NAG Library Function Document

nag_pde_parab_1d_fd_ode (d03phc)

1

Purpose

2

Specification

3

Description

4

References

5

Arguments

1:    $\mathbf{npde}$ – IntegerInput

On entry: the number of PDEs to be solved.

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

2:    $\mathbf{m}$ – IntegerInput

On entry: the coordinate system used:

$\mathbf{m}=0$

Indicates Cartesian coordinates.

$\mathbf{m}=1$

Indicates cylindrical polar coordinates.

$\mathbf{m}=2$

Indicates spherical polar coordinates.

Constraint: $\mathbf{m}=0$, $1$ or $2$.

3:    $\mathbf{ts}$ – double *Input/Output

On entry: the initial value of the independent variable $t$.

On exit: the value of $t$ corresponding to the solution values in u. Normally $\mathbf{ts}=\mathbf{tout}$.

Constraint: $\mathbf{ts}<\mathbf{tout}$.

4:    $\mathbf{tout}$ – doubleInput

On entry: the final value of $t$ to which the integration is to be carried out.

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

pdedef must evaluate the functions $P_{i,j}$, $Q_i$ and $R_i$ which define the system of PDEs. The functions may depend on $x$, $t$, $U$, $U_x$ and $V$. $Q_i$ may depend linearly on $\stackrel{.}{V}$. pdedef is called approximately midway between each pair of mesh points in turn by nag_pde_parab_1d_fd_ode (d03phc).

The specification of pdedef is:

void  pdedef (Integer npde, double t, double x, const double u[], const double ux[], Integer nv, const double v[], const double vdot[], double p[], double q[], double r[], Integer *ires, Nag_Comm *comm)

1:    $\mathbf{npde}$ – IntegerInput

On entry: the number of PDEs in the system.

2:    $\mathbf{t}$ – doubleInput

On entry: the current value of the independent variable $t$.

3:    $\mathbf{x}$ – doubleInput

On entry: the current value of the space variable $x$.

4:    $\mathbf{u}\left[\mathbf{npde}\right]$ – const doubleInput

On entry: $\mathbf{u}\left[\mathit{i}-1\right]$ contains the value of the component $U_{\mathit{i}}\left(x,t\right)$, for $\mathit{i}=1,2,\dots ,\mathbf{npde}$.

5:    $\mathbf{ux}\left[\mathbf{npde}\right]$ – const doubleInput

On entry: $\mathbf{ux}\left[\mathit{i}-1\right]$ contains the value of the component $\frac{\partial U_{\mathit{i}}\left(x,t\right)}{\partial x}$, for $\mathit{i}=1,2,\dots ,\mathbf{npde}$.

6:    $\mathbf{nv}$ – IntegerInput

On entry: the number of coupled ODEs in the system.

7:    $\mathbf{v}\left[\mathbf{nv}\right]$ – const doubleInput

On entry: if $\mathbf{nv}>0$, $\mathbf{v}\left[\mathit{i}-1\right]$ contains the value of the component $V_{\mathit{i}}\left(t\right)$, for $\mathit{i}=1,2,\dots ,\mathbf{nv}$.

8:    $\mathbf{vdot}\left[\mathbf{nv}\right]$ – const doubleInput

On entry: if $\mathbf{nv}>0$, $\mathbf{vdot}\left[\mathit{i}-1\right]$ contains the value of component ${\stackrel{.}{V}}_{\mathit{i}}\left(t\right)$, for $\mathit{i}=1,2,\dots ,\mathbf{nv}$.

Note: ${\stackrel{.}{V}}_{\mathit{i}}\left(t\right)$, for $\mathit{i}=1,2,\dots ,\mathbf{nv}$, may only appear linearly in $Q_{\mathit{j}}$, for $\mathit{j}=1,2,\dots ,\mathbf{npde}$.

9:    $\mathbf{p}\left[\mathbf{npde}\times \mathbf{npde}\right]$ – doubleOutput

Note: the $\left(i,j\right)$th element of the matrix $P$ is stored in $\mathbf{p}\left[\left(j-1\right)\times \mathbf{npde}+i-1\right]$.

On exit: $\mathbf{p}\left[\left(\mathit{j}-1\right)\times \mathbf{npde}+\mathit{i}-1\right]$ must be set to the value of $P_{\mathit{i},\mathit{j}}\left(x,t,U,U_x,V\right)$, for $\mathit{i}=1,2,\dots ,\mathbf{npde}$ and $\mathit{j}=1,2,\dots ,\mathbf{npde}$.

10:  $\mathbf{q}\left[\mathbf{npde}\right]$ – doubleOutput

On exit: $\mathbf{q}\left[\mathit{i}-1\right]$ must be set to the value of $Q_{\mathit{i}}\left(x,t,U,U_x,V,\stackrel{.}{V}\right)$, for $\mathit{i}=1,2,\dots ,\mathbf{npde}$.

11:  $\mathbf{r}\left[\mathbf{npde}\right]$ – doubleOutput

On exit: $\mathbf{r}\left[\mathit{i}-1\right]$ must be set to the value of $R_{\mathit{i}}\left(x,t,U,U_x,V\right)$, for $\mathit{i}=1,2,\dots ,\mathbf{npde}$.

12:  $\mathbf{ires}$ – Integer *Input/Output

On entry: set to $-1\text{ or }1$.

On exit: should usually remain unchanged. However, you may set ires to force the integration function to take certain actions as described below:

$\mathbf{ires}=2$

Indicates to the integrator that control should be passed back immediately to the calling function with the error indicator set to $\mathbf{fail}\mathbf{.}\mathbf{code}=$ NE_USER_STOP.

$\mathbf{ires}=3$

Indicates to the integrator that the current time step should be abandoned and a smaller time step used instead. You may wish to set $\mathbf{ires}=3$ when a physically meaningless input or output value has been generated. If you consecutively set $\mathbf{ires}=3$, nag_pde_parab_1d_fd_ode (d03phc) returns to the calling function with the error indicator set to $\mathbf{fail}\mathbf{.}\mathbf{code}=$ NE_FAILED_DERIV.

