diff --git a/doc/boost_sandbox_numeric_odeint/concepts.html b/doc/boost_sandbox_numeric_odeint/concepts.html index 4b916cc1..47bf8e5f 100644 --- a/doc/boost_sandbox_numeric_odeint/concepts.html +++ b/doc/boost_sandbox_numeric_odeint/concepts.html @@ -3,8 +3,8 @@ Concepts - - + + @@ -15,7 +15,7 @@
PrevUpHomeNext
-
+

Concepts

@@ -44,7 +44,8 @@
-
Copyright © 2009 -2011 Karsten Ahnert and Mario Mulansky

+

Copyright © 2009-2011 Karsten Ahnert + and Mario Mulansky

Distributed under 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)

diff --git a/doc/boost_sandbox_numeric_odeint/concepts/controlled_stepper.html b/doc/boost_sandbox_numeric_odeint/concepts/controlled_stepper.html index 104f5f29..37af938a 100644 --- a/doc/boost_sandbox_numeric_odeint/concepts/controlled_stepper.html +++ b/doc/boost_sandbox_numeric_odeint/concepts/controlled_stepper.html @@ -3,8 +3,8 @@ Controlled Stepper - - + + @@ -15,7 +15,7 @@
PrevUpHomeNext
-
+

Controlled Stepper @@ -26,7 +26,7 @@ functions.

- + Description

@@ -37,7 +37,7 @@ of the solution the step might be rejected and a smaller step-size is suggested.

- + Notation
@@ -78,7 +78,7 @@
- + Valid Expressions
@@ -179,30 +179,31 @@
- + Models
-
-
Copyright © 2009 -2011 Karsten Ahnert and Mario Mulansky

+

Copyright © 2009-2011 Karsten Ahnert + and Mario Mulansky

Distributed under 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)

diff --git a/doc/boost_sandbox_numeric_odeint/concepts/dense_output_stepper.html b/doc/boost_sandbox_numeric_odeint/concepts/dense_output_stepper.html index 7ead6367..e0c1f1eb 100644 --- a/doc/boost_sandbox_numeric_odeint/concepts/dense_output_stepper.html +++ b/doc/boost_sandbox_numeric_odeint/concepts/dense_output_stepper.html @@ -3,8 +3,8 @@ Dense Output Stepper - - + + @@ -15,7 +15,7 @@
PrevUpHomeNext
-
+

Dense Output Stepper @@ -26,7 +26,7 @@ functions.

- + Description

@@ -38,7 +38,7 @@ <= t' <= t+dt.

- + Associated types
@@ -56,7 +56,7 @@
- + Notation
@@ -93,7 +93,7 @@
- + Valid Expressions
@@ -285,22 +285,23 @@
- + Models
-
-
Copyright © 2009 -2011 Karsten Ahnert and Mario Mulansky

+

Copyright © 2009-2011 Karsten Ahnert + and Mario Mulansky

Distributed under 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)

diff --git a/doc/boost_sandbox_numeric_odeint/concepts/error_stepper.html b/doc/boost_sandbox_numeric_odeint/concepts/error_stepper.html index 8b8ece9e..df087109 100644 --- a/doc/boost_sandbox_numeric_odeint/concepts/error_stepper.html +++ b/doc/boost_sandbox_numeric_odeint/concepts/error_stepper.html @@ -3,8 +3,8 @@ Error Stepper - - + + @@ -15,7 +15,7 @@
PrevUpHomeNext
-
+

Error Stepper @@ -25,7 +25,7 @@ be used within a ControlledErrorStepper.

- + Description

@@ -41,7 +41,7 @@ System.

- + Notation
@@ -91,7 +91,7 @@
- + Valid Expressions
@@ -188,24 +188,25 @@
- + Models
-
-
Copyright © 2009 -2011 Karsten Ahnert and Mario Mulansky

+

Copyright © 2009-2011 Karsten Ahnert + and Mario Mulansky

Distributed under 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)

diff --git a/doc/boost_sandbox_numeric_odeint/concepts/implicit_system.html b/doc/boost_sandbox_numeric_odeint/concepts/implicit_system.html index 4610c182..c25b7a9d 100644 --- a/doc/boost_sandbox_numeric_odeint/concepts/implicit_system.html +++ b/doc/boost_sandbox_numeric_odeint/concepts/implicit_system.html @@ -3,8 +3,8 @@ Implicit System - - + + @@ -15,26 +15,26 @@
PrevUpHomeNext
-
+
- + 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 /dfdx = A(x,t). A + 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 Boost.UBlas, therefore, the state_type implicit routines is ublas::vector and the matrix is defined as ublas::matrix.

- + Notation
@@ -71,7 +71,7 @@
- + Valid Expressions
@@ -162,7 +162,8 @@
-
Copyright © 2009 -2011 Karsten Ahnert and Mario Mulansky

+

Copyright © 2009-2011 Karsten Ahnert + and Mario Mulansky

Distributed under 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)

diff --git a/doc/boost_sandbox_numeric_odeint/concepts/observer.html b/doc/boost_sandbox_numeric_odeint/concepts/observer.html index 0edb1d42..1a8ba9f4 100644 --- a/doc/boost_sandbox_numeric_odeint/concepts/observer.html +++ b/doc/boost_sandbox_numeric_odeint/concepts/observer.html @@ -3,8 +3,8 @@ Observer - - + + @@ -15,12 +15,12 @@
PrevUpHomeNext
-
+
- + Description

@@ -33,7 +33,7 @@ function is used.

