NAG Library Function Document

nag_ode_bvp_fd_nonlin_fixedbc (d02gac)

1  Purpose

2  Specification

3  Description

4  References

5  Arguments

1:    $\mathbf{neq}$ – IntegerInput

On entry: the number of equations.

Constraint: $\mathbf{neq}\ge 2$.

2:    $\mathbf{fcn}$ – function, supplied by the userExternal Function

fcn must evaluate the functions $f_i$ (i.e., the derivatives $y_i^{\prime }$) at the general point $x$.

The specification of fcn is:

void  fcn (Integer neq, double x, const double y[], double f[], Nag_User *comm)

1:    $\mathbf{neq}$ – IntegerInput

On entry: the number of differential equations.

2:    $\mathbf{x}$ – doubleInput

On entry: the value of the argument $x$.

3:    $\mathbf{y}\left[\mathbf{neq}\right]$ – const doubleInput

On entry: $y\left[\mathit{i}-1\right]$ holds the value of the argument $y_{\mathit{i}}$, for $\mathit{i}=1,2,\dots ,\mathbf{neq}$.

4:    $\mathbf{f}\left[\mathbf{neq}\right]$ – doubleOutput

On exit: $f\left[\mathit{i}-1\right]$ must contain the values of $f_{\mathit{i}}$, for $\mathit{i}=1,2,\dots ,\mathbf{neq}$.

5:    $\mathbf{comm}$ – Nag_User *

Pointer to a structure of type Nag_User with the following member:

p – Pointer 

On entry/exit: the pointer $\mathbf{comm}\mathbf{\to }\mathbf{p}$ should be cast to the required type, e.g., struct user *s = (struct user *)comm → p, to obtain the original object's address with appropriate type. (See the argument comm below.)

3:    $\mathbf{a}$ – doubleInput

On entry: the left-hand boundary point, $a$.

4:    $\mathbf{b}$ – doubleInput

On entry: the right-hand boundary point, $b$.

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

5:    $\mathbf{u}\left[\mathbf{neq}\times 2\right]$ – const doubleInput

On entry: $\mathbf{u}\left[\left(i-1\right)\times 2\right]$ must be set to the known (assigned) or estimated values of $y_i$ at $a$ and $\mathbf{u}\left[\left(\mathit{i}-1\right)\times 2+1\right]$ must be set to the known or estimated values of $y_{\mathit{i}}$ at $b$, for $\mathit{i}=1,2,\dots ,\mathbf{neq}$.

6:    $\mathbf{v}\left[\mathbf{neq}\times 2\right]$ – const IntegerInput

On entry: $\mathbf{v}\left[\left(i-1\right)\times 2+j-1\right]$ must be set to 0 if $\mathbf{u}\left[\left(i-1\right)\times 2+j-1\right]$ is a known (assigned) value and to 1 if $\mathbf{u}\left[\left(i-1\right)\times 2+j-1\right]$ is an estimated value, $i=1,2,\dots ,\mathbf{neq}$ and $j=1,2$.

Constraint: precisely neq of the $\mathbf{v}\left[\left(i-1\right)\times 2+j-1\right]$ must be set to 0, i.e., precisely neq of $\mathbf{u}\left[\left(i-1\right)\times 2\right]$ and $\mathbf{u}\left[\left(i-1\right)\times 2+1\right]$ must be known values and these must not be all at $a$ or $b$.

7:    $\mathbf{mnp}$ – IntegerInput

On entry: the maximum permitted number of mesh points.

Constraint: $\mathbf{mnp}\ge 32$.

8:    $\mathbf{np}$ – Integer *Input/Output

On entry: determines whether a default or user-supplied initial mesh is used.

$\mathbf{np}=0$

np is set to a default value of 4 and a corresponding equispaced mesh $\mathbf{x}\left[0\right],\mathbf{x}\left[1\right],\dots ,\mathbf{x}\left[\mathbf{np}-1\right]$ is used.

$\mathbf{np}\ge 4$

You must define an initial mesh using the array x as described.

Constraint: $\mathbf{np}=0$ or $4\le \mathbf{np}\le \mathbf{mnp}$.

On exit: the number of points in the final (returned) mesh.

9:    $\mathbf{x}\left[\mathbf{mnp}\right]$ – doubleInput/Output

On entry: if $\mathbf{np}\ge 4$ (see np above), the first np elements must define an initial mesh. Otherwise the elements of x need not be set.

Constraint:

$$\mathbf{a}=\mathbf{x}\left[0\right]<\mathbf{x}\left[1\right]<\cdots <\mathbf{x}\left[\mathbf{np}-1\right]=\mathbf{b}\text{ for }\mathbf{np}\ge 4$$

(2)

On exit: $\mathbf{x}\left[0\right],\mathbf{x}\left[1\right],\dots ,\mathbf{x}\left[\mathbf{np}-1\right]$ define the final mesh (with the returned value of np) satisfying the relation (2).

10:  $\mathbf{y}\left[\mathbf{neq}\times \mathbf{mnp}\right]$ – doubleOutput

On exit: the approximate solution $z_j\left(x_i\right)$ satisfying (3), on the final mesh, that is

$$\mathbf{y}\left[\left(j-1\right)\times \mathbf{mnp}+i-1\right]=z_j\left(x_i\right),i=1,2,\dots ,\mathbf{np}\text{; }j=1,2,\dots ,\mathbf{neq}\text{,}$$

where np is the number of points in the final mesh.

