75 odepack

75.1 Introduction to ODEPACK

ODEPACK is a collection of Fortran solvers for the initial value problem for ordinary differential equation systems. It consists of nine solvers, namely a basic solver called LSODE and eight variants of it – LSODES, LSODA, LSODAR, LSODPK, LSODKR, LSODI, LSOIBT, and LSODIS. The collection is suitable for both stiff and nonstiff systems. It includes solvers for systems given in explicit form, dy/dt = f(t,y), and also solvers for systems given in linearly implicit form, A(t,y) dy/dt = g(t,y). Two of the solvers use general sparse matrix solvers for the linear systems that arise. Two others use iterative (preconditioned Krylov) methods instead of direct methods for these linear systems. The most recent addition is LSODIS, which solves implicit problems with general sparse treatment of all matrices involved.

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References: [1] Fortran Code is from http://netlib.org/odepack/

75.1.1 Getting Started

Of the eight variants of the solver, maxima currently only has an inteface the dlsode.

Let’s say we have this sytem of equations to solve:

  f1 = -.04d0*y1 + 1d4*y2*y3
  f3 = 3d7*y2*y2
  dy1/dt = f1
  dy2/dt = -f1 - f3
  dy3/dt = f3

The independent variable is t; the dependent variables are y1, y2, and y3,

To start the solution, set up the differential equations to solved:

load("dlsode");
f1: -.04d0*y1 + 1d4*y2*y3$
f3: 3d7*y2*y2$
f2: -f1 - f3$
fex: [f1, f2, f3];

Initialize the solver, where we have selected method 21

(%i6) state : dlsode_init(fex, ['t,y1,y2,y3], 21);
(%o6) [[f, #<Function "LAMBDA ($T $Y1 $Y2 $Y3)" {49DAC061}>], 
[vars, [t, y1, y2, y3]], [mf, 21], [neq, 3], [lrw, 58], [liw, 23], [rwork, {Li\
sp Array: #(0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0
               0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0
               0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0
               0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0)}], 
[iwork, {Lisp Array: #(0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0)}], 
[fjac, #<Function "LAMBDA ($T $Y1 $Y2 $Y3)" {49D52AC9}>]]

The arrays rwork and iwork carry state between calls to dlsode_step, so they should not be modified by the user. In fact, this state should not be modified by the user at all.

Now that the algorithm has been initialized we can compute solutions to the differential equation, using the state returned above.

For this example, we want to compute the solution at times 0.4*10^k for k from 0 to 11, with the initial values of 1, 0, 0 for the dependent variables and with a relative tolerance of 1d-4 and absolute tolerances of 1e-6, 1e-10, and 1d-6 for the dependent variables.

Then

y: [1d0, 0d0, 0d0];
t: 0d0;
rtol : 1d-4;
atol: [1d-6, 1d-10, 1d-6];
istate: 1;
t:0d0;
tout:.4d0;

for k : 1 thru 12 do
  block([],
    result: dlsode_step(y, t, tout, rtol, atol, istate, state),
    printf(true, "At t = ~12,4,2e   y = ~{~14,6,2e~}~%", result[1], result[2]),
    istate : result[3],
    tout : tout * 10);

This produces the output:

At t =   4.0000e-01   y =   9.851726e-01  3.386406e-05  1.479357e-02
At t =   4.0000e+00   y =   9.055142e-01  2.240418e-05  9.446344e-02
At t =   4.0000e+01   y =   7.158050e-01  9.184616e-06  2.841858e-01
At t =   4.0000e+02   y =   4.504846e-01  3.222434e-06  5.495122e-01
At t =   4.0000e+03   y =   1.831701e-01  8.940379e-07  8.168290e-01
At t =   4.0000e+04   y =   3.897016e-02  1.621193e-07  9.610297e-01
At t =   4.0000e+05   y =   4.935213e-03  1.983756e-08  9.950648e-01
At t =   4.0000e+06   y =   5.159269e-04  2.064759e-09  9.994841e-01
At t =   4.0000e+07   y =   5.306413e-05  2.122677e-10  9.999469e-01
At t =   4.0000e+08   y =   5.494530e-06  2.197824e-11  9.999945e-01
At t =   4.0000e+09   y =   5.129458e-07  2.051784e-12  9.999995e-01
At t =   4.0000e+10   y =  -7.170563e-08 -2.868225e-13  1.000000e+00

75.2 Functions and Variables for odepack

Function: dlsode_init (fex, vars, method)

This must be called before running the solver. This function returns a state object for use in the solver. The user must not modify the state.

The ODE to be solved is given in fex, which is a list of the equations. vars is a list of independent variable and the dependent variables. The list of dependent variables must be in the same order as the equations if fex. Finally, method indicates the method to be used by the solver:

10

Nonstiff (Adams) method, no Jacobian used.

21

Stiff (BDF) method, user-supplied full Jacobian.

22

Stiff method, internally generated full Jacobian.

The returned state object is a list of lists. The sublist is a list of two elements:

f

The compiled function for the ODE.

vars

The list independent and dependent variables (vars).

mf

The method to be used (method).

neq

The number of equations.

lrw

Length of the work vector for real values.

liw

Length of the work vector for integer values.

rwork

Lisp array holding the real-valued work vector.

iwork

Lisp array holding the integer-valued work vector.

fjac

Compiled analytical Jacobian of the eqations

Function: dlsode_step (inity, t, tout, rtol, atol, istate, state)

Performs one step of the solver, returning the values of the independent and dependent variables, a succes or error code.

inity

For the first call (when istate = 1), the initial values

t

Current value of the independent value

tout

Next point where output is desired which must not be equal to t.

rtol

relative tolerance parameter

atol

Absolute tolerance parameter, scalar of vector. If scalar, it applies to all dependent variables. Otherwise it must be the tolerance for each dependent variable.

Use rtol = 0 for pure absolute error and use atol = 0 for pure relative error.

istate

1 for the first call to dlsode, 2 for subsequent calls.

state

state returned by dlsode_init.

The output is a list of the following items:

t

independent variable value

y

list of values of the dependent variables at time t.

istate

Integration status:

1

no work because tout = tt

2

successful result

-1

Excess work done on this call

-2

Excess accuracy requested

-3

Illegal input detected

-4

Repeated error test failures

-5

Repeated convergence failures (perhaps bad Jacobian or wrong choice of mf or tolerances)

-6

Error weight becase zero during problem (solution component i vanishded and atol(i) = 0.

info

association list of various bits of information:

n_steps

total steps taken thus far

n_f_eval

total number of function evals

n_j_eval

total number of Jacobian evals

method_order

method order

len_rwork

Actual length used for real work array

len_iwork

Actual length used for integer work array