- + Notation
@@ -66,7 +66,7 @@
- + Valid Expressions
@@ -128,7 +128,8 @@
- @@ -284,13 +281,13 @@
Copyright © 2009 -2011 Karsten Ahnert and Mario Mulansky

+

Copyright © 2009-2011 Karsten Ahnert + and Mario Mulansky

Distributed under 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)

diff --git a/doc/boost_sandbox_numeric_odeint/concepts/simple_symplectic_system.html b/doc/boost_sandbox_numeric_odeint/concepts/simple_symplectic_system.html index 70478d4f..1e830be6 100644 --- a/doc/boost_sandbox_numeric_odeint/concepts/simple_symplectic_system.html +++ b/doc/boost_sandbox_numeric_odeint/concepts/simple_symplectic_system.html @@ -3,8 +3,8 @@ Simple Symplectic System - - + + @@ -15,13 +15,13 @@
PrevUpHomeNext
-
+
- + Description

@@ -45,7 +45,7 @@ stepper. We call this concept Simple_Symplectic_System

- + Notation
@@ -78,7 +78,7 @@
- + Valid Expressions
@@ -166,7 +166,8 @@
-
Copyright © 2009 -2011 Karsten Ahnert and Mario Mulansky

+

Copyright © 2009-2011 Karsten Ahnert + and Mario Mulansky

Distributed under 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)

diff --git a/doc/boost_sandbox_numeric_odeint/concepts/state_algebra_operations.html b/doc/boost_sandbox_numeric_odeint/concepts/state_algebra_operations.html index 8548076d..52379952 100644 --- a/doc/boost_sandbox_numeric_odeint/concepts/state_algebra_operations.html +++ b/doc/boost_sandbox_numeric_odeint/concepts/state_algebra_operations.html @@ -3,8 +3,8 @@ State Algebra Operations - - + + @@ -15,7 +15,7 @@
PrevUpHomeNext
-
+

State Algebra Operations @@ -41,7 +41,7 @@

- + Description

@@ -78,12 +78,12 @@ Algebra and Operations are used together within the stepper implementations.

-
+
- + Notation
@@ -111,7 +111,7 @@ Type that represents a general scale_sum operation, N should be replaced by a number from 1 to 14.

-
scale_sumN
+
scale_sum/N/

Object of type ScaleSumN, N should be replaced by a @@ -143,7 +143,7 @@

- + Valid Expressions
@@ -258,21 +258,18 @@

- Operations::scale_sumN< Value1 - , ... , ValueN > + Operations::scale_sum/N/< Value1 , ... , ValueN >

- ScaleSumN + ScaleSum/N/

- Get the ScaleSumN type - from Operations, - N should be replaced - by a number from 1 to 14. + Get the ScaleSum/N/ type from Operations, N + should be replaced by a number from 1 to 14.

- ScaleSumN< Value1 , ... , ValueN - >( a1 , ... , aN ) + ScaleSum/N/< Value1 , ... , ValueN >( a1 , ... + , aN )

- ScaleSumN + ScaleSum/N/

@@ -310,7 +307,7 @@

- scale_sumN( y , x1 , ... , xN ) + scale_sum/N/( y , x1 , ... , xN )

@@ -413,12 +410,12 @@
-
+
- + Notation
@@ -432,7 +429,7 @@

The algebra type

-
OperationN
+
Operation/N/

An N-ary operation type, N should be a number from @@ -442,7 +439,7 @@

Object of type Algebra

-
operationN
+
operation/N/

Object of type Operation/N

@@ -456,7 +453,7 @@
- + Valid Expressions
@@ -557,8 +554,8 @@

- algebra.for_eachN( y , x1 , ... - , xN , operationN ) + algebra.for_each/N/( y , x1 , ... , xN , operation/N/ + )

@@ -568,8 +565,8 @@

- Calls operationN( y_i , x1_i , - ... , xN_i ) for each element y_i + 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. @@ -579,7 +576,7 @@

-
+

Pre-Defined implementations @@ -746,7 +743,7 @@

-
+

Example expressions @@ -817,7 +814,8 @@

-
Copyright © 2009 -2011 Karsten Ahnert and Mario Mulansky

+

Copyright © 2009-2011 Karsten Ahnert + and Mario Mulansky

Distributed under 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)

diff --git a/doc/boost_sandbox_numeric_odeint/concepts/state_wrapper.html b/doc/boost_sandbox_numeric_odeint/concepts/state_wrapper.html index af174459..9733f965 100644 --- a/doc/boost_sandbox_numeric_odeint/concepts/state_wrapper.html +++ b/doc/boost_sandbox_numeric_odeint/concepts/state_wrapper.html @@ -3,8 +3,8 @@ State Wrapper - - + + @@ -15,13 +15,13 @@
PrevUpHomeNext
-
+
- + Description

@@ -31,7 +31,7 @@ methods.

- + Notation
@@ -57,7 +57,7 @@
- + Valid Expressions
@@ -267,7 +267,8 @@
-
Copyright © 2009 -2011 Karsten Ahnert and Mario Mulansky

+

Copyright © 2009-2011 Karsten Ahnert + and Mario Mulansky

Distributed under 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)

diff --git a/doc/boost_sandbox_numeric_odeint/concepts/stepper.html b/doc/boost_sandbox_numeric_odeint/concepts/stepper.html index cb333d85..3279fc2b 100644 --- a/doc/boost_sandbox_numeric_odeint/concepts/stepper.html +++ b/doc/boost_sandbox_numeric_odeint/concepts/stepper.html @@ -3,8 +3,8 @@ Stepper - - + + @@ -15,7 +15,7 @@
PrevUpHomeNext
-
+
@@ -25,7 +25,7 @@ functions.

- + Description

@@ -41,20 +41,20 @@ System. Note that all error steppers are also basic steppers.

- + Refinement of
-
    -
  • +
      +
    • DefaultConstructable
      • -
    • +
    • CopyConstructable
    - + Notation
    @@ -96,7 +96,7 @@
    - + Valid Expressions
    @@ -188,30 +188,31 @@
- + Models
-
    -
  • +
      +
    • runge_kutta4
      • -
    • +
    • euler
      • -
    • +
    • runge_kutta_cash_karp54
      • -
    • +
    • runge_kutta_dopri5
      • -
    • +
    • runge_kutta_fehlberg78
-
Copyright © 2009 -2011 Karsten Ahnert and Mario Mulansky

+

Copyright © 2009-2011 Karsten Ahnert + and Mario Mulansky

Distributed under 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)

diff --git a/doc/boost_sandbox_numeric_odeint/concepts/symplectic_system.html b/doc/boost_sandbox_numeric_odeint/concepts/symplectic_system.html index 7ee61a09..1f8727b9 100644 --- a/doc/boost_sandbox_numeric_odeint/concepts/symplectic_system.html +++ b/doc/boost_sandbox_numeric_odeint/concepts/symplectic_system.html @@ -3,8 +3,8 @@ Symplectic System - - + + @@ -15,13 +15,13 @@
PrevUpHomeNext
-
+
- + Description

@@ -59,7 +59,7 @@ steppers like symplectic_rkn_sb3a_mclachlan.

