NAG Toolbox: nag_ode_ivp_stiff_blend (d02nw)

Purpose

Syntax

Description

References

Parameters

Compulsory Input Parameters

1:     $\mathrm{neqmax}$ – int64int32nag_int scalar

A bound on the maximum number of differential equations to be solved.

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

2:     $\mathrm{sdysav}$ – int64int32nag_int scalar

The second dimension of the array ysav that will be supplied to the integrator, as declared in the (sub)program from which the integrator is called (e.g., see nag_ode_ivp_stiff_exp_fulljac (d02nb)).

Constraint: $\mathbf{sdysav}\ge \mathbf{maxord}+3$.

3:     $\mathrm{maxord}$ – int64int32nag_int scalar

The maximum order to be used for the BLEND method.

Constraint: $0<\mathbf{maxord}\le 11$.

4:     $\mathrm{con}\left(6\right)$ – double array

Values to be used to control step size choice during integration. If any $\mathbf{con}\left(i\right)=0.0$ on entry, it is replaced by its default value described below. In most cases this is the recommended setting.

$\mathbf{con}\left(1\right)$, $\mathbf{con}\left(2\right)$, and $\mathbf{con}\left(3\right)$ are factors used to bound step size changes. If the current step size $h$ fails, then the modulus of the next step size is bounded by $\mathbf{con}\left(1\right)\times \left|h\right|$. The default value of $\mathbf{con}\left(1\right)$ is $2.0$. Note that the new step size may be used with a method of different order to the failed step. If the initial step size is $h$, then the modulus of the step size on the second step is bounded by $\mathbf{con}\left(3\right)\times \left|h\right|$. At any other stage in the integration, if the current step size is $h$, then the modulus of the next step size is bounded by $\mathbf{con}\left(2\right)\times \left|h\right|$. The default values are $10.0$ for $\mathbf{con}\left(2\right)$ and $1000.0$ for $\mathbf{con}\left(3\right)$.

$\mathbf{con}\left(4\right)$, $\mathbf{con}\left(5\right)$ and $\mathbf{con}\left(6\right)$ are ‘tuning’ constants used in determining the next order and step size. They are used to scale the error estimates used in determining whether to keep the same order of the BLEND method, decrease the order or increase the order respectively. The larger the value of $\mathbf{con}\left(\mathit{i}\right)$, for $\mathit{i}=4,5,6$, the less likely the choice of the corresponding order. The default values are: $\mathbf{con}\left(4\right)=1.2$, $\mathbf{con}\left(5\right)=1.3$, $\mathbf{con}\left(6\right)=1.4$.

Constraints:

These constraints must be satisfied after any zero values have been replaced by their default values.

• $0.0<\mathbf{con}\left(1\right)\le \mathbf{con}\left(2\right)\le \mathbf{con}\left(3\right)$;
• $\mathbf{con}\left(\mathit{i}\right)\ge 1.0$, for $\mathit{i}=2,3,\dots ,6$.

5:     $\mathrm{tcrit}$ – double scalar

A point beyond which integration must not be attempted. The use of tcrit is described under the argument itask in the specification for the integrator (e.g., see nag_ode_ivp_stiff_exp_fulljac (d02nb)). A value, $0.0$ say, must be specified even if itask subsequently specifies that tcrit will not be used.

6:     $\mathrm{hmin}$ – double scalar

The minimum absolute step size to be allowed. Set $\mathbf{hmin}=0.0$ if this option is not required.

7:     $\mathrm{hmax}$ – double scalar

The maximum absolute step size to be allowed. Set $\mathbf{hmax}=0.0$ if this option is not required.

8:     $\mathrm{h0}$ – double scalar

The step size to be attempted on the first step. Set $\mathbf{h0}=0.0$ if the initial step size is calculated internally.

9:     $\mathrm{maxstp}$ – int64int32nag_int scalar

The maximum number of steps to be attempted during one call to the integrator after which it will return with $\mathbf{ifail}=\mathbf{2}$. Set $\mathbf{maxstp}=0$ if no limit is to be imposed.

10:   $\mathrm{mxhnil}$ – int64int32nag_int scalar

The maximum number of warnings printed (if $\mathbf{itrace}\ge 0$, e.g., see nag_ode_ivp_stiff_exp_fulljac (d02nb)) per problem when $t+h=t$ on a step ($h=\text{ current step size}$). If $\mathbf{mxhnil}\le 0$, a default value of $10$ is assumed.

11:   $\mathrm{norm\_p}$ – string (length ≥ 1)

Indicates the type of norm to be used.

$\mathbf{norm\_p}=\text{'M'}$

Maximum norm.

$\mathbf{norm\_p}=\text{'A'}$

Averaged L2 norm.

$\mathbf{norm\_p}=\text{'D'}$

Is the same as 'A'.

If $\mathit{vnorm}$ denotes the norm of the vector $v$ of length $n$, then for the averaged L2 norm

$$\mathit{vnorm}=\sqrt{\frac{1}{n}\sum _{i=1}^n{\left(v_i/w_i\right)}^2}\text{,}$$

while for the maximum norm

$$\mathit{vnorm}=\underset{i}{\max }\left|v_i/w_i\right|\text{.}$$

If you wish to weight the maximum norm or the L2 norm, then rtol and atol should be scaled appropriately on input to the integrator (see under itol in the specification of the integrator for the formulation of the weight vector $w_i$ from rtol and atol, e.g., nag_ode_ivp_stiff_exp_fulljac (d02nb)).

