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Solving ordinary differential equation numerically is usually done iteratively,
that is a given state of an ordinary differential equation is iterated forward
x(t) > x(t+dt) > x(t+2dt). The steppers in odeint
perform one single step. The most general stepper type is described by the
Stepper concept.
The stepper concepts of odeint are described in detail in section Concepts,
here we briefly present the mathematical and numerical details of the steppers.
The Stepper
has two versions of the do_step
method, one with an inplace transform of the current state and one with
an outofplace transform:
do_step(
sys ,
inout ,
t , dt )
do_step(
sys ,
in ,
t , out , dt )
The first parameter is always the system function  a function describing
the ODE. In the first version the second parameter is the step which is here
updated inplace and the third and the fourth parameters are the time and
step size (the time step). After a call to do_step
the state inout
is updated
and now represents an approximate solution of the ODE at time t+dt.
In the second version the second argument is the state of the ODE at time
t, the third argument is t, the fourth argument is the
approximate solution at time t+dt which is filled by
do_step
and the fifth argument
is the time step. Note that these functions do not change the time t
.
System functions
Up to now, we have nothing said about the system function. This function
depends on the stepper. For the explicit RungeKutta steppers this function
can be a simple callable object hence a simple (global) Cfunction or a functor.
The parameter syntax is sys( x ,
dxdt ,
t )
and it is assumed that it calculates dx/dt = f(x,t).
The function structure in most cases looks like:
void sys( const state_type & /*x*/ , state_type & /*dxdt*/ , const double /*t*/ ) { // ... }
Other types of system functions might represent Hamiltonian systems or systems which also compute the Jacobian needed in implicit steppers. For information which stepper uses which system function see the stepper table below. It might be possible that odeint will introduce new system types in near future. Since the system function is strongly related to the stepper type, such an introduction of a new stepper might result in a new type of system function.
A first specialization are the explicit steppers. Explicit means that the new state of the ode can be computed explicitly from the current state without solving implicit equations. Such steppers have in common that they evaluate the system at time t such that the result of f(x,t) can be passed to the stepper. In odeint, the explicit stepper have two additional methods
do_step(
sys ,
inout ,
dxdtin ,
t ,
dt )
do_step(
sys ,
in ,
dxdtin ,
t ,
out ,
dt )
Here, the additional parameter is the value of the function f at state x and time t. An example is the RungeKutta stepper of fourth order:
runge_kutta4< state_type > rk; rk.do_step( sys1 , inout , t , dt ); // Inplace transformation of inout rk.do_step( sys2 , inout , t , dt ); // call with different system: Ok rk.do_step( sys1 , in , t , out , dt ); // Outofplace transformation rk.do_step( sys1 , inout , dxdtin , t , dt ); // Inplace tranformation of inout rk.do_step( sys1 , in , dxdtin , t , out , dt ); // Outofplace transformation
In fact, you do not need to call these two methods. You can always use
the simpler do_step(
sys ,
inout ,
t ,
dt )
,
but sometimes the derivative of the state is needed externally to do some
external computations or to perform some statistical analysis.
A special class of the explicit steppers are the FSAL (firstsameaslast)
steppers, where the last evaluation of the system function is also the
first evaluation of the following step. For such steppers the do_step
method are slightly different:
do_step(
sys ,
inout ,
dxdtinout ,
t ,
dt )
do_step(
sys ,
in ,
dxdtin ,
out ,
dxdtout ,
t ,
dt )
This method takes the derivative at time t
and also stores the derivative at time t+dt. Calling
these functions subsequently iterating along the solution one saves one
function call by passing the result for dxdt into the next function call.
However, when using FSAL steppers without supplying derivatives:
do_step(
sys ,
inout ,
t ,
dt )
the stepper internally satisfies the FSAL property which means it remembers
the last dxdt
and uses
it for the next step. An example for a FSAL stepper is the RungeKuttaDopri5
stepper. The FSAL trick is sometimes also referred as the Fehlberg trick.
An example how the FSAL steppers can be used is
runge_kutta_dopri5< state_type > rk; rk.do_step( sys1 , in , t , out , dt ); rk.do_step( sys2 , in , t , out , dt ); // DONT do this, sys1 is assumed rk.do_step( sys2 , in2 , t , out , dt ); rk.do_step( sys2 , in3 , t , out , dt ); // DONT do this, in2 is assumed rk.do_step( sys1 , inout , dxdtinout , t , dt ); rk.do_step( sys2 , inout , dxdtinout , t , dt ); // Ok, internal derivative is not used, dxdtinout is updated rk.do_step( sys1 , in , dxdtin , t , out , dxdtout , dt ); rk.do_step( sys2 , in , dxdtin , t , out , dxdtout , dt ); // Ok, internal derivative is not used
Caution  

