ceph/src/boost/libs/numeric/odeint/doc/concepts/implicit_system.qbk

   1 [/============================================================================
   2   Boost.odeint
   3
   4   Copyright 2011 Mario Mulansky
   5   Copyright 2011-2012 Karsten Ahnert
   6
   7   Use, modification and distribution is subject to the Boost Software License,
   8   Version 1.0. (See accompanying file LICENSE_1_0.txt or copy at
   9   http://www.boost.org/LICENSE_1_0.txt)
  10 =============================================================================/]
  11
  12
  13
  14 [section Implicit System]
  15
  16 [heading Description]
  17
  18 This concept describes how to define a ODE that can be solved by an implicit routine.
  19 Implicit routines need not only the function /f(x,t)/ but also the Jacobian /df/dx = A(x,t)/.
  20 /A/ is a matrix and implicit routines need to solve the linear problem /Ax = b/.
  21 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].
  22
  23 [heading Notation]
  24
  25 [variablelist
  26   [[`System`] [A type that is a model of `Implicit System`]]
  27   [[`Time`] [A type representing the time of the ODE]]
  28   [[`sys`] [An object of type `System`]]
  29   [[`x`] [Object of type ublas::vector]]
  30   [[`dxdt`] [Object of type ublas::vector]]
  31   [[`jacobi`] [Object of type ublas::matrix]]
  32   [[`t`] [Object of type `Time`]]
  33 ]
  34
  35 [heading Valid Expressions]
  36
  37 [table
  38   [[Name] [Expression] [Type] [Semantics]]
  39   [[Calculate ['dx/dt := f(x,t)]] [`sys.first( x , dxdt , t )`] [`void`] [Calculates `f(x,t)`, the result is stored into dxdt] ]
  40   [[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`] ]
  41 ]
  42
  43 [endsect]