13:  $\mathbf{comm}$ – Nag_Comm *

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

user – double *

iuser – Integer *

p – Pointer 

The type Pointer will be void *. Before calling nag_pde_parab_1d_fd_ode (d03phc) you may allocate memory and initialize these pointers with various quantities for use by pdedef when called from nag_pde_parab_1d_fd_ode (d03phc) (see Section 3.3.1.1 in How to Use the NAG Library and its Documentation).

Note: pdedef should not return floating-point NaN (Not a Number) or infinity values, since these are not handled by nag_pde_parab_1d_fd_ode (d03phc). If your code inadvertently does return any NaNs or infinities, nag_pde_parab_1d_fd_ode (d03phc) is likely to produce unexpected results.

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

bndary must evaluate the functions ${\beta }_i$ and ${\gamma }_i$ which describe the boundary conditions, as given in (4).

The specification of bndary is:

void  bndary (Integer npde, double t, const double u[], const double ux[], Integer nv, const double v[], const double vdot[], Integer ibnd, double beta[], double gamma[], Integer *ires, Nag_Comm *comm)

1:    $\mathbf{npde}$ – IntegerInput

On entry: the number of PDEs in the system.

2:    $\mathbf{t}$ – doubleInput

On entry: the current value of the independent variable $t$.

3:    $\mathbf{u}\left[\mathbf{npde}\right]$ – const doubleInput

On entry: $\mathbf{u}\left[\mathit{i}-1\right]$ contains the value of the component $U_{\mathit{i}}\left(x,t\right)$ at the boundary specified by ibnd, for $\mathit{i}=1,2,\dots ,\mathbf{npde}$.

4:    $\mathbf{ux}\left[\mathbf{npde}\right]$ – const doubleInput

On entry: $\mathbf{ux}\left[\mathit{i}-1\right]$ contains the value of the component $\frac{\partial U_{\mathit{i}}\left(x,t\right)}{\partial x}$ at the boundary specified by ibnd, for $\mathit{i}=1,2,\dots ,\mathbf{npde}$.

5:    $\mathbf{nv}$ – IntegerInput

On entry: the number of coupled ODEs in the system.

6:    $\mathbf{v}\left[\mathbf{nv}\right]$ – const doubleInput

On entry: if $\mathbf{nv}>0$, $\mathbf{v}\left[\mathit{i}-1\right]$ contains the value of the component $V_{\mathit{i}}\left(t\right)$, for $\mathit{i}=1,2,\dots ,\mathbf{nv}$.

7:    $\mathbf{vdot}\left[\mathbf{nv}\right]$ – const doubleInput

On entry: if $\mathbf{nv}>0$, $\mathbf{vdot}\left[\mathit{i}-1\right]$ contains the value of component ${\stackrel{.}{V}}_{\mathit{i}}\left(t\right)$, for $\mathit{i}=1,2,\dots ,\mathbf{nv}$.

Note: ${\stackrel{.}{V}}_{\mathit{i}}\left(t\right)$, for $\mathit{i}=1,2,\dots ,\mathbf{nv}$, may only appear linearly in $Q_{\mathit{j}}$, for $\mathit{j}=1,2,\dots ,\mathbf{npde}$.

8:    $\mathbf{ibnd}$ – IntegerInput

On entry: specifies which boundary conditions are to be evaluated.

$\mathbf{ibnd}=0$

bndary must set up the coefficients of the left-hand boundary, $x=a$.

$\mathbf{ibnd}\ne 0$

bndary must set up the coefficients of the right-hand boundary, $x=b$.

9:    $\mathbf{beta}\left[\mathbf{npde}\right]$ – doubleOutput

On exit: $\mathbf{beta}\left[\mathit{i}-1\right]$ must be set to the value of ${\beta }_{\mathit{i}}\left(x,t\right)$ at the boundary specified by ibnd, for $\mathit{i}=1,2,\dots ,\mathbf{npde}$.

10:  $\mathbf{gamma}\left[\mathbf{npde}\right]$ – doubleOutput

On exit: $\mathbf{gamma}\left[\mathit{i}-1\right]$ must be set to the value of ${\gamma }_{\mathit{i}}\left(x,t,U,U_x,V,\stackrel{.}{V}\right)$ at the boundary specified by ibnd, for $\mathit{i}=1,2,\dots ,\mathbf{npde}$.

11:  $\mathbf{ires}$ – Integer *Input/Output

On entry: set to $-1\text{ or }1$.

On exit: should usually remain unchanged. However, you may set ires to force the integration function to take certain actions as described below:

$\mathbf{ires}=2$

Indicates to the integrator that control should be passed back immediately to the calling function with the error indicator set to $\mathbf{fail}\mathbf{.}\mathbf{code}=$ NE_USER_STOP.

$\mathbf{ires}=3$

Indicates to the integrator that the current time step should be abandoned and a smaller time step used instead. You may wish to set $\mathbf{ires}=3$ when a physically meaningless input or output value has been generated. If you consecutively set $\mathbf{ires}=3$, nag_pde_parab_1d_fd_ode (d03phc) returns to the calling function with the error indicator set to $\mathbf{fail}\mathbf{.}\mathbf{code}=$ NE_FAILED_DERIV.

12:  $\mathbf{comm}$ – Nag_Comm *

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

user – double *

iuser – Integer *

p – Pointer 

The type Pointer will be void *. Before calling nag_pde_parab_1d_fd_ode (d03phc) you may allocate memory and initialize these pointers with various quantities for use by bndary when called from nag_pde_parab_1d_fd_ode (d03phc) (see Section 3.3.1.1 in How to Use the NAG Library and its Documentation).

Note: bndary should not return floating-point NaN (Not a Number) or infinity values, since these are not handled by nag_pde_parab_1d_fd_ode (d03phc). If your code inadvertently does return any NaNs or infinities, nag_pde_parab_1d_fd_ode (d03phc) is likely to produce unexpected results.

7:    $\mathbf{u}\left[\mathbf{neqn}\right]$ – doubleInput/Output

On entry: the initial values of the dependent variables defined as follows:

• $\mathbf{u}\left[\mathbf{npde}\times \left(\mathit{j}-1\right)+\mathit{i}-1\right]$ contain $U_{\mathit{i}}\left(x_{\mathit{j}},t_0\right)$, for $\mathit{i}=1,2,\dots ,\mathbf{npde}$ and $\mathit{j}=1,2,\dots ,\mathbf{npts}$, and
• $\mathbf{u}\left[\mathbf{npts}\times \mathbf{npde}+\mathit{i}-1\right]$ contain $V_{\mathit{i}}\left(t_0\right)$, for $\mathit{i}=1,2,\dots ,\mathbf{nv}$.

