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Chapter Introduction
NAG Toolbox

# NAG Toolbox: nag_ode_dae_dassl_linalg (d02np)

## Purpose

nag_ode_dae_dassl_linalg (d02np) is a setup function which you must call prior to nag_ode_dae_dassl_gen (d02ne) and after a call to nag_ode_dae_dassl_setup (d02mw), if the Jacobian is to be considered as having a banded structure.

## Syntax

[icom, ifail] = d02np(neq, ml, mu, icom, 'licom', licom)
[icom, ifail] = nag_ode_dae_dassl_linalg(neq, ml, mu, icom, 'licom', licom)

## Description

A call to nag_ode_dae_dassl_linalg (d02np) specifies that the Jacobian to be used is banded in structure. If nag_ode_dae_dassl_linalg (d02np) is not called before a call to nag_ode_dae_dassl_gen (d02ne) then the Jacobian is assumed to be full.

None.

## Parameters

### Compulsory Input Parameters

1:     neq – int64int32nag_int scalar
The number of differential-algebraic equations to be solved.
Constraint: 1neq$1\le {\mathbf{neq}}$.
2:     ml – int64int32nag_int scalar
mL${m}_{L}$, the number of subdiagonals in the band.
Constraint: 0mlneq1$0\le {\mathbf{ml}}\le {\mathbf{neq}}-1$.
3:     mu – int64int32nag_int scalar
mU${m}_{U}$, the number of superdiagonals in the band.
Constraint: 0muneq1$0\le {\mathbf{mu}}\le {\mathbf{neq}}-1$.
4:     icom(licom) – int64int32nag_int array
licom, the dimension of the array, must satisfy the constraint licom50 + neq${\mathbf{licom}}\ge 50+{\mathbf{neq}}$.
icom is used to communicate details of the integration from nag_ode_dae_dassl_setup (d02mw) and details of the banded structure of the Jacobian to nag_ode_dae_dassl_gen (d02ne).

### Optional Input Parameters

1:     licom – int64int32nag_int scalar
Default: The dimension of the array icom.
The dimension of the array icom as declared in the (sub)program from which nag_ode_dae_dassl_linalg (d02np) is called.
Constraint: licom50 + neq${\mathbf{licom}}\ge 50+{\mathbf{neq}}$.

None.

### Output Parameters

1:     icom(licom) – int64int32nag_int array
2:     ifail – int64int32nag_int scalar
${\mathrm{ifail}}={\mathbf{0}}$ unless the function detects an error (see [Error Indicators and Warnings]).

## Error Indicators and Warnings

Errors or warnings detected by the function:
ifail = 1${\mathbf{ifail}}=1$
 On entry, neq < 1${\mathbf{neq}}<1$.
ifail = 2${\mathbf{ifail}}=2$
 On entry, ml < 0${\mathbf{ml}}<0$ or ml > neq − 1${\mathbf{ml}}>{\mathbf{neq}}-1$.
ifail = 3${\mathbf{ifail}}=3$
 On entry, mu < 0${\mathbf{mu}}<0$ or mu > neq − 1${\mathbf{mu}}>{\mathbf{neq}}-1$.
ifail = 4${\mathbf{ifail}}=4$
Either nag_ode_dae_dassl_setup (d02mw) has not been called before this call or the communication array icom has been corrupted.
ifail = 5${\mathbf{ifail}}=5$
 On entry, licom < 50 + neq${\mathbf{licom}}<50+{\mathbf{neq}}$.

Not applicable.

None.

## Example

```function nag_ode_dae_dassl_linalg_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);

% 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

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

```
```function d02np_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);

% 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

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