NAG C Library Function Document

nag_inteq_fredholm2_smooth (d05abc)

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 which approximates the solution $f\left(x\right)$.

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

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

k must compute the value of the kernel $k\left(x,s\right)$ of the integral equation over the square $a\le x\le b$, $a\le s\le b$.

The specification of k is:

double  k (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 $k\left(x,s\right)$ is to be calculated.

3:    $\mathbf{comm}$ – Nag_Comm *

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

user – double *

iuser – Integer *

p – Pointer 

The type Pointer will be void *. Before calling nag_inteq_fredholm2_smooth (d05abc) you may allocate memory and initialize these pointers with various quantities for use by k when called from nag_inteq_fredholm2_smooth (d05abc) (see Section 3.3.1.1 in How to Use the NAG Library and its Documentation).

Note: k should not return floating-point NaN (Not a Number) or infinity values, since these are not handled by nag_inteq_fredholm2_smooth (d05abc). If your code inadvertently does return any NaNs or infinities, nag_inteq_fredholm2_smooth (d05abc) is likely to produce unexpected results.

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

g must compute the value of the function $g\left(x\right)$ of the integral equation in the interval $a\le x\le b$.

The specification of g is:

double  g (double x, Nag_Comm *comm)

1:    $\mathbf{x}$ – doubleInput

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

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_smooth (d05abc) you may allocate memory and initialize these pointers with various quantities for use by g when called from nag_inteq_fredholm2_smooth (d05abc) (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_smooth (d05abc). If your code inadvertently does return any NaNs or infinities, nag_inteq_fredholm2_smooth (d05abc) is likely to produce unexpected results.

7:    $\mathbf{odorev}$ – Nag_BooleanInput

On entry: indicates whether it is known that the solution $f\left(x\right)$ is odd or even about the mid-point of the range of integration. If odorev is Nag_TRUE then an odd or even solution is sought depending upon the value of ev.

8:    $\mathbf{ev}$ – Nag_BooleanInput

On entry: is ignored if odorev is Nag_FALSE. Otherwise, if ev is Nag_TRUE, an even solution is sought, whilst if ev is Nag_FALSE, an odd solution is sought.

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 the function $f\left(x\right)$ at the first n of $m+1$ Chebyshev points (see Section 3), where

$m=2\mathbf{n}-1$	if $\mathbf{odorev}=\mathrm{Nag\_TRUE}$ and $\mathbf{ev}=\mathrm{Nag\_TRUE}$.
$m=2\mathbf{n}$	if $\mathbf{odorev}=\mathrm{Nag\_TRUE}$ and $\mathbf{ev}=\mathrm{Nag\_FALSE}$.
$m=\mathbf{n}-1$	if $\mathbf{odorev}=\mathrm{Nag\_FALSE}$.

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 $f\left(x\right)$. When odorev is Nag_TRUE, this series contains polynomials of even order only or of odd order only, according to ev being Nag_TRUE or Nag_FALSE respectively.

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 (d05abce.c)

10.2

Program Data

10.3

Program Results

Program Results (d05abce.r)