The FSALsteppers save the derivative at time t+dt
internally if they are called via 
As mentioned above symplectic solvers are used for Hamiltonian systems. Symplectic solvers conserve the phase space volume exactly and if the Hamiltonian system is energy conservative they also conserve the energy approximately. A special class of symplectic systems are separable systems which can be written in the form dqdt/dt = f1(p), dpdt/dt = f2(q), where (q,p) are the state of system. The space of (q,p) is sometimes referred as the phase space and q and p are said the be the phase space variables. Symplectic systems in this special form occur widely in nature. For example the complete classical mechanics as written down by Newton, Lagrange and Hamilton can be formulated in this framework. The separability of the system depends on the specific choice of coordinates.
Symplectic systems can be solved by odeint by means of the symplectic_euler
stepper and a symplectic RungeKuttaNystrom method of fourth order. These
steppers assume that the system is autonomous, hence the time will not
explicitly occur. Further they fulfill in principle the default Stepper
concept, but they expect the system to be a pair of callable objects. The
first entry of this pair calculates f1(p) while the
second calculates f2(q). The syntax is sys.first(p,dqdt)
and sys.second(q,dpdt)
,
where the first and second part can be again simple Cfunctions of functors.
An example is the harmonic oscillator:
typedef boost::array< double , 1 > vector_type; struct harm_osc_f1 { void operator()( const vector_type &p , vector_type &dqdt ) { dqdt[0] = p[0]; } }; struct harm_osc_f2 { void operator()( const vector_type &q , vector_type &dpdt ) { dpdt[0] = q[0]; } };
The state of such an ODE consist now also of two parts, the part for q (also called the coordinates) and the part for p (the momenta). The full example for the harmonic oscillator is now:
pair< vector_type , vector_type > x; x.first[0] = 1.0; x.second[0] = 0.0; symplectic_rkn_sb3a_mclachlan< vector_type > rkn; rkn.do_step( make_pair( harm_osc_f1() , harm_osc_f2() ) , x , t , dt );
If you like to represent the system with one class you can easily bind two public method:
struct harm_osc { void f1( const vector_type &p , vector_type &dqdt ) const { dqdt[0] = p[0]; } void f2( const vector_type &q , vector_type &dpdt ) const { dpdt[0] = q[0]; } };
harm_osc h; rkn.do_step( make_pair( boost::bind( &harm_osc::f1 , h , _1 , _2 ) , boost::bind( &harm_osc::f2 , h , _1 , _2 ) ) , x , t , dt );
Many Hamiltonian system can be written as dq/dt=p,
dp/dt=f(q) which is computationally much easier than
the full separable system. Very often, it is also possible to transform
the original equations of motion to bring the system in this simplified
form. This kind of system can be used in the symplectic solvers, by simply
passing f(p) to the do_step
method, again f(p) will be represented by a simple
Cfunction or a functor. Here, the above example of the harmonic oscillator
can be written as
pair< vector_type , vector_type > x; x.first[0] = 1.0; x.second[0] = 0.0; symplectic_rkn_sb3a_mclachlan< vector_type > rkn; rkn.do_step( harm_osc_f1() , x , t , dt );
In this example the function harm_osc_f1
is exactly the same function as in the above examples.
Note, that the state of the ODE must not be constructed explicitly via
pair<
vector_type ,
vector_type >
x
. One can also use a combination
of make_pair
and ref
. Furthermore, a convenience version
of do_step
exists which
takes q and p without combining them into a pair:
rkn.do_step( harm_osc_f1() , make_pair( boost::ref( q ) , boost::ref( p ) ) , t , dt ); rkn.do_step( harm_osc_f1() , q , p , t , dt ); rkn.do_step( make_pair( harm_osc_f1() , harm_osc_f2() ) , q , p , t , dt );
Caution  