On exit: the computed solution $U_{\mathit{i}}\left(x_{\mathit{j}},t\right)$, for $\mathit{i}=1,2,\dots ,\mathbf{npde}$ and $\mathit{j}=1,2,\dots ,\mathbf{npts}$, and $V_{\mathit{k}}\left(t\right)$, for $\mathit{k}=1,2,\dots ,\mathbf{nv}$, evaluated at $t=\mathbf{ts}$.

8:    $\mathbf{npts}$ – IntegerInput

On entry: the number of mesh points in the interval $\left[a,b\right]$.

Constraint: $\mathbf{npts}\ge 3$.

9:    $\mathbf{x}\left[\mathbf{npts}\right]$ – const doubleInput

On entry: the mesh points in the space direction. $\mathbf{x}\left[0\right]$ must specify the left-hand boundary, $a$, and $\mathbf{x}\left[\mathbf{npts}-1\right]$ must specify the right-hand boundary, $b$.

Constraint: $\mathbf{x}\left[0\right]<\mathbf{x}\left[1\right]<\cdots <\mathbf{x}\left[\mathbf{npts}-1\right]$.

10:  $\mathbf{nv}$ – IntegerInput

On entry: the number of coupled ODE components.

Constraint: $\mathbf{nv}\ge 0$.

11:  $\mathbf{odedef}$ – function, supplied by the userExternal Function

odedef must evaluate the functions $F$, which define the system of ODEs, as given in (3).

If $\mathbf{nv}=0$, odedef will never be called and the NAG defined null void function pointer, NULLFN, can be supplied in the call to nag_pde_parab_1d_fd_ode (d03phc).

The specification of odedef is:

void  odedef (Integer npde, double t, Integer nv, const double v[], const double vdot[], Integer nxi, const double xi[], const double ucp[], const double ucpx[], const double rcp[], const double ucpt[], const double ucptx[], double f[], Integer *ires, Nag_Comm *comm)

1:    $\mathbf{npde}$ – IntegerInput

On entry: the number of PDEs in the system.

2:    $\mathbf{t}$ – doubleInput

On entry: the current value of the independent variable $t$.

3:    $\mathbf{nv}$ – IntegerInput

On entry: the number of coupled ODEs in the system.

4:    $\mathbf{v}\left[\mathbf{nv}\right]$ – const doubleInput

On entry: if $\mathbf{nv}>0$, $\mathbf{v}\left[\mathit{i}-1\right]$ contains the value of the component $V_{\mathit{i}}\left(t\right)$, for $\mathit{i}=1,2,\dots ,\mathbf{nv}$.

5:    $\mathbf{vdot}\left[\mathbf{nv}\right]$ – const doubleInput

On entry: if $\mathbf{nv}>0$, $\mathbf{vdot}\left[\mathit{i}-1\right]$ contains the value of component ${\stackrel{.}{V}}_{\mathit{i}}\left(t\right)$, for $\mathit{i}=1,2,\dots ,\mathbf{nv}$.

6:    $\mathbf{nxi}$ – IntegerInput

On entry: the number of ODE/PDE coupling points.

7:    $\mathbf{xi}\left[\mathbf{nxi}\right]$ – const doubleInput

On entry: if $\mathbf{nxi}>0$, $\mathbf{xi}\left[\mathit{i}-1\right]$ contains the ODE/PDE coupling points, ${\xi }_{\mathit{i}}$, for $\mathit{i}=1,2,\dots ,\mathbf{nxi}$.

8:    $\mathbf{ucp}\left[\mathbf{npde}\times \mathbf{nxi}\right]$ – const doubleInput

Note: the $\left(i,j\right)$th element of the matrix is stored in $\mathbf{ucp}\left[\left(j-1\right)\times \mathbf{npde}+i-1\right]$.

On entry: if $\mathbf{nxi}>0$, $\mathbf{ucp}\left[\left(\mathit{j}-1\right)\times \mathbf{npde}+\mathit{i}-1\right]$ contains the value of $U_{\mathit{i}}\left(x,t\right)$ at the coupling point $x={\xi }_{\mathit{j}}$, for $\mathit{i}=1,2,\dots ,\mathbf{npde}$ and $\mathit{j}=1,2,\dots ,\mathbf{nxi}$.

9:    $\mathbf{ucpx}\left[\mathbf{npde}\times \mathbf{nxi}\right]$ – const doubleInput

Note: the $\left(i,j\right)$th element of the matrix is stored in $\mathbf{ucpx}\left[\left(j-1\right)\times \mathbf{npde}+i-1\right]$.

On entry: if $\mathbf{nxi}>0$, $\mathbf{ucpx}\left[\left(\mathit{j}-1\right)\times \mathbf{npde}+\mathit{i}-1\right]$ contains the value of $\frac{\partial U_{\mathit{i}}\left(x,t\right)}{\partial x}$ at the coupling point $x={\xi }_{\mathit{j}}$, for $\mathit{i}=1,2,\dots ,\mathbf{npde}$ and $\mathit{j}=1,2,\dots ,\mathbf{nxi}$.

10:  $\mathbf{rcp}\left[\mathbf{npde}\times \mathbf{nxi}\right]$ – const doubleInput

Note: the $\left(i,j\right)$th element of the matrix is stored in $\mathbf{rcp}\left[\left(j-1\right)\times \mathbf{npde}+i-1\right]$.

On entry: $\mathbf{rcp}\left[\left(\mathit{j}-1\right)\times \mathbf{npde}+\mathit{i}-1\right]$ contains the value of the flux $R_{\mathit{i}}$ at the coupling point $x={\xi }_{\mathit{j}}$, for $\mathit{i}=1,2,\dots ,\mathbf{npde}$ and $\mathit{j}=1,2,\dots ,\mathbf{nxi}$.

