[/============================================================================ Boost.odeint Copyright 2011 Mario Mulansky Copyright 2012 Karsten Ahnert Copyright 2013 Pascal Germroth 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 State Algebra Operations] [note The following does not apply to implicit steppers like implicit_euler or Rosenbrock 4 as there the `state_type` can not be changed from `ublas::vector` and no algebra/operations are used.] [heading Description] The `State`, `Algebra` and `Operations` together define a concept describing how the mathematical vector operations required for the stepper algorithms are performed. The typical vector operation done within steppers is ['*y* = __Sigma __alpha[sub i] [*x[sub i]]]. The `State` represents the state variable of an ODE, usually denoted with /x/. Algorithmically, the state is often realized as a `vector< double >` or `array< double , N >`, however, the genericity of odeint enables you to basically use anything as a state type. The algorithmic counterpart of such mathematical expressions is divided into two parts. First, the `Algebra` is used to account for the vector character of the equation. In the case of a `vector` as state type this means the `Algebra` is responsible for iteration over all vector elements. Second, the `Operations` are used to represent the actual operation applied to each of the vector elements. So the `Algebra` iterates over all elements of the `State`s and calls an operation taken from the `Operations` for each element. This is where `State`, `Algebra` and `Operations` have to work together to make odeint running. Please have a look at the `range_algebra` and `default_operations` to see an example how this is implemented. In the following we describe how `State`, `Algebra` and `Operations` are used together within the stepper implementations. [section Operations] [heading Notation] [variablelist [[`Operations`] [The operations type]] [/[`Time`] [A type representing the time type of steppers]] [[`Value1`, ... , `ValueN`] [Types representing the value or time type of stepper]] [[`Scale`] [Type of the scale operation]] [[`scale`] [Object of type `Scale`]] [[[^ScaleSum['N]]] [Type that represents a general scale_sum operation, [^/N/] should be replaced by a number from 1 to 14.]] [[[^scale_sum['N]]] [Object of type [^ScaleSum['N]], [^/N/] should be replaced by a number from 1 to 14.]] [[`ScaleSumSwap2`] [Type of the scale sum swap operation]] [[`scale_sum_swap2`] [Object of type `ScaleSumSwap2`]] [[`a1, a2, ...`] [Objects of type `Value1`, `Value2`, ...]] [[`y, x1, x2, ...`] [Objects of `State`'s value type]] ] [heading Valid Expressions] [table [[Name] [Expression] [Type] [Semantics]] [[Get scale operation] [`Operations::scale< Value >`] [`Scale`] [Get `Scale` from `Operations`]] [[`Scale` constructor] [`Scale< Value >( a )`] [`Scale`] [Constructs a `Scale` object]] [[`Scale` operation] [`scale( x )`] [`void`] [Calculates `x *= a`]] [[Get general `scale_sum` operation] [[^Operations::scale_sum['N]< Value1 , ... , ValueN >]] [[^ScaleSum['N]]] [Get the [^ScaleSum['N]] type from `Operations`, [^/N/] should be replaced by a number from 1 to 14.]] [[`scale_sum` constructor] [[^ScaleSum['N]< Value1 , ... , ValueN >( a1 , ... , aN )]] [[^ScaleSum['N]]] [Constructs a `scale_sum` object given [^/N/] parameter values with [^/N/] between 1 and 14.]] [[`scale_sum` operation] [[^scale_sum['N]( y , x1 , ... , xN )]] [`void`] [Calculates `y = a1*x1 + a2*x2 + ... + aN*xN`. Note that this is an [^/N/+1]-ary function call.]] [[Get scale sum swap operation] [`Operations::scale_sum_swap2< Value1 , Value2 >`] [`ScaleSumSwap2`] [Get scale sum swap from operations]] [[`ScaleSumSwap2` constructor] [`ScaleSumSwap2< Value1 , Value2 >( a1 , a2 )`] [`ScaleSumSwap2`] [Constructor]] [[`ScaleSumSwap2` operation] [`scale_sum_swap2( x1 , x2 , x3 )`] [`void`] [Calculates `tmp = x1`, `x1 = a1*x2 + a2*x3` and `x2 = tmp`.]] ] [endsect] [section Algebra] [heading Notation] [variablelist [[`State`] [The state type]] [[`Algebra`] [The algebra type]] [[[^Operation['N]]] [An [^/N/]-ary operation type, [^/N/] should be a number from 1 to 14.]] [[`algebra`] [Object of type `Algebra`]] [[[^operation['N]]] [Object of type [^Operation['N]]]] [[`y, x1, x2, ...`] [Objects of type `State`]] ] [heading Valid Expressions] [table [[Name] [Expression] [Type] [Semantics]] [[Vector Operation with arity 2] [`algebra.for_each2( y , x , operation2 )`] [void] [Calls `operation2( y_i , x_i )` for each element `y_i` of `y` and `x_i` of `x`.]] [[Vector Operation with arity 3] [`algebra.for_each3( y , x1 , x2 , operation3 )`] [void] [Calls `operation3( y_i , x1_i , x2_i )` for each element `y_i` of `y` and `x1_i` of `x1` and `x2_i` of `x2`.]] [[Vector Operation with arity [^/N/]] [[^algebra.for_each['N]( y , x1 , ... , xN , operation['N] )]] [void] [Calls [^operation['N]( y_i , x1_i , ... , xN_i )] for each element `y_i` of `y` and `x1_i` of `x1` and so on. [^/N/] should be replaced by a number between 1 and 14.]] ] [endsect] [section Pre-Defined implementations] As standard configuration odeint uses the `range_algebra` and `default_operations` which suffices most situations. However, a few more possibilities exist either to gain better performance or to ensure interoperability with other libraries. In the following we list the existing `Algebra`/`Operations` configurations that can be used in the steppers. [table [[`State`] [`Algebra`] [`Operations`] [Remarks]] [[Anything supporting __boost_range, like `std::vector`, `std::list`, `boost::array`,... based on a `value_type` that supports operators +,* (typically `double`)] [`range_algebra`] [`default_operations`] [Standard implementation, applicable for most typical situations.]] [[`boost::array` based on a `value_type` that supports operators +,*] [`array_algebra`] [`default_operations`] [Special implementation for boost::array with better performance than `range_algebra`]] [[Anything that defines operators + within itself and * with scalar (Mathematically spoken, anything that is a vector space).] [`vector_space_algebra`] [`default_operations`] [For the use of __controlled_stepper, the template `vector_space_reduce` has to be instantiated.]] [[`thrust::device_vector`, `thrust::host_vector`] [`thrust_algebra`] [`thrust_operations`] [For running odeint on CUDA devices by using __thrust]] [[Any RandomAccessRange] [`openmp_range_algebra`] [`default_operations`] [OpenMP-parallelised range algebra]] [[`openmp_state`] [`openmp_algebra`] [`default_operations`] [OpenMP-parallelised algebra for split data]] [[`boost::array` or anything which allocates the elements in a C-like manner] [`vector_space_algebra`] [`mkl_operations`] [Using the __intel_mkl in odeint for maximum performance. Currently, only the RK4 stepper is supported.]] ] [endsect] [section Example expressions] [table [[Name] [Expression] [Type] [Semantics]] [[Vector operation] [`algebra.for_each3( y , x1 , x2 , Operations::scale_sum2< Value1 , Value2 >( a1 , a2 ) )`] [void] [Calculates ['*y* = a1 *x1* + a2 *x2*]]] ] [endsect] [endsect]