[/============================================================================
  Boost.odeint

  Copyright 2011-2012 Karsten Ahnert
  Copyright 2011-2013 Mario Mulansky
  Copyright 2012 Sylwester Arabas

  Use, modification and distribution is subject to the Boost Software License,
  Version 1.0. (See accompanying file LICENSE_1_0.txt or copy at
  http://www.boost.org/LICENSE_1_0.txt)
=============================================================================/]


[/ [section Special topics] /]

[section Complex state types]

[import ../examples/stuart_landau.cpp]

Thus far we have seen several examples defined for real values. 
odeint can handle complex state types, hence ODEs which are defined on complex
vector spaces, as well. An example is the Stuart-Landau oscillator

['d __Psi / dt = ( 1 + i __eta ) __Psi + ( 1 + i __alpha ) | __Psi |[super 2] __Psi ]

where ['__Psi] and ['i] is a complex variable. The definition of this ODE in C++
using complex< double > as a state type may look as follows

[stuart_landau_system_function]

One can also use a function instead of a functor to implement it

[stuart_landau_system_function_alternative]

We strongly recommend to use the first ansatz. In this case you have explicit control over the parameters of the system and are not restricted to use global variables to parametrize the oscillator.

When choosing the stepper type one has to account for the "unusual" state type:
it is a single `complex<double>` opposed to the vector types used in the
previous examples. This means that no iterations over vector elements have to
be performed inside the stepper algorithm. Odeint already detects that and
automatically uses the `vector_space_algebra` for computation.
You can enforce this by supplying additional template arguments to the stepper
including the `vector_space_algebra`. Details on the usage of algebras can be
found in the section __adapt_state_types.

[stuart_landau_integration]

The full cpp file for the Stuart-Landau example can be found here [github_link
examples/stuart_landau.cpp stuart_landau.cpp]

[endsect]

[section Lattice systems]

[import ../examples/fpu.cpp]


odeint can also be used to solve ordinary differential equations defined on lattices. A prominent example is the Fermi-Pasta-Ulam system __fpu_scholarpedia_ref. It is a Hamiltonian system of nonlinear coupled harmonic oscillators. The Hamiltonian is

[' H = __Sigma[subl i] p[subl i][super 2]/2 + 1/2 ( q[subl i+1] - q[subl i] )^2 + __beta / 4 ( q[subl i+1] - q[subl i] )^4 ]

Remarkably, the Fermi-Pasta-Ulam system was the first numerical experiment to be implemented on a computer. It was studied at Los Alamos in 1953 on one of the first computers (a MANIAC I) and it triggered a whole new tree of mathematical and physical science.
 
Like the __tut_solar_system, the FPU is solved again by a symplectic solver, but in this case we can speed up the computation because the ['q] components trivially reduce to ['dq[subl i] / dt = p[subl i]]. odeint is capable of doing this performance improvement. All you have to do is to call the symplectic solver with an state function for the ['p] components. Here is how this function looks like

[fpu_system_function]

You can also use `boost::array< double , N >` for the state type.

Now, you have to define your initial values and perform the integration:

[fpu_integration]

The observer uses a reference to the system object to calculate the local energies:

[fpu_observer]

The full cpp file for this FPU example can be found here [github_link examples/fpu.cpp fpu.cpp]

[endsect]

[section Ensembles of oscillators]

[import ../examples/phase_oscillator_ensemble.cpp]

Another important high dimensional system of coupled ordinary differential equations is an ensemble of ['N] all-to-all coupled phase oscillators __synchronization_pikovsky_ref. It is defined as

[' d__phi[subl k] / dt = __omega[subl k] + __epsilon / N __Sigma[subl j] sin( __phi[subl j] - __phi[subl k] )]

The natural frequencies ['__omega[subl i]] of each oscillator follow some distribution and ['__epsilon] is the coupling strength. We choose here a Lorentzian distribution for ['__omega[subl i]]. Interestingly a phase transition can be observed if the coupling strength exceeds a critical value. Above this value synchronization sets in and some of the oscillators oscillate with the same frequency despite their different natural frequencies. The transition is also called Kuramoto transition. Its behavior can be analyzed by employing the mean field of the phase

['Z = K e[super i __Theta] = 1 / N __Sigma[subl k]e[super i __phi[subl k]]]

The definition of the system function is now a bit more complex since we also need to store the individual frequencies of each oscillator.

[phase_oscillator_ensemble_system_function]

Note, that we have used ['Z] to simplify the equations of motion. Next, we create an observer which computes the value of ['Z] and we record ['Z] for different values of ['__epsilon].

[phase_oscillator_ensemble_observer]

Now, we do several integrations for different values of ['__epsilon] and record ['Z]. The result nicely confirms the analytical result of the phase transition, i.e. in our example the standard deviation of the Lorentzian is 1 such that the transition will be observed at ['__epsilon = 2].

[phase_oscillator_ensemble_integration]

The full cpp file for this example can be found here [github_link examples/phase_oscillator_ensemble.cpp phase_oscillator_ensemble.cpp]

[endsect]

[section Using boost::units]

[import ../examples/harmonic_oscillator_units.cpp]

odeint also works well with __boost_units - a library for compile time unit
and dimension analysis. It works by decoding unit information into the types
of values. For a one-dimensional unit you can just use the Boost.Unit types as
state type, deriv type and time type and hand the `vector_space_algebra` to
the stepper definition and everything works just fine:

```
typedef units::quantity< si::time , double > time_type;
typedef units::quantity< si::length , double > length_type;
typedef units::quantity< si::velocity , double > velocity_type;

typedef runge_kutta4< length_type , double , velocity_type , time_type ,
                      vector_space_algebra > stepper_type;
```
 
If you want to solve more-dimensional problems the individual entries
typically have different units. That means that the `state_type` is now
possibly heterogeneous, meaning that every entry might have a different type.
To solve this problem, compile-time sequences from __boost_fusion can be used.