11:  $\mathbf{ucpt}\left[\mathbf{npde}\times \mathbf{nxi}\right]$ – const doubleInput

Note: the $\left(i,j\right)$th element of the matrix is stored in $\mathbf{ucpt}\left[\left(j-1\right)\times \mathbf{npde}+i-1\right]$.

On entry: if $\mathbf{nxi}>0$, $\mathbf{ucpt}\left[\left(\mathit{j}-1\right)\times \mathbf{npde}+\mathit{i}-1\right]$ contains the value of $\frac{\partial U_{\mathit{i}}}{\partial t}$ at the coupling point $x={\xi }_{\mathit{j}}$, for $\mathit{i}=1,2,\dots ,\mathbf{npde}$ and $\mathit{j}=1,2,\dots ,\mathbf{nxi}$.

12:  $\mathbf{ucptx}\left[\mathbf{npde}\times \mathbf{nxi}\right]$ – const doubleInput

Note: the $\left(i,j\right)$th element of the matrix is stored in $\mathbf{ucptx}\left[\left(j-1\right)\times \mathbf{npde}+i-1\right]$.

On entry: $\mathbf{ucptx}\left[\left(\mathit{j}-1\right)\times \mathbf{npde}+\mathit{i}-1\right]$ contains the value of $\frac{{\partial }^2{U}_{\mathit{i}}}{\partial x\partial t}$ at the coupling point $x={\xi }_{\mathit{j}}$, for $\mathit{i}=1,2,\dots ,\mathbf{npde}$ and $\mathit{j}=1,2,\dots ,\mathbf{nxi}$.

13:  $\mathbf{f}\left[\mathbf{nv}\right]$ – doubleOutput

On exit: $\mathbf{f}\left[\mathit{i}-1\right]$ must contain the $\mathit{i}$th component of $F$, for $\mathit{i}=1,2,\dots ,\mathbf{nv}$, where $F$ is defined as

$$F=G-A\stackrel{.}{V}-B\left(\begin{array}{c}{U}_t^{*}\\ U_{xt}^{*}\end{array}\right)\text{,}$$

(5)

or

$$F=-A\stackrel{.}{V}-B\left(\begin{array}{c}{U}_t^{*}\\ U_{xt}^{*}\end{array}\right)\text{.}$$

(6)

The definition of $F$ is determined by the input value of ires.

14:  $\mathbf{ires}$ – Integer *Input/Output

On entry: the form of $F$ that must be returned in the array f.

$\mathbf{ires}=1$

Equation (5) must be used.

$\mathbf{ires}=-1$

Equation (6) must be used.

On exit: should usually remain unchanged. However, you may reset ires to force the integration function to take certain actions as described below:

$\mathbf{ires}=2$

Indicates to the integrator that control should be passed back immediately to the calling function with the error indicator set to $\mathbf{fail}\mathbf{.}\mathbf{code}=$ NE_USER_STOP.

$\mathbf{ires}=3$

Indicates to the integrator that the current time step should be abandoned and a smaller time step used instead. You may wish to set $\mathbf{ires}=3$ when a physically meaningless input or output value has been generated. If you consecutively set $\mathbf{ires}=3$, nag_pde_parab_1d_fd_ode (d03phc) returns to the calling function with the error indicator set to $\mathbf{fail}\mathbf{.}\mathbf{code}=$ NE_FAILED_DERIV.

15:  $\mathbf{comm}$ – Nag_Comm *

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

user – double *

iuser – Integer *

p – Pointer 

The type Pointer will be void *. Before calling nag_pde_parab_1d_fd_ode (d03phc) you may allocate memory and initialize these pointers with various quantities for use by odedef when called from nag_pde_parab_1d_fd_ode (d03phc) (see Section 3.3.1.1 in How to Use the NAG Library and its Documentation).

Note: odedef should not return floating-point NaN (Not a Number) or infinity values, since these are not handled by nag_pde_parab_1d_fd_ode (d03phc). If your code inadvertently does return any NaNs or infinities, nag_pde_parab_1d_fd_ode (d03phc) is likely to produce unexpected results.

12:  $\mathbf{nxi}$ – IntegerInput

On entry: the number of ODE/PDE coupling points.

Constraints:

• if $\mathbf{nv}=0$, $\mathbf{nxi}=0$;
• if $\mathbf{nv}>0$, $\mathbf{nxi}\ge 0$.

13:  $\mathbf{xi}\left[\mathbf{nxi}\right]$ – const doubleInput

On entry: if $\mathbf{nxi}>0$, $\mathbf{xi}\left[\mathit{i}-1\right]$, for $\mathit{i}=1,2,\dots ,\mathbf{nxi}$, must be set to the ODE/PDE coupling points.

Constraint: $\mathbf{x}\left[0\right]\le \mathbf{xi}\left[0\right]<\mathbf{xi}\left[1\right]<\cdots <\mathbf{xi}\left[\mathbf{nxi}-1\right]\le \mathbf{x}\left[\mathbf{npts}-1\right]$.

14:  $\mathbf{neqn}$ – IntegerInput

On entry: the number of ODEs in the time direction.

Constraint: $\mathbf{neqn}=\mathbf{npde}\times \mathbf{npts}+\mathbf{nv}$.

15:  $\mathbf{rtol}\left[\mathit{dim}\right]$ – const doubleInput

Note: the dimension, dim, of the array rtol must be at least

• $1$ when $\mathbf{itol}=1$ or $2$;
• $\mathbf{neqn}$ when $\mathbf{itol}=3$ or $4$.

On entry: the relative local error tolerance.

Constraint: $\mathbf{rtol}\left[i-1\right]\ge 0.0$ for all relevant $i$.

16:  $\mathbf{atol}\left[\mathit{dim}\right]$ – const doubleInput

Note: the dimension, dim, of the array atol must be at least

• $1$ when $\mathbf{itol}=1$ or $3$;
• $\mathbf{neqn}$ when $\mathbf{itol}=2$ or $4$.

On entry: the absolute local error tolerance.

Constraint: $\mathbf{atol}\left[i-1\right]\ge 0.0$ for all relevant $i$.

Note: corresponding elements of rtol and atol cannot both be $0.0$.

