hide long namesshow long names
hide short namesshow short names
Integer type:  int32  int64  nag_int  show int32  show int32  show int64  show int64  show nag_int  show nag_int

PDF version (NAG web site, 64-bit version, 64-bit version)
Chapter Contents
Chapter Introduction
NAG Toolbox

NAG Toolbox: nag_ode_dae_dassl_setup (d02mw)

Purpose

nag_ode_dae_dassl_setup (d02mw) is a setup function which must be called prior to the integrator nag_ode_dae_dassl_gen (d02ne), if the DASSL implementation of Backward Differentiation Formulae (BDF) is to be used.

Syntax

[icom, com, ifail] = d02mw(neq, maxord, jceval, hmax, h0, itol, lcom)
[icom, com, ifail] = nag_ode_dae_dassl_setup(neq, maxord, jceval, hmax, h0, itol, lcom)

Description

This integrator setup function must be called before the first call to the integrator nag_ode_dae_dassl_gen (d02ne). This setup function nag_ode_dae_dassl_setup (d02mw) permits you to define options for the DASSL integrator, such as: whether the Jacobian is to be provided or is to be approximated numerically by the integrator; the initial and maximum step-sizes for the integration; whether relative and absolute tolerances are system wide or per system equation; and the maximum order of BDF method permitted.

References

None.

Parameters

Compulsory Input Parameters

1:     neq – int64int32nag_int scalar
The number of differential-algebraic equations to be solved.
Constraint: neq1neq1.
2:     maxord – int64int32nag_int scalar
The maximum order to be used for the BDF method. Orders up to 5th order are available; setting maxord > 5maxord>5 means that the maximum order used will be 55.
Constraint: 1maxord1maxord.
3:     jceval – string (length ≥ 1)
Specifies the technique to be used to compute the Jacobian.
jceval = 'N'jceval='N'
The Jacobian is to be evaluated numerically by the integrator.
jceval = 'A'jceval='A'
You must supply a function to evaluate the Jacobian on a call to the integrator.
Only the first character of the actual paramater jceval is passed to nag_ode_dae_dassl_setup (d02mw); hence it is permissible for the actual argument to be more descriptive, e.g., ‘Numerical’ or ‘Analytical’, on a call to nag_ode_dae_dassl_setup (d02mw).
Constraint: jceval = 'N'jceval='N' or 'A''A'.
4:     hmax – double scalar
The maximum absolute step size to be allowed. Set hmax = 0.0hmax=0.0 if this option is not required.
Constraint: hmax0.0hmax0.0.
5:     h0 – double scalar
The step size to be attempted on the first step. Set h0 = 0.0h0=0.0 if the initial step size is calculated internally.
6:     itol – int64int32nag_int scalar
A value to indicate the form of the local error test.
itol = 0itol=0
rtol and atol are single element vectors.
itol = 1itol=1
rtol and atol are vectors. This should be chosen if you want to apply different tolerances to each equation in the system.
Note: the tolerances must either both be single element vectors or both be vectors of length neq.
Constraint: itol = 0itol=0 or 11.
7:     lcom – int64int32nag_int scalar
The dimension of the array com as declared in the (sub)program from which nag_ode_dae_dassl_setup (d02mw) is called.
Constraints:
the array com must be large enough for the requirements of nag_ode_dae_dassl_gen (d02ne). That is:
  • if the system Jacobian is dense, lcom 40 + (maxord + 4) × neq + neq2 lcom 40 + (maxord+4) × neq + neq2 ;
  • if the system Jacobian is banded,   lcom 40 + (maxord + 4) × neq + (2 × ml + mu + 1) × neq + 2 × (neq / (ml + mu + 1) + 1) lcom 40 + (maxord+4) × neq + ( 2×ml+mu+1 ) × neq + 2 × ( neq / ( ml + mu + 1 ) + 1 ) .
Here ml and mu are the lower and upper bandwidths respectively that are to be specified in a subsequent call to nag_ode_dae_dassl_linalg (d02np).

Optional Input Parameters

None.

Input Parameters Omitted from the MATLAB Interface

licom

Output Parameters

1:     icom(licom) – int64int32nag_int array
licomneq + 50licomneq+50.
Used to communicate details of the task to be carried out to the integration function nag_ode_dae_dassl_gen (d02ne).
2:     com(lcom) – double array
Used to communicate problem parameters to the integration function nag_ode_dae_dassl_gen (d02ne). This must be the same communication array as the array com supplied to nag_ode_dae_dassl_gen (d02ne). In particular, the values of hmax and h0 are contained in com.
3:     ifail – int64int32nag_int scalar
ifail = 0ifail=0 unless the function detects an error (see [Error Indicators and Warnings]).