To illustrate how odeint works with __boost_units we use the harmonic oscillator as primary example. We start with defining all quantities

[units_define_basic_quantities]

Note, that the `state_type` and the `deriv_type` are now a compile-time fusion
sequences. `deriv_type` represents ['x'] and is now different from the state
type as it has different unit definitions. Next, we define the ordinary
differential equation which is completely equivalent to the example in __tut_harmonic_oscillator: 

[units_define_ode]

Next, we instantiate an appropriate stepper. We must explicitly parametrize
the stepper with the `state_type`, `deriv_type`, `time_type`. 

[units_define_stepper]

[note When using compile-time sequences, the iteration over vector elements is
done by the `fusion_algebra`, which is automatically chosen by odeint. For
more on the state types / algebras see chapter __adapt_state_types.]

It is quite easy but the compilation time might take very long. Furthermore, the observer is defined a bit different

[units_observer]

[caution Using __boost_units works nicely but compilation can be very time and
memory consuming. For example the unit test for the usage of __boost_units in odeint take up to 4 GB
of memory at compilation.]

The full cpp file for this example can be found here [github_link examples/harmonic_oscillator_units.cpp harmonic_oscillator_units.cpp].

[endsect] 

[section Using matrices as state types]

[import ../examples/two_dimensional_phase_lattice.cpp]

odeint works well with a variety of different state types. It is not restricted to pure vector-wise types, like `vector< double >`, `array< double , N >`, `fusion::vector< double , double >`, etc. but also works with types having a different topology then simple vectors. Here, we show how odeint can be used with matrices as states type, in the next section we will show how can be used to solve ODEs defined on complex networks.

By default, odeint can be used with `ublas::matrix< T >` as state type for matrices. A simple example is a two-dimensional lattice of coupled phase oscillators. Other matrix types like `mtl::dense_matrix` or blitz arrays and matrices can used as well but need some kind of activation in order to work with odeint. This activation is described in following sections,

The definition of the system is

[two_dimensional_phase_lattice_definition]

In principle this is all. Please note, that the above code is far from being optimal. Better performance can be achieved if every interaction is only calculated once and iterators for columns and rows are used. Below are some visualizations of the evolution of this lattice equation.

[$phase_lattice_2d_0000.jpg] [$phase_lattice_2d_0100.jpg] [$phase_lattice_2d_1000.jpg]

The full cpp for this example can be found here [github_link examples/two_dimensional_phase_lattice.cpp two_dimensional_phase_lattice.cpp].

[endsect]

[/
[section Partial differential equations]
To be continued:
*Wave equation
*KdV
*Ginzburg-Landau
[endsect]
[section Ordinary differential equations on networks]
to be continued
[endsect]
]

[section Using arbitrary precision floating point types]

[import ../examples/multiprecision/lorenz_mp.cpp]

Sometimes one needs results with higher precision than provided by the
standard floating point types.
As odeint allows to configure the fundamental numerical type, it is well
suited to be run with arbitrary precision types.
Therefore, one only needs a library that provides a type representing values
with arbitrary precision and the fundamental operations for those values.
__boost_multiprecision is a boost library that does exactly this.
Making use of __boost_multiprecision to solve odes with odeint is very simple,
as the following example shows.

Here we use `cpp_dec_float_50` as the fundamental value type, which ensures
exact computations up to 50 decimal digits.


[mp_lorenz_defs]

As exemplary ODE again the lorenz system is chosen, but here we have to make
sure all constants are initialized as high precision values.

[mp_lorenz_rhs]

The actual integration then is straight forward:

[mp_lorenz_int]

The full example can be found at [github_link examples/multiprecision/lorenz_mp.cpp lorenz_mp.cpp].
Another example that compares the accuracy of the high precision type with
standard double can be found at [github_link examples/multiprecision/cmp_precision.cpp cmp_precision.cpp].

Furthermore, odeint can also be run with other multiprecision libraries,
e.g. [@http://gmplib.org/ gmp].
An example for this is given in [github_link examples/gmpxx/lorenz_gmpxx.cpp lorenz_gmpxx.cpp].

[endsect]

[section Self expanding lattices]

[import ../examples/resizing_lattice.cpp]

odeint supports changes of the state size during integration if a state_type
is used which can be resized, like `std::vector`.
The adjustment of the state's size has to be done from outside and the stepper
has to be instantiated with `always_resizer` as the template argument for the
`resizer_type`.
In this configuration, the stepper checks for changes in the state size and
adjust it's internal storage accordingly.

We show this for a Hamiltonian system of nonlinear, disordered oscillators with nonlinear nearest neighbor coupling.

The system function is implemented in terms of a class that also provides functions for calculating the energy.
Note, that this class stores the random potential internally which is not resized, but rather a start index is kept which should be changed whenever the states' size change.

[resizing_lattice_system_class]

The total size we allow is 1024 and we start with an initial state size of 60.

[resizing_lattice_initialize]

The lattice gets resized whenever the energy distribution comes close to the borders `distr[10] > 1E-150`, `distr[distr.size()-10] > 1E-150`.
If we increase to the left, `q` and `p` have to be rotated because their resize function always appends at the end.
Additionally, the start index of the potential changes in this case.

[resizing_lattice_steps_loop]

The `do_resize` function simply calls `vector.resize` of `q` , `p` and `distr`.

[resizing_lattice_resize_function]

The full example can be found in [github_link examples/resizing_lattice.cpp resizing_lattice.cpp]

[endsect]

[/ [endsect] /]
