NAG CL Interface
d03pkc (dim1_​parab_​dae_​keller)

1 Purpose

2 Specification

3 Description

4 References

5 Arguments

1: $\mathbf{npde}$ – Integer Input

On entry: the number of PDEs to be solved.

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

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

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

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

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

3: $\mathbf{tout}$ – double Input

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

4: $\mathbf{pdedef}$ – function, supplied by the user External Function

pdedef must evaluate the functions $G_i$ which define the system of PDEs. pdedef is called approximately midway between each pair of mesh points in turn by d03pkc.

The specification of pdedef is:

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

1: $\mathbf{npde}$ – Integer Input

On entry: the number of PDEs in the system.

2: $\mathbf{t}$ – double Input

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

3: $\mathbf{x}$ – double Input

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

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

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

5: $\mathbf{ut}\left[\mathbf{npde}\right]$ – const double Input

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

6: $\mathbf{ux}\left[\mathbf{npde}\right]$ – const double Input

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

7: $\mathbf{nv}$ – Integer Input

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

8: $\mathbf{v}\left[\mathbf{nv}\right]$ – const double Input

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 $\mathit{i}=1,2,\dots ,\mathbf{nv}$.

9: $\mathbf{vdot}\left[\mathbf{nv}\right]$ – const double Input

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 $\mathit{i}=1,2,\dots ,\mathbf{nv}$.

10: $\mathbf{res}\left[\mathbf{npde}\right]$ – double Output

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

$$G_{\mathit{i}}=\sum _{\mathit{j}=1}^{\mathbf{npde}}{P}_{\mathit{i},\mathit{j}}\frac{\partial U_{\mathit{j}}}{\partial t}+\sum _{\mathit{j}=1}^{\mathbf{nv}}{Q}_{\mathit{i},\mathit{j}}{\stackrel{.}{V}}_{\mathit{j}}\text{,}$$

(9)

i.e., only terms depending explicitly on time derivatives, or

$$G_{\mathit{i}}=\sum _{\mathit{j}=1}^{\mathbf{npde}}{P}_{\mathit{i},\mathit{j}}\frac{\partial U_{\mathit{j}}}{\partial t}+\sum _{\mathit{j}=1}^{\mathbf{nv}}{Q}_{\mathit{i},\mathit{j}}{\stackrel{.}{V}}_{\mathit{j}}+S_{\mathit{i}}\text{,}$$

(10)

i.e., all terms in equation (3).

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

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

On entry: the form of $G_i$ that must be returned in the array res.

$\mathbf{ires}=\mathrm{-1}$

Equation (9) must be used.

$\mathbf{ires}=1$

Equation (10) must be used.

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 $\mathbf{ires}=3$, d03pkc 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 pdedef.

user – double *

iuser – Integer *

p – Pointer 

The type Pointer will be void *. Before calling d03pkc you may allocate memory and initialize these pointers with various quantities for use by pdedef when called from d03pkc (see Section 3.1.1 in the Introduction to the NAG Library CL Interface).

Note: pdedef should not return floating-point NaN (Not a Number) or infinity values, since these are not handled by d03pkc. If your code inadvertently does return any NaNs or infinities, d03pkc is likely to produce unexpected results.

5: $\mathbf{bndary}$ – function, supplied by the user External Function

bndary must evaluate the functions $G_i^L$ and $G_i^R$ which describe the boundary conditions, as given in (5) and (6).

The specification of bndary is:

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void  bndary (Integer npde, double t, Integer ibnd, Integer nobc, const double u[], const double ut[], Integer nv, const double v[], const double vdot[], double res[], Integer *ires, Nag_Comm *comm)

1: $\mathbf{npde}$ – Integer Input

On entry: the dimension of the array u and the dimension of the array ut. The number of PDEs in the system.

2: $\mathbf{t}$ – double Input

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

3: $\mathbf{ibnd}$ – Integer Input

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

$\mathbf{ibnd}=0$

bndary must compute the left-hand boundary condition at $x=a$.

$\mathbf{ibnd}\ne 0$

bndary must compute the right-hand boundary condition at $x=b$.

4: $\mathbf{nobc}$ – Integer Input

On entry: specifies the number of boundary conditions at the boundary specified by ibnd.

5: $\mathbf{u}\left[\mathbf{npde}\right]$ – const double Input

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

6: $\mathbf{ut}\left[\mathbf{npde}\right]$ – const double Input

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

7: $\mathbf{nv}$ – Integer Input

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

8: $\mathbf{v}\left[\mathbf{nv}\right]$ – const double Input

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 $\mathit{i}=1,2,\dots ,\mathbf{nv}$.