This section is not uptodate. 
For some kind of systems the stability properties of the classical RungeKutta are not sufficient, especially if the system is said to be stiff. A stiff system possesses two or more time scales of very different order. Solvers for stiff systems are usually implicit, meaning that they solve equations like x(t+dt) = x(t) + dt * f(x(t+1)). This particular scheme is the implicit Euler method. Implicit methods usually solve the system of equations by a root finding algorithm like the Newton method and therefore need to know the Jacobian of the system J_{ij} = df_{i} / dx_{j}.
For implicit solvers the system is again a pair, where the first component
computes f(x,t) and the second the Jacobian. The syntax
is sys.first( x , dxdt , t )
and
sys.second( x , J , t )
.
For the implicit solver the state_type
is ublas::vector
and the Jacobian is represented
by ublas::matrix
.
Important  

Implicit solvers only work with ublas::vector as state type. At the moment, no other state types are supported. 
Another large class of solvers are multistep method. They save a small part of the history of the solution and compute the next step with the help of this history. Since multistep methods know a part of their history they do not need to compute the system function very often, usually it is only computed once. This makes multistep methods preferable if a call of the system function is expensive. Examples are ODEs defined on networks, where the computation of the interaction is usually where expensive (and might be of order O(N^2)).
Multistep methods differ from the normal steppers. They save a part of their history and this part has to be explicitly calculated and initialized. In the following example an AdamsBashforthstepper with a history of 5 steps is instantiated and initialized;
adams_bashforth_moulton< 5 , state_type > abm; abm.initialize( sys , inout , t , dt ); abm.do_step( sys , inout , t , dt );
The initialization uses a fourthorder RungeKutta stepper and after the
call of initialize
the
state of inout
has changed
to the current state, such that it can be immediately used by passing it
to following calls of do_step
.
You can also use you own steppers to initialize the internal state of the
AdamsBashforthStepper:
abm.initialize( runge_kutta_fehlberg78< state_type >() , sys , inout , t , dt );
Many multistep methods are also explicit steppers, hence the parameter
of do_step
method do not
differ from the explicit steppers.
Caution  

The multistep methods have some internal variables which depend on the explicit solution. Hence after any external changes of your state (e.g. size) or system the initialize function has to be called again to adjust the internal state of the stepper. If you use the integrate functions this will be taken into account. See the Using steppers section for more details. 
Many of the above introduced steppers possess the possibility to use adaptive stepsize control. Adaptive step size integration works in principle as follows:
The class of controlled steppers has their own concept in odeint  the Controlled Stepper concept. They are usually constructed from the underlying error steppers. An example is the controller for the explicit RungeKutta steppers. The RungeKutta steppers enter the controller as a template argument. Additionally one can pass the RungeKutta stepper to the constructor, but this step is not necessary; the stepper is defaultconstructed if possible.
Different step size controlling mechanism exist. They all have in common that they somehow compare predefined error tolerance against the error and that they might reject or accept a step. If a step is rejected the step size is usually decreased and the step is made again with the reduced step size. This procedure is repeated until the step is accepted. This algorithm is implemented in the integration functions.
A classical way to decide whether a step is rejected or accepted is to calculate
val =   err_{i}  / ( ε_{abs} + ε_{rel} * ( a_{x}  x_{i}  + a_{dxdt}   dxdt_{i}  )
ε_{abs} and ε_{rel} are the absolute and the relative error tolerances, and  x  is a norm, typically x=(Σ_{i} x_{i}^{2})^{1/2} or the maximum norm. The step is rejected if val is greater then 1, otherwise it is accepted. For details of the used norms and error tolerance see the table below.
For the controlled_runge_kutta
stepper the new step size is then calculated via
val > 1 : dt_{new} = dt_{current} max( 0.9 pow( val , 1 / ( O_{E}  1 ) ) , 0.2 )
val < 0.5 : dt_{new} = dt_{current} min( 0.9 pow( val , 1 / O_{S} ) , 5 )
else : dt_{new} = dt_{current}
Here, O_{S} and O_{E} are the order of the stepper and the error stepper. These formulas also contain some safety factors, avoiding that the step size is reduced or increased to much. For details of the implementations of the controlled steppers in odeint see the table below.
Table 1.5. Adaptive step size algorithms
Stepper 
Tolerance formula 
Norm 
Step size adaption 