- + Notation
@@ -108,7 +108,7 @@
- + Valid expressions
@@ -220,7 +220,8 @@
-
Copyright © 2009 -2011 Karsten Ahnert and Mario Mulansky

+

Copyright © 2009-2011 Karsten Ahnert + and Mario Mulansky

Distributed under 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)

diff --git a/doc/boost_sandbox_numeric_odeint/concepts/system.html b/doc/boost_sandbox_numeric_odeint/concepts/system.html index 0f6478b7..8c66523e 100644 --- a/doc/boost_sandbox_numeric_odeint/concepts/system.html +++ b/doc/boost_sandbox_numeric_odeint/concepts/system.html @@ -3,8 +3,8 @@ System - - + + @@ -15,12 +15,12 @@
PrevUpHomeNext
-
+
- + Description

@@ -35,7 +35,7 @@ System.

- + Notation
@@ -76,7 +76,7 @@
- + Valid expressions
@@ -138,7 +138,8 @@
-
Copyright © 2009 -2011 Karsten Ahnert and Mario Mulansky

+

Copyright © 2009-2011 Karsten Ahnert + and Mario Mulansky

Distributed under 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)

diff --git a/doc/boost_sandbox_numeric_odeint/extend_odeint.html b/doc/boost_sandbox_numeric_odeint/extend_odeint.html index 561f0252..2cc175a4 100644 --- a/doc/boost_sandbox_numeric_odeint/extend_odeint.html +++ b/doc/boost_sandbox_numeric_odeint/extend_odeint.html @@ -3,8 +3,8 @@ Extend odeint - - + + @@ -15,7 +15,7 @@
PrevUpHomeNext
-
+
@@ -33,7 +33,8 @@
-
Copyright © 2009 -2011 Karsten Ahnert and Mario Mulansky

+

Copyright © 2009-2011 Karsten Ahnert + and Mario Mulansky

Distributed under 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)

diff --git a/doc/boost_sandbox_numeric_odeint/extend_odeint/adapt_your_own_operations.html b/doc/boost_sandbox_numeric_odeint/extend_odeint/adapt_your_own_operations.html index 36e2bd8a..c76bbb5f 100644 --- a/doc/boost_sandbox_numeric_odeint/extend_odeint/adapt_your_own_operations.html +++ b/doc/boost_sandbox_numeric_odeint/extend_odeint/adapt_your_own_operations.html @@ -3,8 +3,8 @@ Adapt your own operations - - + + @@ -15,7 +15,7 @@
PrevUpHomeNext
-
+

Adapt your own operations @@ -29,7 +29,8 @@

-
Copyright © 2009 -2011 Karsten Ahnert and Mario Mulansky

+

Copyright © 2009-2011 Karsten Ahnert + and Mario Mulansky

Distributed under 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)

diff --git a/doc/boost_sandbox_numeric_odeint/extend_odeint/adapt_your_own_state_types.html b/doc/boost_sandbox_numeric_odeint/extend_odeint/adapt_your_own_state_types.html index 0df5f409..62e2ad3c 100644 --- a/doc/boost_sandbox_numeric_odeint/extend_odeint/adapt_your_own_state_types.html +++ b/doc/boost_sandbox_numeric_odeint/extend_odeint/adapt_your_own_state_types.html @@ -3,8 +3,8 @@ Adapt your own state types - - + + @@ -15,7 +15,7 @@
PrevUpHomeNext
-
+

Adapt your own state types @@ -44,14 +44,14 @@ are to be performed on state types. So when introducing new state types to odeint, the following points have to be considered:

-
    -
  • +
      +
    • construction/destruction
      • -
    • +
    • resizing (if possible/required)
      • -
    • +
    • algebraic operations
    @@ -61,7 +61,7 @@ or a boost::array as state type no additional work is required, they just work out of the box.

    -
    +
    @@ -366,7 +366,7 @@

-
+
Vectors derived from std::vector @@ -394,11 +394,11 @@ : std::vector< double >() { } -// ... -}; +// ... +}; + +// define my_vector as reizeable -// define my_vector as reizeable - namespace boost { namespace numeric { namespace odeint { template<> @@ -420,7 +420,7 @@ can be found in my_vector.cpp

-
+
@@ -523,15 +523,15 @@ template< > struct is_resizeable< state_type > -{ // declare resizeablility - typedef boost::true_type type; +{ // declare resizeablility + typedef boost::true_type type; const static bool value = type::value; }; template< > struct same_size_impl< state_type , state_type > -{ // define how to check size - static bool same_size( const state_type &v1 , +{ // define how to check size + static bool same_size( const state_type &v1 , const state_type &v2 ) { return v1.size() == v2.size(); @@ -540,8 +540,8 @@ template< > struct resize_impl< state_type , state_type > -{ // define how to resize - static void resize( state_type &v1 , +{ // define how to resize + static void resize( state_type &v1 , const state_type &v2 ) { v1.resize( v2.size() ); @@ -559,212 +559,39 @@

-
+
+
GSL + Vector
Vector Space Algebra
Boost.Ublas
-
GSL - Vector
+
Point + type

If your state type is not supported by Boost.Range there are several possibilities to tell odeint how to do algebraic operations:

-
    -
  • +
      +
    • Implement boost::begin and boost::end for your state type so it works with Boost.Range.
      • -
    • +
    • Implement vector-vector addition operator + and scalar-vector multiplication operator * and use the non-standard vector_space_algebra.
      • -
    • +
    • Implement your own algebra that defines the required operations.
    -
    -
    -

    - The standard way of performing algebraic operations on container-like - state types in odeint is to iterate through the elements of the container - and perform the operations element-wise on the underlying value type. - This is realized by means of the range_algebra - that uses Boost.Range - for obtaining iterators of the state types. However, there are other - ways to implement the algebraic operations on containers, one of which - is defining the addition/multiplication operators for the containers - directly and then using the vector_space_algebra. - If you use this algebra, the following operators have to be defined for - the state_type: -