9: $\mathbf{vdot}\left[\mathbf{nv}\right]$ – const double Input

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: $\mathbf{vdot}\left[\mathit{i}-1\right]$, for $\mathit{i}=1,2,\dots ,\mathbf{nv}$, may only appear linearly as in (7) and (8).

10: $\mathbf{res}\left[\mathit{dim}\right]$ – double Output

On exit: $\mathbf{res}\left[\mathit{i}-1\right]$ must contain the $\mathit{i}$th component of $G^L$ or $G^R$, depending on the value of ibnd, for $\mathit{i}=1,2,\dots ,\mathbf{nobc}$, where $G^L$ is defined as

$$G_{\mathit{i}}^L=\sum _{\mathit{j}=1}^{\mathbf{npde}}{E}_{\mathit{i},\mathit{j}}^L\frac{\partial U_{\mathit{j}}}{\partial t}+\sum _{\mathit{j}=1}^{\mathbf{nv}}{H}_{\mathit{i},\mathit{j}}^L{\stackrel{.}{V}}_{\mathit{j}}\text{,}$$

(11)

i.e., only terms depending explicitly on time derivatives, or

$$G_{\mathit{i}}^L=\sum _{\mathit{j}=1}^{\mathbf{npde}}{E}_{\mathit{i},\mathit{j}}^L\frac{\partial U_{\mathit{j}}}{\partial t}+\sum _{\mathit{j}=1}^{\mathbf{nv}}{H}_{\mathit{i},\mathit{j}}^L{\stackrel{.}{V}}_{\mathit{j}}+K_{\mathit{i}}^L\text{,}$$

(12)

i.e., all terms in equation (7), and similarly for $G_{\mathit{i}}^R$.

The definitions of $G^L$ and $G^R$ are determined by the input value of ires.

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

On entry: the form of $G_i^L$ (or $G_i^R$) that must be returned in the array res.

$\mathbf{ires}=\mathrm{-1}$

Equation (11) must be used.

$\mathbf{ires}=1$

Equation (12) must be used.

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 $\mathbf{ires}=3$, d03pkc 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 d03pkc you may allocate memory and initialize these pointers with various quantities for use by bndary when called from d03pkc (see Section 3.1.1 in the Introduction to the NAG Library CL Interface).

Note: bndary should not return floating-point NaN (Not a Number) or infinity values, since these are not handled by d03pkc. If your code inadvertently does return any NaNs or infinities, d03pkc is likely to produce unexpected results.

6: $\mathbf{u}\left[\mathbf{neqn}\right]$ – double Input/Output

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

• $\mathbf{u}\left[\mathbf{npde}\times (\mathit{j}-1)+\mathit{i}-1\right]$ contain $U_{\mathit{i}}(x_{\mathit{j}},t_0)$, 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 $U_{\mathit{i}}(x_{\mathit{j}},t)$, 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 $t=\mathbf{ts}$, as follows:

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

7: $\mathbf{npts}$ – Integer Input

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

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

8: $\mathbf{x}\left[\mathbf{npts}\right]$ – const double Input

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

9: $\mathbf{nleft}$ – Integer Input

On entry: the number $n_a$ of boundary conditions at the left-hand mesh point $\mathbf{x}\left[0\right]$.

Constraint: $0\le \mathbf{nleft}\le \mathbf{npde}$.

10: $\mathbf{nv}$ – Integer Input

On entry: the number of coupled ODE components.

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

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

odedef must evaluate the functions $R$, which define the system of ODEs, as given in (4).

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 d03pkc.

The specification of odedef is:

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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 ucpt[], double r[], Integer *ires, Nag_Comm *comm)

1: $\mathbf{npde}$ – Integer Input

On entry: the number of PDEs in the system.

2: $\mathbf{t}$ – double Input

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

3: $\mathbf{nv}$ – Integer Input

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

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

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 double Input

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: $\mathbf{nxi}$ – Integer Input

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

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

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 double Input

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

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 $U_{\mathit{i}}(x,t)$ at the coupling point $x={\xi }_{\mathit{j}}$, for $\mathit{i}=1,2,\dots ,\mathbf{npde}$ and $\mathit{j}=1,2,\dots ,\mathbf{nxi}$.

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

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

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 $\frac{\partial U_{\mathit{i}}(x,t)}{\partial x}$ 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: $\mathbf{ucpt}\left[\mathbf{npde}\times \mathbf{nxi}\right]$ – const double Input

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

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 $\mathit{j}=1,2,\dots ,\mathbf{nxi}$.