val =   err_{i}  / ( ε_{abs} + ε_{rel} * ( a_{x}  x_{i}  + a_{dxdt}   dxdt_{i}  ) 
x = max( x_{i} ) 
val > 1 : dt_{new} = dt_{current} max( 0.9 pow( val , 1 / ( O_{E}  1 ) ) , 0.2 ) val < 0.5 : dt_{new} = dt_{current} min( 0.9 pow( val , 1 / O_{S} ) , 5 ) else : dt_{new} = dt_{current} 

val =  err_{i} / ( ε_{abs} + ε_{rel} max(  x_{i}  ,  xold_{i}  ) )  
x=(Σ_{i} x_{i}^{2})^{1/2} 
fac = max( 1 / 6 , min( 5 , pow( val , 1 / 4 ) / 0.9 ) fac2 = max( 1 / 6 , min( 5 , dt_{old} / dt_{current} pow( val^{2} / val_{old} , 1 / 4 ) / 0.9 ) val > 1 : dt_{new} = dt_{current} / fac val < 1 : dt_{new} = dt_{current} / max( fac , fac2 ) 
bulirsch_stoer 
tol=1/2 
 
dt_{new} = dt_{old}^{1/a} 
To ease to generation of the controlled stepper, generation functions exist which take the absolute and relative error tolerances and a predefined error stepper and construct from this knowledge an appropriate controlled stepper. The generation functions are explained in detail in Generation functions.
A fourth class of stepper exists which are the so called dense output steppers.
Denseoutput steppers might take larger steps and interpolate the solution
between two consecutive points. This interpolated points have usually the
same order as the order of the stepper. Denseoutput steppers are often
composite stepper which take the underlying method as a template parameter.
An example is the dense_output_runge_kutta
stepper which takes a RungeKutta stepper with denseoutput facilities
as argument. Not all RungeKutta steppers provide denseoutput calculation;
at the moment only the DormandPrince 5 stepper provides dense output.
An example is
dense_output_runge_kutta< controlled_runge_kutta< runge_kutta_dopri5< state_type > > > dense; dense.initialize( in , t , dt ); pair< double , double > times = dense.do_step( sys ); (void)times;
Dense output stepper have their own concept. The main difference to usual
steppers is that they manage the state and time internally. If you call
do_step
, only the ODE is
passed as argument. Furthermore do_step
return the last time interval: t
and t+dt
, hence you can interpolate the solution
between these two times points. Another difference is that they must be
initialized with initialize
,
otherwise the internal state of the stepper is default constructed which
might produce funny errors or bugs.
The construction of the dense output stepper looks a little bit nasty,
since in the case of the dense_output_runge_kutta
stepper a controlled stepper and an error stepper have to be nested. To
simplify the generation of the dense output stepper generation functions
exist:
typedef boost::numeric::odeint::result_of::make_dense_output< runge_kutta_dopri5< state_type > >::type dense_stepper_type; dense_stepper_type dense2 = make_dense_output( 1.0e6 , 1.0e6 , runge_kutta_dopri5< state_type >() ); (void)dense2;
This statement is also lengthy; it demonstrates how make_dense_output
can be used with the result_of
protocol. The parameters to make_dense_output
are the absolute error tolerance, the relative error tolerance and the
stepper. This explicitly assumes that the underlying stepper is a controlled
stepper and that this stepper has an absolute and a relative error tolerance.
For details about the generation functions see Generation
functions. The generation functions have been designed for easy
use with the integrate functions:
integrate_const( make_dense_output( 1.0e6 , 1.0e6 , runge_kutta_dopri5< state_type >() ) , sys , inout , t_start , t_end , dt );
This section contains some general information about the usage of the steppers in odeint.
Steppers are copied by value
The stepper in odeint are always copied by values. They are copied for the creation of the controlled steppers or the dense output steppers as well as in the integrate functions.
Steppers might have a internal state
Caution  