17:  $\mathbf{itol}$ – IntegerInput

On entry: a value to indicate the form of the local error test. itol indicates to nag_pde_parab_1d_fd_ode (d03phc) whether to interpret either or both of rtol or atol as a vector or scalar. The error test to be satisfied is $‖e_i/w_i‖<1.0$, where $w_i$ is defined as follows:

itol	rtol	atol	$w_i$
1	scalar	scalar	$\mathbf{rtol}\left[0\right]\times \left|U_i\right|+\mathbf{atol}\left[0\right]$
2	scalar	vector	$\mathbf{rtol}\left[0\right]\times \left|U_i\right|+\mathbf{atol}\left[i-1\right]$
3	vector	scalar	$\mathbf{rtol}\left[i-1\right]\times \left|U_i\right|+\mathbf{atol}\left[0\right]$
4	vector	vector	$\mathbf{rtol}\left[i-1\right]\times \left|U_i\right|+\mathbf{atol}\left[i-1\right]$

In the above, $e_{\mathit{i}}$ denotes the estimated local error for the $\mathit{i}$th component of the coupled PDE/ODE system in time, $\mathbf{u}\left[\mathit{i}-1\right]$, for $\mathit{i}=1,2,\dots ,\mathbf{neqn}$.

The choice of norm used is defined by the argument norm.

Constraint: $1\le \mathbf{itol}\le 4$.

18:  $\mathbf{norm}$ – Nag_NormTypeInput

On entry: the type of norm to be used.

$\mathbf{norm}=\mathrm{Nag\_MaxNorm}$

Maximum norm.

$\mathbf{norm}=\mathrm{Nag\_TwoNorm}$

Averaged $L_2$ norm.

If ${\mathbf{u}}_{\mathrm{norm}}$ denotes the norm of the vector u of length neqn, then for the averaged $L_2$ norm

$${\mathbf{u}}_{\mathrm{norm}}=\sqrt{\frac{1}{\mathbf{neqn}}\sum _{i=1}^{\mathbf{neqn}}{\left(\mathbf{u}\left[i-1\right]/w_i\right)}^2}\text{,}$$

while for the maximum norm

$${\mathbf{u}}_{\mathrm{norm}}=\underset{i}{\max }\left|\mathbf{u}\left[i-1\right]/w_i\right|\text{.}$$

See the description of itol for the formulation of the weight vector $w$.

Constraint: $\mathbf{norm}=\mathrm{Nag\_MaxNorm}$ or $\mathrm{Nag\_TwoNorm}$.

19:  $\mathbf{laopt}$ – Nag_LinAlgOptionInput

On entry: the type of matrix algebra required.

$\mathbf{laopt}=\mathrm{Nag\_LinAlgFull}$

Full matrix methods to be used.

$\mathbf{laopt}=\mathrm{Nag\_LinAlgBand}$

Banded matrix methods to be used.

$\mathbf{laopt}=\mathrm{Nag\_LinAlgSparse}$

Sparse matrix methods to be used.

Constraint: $\mathbf{laopt}=\mathrm{Nag\_LinAlgFull}$, $\mathrm{Nag\_LinAlgBand}$ or $\mathrm{Nag\_LinAlgSparse}$.

Note: you are recommended to use the banded option when no coupled ODEs are present (i.e., $\mathbf{nv}=0$).

20:  $\mathbf{algopt}\left[30\right]$ – const doubleInput

On entry: may be set to control various options available in the integrator. If you wish to employ all the default options, $\mathbf{algopt}\left[0\right]$ should be set to $0.0$. Default values will also be used for any other elements of algopt set to zero. The permissible values, default values, and meanings are as follows:

$\mathbf{algopt}\left[0\right]$

Selects the ODE integration method to be used. If $\mathbf{algopt}\left[0\right]=1.0$, a BDF method is used and if $\mathbf{algopt}\left[0\right]=2.0$, a Theta method is used. The default value is $\mathbf{algopt}\left[0\right]=1.0$.

If $\mathbf{algopt}\left[0\right]=2.0$, $\mathbf{algopt}\left[\mathit{i}-1\right]$, for $\mathit{i}=2,3,4$ are not used.

$\mathbf{algopt}\left[1\right]$

Specifies the maximum order of the BDF integration formula to be used. $\mathbf{algopt}\left[1\right]$ may be $1.0$, $2.0$, $3.0$, $4.0$ or $5.0$. The default value is $\mathbf{algopt}\left[1\right]=5.0$.

$\mathbf{algopt}\left[2\right]$

Specifies what method is to be used to solve the system of nonlinear equations arising on each step of the BDF method. If $\mathbf{algopt}\left[2\right]=1.0$ a modified Newton iteration is used and if $\mathbf{algopt}\left[2\right]=2.0$ a functional iteration method is used. If functional iteration is selected and the integrator encounters difficulty, there is an automatic switch to the modified Newton iteration. The default value is $\mathbf{algopt}\left[2\right]=1.0$.

$\mathbf{algopt}\left[3\right]$

Specifies whether or not the Petzold error test is to be employed. The Petzold error test results in extra overhead but is more suitable when algebraic equations are present, such as $P_{i,\mathit{j}}=0.0$, for $\mathit{j}=1,2,\dots ,\mathbf{npde}$, for some $i$ or when there is no ${\stackrel{.}{V}}_i\left(t\right)$ dependence in the coupled ODE system. If $\mathbf{algopt}\left[3\right]=1.0$, the Petzold test is used. If $\mathbf{algopt}\left[3\right]=2.0$, the Petzold test is not used. The default value is $\mathbf{algopt}\left[3\right]=1.0$.

If $\mathbf{algopt}\left[0\right]=1.0$, $\mathbf{algopt}\left[\mathit{i}-1\right]$, for $\mathit{i}=5,6,7$, are not used.

$\mathbf{algopt}\left[4\right]$

Specifies the value of Theta to be used in the Theta integration method. $0.51\le \mathbf{algopt}\left[4\right]\le 0.99$. The default value is $\mathbf{algopt}\left[4\right]=0.55$.

$\mathbf{algopt}\left[5\right]$

Specifies what method is to be used to solve the system of nonlinear equations arising on each step of the Theta method. If $\mathbf{algopt}\left[5\right]=1.0$, a modified Newton iteration is used and if $\mathbf{algopt}\left[5\right]=2.0$, a functional iteration method is used. The default value is $\mathbf{algopt}\left[5\right]=1.0$.