    -
    ------ - - - - - - - - - - - - - - - - - - - - - - - - - - -
    -

    - Name -

    -     		-

    - Expression -

    -     		-

    - Type -

    -     		-

    - Semantics -

    -
    -

    - Addition -

    -     		-

    - x + - y -

    -     		-

    - state_type -

    -     		-

    - Calculates the vector sum 'x+y'. -

    -
    -

    - Assign addition -

    -     		-

    - x += - y -

    -     		-

    - state_type -

    -     		-

    - Performs x+y in place. -

    -
    -

    - Scalar multiplication -

    -     		-

    - a * - x -

    -     		-

    - state_type -

    -     		-

    - Performs multiplication of vector x with scalar a. -

    -
    -
    -
    -
    -

    - As an example for the employment of the vector_space_algebra - we will adopt ublas::vector from Boost.UBlas - to work as a state type in odeint. This is particularily easy because - ublas::vector supports vector-vector addition - and scalar-vector multiplication as well as boost::size. - It also has a resize member function so all that has to be done in this - case is to declare resizability: -

    -

    - -

    -
    typedef boost::numeric::ublas::vector< double > state_type;
-
-namespace boost { namespace numeric { namespace odeint {
-
-template<>
-struct is_resizeable< state_type >
-{
-    typedef boost::true_type type;
-    const static bool value = type::value;
-};
-
-} } }
-
    -

    -

    -

    - Now ublas::vector can be used as state type in odeint by specifying the - vector_space_algebra - as algebra in the template parameter list of the stepper. The following - code shows the corresponding definitions: -

    -

    - -

    -
    int main()
-{
-    state_type x(3);
-    x[0] = 10.0; x[1] = 5.0 ; x[2] = 0.0;
-    typedef runge_kutta4< state_type , double , state_type , double , vector_space_algebra > stepper;
-    integrate_const( stepper() , lorenz , x , 
-                     0.0 , 10.0 , 0.1 );
-}
-
    -

    -

    -

    - You can find the full example in lorenz_ublas.cpp. -

    -
    -
    +
    GSL Vector @@ -912,14 +739,14 @@ Finally, the bindings for Boost.Range are added:

    -
    // template<>
-inline gsl_vector_iterator range_begin( gsl_vector *x )
+
    // template<>
+inline gsl_vector_iterator range_begin( gsl_vector *x )
 {
     return gsl_vector_iterator( x );
 }
 
-// template<>
-inline gsl_vector_iterator range_end( gsl_vector *x )
+// template<>
+inline gsl_vector_iterator range_end( gsl_vector *x )
 {
     return end_iterator( x );
 }
@@ -933,11 +760,355 @@
             working at all in the first place.
    +
    +
    +

    + As seen above, the standard way of performing algebraic operations on + container-like state types in odeint is to iterate through the elements + of the container and perform the operations element-wise on the underlying + value type. This is realized by means of the range_algebra + that uses Boost.Range + for obtaining iterators of the state types. However, there are other + ways to implement the algebraic operations on containers, one of which + is defining the addition/multiplication operators for the containers + directly and then using the vector_space_algebra. + If you use this algebra, the following operators have to be defined for + the state_type: +

    +
    ++++++ + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + +
    +

    + Name +

    +     		+

    + Expression +

    +     		+

    + Type +

    +     		+

    + Semantics +

    +
    +

    + Addition +

    +     		+

    + x + + y +

    +     		+

    + state_type +

    +     		+

    + Calculates the vector sum 'x+y'. +

    +
    +

    + Assign addition +

    +     		+

    + x += + y +

    +     		+

    + state_type +

    +     		+

    + Performs x+y in place. +

    +
    +

    + Scalar multiplication +

    +     		+

    + a * + x +

    +     		+

    + state_type +

    +     		+

    + Performs multiplication of vector x with scalar a. +

    +
    +

    + Assign scalar multiplication +

    +     		+

    + x *= + a +

    +     		+

    + state_type +

    +     		+

    + Performs in-place multiplication of vector x with scalar a. +

    +
    +
    +
    +
    +

    + As an example for the employment of the vector_space_algebra + we will adopt ublas::vector from Boost.UBlas + to work as a state type in odeint. This is particularily easy because + ublas::vector supports vector-vector addition + and scalar-vector multiplication described above as well as boost::size. It also has a resize member function + so all that has to be done in this case is to declare resizability: +

    +

    + +

    +
    typedef boost::numeric::ublas::vector< double > state_type;
+
+namespace boost { namespace numeric { namespace odeint {
+
+template<>
+struct is_resizeable< state_type >
+{
+    typedef boost::true_type type;
+    const static bool value = type::value;
+};
+
+} } }
+
    +

    +

    +

    + Now ublas::vector can be used as state type in odeint by specifying the + vector_space_algebra + as algebra in the template parameter list of the stepper. The following + code shows the corresponding definitions: +

    +

    + +

    +
    int main()
+{
+    state_type x(3);
+    x[0] = 10.0; x[1] = 5.0 ; x[2] = 0.0;
+    typedef runge_kutta4< state_type , double , state_type , double , vector_space_algebra > stepper;
+    integrate_const( stepper() , lorenz , x , 
+                     0.0 , 10.0 , 0.1 );
+}
+
    +

    +

    +

    + You can find the full example in lorenz_ublas.cpp. +

    +
    +
    +
    +

    + Here we show how to implement the required operators on a state type. + As example we define a new class point3D + representing a three-dimensional vector with components x,y,z and define + addition and scalar multiplication operators for it. We use Boost.Operators + to reduce the amount of code required. The class for the point type looks + as follows: +

    +

    + +

    +
    class point3D :     
+    boost::additive1< point3D ,
+    boost::additive2< point3D , double ,
+    boost::multiplicative2< point3D , double > > >
+{
+public:
+
+    double x , y , z;
+
+    point3D()
+        : x( 0.0 ) , y( 0.0 ) , z( 0.0 )
+    { }
+    
+    point3D( const double val )
+        : x( val ) , y( val ) , z( val )
+    { }
+
+    point3D( const double _x , const double _y , const double _z  )
+        : x( x ) , y( y ) , z( z )
+    { }
+
+    point3D& operator+=( const point3D &p )
+    {
+        x += p.x; y += p.y; z += p.z;
+        return *this;
+    }
+
+    point3D& operator*=( const double a )
+    {
+        x *= a; y *= a; z *= a;
+        return *this;
+    }
+
+};
+
    +

    +

    +

    + By deriving from Boost.Operators + classes we don't have to define outer class operators like operator+( point3D , + point3D ) + because that is taken care of by the operators library. Note that for + simple Runge-Kutta schemes (like runge_kutta4) + only the + and * operators are required. If, however, + a controlled stepper is used one also needs to specify the division operator + / because calculation of + the error term involves an element wise division of the state types. + Additionally, controlled steppers require an abs + function calculating the element-wise absolute value for the state type: +