Some of the features described in this section are not yet implemented 
Some steppers require to store some information about the state of the
ODE between two steps. Examples are the multistep methods which store
a part of the solution during the evolution of the ODE, or the FSAL steppers
which store the last derivative at time t+dt, to be
used in the next step. In both cases the steppers expect that consecutive
calls of do_step
are from
the same solution and the same ODE. In this case it is absolutely necessary
that you call do_step
with
the same system function and the same state, see also the examples for
the FSAL steppers above.
Stepper with an internal state support two additional methods: reset
which resets the state and initialize
which initializes the internal
state. The parameters of initialize
depend on the specific stepper. For example the AdamsBashforthMoulton
stepper provides two initialize methods: initialize( system , inout , t , dt )
which initializes the internal states
with the help of the RungeKutta 4 stepper, and initialize( stepper , system , inout , t , dt )
which initializes with the help of stepper
. For the case of the FSAL steppers,
initialize
is initialize(
sys ,
in ,
t )
which simply calculates the r.h.s. of the ODE and assigns its value to
the internal derivative.
All these steppers have in common, that they initially fill their internal
state by themselves. Hence you are not required to call initialize. See
how this works for the AdamsBashforthMoulton stepper: in the example
we instantiate a fourth order AdamsBashforthMoulton stepper, meaning
that it will store 4 internal derivatives of the solution at times (tdt,t2*dt,t3*dt,t4*dt)
.
adams_bashforth_moulton< 4 , state_type > stepper; stepper.do_step( sys , x , t , dt ); // make one step with the classical RungeKutta stepper and initialize the first internal state // the internal array is now [x(tdt)] stepper.do_step( sys , x , t , dt ); // make one step with the classical RungeKutta stepper and initialize the second internal state // the internal state array is now [x(tdt), x(t2*dt)] stepper.do_step( sys , x , t , dt ); // make one step with the classical RungeKutta stepper and initialize the third internal state // the internal state array is now [x(tdt), x(t2*dt), x(t3*dt)] stepper.do_step( sys , x , t , dt ); // make one step with the classical RungeKutta stepper and initialize the fourth internal state // the internal state array is now [x(tdt), x(t2*dt), x(t3*dt), x(t4*dt)] stepper.do_step( sys , x , t , dt ); // make one step with AdamBashforthMoulton, the internal array of states is now rotated
In the stepper table at the bottom of this page one can see which stepper
have an internal state and hence provide the reset
and initialize
methods.
Stepper might be resizable
Nearly all steppers in odeint need to store some intermediate results of
the type state_type
or
deriv_type
. To do so odeint
need some memory management for the internal temporaries. As this memory
management is typically related to adjusting the size of vectorlike types,
it is called resizing in odeint. So, most steppers in odeint provide an
additional template parameter which controls the size adjustment of the
internal variables  the resizer. In detail odeint provides three policy
classes (resizers) always_resizer
,
initially_resizer
, and
never_resizer
. Furthermore,
all stepper have a method adjust_size
which takes a parameter representing a state type and which manually adjusts
the size of the internal variables matching the size of the given instance.
Before performing the actual resizing odeint always checks if the sizes
of the state and the internal variable differ and only resizes if they
are different.
Note  