11: $\mathbf{r}\left[\mathbf{nv}\right]$ – double Output

On exit: if $\mathbf{nv}>0$, $\mathbf{r}\left[\mathit{i}-1\right]$ must contain the $\mathit{i}$th component of $R$, for $\mathit{i}=1,2,\dots ,\mathbf{nv}$, where $R$ is defined as

$$R=-B\stackrel{.}{V}-C{U}_t^{*}\text{,}$$

(13)

i.e., only terms depending explicitly on time derivatives, or

$$R=A-B\stackrel{.}{V}-C{U}_t^{*}\text{,}$$

(14)

i.e., all terms in equation (4). The definition of $R$ is determined by the input value of ires.

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

On entry: the form of $R$ that must be returned in the array r.

$\mathbf{ires}=\mathrm{-1}$

Equation (13) must be used.

$\mathbf{ires}=1$

Equation (14) 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 $\mathbf{ires}=3$, d03pkc 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 odedef.

user – double *

iuser – Integer *

p – Pointer 

The type Pointer will be void *. Before calling d03pkc you may allocate memory and initialize these pointers with various quantities for use by odedef when called from d03pkc (see Section 3.1.1 in the Introduction to the NAG Library CL Interface).

Note: odedef should not return floating-point NaN (Not a Number) or infinity values, since these are not handled by d03pkc. If your code inadvertently does return any NaNs or infinities, d03pkc is likely to produce unexpected results.

12: $\mathbf{nxi}$ – Integer Input

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 double Input

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

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: $\mathbf{neqn}$ – Integer Input

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 double Input

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 double Input

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: $\mathbf{itol}$ – Integer Input

On entry: a value to indicate the form of the local error test. itol indicates to d03pkc whether to interpret either or both of rtol or atol as a vector or scalar. The error test to be satisfied is $\Vert e_i/w_i\Vert <1.0$, where $w_i$ is defined as follows:

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

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_NormType Input

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

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

while for the maximum norm

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

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_LinAlgOption Input

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 double Input

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 $\mathbf{algopt}\left[0\right]=2.0$, then $\mathbf{algopt}\left[\mathit{i}-1\right]$, 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, i.e., $\mathbf{laopt}=\mathrm{Nag\_LinAlgSparse}$.

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

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 $\mathbf{algopt}\left[28\right]=0.1$.

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

Used as a relative pivot threshold during subsequent Jacobian decompositions (see $\mathbf{algopt}\left[28\right]$) below which an internal error is invoked. $\mathbf{algopt}\left[29\right]$ must be greater than zero, otherwise the default value is used. 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: $\mathbf{rsave}\left[\mathbf{lrsave}\right]$ – double Communication 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: $\mathbf{lrsave}$ – Integer Input

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

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

If $\mathbf{laopt}=\mathrm{Nag\_LinAlgBand}$, $\mathbf{lrsave}\ge (2\mathit{ml}+\mathit{mu}+2)\times \mathbf{neqn}+\mathit{nwkres}+\mathit{lenode}$.

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

Where $\mathit{ml}$ and $\mathit{mu}$ are the lower and upper half bandwidths given by $\mathit{ml}=\mathbf{npde}+\mathbf{nleft}-1$ such that

for problems involving PDEs only,

$\mathit{mu}=2\mathbf{npde}-\mathbf{nleft}-1\text{;}$

for coupled PDE/ODE problems,

$\mathit{ml}=\mathit{mu}=\mathbf{neqn}-1\text{.}$

Where $\mathit{nwkres}$ is defined by

if $\mathbf{nv}>0\text{ and }\mathbf{nxi}>0$,

$\mathit{nwkres}=\mathbf{npde}(3\mathbf{npde}+6\mathbf{nxi}+\mathbf{npts}+15)+\mathbf{nxi}+\mathbf{nv}+7\mathbf{npts}+2\text{;}$

if $\mathbf{nv}>0\text{ and }\mathbf{nxi}=0$,

$\mathit{nwkres}=\mathbf{npde}(3\mathbf{npde}+\mathbf{npts}+21)+\mathbf{nv}+7\mathbf{npts}+3\text{;}$

if $\mathbf{nv}=0$,

$\mathit{nwkres}=\mathbf{npde}(3\mathbf{npde}+\mathbf{npts}+21)+7\mathbf{npts}+4\text{.}$

Where $\mathit{lenode}$ is defined by

if the BDF method is used,

$\mathit{lenode}=(6+\mathrm{int}\left(\mathbf{algopt}\left[1\right]\right))\times \mathbf{neqn}+50\text{;}$

if the Theta method is used,

$\mathit{lenode}=9\mathbf{neqn}+50\text{.}$

Note: when using the sparse option, 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: $\mathbf{isave}\left[\mathbf{lisave}\right]$ – Integer Communication Array

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

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 the following components of the array isave concern the efficiency of the integration:

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

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 evaluating 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 ODE method last used in the time integration.

