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[/============================================================================
Boost.odeint
Copyright 2011 Mario Mulansky
Copyright 2011-2012 Karsten Ahnert
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 Implicit System]
[heading Description]
This concept describes how to define a ODE that can be solved by an implicit routine.
Implicit routines need not only the function /f(x,t)/ but also the Jacobian /df/dx = A(x,t)/.
/A/ is a matrix and implicit routines need to solve the linear problem /Ax = b/.
In odeint this is implemented with use of __ublas, therefore, the ['state_type] implicit routines is ['ublas::vector] and the matrix is defined as ['ublas::matrix].
[heading Notation]
[variablelist
[[`System`] [A type that is a model of `Implicit System`]]
[[`Time`] [A type representing the time of the ODE]]
[[`sys`] [An object of type `System`]]
[[`x`] [Object of type ublas::vector]]
[[`dxdt`] [Object of type ublas::vector]]
[[`jacobi`] [Object of type ublas::matrix]]
[[`t`] [Object of type `Time`]]
]
[heading Valid Expressions]
[table
[[Name] [Expression] [Type] [Semantics]]
[[Calculate ['dx/dt := f(x,t)]] [`sys.first( x , dxdt , t )`] [`void`] [Calculates `f(x,t)`, the result is stored into dxdt] ]
[[Calculate ['A := df/dx (x,t)]] [`sys.second( x , jacobi , t )`] [`void`] [Calculates the Jacobian of /f/ at /x/,/t/, the result is stored into `jacobi`] ]
]
[endsect]
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