# Eulix - a versatile numerical ODE solver of Maxima

## Properties

Eulix is an extrapolated, linearly implicit (by default) Euler method. It can approximate the solution of an initial value problem for a system of ordinary differential equations or for differential algebraic equations (DAEs) of index 1. More generally, it can solve an implicit system like M y'(t)= f(t,y) where M is a constant mass matrix.

It uses variable step sizes and variable orders. It offers dense output and root finding, i.e., an implicitly defined final time. Furthermore, it can deliver an accurate spline approximation of the solution.

The linearly implicit version is A-stable up to order 2 and A(alpha) stable with alpha >= 89.81 up to order 8 and probably above, where A-stable = A(alpha)-stable with alpha = 90 degrees

Eulix is written in pure Maxima and all of its coefficients are rational numbers. Therefore, it can approximate the solution to arbitrary precision if fpprec is set sufficiently high and if all floating point constants in the right-hand-sides and in the initial values are specified as bfloat constants.

## Credits

An essential part of Eulix consists of a simplified version of the FORTRAN code SEULEX which has been published in:

    E. Hairer, G.Wanner
    Solving Ordinary Differential Equations II
    Stiff and Differential-Algebraic Problems
    Springer 1991

## Features

Eulix offers four APIs - one high level one (Eulix) which is an extension of the interface provided by rk and rkf45, and two intermediate level interfaces, Eulix_Table and Eulix_Spline. Lastly, there is a low-level interface Eulix_Step which computes a single time step.
All four interfaces allow for a number of options, see the chapter on Options below.

## Eulix

Eulix is called as:

    Eulix(Expressions,Vars,initials,range,[options])

Expressions: | is a list of expressions (or a single expression) which give the r.h.s of the ODE system. As a special case there might be an equation with r.h.s. or l.h.s equal to 0 to designate an algebraic equation.
Vars: | is a list of (dependent) variables which occur in these expressions. For each variable there has to be an initial value specified by the list initials.
initials: | initial values for each of the independent variables.

range:

is a list of 3 to 5 elements.

`range[1]`: | specifies the independent variable which will be called time in the following.
`range[2]`: | and `range[3]` specify the initial and final time, respectively. If `range[3]` < `range[2]` the ODE is integrated backwards in time.
`range[4]`: | is called `delta_t` (if present). Depending on an option given below, this determines the exact or maximum distance of time values in the list of results which Eulix returns. A small value of delta_t isn't expensive since it has to evaluate a polynomial only. If `delta_t=0` only the natural time steps are returned.

`range[5]`:

is a single root condition or a list of root conditions (if present).
A root condition is a real-valued expression which depends on the dependent variables (see Vars) as well as on the independent variable `range[1]`. The integration stops if either one of the root conditions evaluates to zero or if the final time `range[3]` is reached. A root condition implies dense output. If `delta_t` (`range[4]`) is zero it is reset to `(range[3]-range[2])/100` .
Hint: A root condition might need quite a small value of `delta_t` since `Eulix_Table` might miss the sign change in a root condition otherwise.

Eulix has four different return values.

If there are no root conditions, Eulix returns a list of lists. Each of these lists consist of a time value and the values of the dependent variables at this point of time - except if the option `tabular=none` (see Options below).
If the option combined_t_y_list=false is given, Eulix returns a list of two lists, the first of which is a list of time values, the second of which is a list of lists of the values of the dependent variables at the corresponding times.

If there are root conditions, Eulix returns a list of a solution list and what is returned if there are no root conditions. The solution list contains three items: the index of the root condition (starting with 1) which has triggered (i.e. went to zero), the time at which this has happened and the state (a list of the values of the dependent variables) at this point of time.
If root conditions have been specified but none of which has triggered before the final time `range[3]` has been reached, the solution list is a list of three zeros.

In the following we call a matrix with a single column a vector.

For the remaining three interfaces you have to specify the right-hand-sides, as well as the root conditions as vector-valued functiona of two arguments: time and the state vector. Furthermore, you have to give a vector-valued function which returns the value of the derivative of the right-hand-side w.r.t. time. In addition, you have to give a Jacobian matrix of the right-hand-side; this might be an approximation only. The mass-matrix (even if trivial) has to specified by the option `mass_matrix` (see Options below).