You only have to worry about memory allocation when using dynamically
sized vector types. If your state type is heap allocated, like 
By default the resizing parameter is initially_resizer
,
meaning that the first call to do_step
performs the resizing, hence memory allocation. If you have changed the
size of your system and your state you have to call adjust_size
by hand in this case. The second resizer is the always_resizer
which tries to resize the internal variables at every call of do_step
. Typical use cases for this kind
of resizer are self expanding lattices like shown in the tutorial ( Self expanding
lattices) or partial differential equations with an adaptive grid.
Here, no calls of adjust_size
are required, the steppers manage everything themselves. The third class
of resizer is the never_resizer
which means that the internal variables are never adjusted automatically
and always have to be adjusted by hand .
There is a second mechanism which influences the resizing and which controls
if a state type is at least resizeable  a metafunction is_resizeable
. This metafunction returns
a static Boolean value if any type is resizable. For example it will return
true
for std::vector< T >
but false
for boost::array< T >
.
By default and for unknown types is_resizeable
returns false
, so if you have
your own type you need to specialize this metafunction. For more details
on the resizing mechanism see the section Adapt
your own state types.
Which steppers should be used in which situation
odeint provides a quite large number of different steppers such that the user is left with the question of which stepper fits his needs. Our personal recommendations are:
runge_kutta_dopri5
is maybe the best default stepper. It has step size control as well
as denseoutput functionality. Simple create a denseoutput stepper
by make_dense_output( 1.0e6 , 1.0e5 , runge_kutta_dopri5< state_type
>() )
.
runge_kutta4
is a good
stepper for constant step sizes. It is widely used and very well known.
If you need to create artificial time series this stepper should be
the first choice.
adams_bashforth_moulton
is very well suited for ODEs where the r.h.s. is expensive (in terms
of computation time). It will calculate the system function only once
during each step.
Table 1.6. Stepper Algorithms
Algorithm 
Class 
Concept 
System Concept 
Order 
Error Estimation 
Dense Output 
Internal state 
Remarks 

Explicit Euler 

1 
No 
Yes 
No 
Very simple, only for demonstrating purpose 

Modified Midpoint 

configurable (2) 
No 
No 
No 
Used in BulirschStoer implementation 

RungeKutta 4 

4 
No 
No 
No 
The classical RungeKutta scheme, good general scheme without error control 

CashKarp 

5 
Yes (4) 
No 
No 
Good general scheme with error estimation, to be used in controlled_error_stepper 

DormandPrince 5 

5 
Yes (4) 
Yes 
Yes 
Standard method with error control and dense output, to be used in controlled_error_stepper and in dense_output_controlled_explicit_fsal. 

Fehlberg 78 

8 
Yes (7) 
No 
No 
Good high order method with error estimation, to be used in controlled_error_stepper. 

Adams Bashforth 

configurable 
No 
No 
Yes 
Multistep method 

Adams Moulton 

configurable 
No 
No 
Yes 
Multistep method 

Adams Bashforth Moulton 

configurable 
No 
No 
Yes 
Combined multistep method 

Controlled RungeKutta 

depends 
Yes 
No 
depends 
Error control for Error Stepper. Requires an Error Stepper from above. Order depends on the given ErrorStepper 

Dense Output RungeKutta 

depends 
No 
Yes 
Yes 
Dense output for Stepper
and Error
Stepper from above if they provide dense output functionality
(like 

BulirschStoer 

variable 
Yes 
No 
No 
Stepper with step size and order control. Very good if high precision is required. 

BulirschStoer Dense Output 

variable 
Yes 
Yes 
No 
Stepper with step size and order control as well as dense output. Very good if high precision and dense output is required. 