Only the first character of the actual argument norm_p is passed to nag_ode_ivp_stiff_blend (d02nw); hence it is permissible for the actual argument to be more descriptive e.g., ‘Maximum’, ‘Average L2’ or ‘Default’ in a call to nag_ode_ivp_stiff_blend (d02nw).

Constraint: $\mathbf{norm\_p}=\text{'M'}$, $\text{'A'}$ or $\text{'D'}$.

12:   $\mathrm{rwork}\left(50+4\times \mathbf{neqmax}\right)$ – double array

This must be the same workspace array as the array rwork supplied to the integrator. It is used to pass information from the setup function to the integrator and therefore the contents of this array must not be changed before calling the integrator.

Optional Input Parameters

None.

Output Parameters

1:     $\mathrm{con}\left(6\right)$ – double array

The values actually used by nag_ode_ivp_stiff_blend (d02nw).

2:     $\mathrm{rwork}\left(50+4\times \mathbf{neqmax}\right)$ – double array

3:     $\mathrm{ifail}$ – int64int32nag_int scalar

$\mathbf{ifail}=\mathbf{0}$ unless the function detects an error (see Error Indicators and Warnings).

Error Indicators and Warnings

   $\mathbf{ifail}=1$

On entry, an illegal input was detected.

   $\mathbf{ifail}=-99$

An unexpected error has been triggered by this routine. Please contact NAG.

   $\mathbf{ifail}=-399$

Your licence key may have expired or may not have been installed correctly.

   $\mathbf{ifail}=-999$

Dynamic memory allocation failed.

Accuracy

Further Comments

Example

Open in the MATLAB editor: d02nw_example

function d02nw_example


fprintf('d02nw example results\n\n');

% Initialize variables and arrays for setup routine
n      = int64(3);
ord    = int64(11);
sdy    = int64(ord + 3);
petzld = false;
con    = zeros(6);
tcrit  = 0;
hmin   = 1.0e-10;
hmax   = 10;
h0     = 0;
maxstp = int64(200);
mxhnil = int64(5);
lrwork = 50 + 4*n;
rwork  = zeros(lrwork);
[const, rwork, ifail] = d02nw(n, sdy, ord, con, tcrit, hmin, hmax, h0, ...
                              maxstp, mxhnil, 'Average-L2', rwork);

% Setup for banded Jacbian using d02nt
ml     = int64(1);
mu     = int64(2);
nwkjac = int64(15);
[rwork, ifail] = d02nt(n, n, 'Analytical', ml, mu, nwkjac, n, rwork);

% Initialize variables and arrays for integration
t      = 0;
tout   = 5;
y      = [1; 0; 0];
rtol   = [0.0001];
atol   = [1e-07; 1e-08; 1e-07];
itol   = int64(2);
inform = zeros(23, 1, 'int64');
ysave  = zeros(n, sdy);
wkjac  = zeros(nwkjac, 1);
jacpvt = zeros(n, 1, 'int64');
itask  = int64(1);
itrace = int64(0);

% Integrate ODE from t=0 to t=tout, no monitoring, using d02nc
[t, y, ydot, rwork, inform, ysave, wkjac, jacpvt, ifail] = ...
    d02nc(t, tout, y, rwork, rtol, atol, itol, inform, @fcn, ysave, @jac, ...
          wkjac, jacpvt, 'd02nby', itask, itrace);

fprintf('Solution y and derivative y'' at t = %7.4f is:\n',t);
fprintf('\n %10s %10s\n','y','y''');
for i=1:n
  fprintf(' %10.4f %10.4f\n',y(i),ydot(i));
end



function [f, ires] = fcn(neq, t, y, ires)
% Evaluate derivative vector.
f = zeros(3,1);
f(1) = -0.04d0*y(1) + 1.0d4*y(2)*y(3);
f(2) =  0.04d0*y(1) - 1.0d4*y(2)*y(3) - 3.0d7*y(2)*y(2);
f(3) =                                  3.0d7*y(2)*y(2);

function p = jac(neq, t, y, h, d, ml, mu, p)
% Evaluate the Jacobian.
p = zeros(ml+mu+1, neq);
hxd = h*d;
p(1,1) = -hxd*(-0.04d0    ) + 1;
p(2,1) = -hxd*( 1.0d4*y(3));
p(3,1) = -hxd*( 1.0d4*y(2));
p(1,2) = -hxd*( 0.04d0    );
p(2,2) = -hxd*(-1.0d4*y(3)-6.0d7*y(2)) + 1;
p(3,2) = -hxd*(-1.0d4*y(2));
p(1,3) = -hxd*( 6.0d7*y(2));
p(2,3) = -hxd*( 0.0d0     ) + 1;

d02nw example results

Solution y and derivative y' at t =  5.0000 is:

          y         y'
     0.8915    -0.0124
     0.0000    -0.0000
     0.1085     0.0124