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

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

24: $\mathbf{lisave}$ – Integer Input

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: $\mathbf{itask}$ – Integer Input

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

$\mathbf{itask}=1$

Normal computation of output values u at $t=\mathbf{tout}$ (by overshooting and interpolating).

$\mathbf{itask}=2$

Take one step in the time direction 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: $\mathbf{itrace}$ – Integer Input

On entry: the level of trace information required from d03pkc and the underlying ODE solver as follows:

$\mathbf{itrace}\le \mathrm{-1}$

No output is generated.

$\mathbf{itrace}=0$

Only warning messages from the PDE solver are printed.

$\mathbf{itrace}=1$

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.

$\mathbf{itrace}=2$

Output from the underlying ODE solver is similar to that produced when $\mathbf{itrace}=1$, except that the advisory messages are given in greater detail.

$\mathbf{itrace}\ge 3$

Output from the underlying ODE solver is similar to that produced when $\mathbf{itrace}=2$, except that the advisory messages are given in greater detail.

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 argument tout should be reset between calls to d03pkc.

Constraint: $\mathbf{ind}=0$ or $1$.

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

29: $\mathbf{comm}$ – Nag_Comm *

The NAG communication argument (see Section 3.1.1 in the Introduction to the NAG Library CL Interface).

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 7 in the Introduction to the NAG Library CL Interface).

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.

The required task has been completed, but it is estimated that a small change in atol and rtol is unlikely to produce any change in the computed solution. (Only applies when you are not operating in one step mode, that is when $\mathbf{itask}\ne 2$ or $5$.)

NE_ALLOC_FAIL

Dynamic memory allocation failed.
See Section 3.1.2 in the Introduction to the NAG Library CL Interface 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: $\mathbf{ts}=⟨\mathit{\text{value}}⟩$.

In the underlying ODE solver, there were repeated error test failures on an attempted step, before completing the requested task, but the integration was successful as far as $t=\mathbf{ts}$. The problem may have a singularity, or the error requirement may be inappropriate. Incorrect positioning of boundary conditions may also result in this error.

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_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, $\mathbf{nleft}=⟨\mathit{\text{value}}⟩$.
Constraint: $\mathbf{nleft}\ge 0$.

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, $\mathit{i}=⟨\mathit{\text{value}}⟩$ and $\mathit{j}=⟨\mathit{\text{value}}⟩$.
Constraint: corresponding elements $\mathbf{atol}\left[\mathit{i}-1\right]$ and $\mathbf{rtol}\left[\mathit{j}-1\right]$ cannot both be $0.0$.

On entry, $\mathbf{lisave}=⟨\mathit{\text{value}}⟩$.
Constraint: $\mathbf{lisave}\ge ⟨\mathit{\text{value}}⟩$.

On entry, $\mathbf{lrsave}=⟨\mathit{\text{value}}⟩$.
Constraint: $\mathbf{lrsave}\ge ⟨\mathit{\text{value}}⟩$.

On entry, $\mathbf{nleft}=⟨\mathit{\text{value}}⟩$, $\mathbf{npde}=⟨\mathit{\text{value}}⟩$.
Constraint: $\mathbf{nleft}\le \mathbf{npde}$.

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 7.5 in the Introduction to the NAG Library CL Interface 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 8 in the Introduction to the NAG Library CL Interface for further information.

NE_NOT_CLOSE_FILE

Cannot close file $⟨\mathit{\text{value}}⟩$.

NE_NOT_STRICTLY_INCREASING

On entry, $\mathit{i}=⟨\mathit{\text{value}}⟩$, $\mathbf{x}\left[\mathit{i}-1\right]=⟨\mathit{\text{value}}⟩$, $\mathit{j}=⟨\mathit{\text{value}}⟩$ and $\mathbf{x}\left[\mathit{j}-1\right]=⟨\mathit{\text{value}}⟩$.
Constraint: $\mathbf{x}\left[0\right]<\mathbf{x}\left[1\right]<\cdots <\mathbf{x}\left[\mathbf{npts}-1\right]$.

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: $\mathbf{xi}\left[\mathit{i}\right]>\mathbf{xi}\left[\mathit{i}-1\right]$.

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 $[\mathbf{x}\left[0\right],\mathbf{x}\left[\mathbf{npts}-1\right]]$: $\mathbf{x}\left[0\right]=⟨\mathit{\text{value}}⟩$ 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_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.

Some error weights $w_i$ became zero during the time integration (see the description of itol). Pure relative error control $(\mathbf{atol}\left[i-1\right]=0.0)$ was requested on a variable (the $i$th) which has become zero. The integration was successful as far as $t=\mathbf{ts}$.

7 Accuracy

8 Parallelism and Performance

9 Further Comments

10 Example

10.1 Program Text

10.2 Program Data

10.3 Program Results