Implicit Euler 

1 
No 
No 
No 
Basic implicit routine. Requires the Jacobian. Works only with Boost.uBLAS vectors as state types. 

Rosenbrock 4 

4 
Yes 
Yes 
No 
Good for stiff systems. Works only with Boost.uBLAS vectors as state types. 

Controlled Rosenbrock 4 

4 
Yes 
Yes 
No 
Rosenbrock 4 with error control. Works only with Boost.uBLAS vectors as state types. 

Dense Output Rosenbrock 4 

4 
Yes 
Yes 
No 
Controlled Rosenbrock 4 with dense output. Works only with Boost.uBLAS vectors as state types. 

Symplectic Euler 

1 
No 
No 
No 
Basic symplectic solver for separable Hamiltonian system 

Symplectic RKN McLachlan 

4 
No 
No 
No 
Symplectic solver for separable Hamiltonian system with 6 stages and order 4. 

Symplectic RKN McLachlan 

4 
No 
No 
No 
Symplectic solver with 5 stages and order 4, can be used with arbitrary precision types. 

Velocity Verlet 

1 
No 
No 
Yes 
Velocity verlet method suitable for molecular dynamics simulation. 
Finally, one can also write new steppers which are fully compatible with odeint. They only have to fulfill one or several of the stepper Concepts of odeint.
We will illustrate how to write your own stepper with the example of the stochastic Euler method. This method is suited to solve stochastic differential equations (SDEs). A SDE has the form
dx/dt = f(x) + g(x) ξ(t)
where ξ is Gaussian white noise with zero mean and a standard deviation σ(t). f(x) is said to be the deterministic part while g(x) ξ is the noisy part. In case g(x) is independent of x the SDE is said to have additive noise. It is not possible to solve SDE with the classical solvers for ODEs since the noisy part of the SDE has to be scaled differently then the deterministic part with respect to the time step. But there exist many solvers for SDEs. A classical and easy method is the stochastic Euler solver. It works by iterating
x(t+Δ t) = x(t) + Δ t f(x(t)) + Δ t^{1/2} g(x) ξ(t)
where ξ(t) is an independent normal distributed random variable.
Now we will implement this method. We will call the stepper stochastic_euler
. It models the Stepper concept.
For simplicity, we fix the state type to be an array< double , N >
The class definition looks like
template< size_t N > class stochastic_euler { public: typedef boost::array< double , N > state_type; typedef boost::array< double , N > deriv_type; typedef double value_type; typedef double time_type; typedef unsigned short order_type; typedef boost::numeric::odeint::stepper_tag stepper_category; static order_type order( void ) { return 1; } // ... };
The types are needed in order to fulfill the stepper concept. As internal
state and deriv type we use simple arrays in the stochastic Euler, they
are needed for the temporaries. The stepper has the order one which is
returned from the order()
function.
The system functions needs to calculate the deterministic and the stochastic
part of our stochastic differential equation. So it might be suitable that
the system function is a pair of functions. The first element of the pair
computes the deterministic part and the second the stochastic one. Then,
the second part also needs to calculate the random numbers in order to
simulate the stochastic process. We can now implement the do_step
method
template< size_t N > class stochastic_euler { public: // ... template< class System > void do_step( System system , state_type &x , time_type t , time_type dt ) const { deriv_type det , stoch ; system.first( x , det ); system.second( x , stoch ); for( size_t i=0 ; i<x.size() ; ++i ) x[i] += dt * det[i] + sqrt( dt ) * stoch[i]; } };
This is all. It is quite simple and the stochastic Euler stepper implement here is quite general. Of course it can be enhanced, for example
boost::ref
for the system functions
boost::range
for the state type in the
do_step
method
Now, lets look how we use the new stepper. A nice example is the OrnsteinUhlenbeck process. It consists of a simple Brownian motion overlapped with an relaxation process. Its SDE reads
dx/dt =  x + ξ
where ξ is Gaussian white noise with standard deviation σ. Implementing the OrnsteinUhlenbeck process is quite simple. We need two functions or functors  one for the deterministic and one for the stochastic part:
const static size_t N = 1; typedef boost::array< double , N > state_type; struct ornstein_det { void operator()( const state_type &x , state_type &dxdt ) const { dxdt[0] = x[0]; } }; struct ornstein_stoch { boost::mt19937 m_rng; boost::normal_distribution<> m_dist; ornstein_stoch( double sigma ) : m_rng() , m_dist( 0.0 , sigma ) { } void operator()( const state_type &x , state_type &dxdt ) { dxdt[0] = m_dist( m_rng ); } };