    +

    + +

    +
    // only required for steppers with error control
+point3D operator/( const point3D &p1 , const point3D &p2 )
+{
+    return point3D( p1.x/p2.x , p1.y/p2.y , p1.z/p1.z );
+}
+
+point3D abs( const point3D &p )
+{
+    return point3D( abs(p.x) , abs(p.y) , abs(p.z) );
+}
+
    +

    +

    +

    + Finally, we have to add a specialization for reduce + implementing a reduction over the state type: +

    +

    + +

    +
    namespace boost { namespace numeric { namespace odeint {
+// specialization of vector_space_reduce, only required for steppers with error control
+template<>
+struct vector_space_reduce< point3D >
+{
+    template< class Value , class Op >
+    Value operator() ( const point3D &p , Op op , Value init )
+    {
+        init = op( init , p.x );
+        init = op( init , p.y );
+        init = op( init , p.z );
+        return init;
+    }
+};
+} } } 
+
    +

    +

    +

    + Again, note that the two last steps were only required if you want to + use controlled steppers. For simple steppers definition of the simple + += and *= + operators are sufficient. Having defined such a point type, we can easily + perform the integration on a lorenz system by using the vector_space_algebra again: +

    +

    + +

    +
    const double sigma = 10.0;
+const double R = 28.0;
+const double b = 8.0 / 3.0;
+
+void lorenz( const point3D &x , point3D &dxdt , const double t )
+{
+    dxdt.x = sigma * ( x.y - x.x );
+    dxdt.y = R * x.x - x.y - x.x * x.z;
+    dxdt.z = -b * x.z + x.x * x.y;
+}
+
+using namespace boost::numeric::odeint;
+
+int main()
+{
+
+    point3D x( 10.0 , 5.0 , 5.0 );
+    //typedef runge_kutta4< point3D , double , point3D , 
+    //                      double , vector_space_algebra > stepper;
+    typedef runge_kutta_dopri5< point3D , double , point3D , 
+                                double , vector_space_algebra > stepper;
+    integrate_adaptive( make_controlled<stepper>( 1E-8 , 1E-8 ) , lorenz , x , 
+                        0.0 , 10.0 , 0.1 );
+    std::cout << x << std::endl;
+}
+
    +

    +

    +
    -
    Copyright © 2009 -2011 Karsten Ahnert and Mario Mulansky

    +

    Copyright © 2009-2011 Karsten Ahnert + and Mario Mulansky

    Distributed under 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)

    diff --git a/doc/boost_sandbox_numeric_odeint/extend_odeint/write_own_steppers.html b/doc/boost_sandbox_numeric_odeint/extend_odeint/write_own_steppers.html index b2fec6f7..d025a5b3 100644 --- a/doc/boost_sandbox_numeric_odeint/extend_odeint/write_own_steppers.html +++ b/doc/boost_sandbox_numeric_odeint/extend_odeint/write_own_steppers.html @@ -3,8 +3,8 @@ Write own steppers - - + + @@ -15,13 +15,14 @@
    PrevUpHomeNext
    -

    +
    -
    Copyright © 2009 -2011 Karsten Ahnert and Mario Mulansky

    +

    Copyright © 2009-2011 Karsten Ahnert + and Mario Mulansky

    Distributed under 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)

    diff --git a/doc/boost_sandbox_numeric_odeint/getting_started.html b/doc/boost_sandbox_numeric_odeint/getting_started.html index a92d20fa..4bb7bb1b 100644 --- a/doc/boost_sandbox_numeric_odeint/getting_started.html +++ b/doc/boost_sandbox_numeric_odeint/getting_started.html @@ -3,8 +3,8 @@ Getting started - - + + @@ -15,7 +15,7 @@
    PrevUpHomeNext
    -
    +
    @@ -29,7 +29,8 @@
    -
    Copyright © 2009 -2011 Karsten Ahnert and Mario Mulansky

    +

    Copyright © 2009-2011 Karsten Ahnert + and Mario Mulansky

    Distributed under 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)

    diff --git a/doc/boost_sandbox_numeric_odeint/getting_started/overview.html b/doc/boost_sandbox_numeric_odeint/getting_started/overview.html index 1f1578e5..963854c8 100644 --- a/doc/boost_sandbox_numeric_odeint/getting_started/overview.html +++ b/doc/boost_sandbox_numeric_odeint/getting_started/overview.html @@ -3,8 +3,8 @@ Overview - - + + @@ -15,7 +15,7 @@
    PrevUpHomeNext
    -
    +
    @@ -57,33 +57,33 @@ or a boost::array, we provide native support for the following libraries:

    -
      -
    • +
      -
      Copyright © 2009 -2011 Karsten Ahnert and Mario Mulansky

      +

      Copyright © 2009-2011 Karsten Ahnert + and Mario Mulansky

      Distributed under 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)

      diff --git a/doc/boost_sandbox_numeric_odeint/getting_started/short_example.html b/doc/boost_sandbox_numeric_odeint/getting_started/short_example.html index b79da97e..2228edea 100644 --- a/doc/boost_sandbox_numeric_odeint/getting_started/short_example.html +++ b/doc/boost_sandbox_numeric_odeint/getting_started/short_example.html @@ -3,8 +3,8 @@ Short Example - - + + @@ -15,7 +15,7 @@
      PrevUpHomeNext
      -
      +

      Short Example @@ -63,8 +63,8 @@ 

      state_type x(2);
-x[0] = 1.0; // start at x=1.0, p=0.0
-x[1] = 0.0;
+x[0] = 1.0; // start at x=1.0, p=0.0
+x[1] = 0.0;

      @@ -186,7 +186,8 @@

      -
      Copyright © 2009 -2011 Karsten Ahnert and Mario Mulansky

      +

      Copyright © 2009-2011 Karsten Ahnert + and Mario Mulansky

      Distributed under 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)

      diff --git a/doc/boost_sandbox_numeric_odeint/getting_started/usage__compilation__headers.html b/doc/boost_sandbox_numeric_odeint/getting_started/usage__compilation__headers.html index 3cf5f598..04b5f298 100644 --- a/doc/boost_sandbox_numeric_odeint/getting_started/usage__compilation__headers.html +++ b/doc/boost_sandbox_numeric_odeint/getting_started/usage__compilation__headers.html @@ -3,8 +3,8 @@ Usage, Compilation, Headers - - + + @@ -15,7 +15,7 @@
      PrevUpHomeNext
      -
      +