$\mathbf{algopt}\left[6\right]$

Specifies whether or not the integrator is allowed to switch automatically between modified Newton and functional iteration methods in order to be more efficient. If $\mathbf{algopt}\left[6\right]=1.0$, switching is allowed and if $\mathbf{algopt}\left[6\right]=2.0$, switching is not allowed. The default value is $\mathbf{algopt}\left[6\right]=1.0$.

$\mathbf{algopt}\left[10\right]$

Specifies a point in the time direction, $t_{\mathrm{crit}}$, beyond which integration must not be attempted. The use of $t_{\mathrm{crit}}$ is described under the argument itask. If $\mathbf{algopt}\left[0\right]\ne 0.0$, a value of $0.0$ for $\mathbf{algopt}\left[10\right]$, say, should be specified even if itask subsequently specifies that $t_{\mathrm{crit}}$ will not be used.

$\mathbf{algopt}\left[11\right]$

Specifies the minimum absolute step size to be allowed in the time integration. If this option is not required, $\mathbf{algopt}\left[11\right]$ should be set to $0.0$.

$\mathbf{algopt}\left[12\right]$

Specifies the maximum absolute step size to be allowed in the time integration. If this option is not required, $\mathbf{algopt}\left[12\right]$ should be set to $0.0$.

$\mathbf{algopt}\left[13\right]$

Specifies the initial step size to be attempted by the integrator. If $\mathbf{algopt}\left[13\right]=0.0$, the initial step size is calculated internally.

$\mathbf{algopt}\left[14\right]$

Specifies the maximum number of steps to be attempted by the integrator in any one call. If $\mathbf{algopt}\left[14\right]=0.0$, no limit is imposed.

$\mathbf{algopt}\left[22\right]$

Specifies what method is to be used to solve the nonlinear equations at the initial point to initialize the values of $U$, $U_t$, $V$ and $\stackrel{.}{V}$. If $\mathbf{algopt}\left[22\right]=1.0$, a modified Newton iteration is used and if $\mathbf{algopt}\left[22\right]=2.0$, functional iteration is used. The default value is $\mathbf{algopt}\left[22\right]=1.0$.

$\mathbf{algopt}\left[28\right]$ and $\mathbf{algopt}\left[29\right]$ are used only for the sparse matrix algebra option, $\mathbf{laopt}=\mathrm{Nag\_LinAlgSparse}$.

$\mathbf{algopt}\left[28\right]$

Governs the choice of pivots during the decomposition of the first Jacobian matrix. It should lie in the range $0.0<\mathbf{algopt}\left[28\right]<1.0$, with smaller values biasing the algorithm towards maintaining sparsity at the expense of numerical stability. If $\mathbf{algopt}\left[28\right]$ lies outside this range then the default value is used. If the functions regard the Jacobian matrix as numerically singular then increasing $\mathbf{algopt}\left[28\right]$ towards $1.0$ may help, but at the cost of increased fill-in. The default value is $\mathbf{algopt}\left[28\right]=0.1$.

$\mathbf{algopt}\left[29\right]$

Is used as a relative pivot threshold during subsequent Jacobian decompositions (see $\mathbf{algopt}\left[28\right]$) below which an internal error is invoked. If $\mathbf{algopt}\left[29\right]$ is greater than $1.0$ no check is made on the pivot size, and this may be a necessary option if the Jacobian is found to be numerically singular (see $\mathbf{algopt}\left[28\right]$). The default value is $\mathbf{algopt}\left[29\right]=0.0001$.

21:  $\mathbf{rsave}\left[\mathbf{lrsave}\right]$ – doubleCommunication Array

If $\mathbf{ind}=0$, rsave need not be set on entry.

If $\mathbf{ind}=1$, rsave must be unchanged from the previous call to the function because it contains required information about the iteration.

22:  $\mathbf{lrsave}$ – IntegerInput

On entry: the dimension of the array rsave.

Constraint:

If $\mathbf{laopt}=\mathrm{Nag\_LinAlgFull}$, $\mathbf{lrsave}\ge \mathbf{neqn}\times \mathbf{neqn}+\mathbf{neqn}+\mathit{nwkres}+\mathit{lenode}$.

If $\mathbf{laopt}=\mathrm{Nag\_LinAlgBand}$, $\mathbf{lrsave}\ge \left(3\times \mathit{mlu}+1\right)\times \mathbf{neqn}+\mathit{nwkres}+\mathit{lenode}$.

If $\mathbf{laopt}=\mathrm{Nag\_LinAlgSparse}$, $\mathbf{lrsave}\ge 4\times \mathbf{neqn}+11\times \mathbf{neqn}/2+1+\mathit{nwkres}+\mathit{lenode}$.

Where

	$\mathit{mlu}$ is the lower or upper half bandwidths such that $\mathit{mlu}=3\times \mathbf{npde}-1$, for PDE problems only (no coupled ODEs); or $\mathit{mlu}=\mathbf{neqn}-1$, for coupled PDE/ODE problems.
	$\mathit{nwkres}=\left\{\begin{array}{ll}\mathbf{npde}\times \left(2\times \mathbf{npts}+6\times \mathbf{nxi}+3\times \mathbf{npde}+26\right)+\mathbf{nxi}+\mathbf{nv}+7\times \mathbf{npts}+2\text{,}& \text{when }\mathbf{nv}>0\text{ and }\mathbf{nxi}>0\text{;}\\ \mathbf{npde}\times \left(2\times \mathbf{npts}+3\times \mathbf{npde}+32\right)+\mathbf{nv}+7\times \mathbf{npts}+3\text{,}& \text{when }\mathbf{nv}>0\text{ and }\mathbf{nxi}=0\text{;}\\ \mathbf{npde}\times \left(2\times \mathbf{npts}+3\times \mathbf{npde}+32\right)+7\times \mathbf{npts}+4\text{,}& \text{when }\mathbf{nv}=0\text{.}\end{array}\right.$
	$\mathit{lenode}=\left\{\begin{array}{ll}\left(6+\mathrm{int}\left(\mathbf{algopt}\left[1\right]\right)\right)\times \mathbf{neqn}+50\text{,}& \text{when the BDF method is used; or}\\ 9\times \mathbf{neqn}+50\text{,}& \text{when the Theta method is used.}\end{array}\right.$

Note: when $\mathbf{laopt}=\mathrm{Nag\_LinAlgSparse}$, the value of lrsave may be too small when supplied to the integrator. An estimate of the minimum size of lrsave is printed on the current error message unit if $\mathbf{itrace}>0$ and the function returns with $\mathbf{fail}\mathbf{.}\mathbf{code}=$ NE_INT_2.