In the stochastic part we have used the Mersenne twister for the random
number generation and a Gaussian white noise generator normal_distribution
with standard deviation σ. Now, we can use the stochastic
Euler stepper with the integrate functions:
double dt = 0.1; state_type x = {{ 1.0 }}; integrate_const( stochastic_euler< N >() , make_pair( ornstein_det() , ornstein_stoch( 1.0 ) ) , x , 0.0 , 10.0 , dt , streaming_observer() );
Note, how we have used the make_pair
function for the generation of the system function.
odeint provides a C++ template metaalgorithm for constructing arbitrary RungeKutta schemes ^{[1]}. Some schemes are predefined in odeint, for example the classical RungeKutta of fourth order, or the RungeKuttaCashKarp 54 and the RungeKuttaFehlberg 78 method. You can use this meta algorithm to construct you own solvers. This has the advantage that you can make full use of odeint's algebra and operation system.
Consider for example the method of Heun, defined by the following Butcher tableau:
c1 = 0 c2 = 1/3, a21 = 1/3 c3 = 2/3, a31 = 0 , a32 = 2/3 b1 = 1/4, b2 = 0 , b3 = 3/4
Implementing this method is very easy. First you have to define the constants:
template< class Value = double > struct heun_a1 : boost::array< Value , 1 > { heun_a1( void ) { (*this)[0] = static_cast< Value >( 1 ) / static_cast< Value >( 3 ); } }; template< class Value = double > struct heun_a2 : boost::array< Value , 2 > { heun_a2( void ) { (*this)[0] = static_cast< Value >( 0 ); (*this)[1] = static_cast< Value >( 2 ) / static_cast< Value >( 3 ); } }; template< class Value = double > struct heun_b : boost::array< Value , 3 > { heun_b( void ) { (*this)[0] = static_cast<Value>( 1 ) / static_cast<Value>( 4 ); (*this)[1] = static_cast<Value>( 0 ); (*this)[2] = static_cast<Value>( 3 ) / static_cast<Value>( 4 ); } }; template< class Value = double > struct heun_c : boost::array< Value , 3 > { heun_c( void ) { (*this)[0] = static_cast< Value >( 0 ); (*this)[1] = static_cast< Value >( 1 ) / static_cast< Value >( 3 ); (*this)[2] = static_cast< Value >( 2 ) / static_cast< Value >( 3 ); } };
While this might look cumbersome, packing all parameters into a templatized
class which is not immediately evaluated has the advantage that you can
change the value_type
of
your stepper to any type you like  presumably arbitrary precision types.
One could also instantiate the coefficients directly
const boost::array< double , 1 > heun_a1 = {{ 1.0 / 3.0 }}; const boost::array< double , 2 > heun_a2 = {{ 0.0 , 2.0 / 3.0 }}; const boost::array< double , 3 > heun_b = {{ 1.0 / 4.0 , 0.0 , 3.0 / 4.0 }}; const boost::array< double , 3 > heun_c = {{ 0.0 , 1.0 / 3.0 , 2.0 / 3.0 }};
But then you are nailed down to use doubles.
Next, you need to define your stepper, note that the Heun method has 3 stages and produces approximations of order 3:
template< class State , class Value = double , class Deriv = State , class Time = Value , class Algebra = boost::numeric::odeint::range_algebra , class Operations = boost::numeric::odeint::default_operations , class Resizer = boost::numeric::odeint::initially_resizer > class heun : public boost::numeric::odeint::explicit_generic_rk< 3 , 3 , State , Value , Deriv , Time , Algebra , Operations , Resizer > { public: typedef boost::numeric::odeint::explicit_generic_rk< 3 , 3 , State , Value , Deriv , Time , Algebra , Operations , Resizer > stepper_base_type; typedef typename stepper_base_type::state_type state_type; typedef typename stepper_base_type::wrapped_state_type wrapped_state_type; typedef typename stepper_base_type::value_type value_type; typedef typename stepper_base_type::deriv_type deriv_type; typedef typename stepper_base_type::wrapped_deriv_type wrapped_deriv_type; typedef typename stepper_base_type::time_type time_type; typedef typename stepper_base_type::algebra_type algebra_type; typedef typename stepper_base_type::operations_type operations_type; typedef typename stepper_base_type::resizer_type resizer_type; typedef typename stepper_base_type::stepper_type stepper_type; heun( const algebra_type &algebra = algebra_type() ) : stepper_base_type( fusion::make_vector( heun_a1<Value>() , heun_a2<Value>() ) , heun_b<Value>() , heun_c<Value>() , algebra ) { } };
That's it. Now, we have a new stepper method and we can use it, for example with the Lorenz system:
typedef boost::array< double , 3 > state_type; heun< state_type > h; state_type x = {{ 10.0 , 10.0 , 10.0 }}; integrate_const( h , lorenz() , x , 0.0 , 100.0 , 0.01 , streaming_observer( std::cout ) );
^{[1] } M. Mulansky, K. Ahnert, TemplateMetaprogramming applied to numerical problems, arxiv:1110.3233