The remaining columns of y are not used.

11:  $\mathbf{tol}$ – doubleInput

On entry: a positive absolute error tolerance. If

$$a=x_1<x_2<\cdots <x_{\mathbf{np}}=b$$

is the final mesh, $z_j\left(x_i\right)$ is the $j$th component of the approximate solution at $x_i$, and $y_j\left(x_i\right)$ is the $j$th component of the true solution of equation (1) (see Section 3) and the boundary conditions, then, except in extreme cases, it is expected that

$$\left|z_j\left(x_i\right)-y_j\left(x_i\right)\right|\le \mathbf{tol},i=1,2,\dots ,\mathbf{np}\text{; }j=1,2,\dots ,\mathbf{neq}$$

(3)

Constraint: $\mathbf{tol}>0.0$.

12:  $\mathbf{comm}$ – Nag_User *

Pointer to a structure of type Nag_User with the following member:

p – Pointer 

On entry/exit: the pointer $\mathbf{comm}\mathbf{\to }\mathbf{p}$, of type Pointer, allows you to communicate information to and from fcn. An object of the required type should be declared, e.g., a structure, and its address assigned to the pointer $\mathbf{comm}\mathbf{\to }\mathbf{p}$ by means of a cast to Pointer in the calling program, e.g., comm.p = (Pointer)&s. The type pointer will be void * with a C compiler that defines void * and char * otherwise.

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

The NAG error argument (see Section 2.7 in How to Use the NAG Library and its Documentation).

6  Error Indicators and Warnings

NE_2_REAL_ARG_LE

On entry, $\mathbf{b}=〈\mathit{\text{value}}〉$ while $\mathbf{a}=〈\mathit{\text{value}}〉$. These arguments must satisfy $\mathbf{b}>\mathbf{a}$.

NE_ALLOC_FAIL

Dynamic memory allocation failed.

NE_CONV_MESH

A finer mesh is required for the accuracy requested; that is mnp is not large enough.

NE_CONV_MESH_INIT

The Newton iteration failed to converge on the initial mesh. This may be due to the initial mesh having too few points or the initial approximate solution being too inaccurate. Try using nag_ode_bvp_fd_nonlin_gen (d02rac).

NE_CONV_ROUNDOFF

Solution cannot be improved due to roundoff error. Too much accuracy might have been requested.

NE_INT_ARG_LT

On entry, $\mathbf{mnp}=〈\mathit{\text{value}}〉$.
Constraint: $\mathbf{mnp}\ge 32$.

On entry, $\mathbf{neq}=〈\mathit{\text{value}}〉$.
Constraint: $\mathbf{neq}\ge 2$.

NE_INT_RANGE_CONS_2

On entry, $\mathbf{np}=〈\mathit{\text{value}}〉$ and $\mathbf{mnp}=〈\mathit{\text{value}}〉$. The argument np must satisfy either $4\le \mathbf{np}\le \mathbf{mnp}$ or $\mathbf{np}=0$.

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.

NE_LF_B_MESH

On entry, the left boundary value a, has not been set to $\mathbf{x}\left[0\right]$: $\mathbf{a}=〈\mathit{\text{value}}〉$, $\mathbf{x}\left[0\right]=〈\mathit{\text{value}}〉$.

NE_LF_B_VAL

The number of known left boundary values must be less than the number of equations:
The number of known left boundary values $=〈\mathit{\text{value}}〉$:
The number of equations $=〈\mathit{\text{value}}〉$.

NE_LFRT_B_VAL

The sum of known left and right boundary values must equal the number of equations:
The number of known left boundary values $=〈\mathit{\text{value}}〉$:
The number of known right boundary values $=〈\mathit{\text{value}}〉$:
The number of equations $=〈\mathit{\text{value}}〉$.

NE_NOT_STRICTLY_INCREASING

The sequence x is not strictly increasing: $\mathbf{x}\left[〈\mathit{\text{value}}〉\right]=〈\mathit{\text{value}}〉$, $\mathbf{x}\left[〈\mathit{\text{value}}〉\right]=〈\mathit{\text{value}}〉$.

NE_REAL_ARG_LE

On entry, tol must not be less than or equal to 0.0: $\mathbf{tol}=〈\mathit{\text{value}}〉$.

NE_RT_B_MESH

On entry, the right boundary value b, has not been set to $\mathbf{x}\left[\mathbf{np}-1\right]$: $\mathbf{b}=〈\mathit{\text{value}}〉$, $\mathbf{x}\left[\mathbf{np}-1\right]=〈\mathit{\text{value}}〉$.

NE_RT_B_VAL

The number of known right boundary values must be less than the number of equations:
The number of known right boundary values $=〈\mathit{\text{value}}〉$:
The number of equations $=〈\mathit{\text{value}}〉$.

7  Accuracy

8  Parallelism and Performance

9  Further Comments

10  Example

10.1  Program Text

Program Text (d02gace.c)

10.2  Program Data

10.3  Program Results

Program Results (d02gace.r)