K, G

3 Description

D05ABF uses the method of El–Gendi (1969) to solve an integral equation of the form

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

for the function

f (x)

in the range

a \leq x \leq 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) \times \cos [(i - 1) \times π / m]), i = 1, 2, \dots, m + 1 .

The values

f_{i}

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

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 Parameters

1: K – REAL (KIND=nag_wp) FUNCTION, supplied by the user.External Procedure

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

The specification of K is:

FUNCTION K (

X, S)

REAL (KIND=nag_wp) K

REAL (KIND=nag_wp)

X, 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 $k (x, s)$ is to be calculated.

K must either be a module subprogram USEd by, or declared as EXTERNAL in, the (sub)program from which D05ABF is called. Parameters denoted as Input must not be changed by this procedure.

2: G – REAL (KIND=nag_wp) FUNCTION, supplied by the user.External Procedure

G must compute the value of the function

g (x)

of the integral equation in the interval

a \leq x \leq b

The specification of G is:

FUNCTION G (

REAL (KIND=nag_wp) G

REAL (KIND=nag_wp)

1: X – REAL (KIND=nag_wp)Input: On entry: the value of $x$ at which $g (x)$ is to be calculated.

G must either be a module subprogram USEd by, or declared as EXTERNAL in, the (sub)program from which D05ABF is called. Parameters denoted as Input must not be changed by this procedure.

3: LAMBDA – REAL (KIND=nag_wp)Input

On entry: the value of the parameter

λ

of the integral equation.

4: A – REAL (KIND=nag_wp)Input

On entry:

a

, the lower limit of integration.

5: B – REAL (KIND=nag_wp)Input

On entry:

b

, the upper limit of integration.

Constraint:

B > A

6: ODOREV – LOGICALInput

On entry: 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 – LOGICALInput

On entry: 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 – INTEGERInput

On entry: the number of terms in the Chebyshev series which approximates the solution

f (x)

Constraint:

N \geq 1

9: CM(LDCM,LDCM) – REAL (KIND=nag_wp) arrayWorkspace

10: F1(LDCM, $1$ ) – REAL (KIND=nag_wp) arrayWorkspace

11: WK( $2$ ,NT2P1) – REAL (KIND=nag_wp) arrayWorkspace

12: LDCM – INTEGERInput

On entry: the first dimension of the arrays CM and F1 and the second dimension of the array CM as declared in the (sub)program from which D05ABF is called.

Constraint:

LDCM \geq N

13: NT2P1 – INTEGERInput

On entry: the second dimension of the array WK as declared in the (sub)program from which D05ABF is called. The value

2 \times N + 1

14: F(N) – REAL (KIND=nag_wp) arrayOutput

On exit: 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 Section 3), where

$m = 2 N - 1$	if $ODOREV = .TRUE.$ and $EV = .TRUE.$ .
$m = 2 N$	if $ODOREV = .TRUE.$ and $EV = .FALSE.$ .
$m = N - 1$	if $ODOREV = .FALSE.$ .

15: 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)

. 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.

16: IFAIL – INTEGERInput/Output

On entry: IFAIL must be set to

0

- 1 ​ or ​ 1

. If you are unfamiliar with this parameter you should refer to Section 3.3 in the Essential Introduction 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 parameter, 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.

7 Accuracy

No explicit error estimate is provided by the routine 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.

8 Further Comments

The time taken by D05ABF depends upon the value of N and upon the complexity of the kernel function

k (x, s)

9 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 D05ABF, to obtain the

f_{i} ≃ f (x_{i})

and the

c_{i}

, and in subsequent applications of C06DCF to obtain

f (x)

at selected points.

The program runs for

N = 5

and

N = 10

9.1 Program Text

Program Text (d05abfe.f90)

9.2 Program Data

None.

9.3 Program Results

Program Results (d05abfe.r)

D05ABF (PDF version)

D05 Chapter Contents

D05 Chapter Introduction

NAG Library Manual

NAG Library Routine DocumentD05ABF

+− Contents

1 Purpose

2 Specification

3 Description

4 References

5 Parameters

6 Error Indicators and Warnings

7 Accuracy

8 Further Comments

9 Example

9.1 Program Text

9.2 Program Data

9.3 Program Results

NAG Library Routine Document

D05ABF