## Eulix_Table

Eulix_Table is called as:

    Eulix_Table(t0,y0,delta_t,t_end,Rhs,Rhs_t,Rhs_jac,Roots,[options])

t0: | initial time
y0: | initial state (a vector)
delta_t: | time increment for the generated list of solutions values. Due to dense output this is independent of the time step used by the integrator. A small value of delta_t isn't expensive since it has to evaluate a polynomial only. If delta_t is zero, only the natural time steps or even only the final state are output, depending on the option `tabular` (see Options below).
t_end: | final time. If `t_end` < `t0` a negative time step is used. In that case `delta_t` must be negative, as well.
Rhs: | a vector-valued function of (t,y), where y is the state vector. It defines the right-hand-side of the ODE system
Rhs_t: | a vector-valued function of (t,y), which gives the partial derivative w.r.t. t
Rhs_jac: | a matrix-valued function of (t,y), which gives the exact or approximate Jacobian of the right-hand-side w.r.t. the state vector y
Roots: | a list of scalar-valued functions R(t,y). If R(t,y(t))=0 the integration stops. A root condition implies dense output. If `delta_t` isn't specified or is set to zero, it is reset to `(range[3]-range[2])/100` . Hint: A root condition might need quite a small value of `delta_t` since Eulix_Table might miss the sign change in a root condition otherwise.
options: | see Options below

Output:

if Roots is an empty list, Eulix_Table returns a list of two lists, `tlist` and `ylist` which contain the points of time and the state vectors at these times, resp. - except if the option `tabular=none`, when only y_final is returned.
if Roots is not empty, it returns a list [Solution,[tlist,ylist]] where Solution is a list of three values: the index of the root condition which triggered the event, the time and state of this event. If none of the root conditions has triggered, the first element of Solution is zero - except if the option `tabular=none` when only [Solution,y_final] is returned

## Eulix_Spline

Eulix_Spline is called as:

    Eulix_Spline(t0,y0,t_end,Rhs,Rhs_t,Rhs_jac,[options])

parameters: | of the same name are the same as those in Eulix_Table above.
options: | see Options below

Output:

Eulix_Spline returns a list of objects of struct `Eulix_Spline_Type`. This is defined as
`defstruct(Eulix_Spline_Type(t_left,t_right,NewtonPoly))`
Each such object defines a polynomial (in `t`) which represents the solution of the ODE / DAE in the time interval `[t_left,t_right]`.
Eulix provides a function `Eulix_SplineEval(t,Spls)` which evaluates the spline at a point `t`. Here, `Spls` is the list returned by Eulix_Spline. It's possible to evaluates the derivative(s) of this spline w.r.t. time. For this, one should write a function similar to Eulix_SplineEval - see for example the function Eulix_Dense_Deriv, below.

## Eulix_Step

Eulix_Step is called as:

    [DO_DQ,tn,hn,me,failed]: Eulix_Step(y,t,Rhs,Rhs_t,Rhs_jac,h,me,[options])

parameters: | of the same name are the same as those in Eulix_Table above.

h:

step size, which must be non-zero. A negative step size is allowed. For the very first step the user must estimate a suitable step size. One might use the technique which is applied by Eulix_Table - see Automatic selection of the initial step size in that function. For any step except the first one, one should supply the value `hn` as new step size except for the very last step (to hit a desired final time)

me: | width of extrapolation tableau (= order). For the very first step the user must estimate a suitable initial order, e.g. 6. For any step except the first one, one should supply the value `me` which is returned by Eulix_Step

DO_DQ:

this value depends on the dense_output option, see Options below. If this is false, `DO_DQ[1]` contains the new state vector at time `tn`.
Otherwiese `DO_DQ` contains the dense output information. This can be passed to `Eulix_Dense_Out(dt,DO_DQ)` or `Eulix_Dense_Deriv(dt,DO_DQ)`.
Both of these take a time difference `dt` relative to the new time `tn`. Therefore, to evaluate the solution at time `t1` in `[t,tn]` with `tn` > `t`, the value `dt` = `t1`-`tn` will be negative. When integrating backwards in time, i.e., `tn` < `t`, this value will be positive.
Eulix_Dense_Out evaluates `y(t)` while Eulix_Dense_Deriv delivers `y'(t)`. This can be used in Newton's method as is done in Eulix_Table for determining zeros of the root conditions.

tn: | is the new time
hn: | is the new optimal size for the next step. This should be passed as parameter `h` to Eulix_Step unless this would exceed the desired final time.
me: | is the new optimal tableau width for the next step. This should be passed unchanged to Eulix_Step for the next call.
failed: | If this is true the output parameter DO_DQ doesn't contain useful information. Eulix_Step had to reject a step more than 10 times in succession. This should only occur if the right-hand-size is discontinuous or if the values delivered by Rhs_t or Rhs_jac are wrong.

