NAG Library Function Document

nag_ode_bvp_fd_lin_gen (d02gbc)

1  Purpose

2  Specification

3  Description

4  References

5  Arguments

1:     neq – IntegerInput

On entry: the number of equations; that is neq is the order of system (1).

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

2:     fcnf – function, supplied by the userExternal Function

fcnf must evaluate the matrix $F\left(x\right)$ in (1) at a general point $x$.

The specification of fcnf is:

void  fcnf (Integer neq, double x, double f[], Nag_User *comm)

1:     neq – IntegerInput

On entry: the number of differential equations.

2:     x – doubleInput

On entry: the value of the independent variable $x$.

3:     f[$\mathbf{neq}\times \mathbf{neq}$] – doubleOutput

On exit: the $\left(i,j\right)$th element of the matrix $F\left(x\right)$, for $i,j=1,2,\dots ,\mathbf{neq}$ where $F_{ij}$ is set by $\mathbf{f}\left[\left(i-1\right)\times \mathbf{neq}+\left(j-1\right)\right]$. (See Section 10 for an example.)

4:     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:     fcng – function, supplied by the userExternal Function

fcng must evaluate the vector $g\left(x\right)$ in (1) at a general point $x$.

The specification of fcng is:

void  fcng (Integer neq, double x, double g[], Nag_User *comm)

1:     neq – IntegerInput

On entry: the number of differential equations.

2:     x – doubleInput

On entry: the value of the independent variable $x$.

3:     g[neq] – doubleOutput

On exit: the $\mathit{i}$th element of the vector $g\left(x\right)$, for $\mathit{i}=1,2,\dots ,\mathbf{neq}$. (See Section 10 for an example.)

4:     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.)

If you do not wish to supply fcng the actual argument fcng must be the NAG defined null function pointer NULLFN.

4:     a – doubleInput

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

5:     b – doubleInput

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

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

6:     c[$\mathbf{neq}\times \mathbf{neq}$] – doubleInput/Output

7:     d[$\mathbf{neq}\times \mathbf{neq}$] – doubleInput/Output

8:     gam[neq] – doubleInput/Output

On entry: the arrays c and d must be set to the matrices $C$ and $D$ in (2). gam must be set to the vector $\gamma$ in (2).

On exit: the rows of c and d and the components of gam are re-ordered so that the boundary conditions are in the order:

(i)	conditions on $y\left(a\right)$ only;
(ii)	condition involving $y\left(a\right)$ and $y\left(b\right)$; and
(iii)	conditions on $y\left(b\right)$ only.

The function will be slightly more efficient if the arrays c, d and gam are ordered in this way before entry, and in this event they will be unchanged on exit.

Note that the boundary conditions must be of boundary value type, that is neither $C$ nor $D$ may be identically zero. Note also that the rank of the matrix $\left[C,D\right]$ must be neq for the problem to be properly posed. Any violation of these conditions will lead to an error exit.

9:     mnp – IntegerInput

On entry: the maximum permitted number of mesh points.

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

10:   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.

11:   x[mnp] – 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:

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

(3)

for $\mathbf{np}\ge 4$.

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 (3).

12:   y[$\mathbf{neq}\times \mathbf{mnp}$] – doubleOutput

On exit: the approximate solution $z_j\left(x_i\right)$ satisfying (4), 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.

13:   tol – doubleInput

On entry: a positive absolute error tolerance.

If

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

(4)

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

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

(5)

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

14:   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 fcnf and fcng. 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.

15:   fail – NagError *Input/Output

The NAG error argument (see Section 3.6 in the Essential Introduction).

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_BOUND_COND_COL

More than neq columns of the neq by $2\times \mathbf{neq}$ matrix $\left[C,D\right]$ are identically zero. i.e., the boundary conditions are rank deficient. The number of non-identically zero columns is $⟨\mathit{\text{value}}⟩$.

NE_BOUND_COND_LC

At least one row of the neq by $2\times \mathbf{neq}$ matrix $\left[C,D\right]$ is a linear combination of the other rows, i.e., the boundary conditions are rank deficient. The index of the first such row is $⟨\mathit{\text{value}}⟩$.

NE_BOUND_COND_MAT

One of the matrices $C$ or $D$ is identically zero, i.e., the problem is of initial value and not of the boundary type.

NE_BOUND_COND_NLC

At least one row of the neq by $2\times \mathbf{neq}$ matrix $\left[C,D\right]$ is a linear combination of the other rows determined up to a numerical tolerance, i.e., the boundary conditions are rank deficient. The index of first such row is $⟨\mathit{\text{value}}⟩$. There is some doubt as to the rank deficiency of the boundary conditions. However even if the boundary conditions are not rank deficient they are not posed in a suitable form for use with this function. For example, if

$$C=\left(\begin{array}{cc}1& 0\\ 1& \epsilon \end{array}\right),D=\left(\begin{array}{cc}1& 0\\ 1& 0\end{array}\right),\gamma =\left(\begin{array}{c}{\gamma }_1\\ {\gamma }_2\end{array}\right)$$

and $\epsilon$ is small enough, this error exit is likely to be taken. A better form for the boundary conditions in this case would be

$$C=\left(\begin{array}{cc}1& 0\\ 0& 1\end{array}\right),D=\left(\begin{array}{cc}1& 0\\ 0& 0\end{array}\right),\gamma =\left(\begin{array}{c}{\gamma }_1\\ {\epsilon }^{-1}\left({\gamma }_2-{\gamma }_1\right)\end{array}\right)$$

NE_BOUND_COND_ROW

Row $⟨\mathit{\text{value}}⟩$ of the array c and the corresponding row of array d are identically zero, i.e., the boundary conditions are rank deficient.

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_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 $\mathbf{x}\left[\mathbf{np}-1\right]$: $\mathbf{b}=⟨\mathit{\text{value}}⟩$, $\mathbf{x}\left[\mathbf{np}-1\right]=⟨\mathit{\text{value}}⟩$.

7  Accuracy

8  Parallelism and Performance

9  Further Comments

10  Example

10.1  Program Text

Program Text (d02gbce.c)

10.2  Program Data

10.3  Program Results

Program Results (d02gbce.r)