      Usage, Compilation, Headers @@ -40,7 +40,8 @@

      -
      Copyright © 2009 -2011 Karsten Ahnert and Mario Mulansky

      +

      Copyright © 2009-2011 Karsten Ahnert + and Mario Mulansky

      Distributed under 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)

      diff --git a/doc/boost_sandbox_numeric_odeint/odeint_in_detail.html b/doc/boost_sandbox_numeric_odeint/odeint_in_detail.html index 4d673c88..0aa2a1b4 100644 --- a/doc/boost_sandbox_numeric_odeint/odeint_in_detail.html +++ b/doc/boost_sandbox_numeric_odeint/odeint_in_detail.html @@ -3,8 +3,8 @@ Odeint in detail - - + + @@ -15,7 +15,7 @@
      PrevUpHomeNext
      -
      +

      Odeint in detail @@ -36,7 +36,8 @@

      -
      Copyright © 2009 -2011 Karsten Ahnert and Mario Mulansky

      +

      Copyright © 2009-2011 Karsten Ahnert + and Mario Mulansky

      Distributed under 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)

      diff --git a/doc/boost_sandbox_numeric_odeint/odeint_in_detail/algebras_and_operations.html b/doc/boost_sandbox_numeric_odeint/odeint_in_detail/algebras_and_operations.html index 4014b28e..03255a30 100644 --- a/doc/boost_sandbox_numeric_odeint/odeint_in_detail/algebras_and_operations.html +++ b/doc/boost_sandbox_numeric_odeint/odeint_in_detail/algebras_and_operations.html @@ -3,8 +3,8 @@ Algebras and operations - - + + @@ -15,7 +15,7 @@
      PrevUpHomeNext
      -
      +

      Algebras and operations @@ -26,7 +26,8 @@

      -
      Copyright © 2009 -2011 Karsten Ahnert and Mario Mulansky

      +

      Copyright © 2009-2011 Karsten Ahnert + and Mario Mulansky

      Distributed under 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)

      diff --git a/doc/boost_sandbox_numeric_odeint/odeint_in_detail/generation_functions.html b/doc/boost_sandbox_numeric_odeint/odeint_in_detail/generation_functions.html index 7be49e9f..ff172bd6 100644 --- a/doc/boost_sandbox_numeric_odeint/odeint_in_detail/generation_functions.html +++ b/doc/boost_sandbox_numeric_odeint/odeint_in_detail/generation_functions.html @@ -3,8 +3,8 @@ Generation functions - - + + @@ -15,7 +15,7 @@
      PrevUpHomeNext
      -
      +

      Generation functions @@ -195,7 +195,8 @@

      -
      Copyright © 2009 -2011 Karsten Ahnert and Mario Mulansky

      +

      Copyright © 2009-2011 Karsten Ahnert + and Mario Mulansky

      Distributed under 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)

      diff --git a/doc/boost_sandbox_numeric_odeint/odeint_in_detail/integrate_functions.html b/doc/boost_sandbox_numeric_odeint/odeint_in_detail/integrate_functions.html index 3161ac2d..70fe3b29 100644 --- a/doc/boost_sandbox_numeric_odeint/odeint_in_detail/integrate_functions.html +++ b/doc/boost_sandbox_numeric_odeint/odeint_in_detail/integrate_functions.html @@ -3,8 +3,8 @@ Integrate functions - - + + @@ -15,7 +15,7 @@
      PrevUpHomeNext
      -
      +

      Integrate functions @@ -36,7 +36,7 @@ functions make use of step-size control or dense output.

      - + Equidistant observer calls
      @@ -81,15 +81,15 @@ steps you should prefer the integrate_n_steps function below.

      -
        -
      • +
          +
        • If stepper is a Stepper or Error Stepper then dt is also the step size used for integration and the observer is called just after every step.
          • -
        • +
        • If stepper is a Controlled Stepper then dt is the initial step size. The actual step size will change due to error @@ -100,7 +100,7 @@ Stepper is reasonable here only if dt is considerably larger than typical step sizes used by the stepper.
          • -
        • +
        • If stepper is a __dense_out_stepper then dt is the initial step size. The actual step size will be adjusted during integration due @@ -109,7 +109,7 @@
        - + Observer calls at each step
        @@ -149,8 +149,8 @@ step of the provided stepper (and before the first step at t0). integrate_adaptive returns the number of steps performed during the integration.

        -
          -
        • +
            +
          • If stepper is a Stepper or Error Stepper then dt @@ -160,7 +160,7 @@ exactly at t1. If provided, the observer is called at each step.
            • -
          • +
          • If stepper is a Controlled Stepper then dt is the initial step size. The actual step size is changed according to @@ -169,7 +169,7 @@ If provided, the observer is called after each time step (and before the first step at t0).
            • -
          • +
          • If stepper is a __dense_out_stepper then dt is the initial step size and integrate_adaptive behaves just like for Controlled @@ -177,7 +177,7 @@
          - + Observer calls at given time points
          @@ -225,8 +225,8 @@ integrate_times returns the number of steps performed during the integration.

          -
            -
          • +
              +
            • If stepper is a Stepper or Error Stepper dt is @@ -235,7 +235,7 @@ will be reduced to obtain the state x(t) exactly at the time point.
              • -
            • +
            • If stepper is a Controlled Stepper then dt is the initial step size. The actual step size is adjusted during integration @@ -243,7 +243,7 @@ is approached the step size is reduced to obtain the state x(t) exactly at the time point.
              • -
            • +
            • If stepper is a __dense_out_stepper then dt is the initial step size. The actual step size is adjusted during integration according @@ -252,7 +252,7 @@
            - + Integrate a given number of steps
            @@ -309,7 +309,7 @@ + n*dt.

            - + Convenience integrate function
            @@ -344,7 +344,8 @@
          -
          Copyright © 2009 -2011 Karsten Ahnert and Mario Mulansky

          +

          Copyright © 2009-2011 Karsten Ahnert + and Mario Mulansky

          Distributed under 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)

          diff --git a/doc/boost_sandbox_numeric_odeint/odeint_in_detail/steppers.html b/doc/boost_sandbox_numeric_odeint/odeint_in_detail/steppers.html index 1d024cc7..8ea7b8d5 100644 --- a/doc/boost_sandbox_numeric_odeint/odeint_in_detail/steppers.html +++ b/doc/boost_sandbox_numeric_odeint/odeint_in_detail/steppers.html @@ -3,8 +3,8 @@ Steppers - - + + @@ -15,7 +15,7 @@
          PrevUpHomeNext
          -
          +
          @@ -94,7 +94,7 @@ 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.