Error Indicators and Warnings

Errors or warnings detected by the function:
  ifail = 1ifail=1
On entry,neq < 1neq<1.
  ifail = 2ifail=2
On entry,maxord < 1maxord<1,
ormaxord > 5maxord>5.
  ifail = 3ifail=3
On entry,jceval'N'jceval'N' or 'A''A'.
  ifail = 4ifail=4
On entry,hmax < 0.0hmax<0.0.
  ifail = 6ifail=6
On entry,itol0itol0 or 11.
  ifail = 8ifail=8
On entry,licom < neq + 50licom<neq+50.

Accuracy

Not applicable.

Further Comments

None.

Example

function nag_ode_dae_dassl_setup_example
neq = 3;
maxord = 5;
mu = 2;
ml = 1;
lcom = 40+(maxord+4)*neq+(2*ml+mu+1)*neq+2*(neq/(ml+mu+1)+1)
itol = int64(1);
rtol = [1e-3; 1e-3; 1e-3];
atol = [1e-6; 1e-6; 1e-6];
ydot = zeros(neq,1);
% Set initial values
y    = [1; 0; 0];
% Initialize the problem, specifying that the Jacobian is to be
% evaluated analytically using the provided routine jac.
jceval = 'Analytic';
hmax = 0;
ho = 0;
t = 0;
tout = 0.02;
[icom, com, ifail] = ...
    nag_ode_dae_dassl_setup(int64(neq), int64(maxord), jceval, hmax, ho, itol, int64(lcom));

% Specify that the Jacobian is banded
if ifail == 0
  [icom, ifail] = nag_ode_dae_dassl_linalg(int64(neq), int64(ml), int64(mu), icom);
end

% Use the user parameter to pass the band dimensions through to jac.
% An alternative would be to hard code values for ml and mu in jac.
user = {ml, mu};

fprintf('\n    t            y(1)        y(2)        y(3)   \n');
fprintf(' %8.4f   %12.6f %12.6f %12.6f\n', t, y);

itask = int64(0);
% Obtain the solution at 5 equally spaced values of T.
for j = 1:5
  if ifail == 0
    [t, y, ydot, rtol, atol, itask, icom, com, user, ifail] = ...
      nag_ode_dae_dassl_gen(t, tout, y, ydot, rtol, atol, itask, @res, @jac, ...
            icom, com, 'user', user);
    fprintf(' %8.4f   %12.6f %12.6f %12.6f\n', t, y);
    tout = tout + 0.02;
    icom = nag_ode_dae_dassl_cont(icom);
  end
end

fprintf('\nThe integrator completed task, ITASK = %d\n', itask);



function [pd, user] = jac(neq, t, y, ydot, pd, cj, user)
  ml = user{1};
  mu = user{2};

  stride = 2*ml+mu+1;
  % Main diagonal pdfull(i,i), i=1,neq
  md = mu + ml + 1;
  pd(md) = -0.04 - cj;
  pd(md+stride) = -1.0e4*y(3) - 6.0e7*y(2) - cj;
  pd(md+2*stride) = -cj;
  % 1 sub-diagonal pdfull(i+1:i), i=1,neq-1
  ms = md + 1;
  pd(ms) = 0.04;
  pd(ms+stride) = 6.0e7*y(2);
  % First super-diagonal pdfull(i-1,i), i=2, neq
  ms = md - 1;
  pd(ms+stride) = 1.0e4*y(3);
  pd(ms+2*stride) = -1.0e4*y(2);
  % Second super-diagonal pdfull(i-2,i), i=3, neq
  ms = md - 2;
  pd(ms+2*stride) = 1.0e4*y(2);

function [r, ires, user] = res(neq, t, y, ydot, ires, user)
  r = zeros(neq, 1);
  r(1) = -0.04*y(1) + 1.0e4*y(2)*y(3) - ydot(1);
  r(2) = 0.04*y(1) - 1.0e4*y(2)*y(3) - 3.0e7*y(2)*y(2) - ydot(2);
  r(3) = 3.0e7*y(2)*y(2) - ydot(3);
 

lcom =

   85.5000


    t            y(1)        y(2)        y(3)   
   0.0000       1.000000     0.000000     0.000000
   0.0200       0.999204     0.000036     0.000760
   0.0400       0.998415     0.000036     0.001549
   0.0600       0.997631     0.000036     0.002333
   0.0800       0.996852     0.000036     0.003112
   0.1000       0.996080     0.000036     0.003884