.

23:  $\mathbf{isave}\left[\mathbf{lisave}\right]$ – IntegerCommunication Array

If $\mathbf{ind}=0$, isave need not be set on entry.

If $\mathbf{ind}=1$, isave must be unchanged from the previous call to the function because it contains required information about the iteration. In particular:

$\mathbf{isave}\left[0\right]$

Contains the number of steps taken in time.

$\mathbf{isave}\left[1\right]$

Contains the number of residual evaluations of the resulting ODE system used. One such evaluation involves computing the PDE functions at all the mesh points, as well as one evaluation of the functions in the boundary conditions.

$\mathbf{isave}\left[2\right]$

Contains the number of Jacobian evaluations performed by the time integrator.

$\mathbf{isave}\left[3\right]$

Contains the order of the last backward differentiation formula method used.

$\mathbf{isave}\left[4\right]$

Contains the number of Newton iterations performed by the time integrator. Each iteration involves an ODE residual evaluation followed by a back-substitution using the $LU$ decomposition of the Jacobian matrix.

24:  $\mathbf{lisave}$ – IntegerInput

On entry: the dimension of the array isave. Its size depends on the type of matrix algebra selected:

• if $\mathbf{laopt}=\mathrm{Nag\_LinAlgFull}$, $\mathbf{lisave}\ge 24$;
• if $\mathbf{laopt}=\mathrm{Nag\_LinAlgBand}$, $\mathbf{lisave}\ge \mathbf{neqn}+24$;
• if $\mathbf{laopt}=\mathrm{Nag\_LinAlgSparse}$, $\mathbf{lisave}\ge 25\times \mathbf{neqn}+24$.

Note: when using the sparse option, the value of lisave may be too small when supplied to the integrator. An estimate of the minimum size of lisave is printed if $\mathbf{itrace}>0$ and the function returns with $\mathbf{fail}\mathbf{.}\mathbf{code}=$ NE_INT_2.

25:  $\mathbf{itask}$ – IntegerInput

On entry: specifies the task to be performed by the ODE integrator.

$\mathbf{itask}=1$

Normal computation of output values u at $t=\mathbf{tout}$.

$\mathbf{itask}=2$

One step and return.

$\mathbf{itask}=3$

Stop at first internal integration point at or beyond $t=\mathbf{tout}$.

$\mathbf{itask}=4$

Normal computation of output values u at $t=\mathbf{tout}$ but without overshooting $t=t_{\mathrm{crit}}$ where $t_{\mathrm{crit}}$ is described under the argument algopt.

$\mathbf{itask}=5$

Take one step in the time direction and return, without passing $t_{\mathrm{crit}}$, where $t_{\mathrm{crit}}$ is described under the argument algopt.

Constraint: $\mathbf{itask}=1$, $2$, $3$, $4$ or $5$.

26:  $\mathbf{itrace}$ – IntegerInput

On entry: the level of trace information required from nag_pde_parab_1d_fd_ode (d03phc) and the underlying ODE solver. itrace may take the value $-1$, $0$, $1$, $2$ or $3$.

$\mathbf{itrace}=-1$

No output is generated.

$\mathbf{itrace}=0$

Only warning messages from the PDE solver are printed.

$\mathbf{itrace}>0$

Output from the underlying ODE solver is printed. This output contains details of Jacobian entries, the nonlinear iteration and the time integration during the computation of the ODE system.

If $\mathbf{itrace}<-1$, $-1$ is assumed and similarly if $\mathbf{itrace}>3$, $3$ is assumed.

The advisory messages are given in greater detail as itrace increases.

27:  $\mathbf{outfile}$ – const char *Input

On entry: the name of a file to which diagnostic output will be directed. If outfile is NULL the diagnostic output will be directed to standard output.

28:  $\mathbf{ind}$ – Integer *Input/Output

On entry: indicates whether this is a continuation call or a new integration.

$\mathbf{ind}=0$

Starts or restarts the integration in time.

$\mathbf{ind}=1$

Continues the integration after an earlier exit from the function. In this case, only the arguments tout and fail should be reset between calls to nag_pde_parab_1d_fd_ode (d03phc).

Constraint: $\mathbf{ind}=0$ or $1$.

On exit: $\mathbf{ind}=1$.

29:  $\mathbf{comm}$ – Nag_Comm *

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

30:  $\mathbf{saved}$ – Nag_D03_Save *Communication Structure

saved must remain unchanged following a previous call to a Chapter d03 function and prior to any subsequent call to a Chapter d03 function.

31:  $\mathbf{fail}$ – 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_ACC_IN_DOUBT

Integration completed, but small changes in atol or rtol are unlikely to result in a changed solution.

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_FAILED_DERIV

In setting up the ODE system an internal auxiliary was unable to initialize the derivative. This could be due to your setting $\mathbf{ires}=3$ in pdedef or bndary.

NE_FAILED_START

atol and rtol were too small to start integration.

NE_FAILED_STEP

Error during Jacobian formulation for ODE system. Increase itrace for further details.

Repeated errors in an attempted step of underlying ODE solver. Integration was successful as far as ts: $\mathbf{ts}=〈\mathit{\text{value}}〉$.

Underlying ODE solver cannot make further progress from the point ts with the supplied values of atol and rtol. $\mathbf{ts}=〈\mathit{\text{value}}〉$.

NE_INCOMPAT_PARAM

On entry, $\mathbf{m}=〈\mathit{\text{value}}〉$ and $\mathbf{x}\left[0\right]=〈\mathit{\text{value}}〉$.
Constraint: $\mathbf{m}\le 0$ or $\mathbf{x}\left[0\right]\ge 0.0$ 

NE_INT

ires set to an invalid value in call to pdedef, bndary, or odedef.