          -
          +

          Explicit steppers @@ -133,12 +133,12 @@ 

          runge_kutta4< state_type > rk;
-rk.do_step( sys1 , inout , t , dt );               // In-place transformation of inout
-rk.do_step( sys2 , inout , t , dt );               // Ok
-rk.do_step( sys1 , in , t , out , dt );            // Out-of-place transformation
-rk.do_step( sys1 , inout , dxdtin , t , dt );      // In-place tranformation of inout
-rk.do_step( sys1 , in , dxdtin , t , out , dt );   // Out-of-place transformation
-
          +rk.do_step( sys1 , inout , t , dt ); // In-place transformation of inout +rk.do_step( sys2 , inout , t , dt ); // Ok +rk.do_step( sys1 , in , t , out , dt ); // Out-of-place transformation +rk.do_step( sys1 , inout , dxdtin , t , dt ); // In-place tranformation of inout +rk.do_step( sys1 , in , dxdtin , t , out , dt ); // Out-of-place transformation +

          @@ -179,17 +179,17 @@ 

          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 , 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( 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( 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
-
          +rk.do_step( sys2 , in , dxdtin , t , out , dxdtout , dt ); // Ok, internal derivative is not used +

          @@ -209,7 +209,7 @@

          -
          +

          Symplectic solvers @@ -348,7 +348,7 @@

          -
          +

          Implicit solvers @@ -383,7 +383,7 @@ by ublas::matrix.

          -
          +

          Multistep methods @@ -450,7 +450,7 @@

      -
      +

      Controlled steppers @@ -460,15 +460,15 @@ stepsize control. Adaptive step size integration works in principle as follows:

      -
        -
      1. +
          +
        1. The error of one step is calculated. This is usually done by performing two steps with different orders. The difference between these two steps is then used as a measure for the error. Stepper which can calculate the error are Error Stepper and they form a own class with an separate concept.
          2. -
        2. +
        3. This error is compared against some predefined error tolerances. Are the tolerance violated the step is reject and the stepsize is decreases. Otherwise the step is accepted and possibly the stepsize is increased. @@ -668,7 +668,7 @@ stepper. The generation functions are explained in detail in XYZ.

        -
        +

        Dense output steppers @@ -737,7 +737,7 @@

        -
        +

        Using steppers @@ -799,12 +799,12 @@ 

        adams_bashforth_moulton< 4 , state_type > stepper;
-stepper.do_step( sys , x , t , dt );   // make one step with the classical Runge-Kutta stepper and initialize the first internal state at time t-4dt
-stepper.do_step( sys , x , t , dt );   // make one step with the classical Runge-Kutta stepper and initialize the first internal state at time t-3dt
-stepper.do_step( sys , x , t , dt );   // make one step with the classical Runge-Kutta stepper and initialize the first internal state at time t-2dt
-stepper.do_step( sys , x , t , dt );   // make one step with the classical Runge-Kutta stepper and initialize the first internal state at time t-dt
-stepper.do_step( sys , x , t , dt );   // make one step with Adam-Bashforth-Moulton, the internal array of states is now rotated
-
        +stepper.do_step( sys , x , t , dt ); // make one step with the classical Runge-Kutta stepper and initialize the first internal state at time t-4dt +stepper.do_step( sys , x , t , dt ); // make one step with the classical Runge-Kutta stepper and initialize the first internal state at time t-3dt +stepper.do_step( sys , x , t , dt ); // make one step with the classical Runge-Kutta stepper and initialize the first internal state at time t-2dt +stepper.do_step( sys , x , t , dt ); // make one step with the classical Runge-Kutta stepper and initialize the first internal state at time t-dt +stepper.do_step( sys , x , t , dt ); // make one step with Adam-Bashforth-Moulton, the internal array of states is now rotated +

        @@ -826,26 +826,26 @@ user is left with the question of which stepper fits his needs. In the following we give our (personal) recommendations

        -
          -
        • +
            +
          • runge_kutta_dopri5 is maybe the best default stepper. It has step size control as well as dense-output functionality. Simple create a dense-output stepper by make_dense_output( 1.0e-6 , 1.0e-5 , 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 artifical time series this stepper should be the first choice.
            • -
          • +
          • 'runge_kutta_fehlberg78' is similar to the 'runge_kutta4' with the advantage that it has higher precision. It can also be used with step size control.
            • -
          • +
          • 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 @@ -853,7 +853,7 @@
        -
        +

        Stepper overview @@ -1857,7 +1857,8 @@

        -
        Copyright © 2009 -2011 Karsten Ahnert and Mario Mulansky

        +

        Copyright © 2009-2011 Karsten Ahnert + and Mario Mulansky

        Distributed under 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)

        diff --git a/doc/boost_sandbox_numeric_odeint/odeint_in_detail/using_boost__range.html b/doc/boost_sandbox_numeric_odeint/odeint_in_detail/using_boost__range.html index 657ea5af..871f3c9e 100644 --- a/doc/boost_sandbox_numeric_odeint/odeint_in_detail/using_boost__range.html +++ b/doc/boost_sandbox_numeric_odeint/odeint_in_detail/using_boost__range.html @@ -3,8 +3,8 @@ Using boost::range - - + + @@ -15,13 +15,14 @@
        PrevUpHomeNext
        -

        +
        -
        Copyright © 2009 -2011 Karsten Ahnert and Mario Mulansky

        +

        Copyright © 2009-2011 Karsten Ahnert + and Mario Mulansky

        Distributed under 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)

        diff --git a/doc/boost_sandbox_numeric_odeint/odeint_in_detail/using_boost__ref.html b/doc/boost_sandbox_numeric_odeint/odeint_in_detail/using_boost__ref.html index e93491f2..c358536c 100644 --- a/doc/boost_sandbox_numeric_odeint/odeint_in_detail/using_boost__ref.html +++ b/doc/boost_sandbox_numeric_odeint/odeint_in_detail/using_boost__ref.html @@ -3,8 +3,8 @@ Using boost::ref - - + + @@ -15,13 +15,14 @@
        PrevUpHomeNext
        -