The integrator completed task, ITASK = 3

function d02mw_example
neq = 3;
maxord = 5;
mu = 2;
ml = 1;
lcom = 40+(maxord+4)*neq+(2*ml+mu+1)*neq+2*(neq/(ml+mu+1)+1)
itol = int64(1);
rtol = [1e-3; 1e-3; 1e-3];
atol = [1e-6; 1e-6; 1e-6];
ydot = zeros(neq,1);
% Set initial values
y    = [1; 0; 0];
% Initialize the problem, specifying that the Jacobian is to be
% evaluated analytically using the provided routine jac.
jceval = 'Analytic';
hmax = 0;
ho = 0;
t = 0;
tout = 0.02;
[icom, com, ifail] = d02mw(int64(neq), int64(maxord), jceval, hmax, ho, itol, int64(lcom));

% Specify that the Jacobian is banded
if ifail == 0
  [icom, ifail] = d02np(int64(neq), int64(ml), int64(mu), icom);
end

% Use the user parameter to pass the band dimensions through to jac.
% An alternative would be to hard code values for ml and mu in jac.
user = {ml, mu};

fprintf('\n    t            y(1)        y(2)        y(3)   \n');
fprintf(' %8.4f   %12.6f %12.6f %12.6f\n', t, y);

itask = int64(0);
% Obtain the solution at 5 equally spaced values of T.
for j = 1:5
  if ifail == 0
    [t, y, ydot, rtol, atol, itask, icom, com, user, ifail] = ...
      d02ne(t, tout, y, ydot, rtol, atol, itask, @res, @jac, ...
            icom, com, 'user', user);
    fprintf(' %8.4f   %12.6f %12.6f %12.6f\n', t, y);
    tout = tout + 0.02;
    icom = d02mc(icom);
  end
end

fprintf('\nThe integrator completed task, ITASK = %d\n', itask);



function [pd, user] = jac(neq, t, y, ydot, pd, cj, user)
  ml = user{1};
  mu = user{2};

  stride = 2*ml+mu+1;
  % Main diagonal pdfull(i,i), i=1,neq
  md = mu + ml + 1;
  pd(md) = -0.04 - cj;
  pd(md+stride) = -1.0e4*y(3) - 6.0e7*y(2) - cj;
  pd(md+2*stride) = -cj;
  % 1 sub-diagonal pdfull(i+1:i), i=1,neq-1
  ms = md + 1;
  pd(ms) = 0.04;
  pd(ms+stride) = 6.0e7*y(2);
  % First super-diagonal pdfull(i-1,i), i=2, neq
  ms = md - 1;
  pd(ms+stride) = 1.0e4*y(3);
  pd(ms+2*stride) = -1.0e4*y(2);
  % Second super-diagonal pdfull(i-2,i), i=3, neq
  ms = md - 2;
  pd(ms+2*stride) = 1.0e4*y(2);

function [r, ires, user] = res(neq, t, y, ydot, ires, user)
  r = zeros(neq, 1);
  r(1) = -0.04*y(1) + 1.0e4*y(2)*y(3) - ydot(1);
  r(2) = 0.04*y(1) - 1.0e4*y(2)*y(3) - 3.0e7*y(2)*y(2) - ydot(2);
  r(3) = 3.0e7*y(2)*y(2) - ydot(3);
 

lcom =

   85.5000


    t            y(1)        y(2)        y(3)   
   0.0000       1.000000     0.000000     0.000000
   0.0200       0.999204     0.000036     0.000760
   0.0400       0.998415     0.000036     0.001549
   0.0600       0.997631     0.000036     0.002333
   0.0800       0.996852     0.000036     0.003112
   0.1000       0.996080     0.000036     0.003884

The integrator completed task, ITASK = 3


PDF version (NAG web site, 64-bit version, 64-bit version)
Chapter Contents
Chapter Introduction
NAG Toolbox

© The Numerical Algorithms Group Ltd, Oxford, UK. 2009–2013