On entry, $\mathbf{ind}=〈\mathit{\text{value}}〉$.
Constraint: $\mathbf{ind}=0$ or $1$.

On entry, $\mathbf{itask}=〈\mathit{\text{value}}〉$.
Constraint: $\mathbf{itask}=1$, $2$, $3$, $4$ or $5$.

On entry, $\mathbf{itol}=〈\mathit{\text{value}}〉$.
Constraint: $\mathbf{itol}=1$, $2$, $3$ or $4$.

On entry, $\mathbf{m}=〈\mathit{\text{value}}〉$.
Constraint: $\mathbf{m}=0$, $1$ or $2$.

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

On entry, $\mathbf{npts}=〈\mathit{\text{value}}〉$.
Constraint: $\mathbf{npts}\ge 3$.

On entry, $\mathbf{nv}=〈\mathit{\text{value}}〉$.
Constraint: $\mathbf{nv}\ge 0$.

On entry, on initial entry $\mathbf{ind}=1$.
Constraint: on initial entry $\mathbf{ind}=0$.

NE_INT_2

On entry, corresponding elements $\mathbf{atol}\left[\mathit{I}-1\right]$ and $\mathbf{rtol}\left[\mathit{J}-1\right]$ are both zero: $\mathit{I}=〈\mathit{\text{value}}〉$ and $\mathit{J}=〈\mathit{\text{value}}〉$.

On entry, lisave is too small: $\mathbf{lisave}=〈\mathit{\text{value}}〉$. Minimum possible dimension: $〈\mathit{\text{value}}〉$.

On entry, lrsave is too small: $\mathbf{lrsave}=〈\mathit{\text{value}}〉$. Minimum possible dimension: $〈\mathit{\text{value}}〉$.

On entry, $\mathbf{nv}=〈\mathit{\text{value}}〉$ and $\mathbf{nxi}=〈\mathit{\text{value}}〉$.
Constraint: $\mathbf{nxi}=0$ when $\mathbf{nv}=0$.

On entry, $\mathbf{nv}=〈\mathit{\text{value}}〉$ and $\mathbf{nxi}=〈\mathit{\text{value}}〉$.
Constraint: $\mathbf{nxi}\ge 0$ when $\mathbf{nv}>0$.

When using the sparse option lisave or lrsave is too small: $\mathbf{lisave}=〈\mathit{\text{value}}〉$, $\mathbf{lrsave}=〈\mathit{\text{value}}〉$.

NE_INT_4

On entry, $\mathbf{neqn}=〈\mathit{\text{value}}〉$, $\mathbf{npde}=〈\mathit{\text{value}}〉$, $\mathbf{npts}=〈\mathit{\text{value}}〉$ and $\mathbf{nv}=〈\mathit{\text{value}}〉$.
Constraint: $\mathbf{neqn}=\mathbf{npde}\times \mathbf{npts}+\mathbf{nv}$.

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.

Serious error in internal call to an auxiliary. Increase itrace for further details.

NE_ITER_FAIL

In solving ODE system, the maximum number of steps $\mathbf{algopt}\left[14\right]$ has been exceeded. $\mathbf{algopt}\left[14\right]=〈\mathit{\text{value}}〉$.

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_NOT_CLOSE_FILE

Cannot close file $〈\mathit{\text{value}}〉$.

NE_NOT_STRICTLY_INCREASING

On entry, $\mathit{I}=〈\mathit{\text{value}}〉$, $\mathbf{xi}\left[\mathit{I}\right]=〈\mathit{\text{value}}〉$ and $\mathbf{xi}\left[\mathit{I}-1\right]=〈\mathit{\text{value}}〉$.
Constraint: $\mathbf{xi}\left[\mathit{I}\right]>\mathbf{xi}\left[\mathit{I}-1\right]$.

On entry, mesh points x appear to be badly ordered: $\mathit{I}=〈\mathit{\text{value}}〉$, $\mathbf{x}\left[\mathit{I}-1\right]=〈\mathit{\text{value}}〉$, $\mathit{J}=〈\mathit{\text{value}}〉$ and $\mathbf{x}\left[\mathit{J}-1\right]=〈\mathit{\text{value}}〉$.

NE_NOT_WRITE_FILE

Cannot open file $〈\mathit{\text{value}}〉$ for writing.

NE_REAL_2

On entry, at least one point in xi lies outside $\left[\mathbf{x}\left[0\right],\mathbf{x}\left[\mathbf{npts}-1\right]\right]$: $\mathbf{x}\left[0\right]=〈\mathit{\text{value}}〉$ and $\mathbf{x}\left[\mathbf{npts}-1\right]=〈\mathit{\text{value}}〉$.

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

On entry, $\mathbf{tout}-\mathbf{ts}$ is too small: $\mathbf{tout}=〈\mathit{\text{value}}〉$ and $\mathbf{ts}=〈\mathit{\text{value}}〉$.

NE_REAL_ARRAY

On entry, $\mathit{I}=〈\mathit{\text{value}}〉$ and $\mathbf{atol}\left[\mathit{I}-1\right]=〈\mathit{\text{value}}〉$.
Constraint: $\mathbf{atol}\left[\mathit{I}-1\right]\ge 0.0$.

On entry, $\mathit{I}=〈\mathit{\text{value}}〉$ and $\mathbf{rtol}\left[\mathit{I}-1\right]=〈\mathit{\text{value}}〉$.
Constraint: $\mathbf{rtol}\left[\mathit{I}-1\right]\ge 0.0$.

NE_SING_JAC

Singular Jacobian of ODE system. Check problem formulation.

NE_TIME_DERIV_DEP

Flux function appears to depend on time derivatives.

NE_USER_STOP

In evaluating residual of ODE system, $\mathbf{ires}=2$ has been set in pdedef, bndary, or odedef. Integration is successful as far as ts: $\mathbf{ts}=〈\mathit{\text{value}}〉$.

NE_ZERO_WTS

Zero error weights encountered during time integration.

7

Accuracy

8

Parallelism and Performance

9

Further Comments

10

Example

10.1

Program Text

Program Text (d03phce.c)

10.2

Program Data

10.3

Program Results

Program Results (d03phce.r)