NAG C Library Function Document

nag_inteq_fredholm2_split (d05aac)

1

Purpose

2

Specification

3

Description

4

References

5

Arguments

1:    $\mathbf{lambda}$ – doubleInput

On entry: the value of the parameter $\lambda$ of the integral equation.

2:    $\mathbf{a}$ – doubleInput

On entry: $a$, the lower limit of integration.

3:    $\mathbf{b}$ – doubleInput

On entry: $b$, the upper limit of integration.

Constraint: $\mathbf{b}>\mathbf{a}$.

4:    $\mathbf{n}$ – IntegerInput

On entry: the number of terms in the Chebyshev series required to approximate $f\left(x\right)$.

Constraint: $\mathbf{n}\ge 1$.

5:    $\mathbf{k1}$ – function, supplied by the userExternal Function

k1 must evaluate the kernel $k\left(x,s\right)=k_1\left(x,s\right)$ of the integral equation for $a\le s\le x$.

The specification of k1 is:

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

1:    $\mathbf{x}$ – doubleInput

2:    $\mathbf{s}$ – doubleInput

On entry: the values of $x$ and $s$ at which $k_1\left(x,s\right)$ is to be evaluated.

3:    $\mathbf{comm}$ – 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 3.3.1.1 in How to Use the NAG Library and its Documentation).

Note: k1 should not return floating-point NaN (Not a Number) or infinity values, since these are not handled by nag_inteq_fredholm2_split (d05aac). If your code inadvertently does return any NaNs or infinities, nag_inteq_fredholm2_split (d05aac) is likely to produce unexpected results.

6:    $\mathbf{k2}$ – function, supplied by the userExternal Function

k2 must evaluate the kernel $k\left(x,s\right)=k_2\left(x,s\right)$ of the integral equation for $x<s\le b$.

The specification of k2 is:

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

1:    $\mathbf{x}$ – doubleInput

2:    $\mathbf{s}$ – doubleInput

On entry: the values of $x$ and $s$ at which $k_2\left(x,s\right)$ is to be evaluated.

3:    $\mathbf{comm}$ – 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 3.3.1.1 in How to Use the NAG Library and its Documentation).

Note: k2 should not return floating-point NaN (Not a Number) or infinity values, since these are not handled by nag_inteq_fredholm2_split (d05aac). If your code inadvertently does return any NaNs or infinities, nag_inteq_fredholm2_split (d05aac) 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 [$a,b$].

7:    $\mathbf{g}$ – function, supplied by the userExternal Function

g must evaluate the function $g\left(x\right)$ for $a\le x\le b$.

The specification of g is:

double  g (double x, Nag_Comm *comm)

1:    $\mathbf{x}$ – doubleInput

On entry: the values of $x$ at which $g\left(x\right)$ is to be evaluated.

2:    $\mathbf{comm}$ – 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 3.3.1.1 in How to Use the NAG Library and its Documentation).

Note: g should not return floating-point NaN (Not a Number) or infinity values, since these are not handled by nag_inteq_fredholm2_split (d05aac). If your code inadvertently does return any NaNs or infinities, nag_inteq_fredholm2_split (d05aac) is likely to produce unexpected results.

8:    $\mathbf{kform}$ – Nag_KernelFormInput

On entry: determines the forms of the kernel, $k\left(x,s\right)$, and the function $g\left(x\right)$.

$\mathbf{kform}=\mathrm{Nag\_NoCentroSymm}$

$k\left(x,s\right)$ is not centro-symmetric (or no account is to be taken of centro-symmetry).

$\mathbf{kform}=\mathrm{Nag\_CentroSymmOdd}$

$k\left(x,s\right)$ is centro-symmetric and $g\left(x\right)$ is odd.

$\mathbf{kform}=\mathrm{Nag\_CentroSymmEven}$

$k\left(x,s\right)$ is centro-symmetric and $g\left(x\right)$ is even.

$\mathbf{kform}=\mathrm{Nag\_CentroSymmNeither}$

$k\left(x,s\right)$ is centro-symmetric but $g\left(x\right)$ is neither odd nor even.

Constraint: $\mathbf{kform}=\mathrm{Nag\_NoCentroSymm}$, $\mathrm{Nag\_CentroSymmOdd}$, $\mathrm{Nag\_CentroSymmEven}$ or $\mathrm{Nag\_CentroSymmNeither}$.

9:    $\mathbf{f}\left[\mathbf{n}\right]$ – doubleOutput

On exit: the approximate values $f_{\mathit{i}}$, for $\mathit{i}=1,2,\dots ,\mathbf{n}$, of $f\left(x\right)$ evaluated at the first n of $m+1$ Chebyshev points $x_i$, (see Section 3).

If $\mathbf{kform}=\mathrm{Nag\_NoCentroSymm}$ or $\mathrm{Nag\_CentroSymmNeither}$, $m=\mathbf{n}-1$.

If $\mathbf{kform}=\mathrm{Nag\_CentroSymmOdd}$, $m=2\times \mathbf{n}$.

If $\mathbf{kform}=\mathrm{Nag\_CentroSymmEven}$, $m=2\times \mathbf{n}-1$.

10:  $\mathbf{c}\left[\mathbf{n}\right]$ – doubleOutput

On exit: the coefficients $c_{\mathit{i}}$, for $\mathit{i}=1,2,\dots ,\mathbf{n}$, of the Chebyshev series approximation to $f\left(x\right)$.

If $\mathbf{kform}=\mathrm{Nag\_CentroSymmOdd}$ this series contains polynomials of odd order only and if $\mathbf{kform}=\mathrm{Nag\_CentroSymmEven}$ the series contains even order polynomials only.

11:  $\mathbf{comm}$ – Nag_Comm *

The NAG communication argument (see Section 3.3.1.1 in How to Use the NAG Library and its Documentation).

12:  $\mathbf{fail}$ – NagError *Input/Output

The NAG error argument (see Section 3.7 in How to Use the NAG Library and its Documentation).

6

Error Indicators and Warnings

NE_ALLOC_FAIL

Dynamic memory allocation failed.
See Section 2.3.1.2 in How to Use the NAG Library and its Documentation for further information.

NE_BAD_PARAM

On entry, argument $〈\mathit{\text{value}}〉$ had an illegal value.

NE_EIGENVALUES

A failure has occurred due to proximity of 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.

NE_INT

On entry, $\mathbf{n}=〈\mathit{\text{value}}〉$.
Constraint: $\mathbf{n}\ge 1$.

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.
See Section 2.7.6 in How to Use the NAG Library and its Documentation for further information.

NE_NO_LICENCE

Your licence key may have expired or may not have been installed correctly.
See Section 2.7.5 in How to Use the NAG Library and its Documentation for further information.

NE_REAL_2

On entry, $\mathbf{a}=〈\mathit{\text{value}}〉$ and $\mathbf{b}=〈\mathit{\text{value}}〉$.
Constraint: $\mathbf{b}>\mathbf{a}$.

7

Accuracy

8

Parallelism and Performance

9

Further Comments

10

Example

10.1

Program Text

Program Text (d05aace.c)

10.2

Program Data

10.3

Program Results

Program Results (d05aace.r)