## Options

Some options are only applicable to some of the interface functions above. The following abbreviations are used to indicate for which function an option is effective: Eul for Eulix, Tab for Eulix_Table, Spl for Eulix_Spline and Stp for Eulix_Step.
The options are given in the form Option Name = default.

absolute_tolerance: | = 1E-6 `[Eul,Tab,Stp,Spl]`

relative_tolerance:

= 1E-6 `[Eul,Tab,Stp,Spl]`
The (local) required tolerance is computed as (with the state vector `y`):

    absolute_tolerance + relative_tolerance *  mat_norm(y,'inf)

root_tolerance:

= 0 `[Eul,Tab]` This defines the tolerance for Newton's method to determine a zero of a root condition.
If this is negative or zero (equivalent to -1E-6), the tolerance is a relative tolerance according to:

    abs(root_eps)*max(mat_norm(y_left,'inf),mat_norm(y_right,'inf))

Here, `y_left` and `y_right` are the state vectors at the left (right, rsp.) start interval for Newton's method.

maximum_order: | = 20 `[Eul,Tab,Stp]` This is the maximum tableau width that is used by Eulix_Step
dense_output: | = true `[Stp,Spl]` This makes Eulix_Step compute the dense output information `DO_DQ`. This can be triggered in a call to Eulix and Eulix_Table by other means.
work_estimates: | = [1,1,0] `[Eul,Tab,Stp,Spl]` For computing the optimal step size and order for the next step, Eulix_Step needs a relative amount of work to evaluate the right-hand-side Rhs and Rhs_t, to compute the Jacobian Rhs_jac and to decompose the Jacobian into a Q-R-decomposition.
check_parameters: | = true `[Eul,Tab,Stp,Spl]` If this is true, several checks to the parameters are performed. The functions `Eulix_Table` and `Eulix_Spline` set this to false after the first call to `Eulix_Step`.

mass_matrix:

= `ident(dim)` `[Stp,Spl]` This parameter should take a matrix of the size of the differential system or `false`.
If it has the value `false`, an extrapolated explicit Euler scheme is used. In this case, the function Rhs_jac will not be called.
If it has the value of a (square) matrix `M`, the (possibly) implicit ODE `M y'(t) = f(t,y(t))` is solved. This matrix need not be regular. Indeed, for an algebraic equation, the corressponding row of `M` should be zero. This can be triggered on a call to Eulix, as well.
If mass_matrix has the value of a matrix, the implicit (extrapolated) Euler scheme will be used.

initial_step_size: | = 0 `[Eul,Tab,Spl]` defines the very first step size; for integration backwards in time, this should have a negative value. If it has the (default) value 0, an estimate is computed. This depends on the right-hand-side as well as on the option `absolute_tolerance` - see Automatic selection of the initial step size in that function Eulix_Step.
initial_order: | = 6 `[Eul,Tab,Spl]` defines the initial width of the extrapolation tableau.
logging: | = `false` `[Eul,Tab,Stp,Spl]` If this is true, some internal information is printed, e.g. rejected steps, the initial step size, options which are in effect and for each time step the values of the current time, step size and tableau width (order).
tabular: | = `all` `[Eul,Tab]` This option can have the values `all` (default), `none` and `tabular`. If it has the value `none`, only the final state is returned or a list [Solution,y_final] when root conditions have been specified. If it has the value `tabular` a list of solutions at equidistant times is produced. For the value `all`, the natural time steps are intermixed, as well.
combined_t_y_list: | = `true` `[Eul]` If this is true, Eulix returns a single list of lists containing the time value and the corresponding solution components at this point of time. If this is false, Eulix returns `[tlist,ylist]` , where `ylist` is a list of lists containing the solution components at the corresponding time in `tlist`.

## Examples

The example files are all named `Eulix_*.mac`. In the following list, we only give the suffix which the star stands for.

T1: | This is a simple 1-d ODE which requires frequent changes of the step size and order. It's partially stiff. The exact solution is known.
T1R: | This solves the same ODE backwards in time
Step_T1: | This solves the same ODE using the low level Eulix_Step interface
T2: | This is a simple 1-d ODE which simulates arbitrary stiffness. The exact solution is known.
T3: | This is the Volterra-Lotka (2-d) system which is rather demanding.
T3N: | This demonstrates the option `tabular=none` for the same ODE
Spline_T3: | This demonstrates the Eulix_Spline interface for this ODE
T4: | Here we solve a differential-algebraic equation (DAE) for a pendulum
T5: | This is a rather demanding AIDS model which requires a good error estimator. Here we demonstrate the root finding feature.
T5N: | Same as above except with `tabular=none`
Table_T5: | This same model as `T5` but this time using the Eulix_Table interface. Here we demonstrate the root finding feature.
T6: | Simple test with a highly non-linear right hand side
T7: | Simple test of the quality of dense output using the exact solution exp(-t*t).
TP: | Here we solve a system of 2 equations with a tolerance of 1E-30. Since the exact solution is known, we can check the tolerance achieved.
TS: | This is an extremely stiff ODE (Shampine's ball of flame)
Step_TP: | A 2-d ODE with a known solution, demonstrating high precision solving using the low level Eulix_Step interface.
Table_TC: | A very stiff system modelling a chemical reaction using the Eulix_Table interface, see Hairer/Wanner Solving ODE II .

The examples Eulix_T1.mac, Eulix_T2.mac, Eulix_T3.mac and Eulix_TS.mac contain comparisons with rkf45.