NAG Library Routine Document

d02jaf (bvp_coll_nth)

1

Purpose

2

Specification

3

Description

4

References

5

Arguments

1:     $\mathbf{n}$ – IntegerInput

On entry: $n$, the order of the differential equation.

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

2:     $\mathbf{cf}$ – real (Kind=nag_wp) Function, supplied by the user.External Procedure

cf defines the differential equation (see Section 3). It must return the value of a function $f_j\left(x\right)$ at a given point $x$, where, for $1\le j\le n+1$, $f_j\left(x\right)$ is the coefficient of $y^{\left(j-1\right)}\left(x\right)$ in the equation, and $f_0\left(x\right)$ is the right-hand side.

The specification of cf is:

Fortran Interface

Function cf ( j, x) Real (Kind=nag_wp) :: cf Integer, Intent (In) :: j Real (Kind=nag_wp), Intent (In) :: x

C Header Interface

#include <nagmk26.h> double  cf (const Integer *j, const double *x)

1:     $\mathbf{j}$ – IntegerInput

On entry: the index of the function $f_j$ to be evaluated.

2:     $\mathbf{x}$ – Real (Kind=nag_wp)Input

On entry: the point at which $f_j$ is to be evaluated.

cf must either be a module subprogram USEd by, or declared as EXTERNAL in, the (sub)program from which d02jaf is called. Arguments denoted as Input must not be changed by this procedure.

Note: cf should not return floating-point NaN (Not a Number) or infinity values, since these are not handled by d02jaf. If your code inadvertently does return any NaNs or infinities, d02jaf is likely to produce unexpected results.

3:     $\mathbf{bc}$ – Subroutine, supplied by the user.External Procedure

bc defines the boundary conditions, each of which has the form $y^{\left(k-1\right)}\left(x_1\right)=s_k$ or $y^{\left(k-1\right)}\left(x_0\right)=s_k$. The boundary conditions may be specified in any order.

The specification of bc is:

Fortran Interface

Subroutine bc ( i, j, rhs) Integer, Intent (In) :: i Integer, Intent (Out) :: j Real (Kind=nag_wp), Intent (Out) :: rhs

C Header Interface

#include <nagmk26.h> void  bc (const Integer *i, Integer *j, double *rhs)

1:     $\mathbf{i}$ – IntegerInput

On entry: the index of the boundary condition to be defined.

2:     $\mathbf{j}$ – IntegerOutput

On exit: must be set to $-k$ if the boundary condition is $y^{\left(k-1\right)}\left(x_0\right)=s_k$, and to $+k$ if it is $y^{\left(k-1\right)}\left(x_1\right)=s_k$.

j must not be set to the same value $k$ for two different values of i.

3:     $\mathbf{rhs}$ – Real (Kind=nag_wp)Output

On exit: must be set to the value $s_k$.

bc must either be a module subprogram USEd by, or declared as EXTERNAL in, the (sub)program from which d02jaf is called. Arguments denoted as Input must not be changed by this procedure.

Note: bc should not return floating-point NaN (Not a Number) or infinity values, since these are not handled by d02jaf. If your code inadvertently does return any NaNs or infinities, d02jaf is likely to produce unexpected results.

4:     $\mathbf{x0}$ – Real (Kind=nag_wp)Input

5:     $\mathbf{x1}$ – Real (Kind=nag_wp)Input

On entry: the left- and right-hand boundaries, $x_0$ and $x_1$, respectively.

Constraint: $\mathbf{x1}>\mathbf{x0}$.

6:     $\mathbf{k1}$ – IntegerInput

On entry: the number of coefficients to be returned in the Chebyshev series representation of the solution (hence the degree of the polynomial approximation is $\mathbf{k1}-1$).

Constraint: $\mathbf{k1}\ge \mathbf{n}+1$.

7:     $\mathbf{kp}$ – IntegerInput

On entry: the number of collocation points to be used.

Constraint: $\mathbf{kp}\ge \mathbf{k1}-\mathbf{n}$.

8:     $\mathbf{c}\left(\mathbf{k1}\right)$ – Real (Kind=nag_wp) arrayOutput

On exit: the computed Chebyshev coefficients; that is, the computed solution is:

$$\underset{i=1}{\overset{\mathbf{k1}}{{\sum }^{\prime }}}\mathbf{c}\left(i\right)T_{i-1}\left(x\right)$$

where $T_i\left(x\right)$ is the $i$th Chebyshev polynomial of the first kind, and ${\sum }^{\prime }$ denotes that the first coefficient, $\mathbf{c}\left(1\right)$, is halved.

9:     $\mathbf{w}\left(\mathbf{lw}\right)$ – Real (Kind=nag_wp) arrayWorkspace

10:   $\mathbf{lw}$ – IntegerInput

On entry: the dimension of the array w as declared in the (sub)program from which d02jaf is called.

Constraint: $\mathbf{lw}\ge 2\times \left(\mathbf{kp}+\mathbf{n}\right)\times \left(\mathbf{k1}+1\right)+7\times \mathbf{k1}$.

11:   $\mathbf{iw}\left(\mathbf{k1}\right)$ – Integer arrayWorkspace

12:   $\mathbf{ifail}$ – IntegerInput/Output

On entry: ifail must be set to $0$, $-1\text{ or }1$. 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 $-1\text{ or }1$ 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 $-\mathbf{1}\text{ or }\mathbf{1}$ is used it is essential to test the value of ifail on exit.

On exit: $\mathbf{ifail}=\mathbf{0}$ unless the routine detects an error or a warning has been flagged (see Section 6).

6

Error Indicators and Warnings

$\mathbf{ifail}=1$

On entry, $\mathbf{k1}=〈\mathit{\text{value}}〉$ and $\mathbf{n}=〈\mathit{\text{value}}〉$.
Constraint: $\mathbf{k1}\ge \mathbf{n}+1$.

On entry, $\mathbf{kp}=〈\mathit{\text{value}}〉$, $\mathbf{n}=〈\mathit{\text{value}}〉$ and $\mathbf{k1}=〈\mathit{\text{value}}〉$.
Constraint: $\mathbf{kp}+\mathbf{n}\ge \mathbf{k1}$.

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

On entry, $\mathbf{x1}=〈\mathit{\text{value}}〉$ and $\mathbf{x0}=〈\mathit{\text{value}}〉$.
Constraint: $\mathbf{x1}>\mathbf{x0}$.

$\mathbf{ifail}=2$

On entry, $\mathbf{lw}=〈\mathit{\text{value}}〉$.
Constraint: $\mathbf{lw}\ge 2\times \left(\mathbf{kp}+\mathbf{n}\right)\times \left(\mathbf{k1}+1\right)+7\times \mathbf{k1}$; that is, $〈\mathit{\text{value}}〉$.

$\mathbf{ifail}=3$

Either the boundary conditions are not linearly independent, or the coefficient matrix is rank deficient. Increasing the number of collocation points may overcome this latter problem.

$\mathbf{ifail}=4$

Iterative refinement in the least squares solution has failed to converge. The coefficient matrix is too ill-conditioned.

$\mathbf{ifail}=-99$

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.

$\mathbf{ifail}=-399$

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.

$\mathbf{ifail}=-999$

Dynamic memory allocation failed.

See Section 3.7 in How to Use the NAG Library and its Documentation for further information.

7

Accuracy

8

Parallelism and Performance

9

Further Comments

10

Example

10.1

Program Text

Program Text (d02jafe.f90)

10.2

Program Data

Program Data (d02jafe.d)

10.3

Program Results

Program Results (d02jafe.r)