        +
        -
        Copyright © 2009 -2011 Karsten Ahnert and Mario Mulansky

        +

        Copyright © 2009-2011 Karsten Ahnert + and Mario Mulansky

        Distributed under 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)

        diff --git a/doc/boost_sandbox_numeric_odeint/old_concepts.html b/doc/boost_sandbox_numeric_odeint/old_concepts.html index 0e3192e9..14ce2958 100644 --- a/doc/boost_sandbox_numeric_odeint/old_concepts.html +++ b/doc/boost_sandbox_numeric_odeint/old_concepts.html @@ -3,8 +3,8 @@ Old Concepts - - + + @@ -15,7 +15,7 @@
        PrevUpHomeNext
        -
        +
        @@ -41,7 +41,8 @@
        -
        Copyright © 2009 -2011 Karsten Ahnert and Mario Mulansky

        +

        Copyright © 2009-2011 Karsten Ahnert + and Mario Mulansky

        Distributed under 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)

        diff --git a/doc/boost_sandbox_numeric_odeint/old_concepts/basic_stepper.html b/doc/boost_sandbox_numeric_odeint/old_concepts/basic_stepper.html index 8fb795ec..d169bea7 100644 --- a/doc/boost_sandbox_numeric_odeint/old_concepts/basic_stepper.html +++ b/doc/boost_sandbox_numeric_odeint/old_concepts/basic_stepper.html @@ -3,8 +3,8 @@ Basic stepper - - + + @@ -15,7 +15,7 @@
        PrevUpHomeNext
        -
        +

        Basic stepper @@ -123,18 +123,18 @@

        Methods

        -
          -
        • +
            +
          • Stepper() Constructor.
            • -
          • +
          • Stepper( container_type &x ) Constructor that allocates internal memory to store intermediate results of the same size as x.
            • -
          • +
          • void do_step( DynamicalSystem &system , container_type @@ -246,7 +246,7 @@ The result of this method is the (approximate) state of the system x(t+dt) and is stored in the variable x (in-place). Note, that the time t is not automatically increased by this method.

            -
            • +
              • void do_step( DynamicalSystem &system , container_type @@ -261,15 +261,15 @@ that represents the derivative x'(t) = f(x,t) at the time t.

                -
                  -
                • +
                    +
                  • void adjust_size( const container_type &x ) Adjusts the internal memory to store intermediate results of the same size as x. This function must be called whenever the system size changes during the integration.
                    • -
                  • +
                  • order_type order_step() Returns the order of the algorithm. If n is the order of a method, then the result of one iteration with the timestep dt @@ -280,21 +280,22 @@

                    Stepper that model this concept

                    -
                      -
                    • +
                        +
                      • stepper_euler
                        • -
                      • +
                      • stepper_rk4
                        • -
                      • +
                      • stepper_rk78_fehlberg
                    -
                    Copyright © 2009 -2011 Karsten Ahnert and Mario Mulansky

                    +

                    Copyright © 2009-2011 Karsten Ahnert + and Mario Mulansky

                    Distributed under 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)

                    diff --git a/doc/boost_sandbox_numeric_odeint/old_concepts/compositestepper.html b/doc/boost_sandbox_numeric_odeint/old_concepts/compositestepper.html index 638b9bb4..2a8227a3 100644 --- a/doc/boost_sandbox_numeric_odeint/old_concepts/compositestepper.html +++ b/doc/boost_sandbox_numeric_odeint/old_concepts/compositestepper.html @@ -3,8 +3,8 @@ CompositeStepper - - + + @@ -15,12 +15,13 @@
                    PrevUpHomeNext
                    -

                    +
                    -
                    Copyright © 2009 -2011 Karsten Ahnert and Mario Mulansky

                    +

                    Copyright © 2009-2011 Karsten Ahnert + and Mario Mulansky

                    Distributed under 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)

                    diff --git a/doc/boost_sandbox_numeric_odeint/old_concepts/controlled_stepper.html b/doc/boost_sandbox_numeric_odeint/old_concepts/controlled_stepper.html index 3bb7951e..c31fc178 100644 --- a/doc/boost_sandbox_numeric_odeint/old_concepts/controlled_stepper.html +++ b/doc/boost_sandbox_numeric_odeint/old_concepts/controlled_stepper.html @@ -3,8 +3,8 @@ Controlled stepper - - + + @@ -15,7 +15,7 @@
                    PrevUpHomeNext
                    -
                    +

                    Controlled stepper @@ -35,7 +35,7 @@

                    Methods

                    -
                    • +
                      • Controlled_Stepper( time_type abs_err, time_type rel_err, @@ -60,8 +60,8 @@ If the estimated error is less than half of the desired error, an increased stepsize will be suggested.

                        -
                          -
                        • +
                            +
                          • Controlled_Stepper( container_type &x, time_type abs_err, time_type rel_err, @@ -70,7 +70,7 @@ Same as above, but with additional allocation of the internal memory to store intermediate results of the same size as x.
                            • -
                          • +
                          • controlled_step_result try_step( DynamicalSystem &system, container_type &x, time_type &t, time_type &dt )
                            • @@ -186,8 +186,8 @@ dt now containes the suggested reduced stepsize that should give an error below the desired level.

                            -
                              -
                            • +
                                +
                              • controlled_step_result try_step( DynamicalSystem &system, container_type &x, const container_type &dxdt, time_type @@ -198,32 +198,33 @@ represents the derivative x'(t) = f(x,t) at the time t.
                                • -
                              • +
                              • void adjust_size( const container_type &x ) Adjusts the internal memory to store intermediate results of the same size as x. This function must be called whenever the system size changes during the integration.
                                • -
                              • +
                              • order_type order_error_step() Returns the order of the result x(t+dt) of the algorithm.

                              Stepper that model this concept

                              -
                                -
                              • +
                                  +
                                • controlled_stepper_standard
                                  • -
                                • +
                                • controlled_stepper_bs
                              -
                              Copyright © 2009 -2011 Karsten Ahnert and Mario Mulansky

                              +

                              Copyright © 2009-2011 Karsten Ahnert + and Mario Mulansky

                              Distributed under 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)

                              diff --git a/doc/boost_sandbox_numeric_odeint/old_concepts/dense_ouput_stepper.html b/doc/boost_sandbox_numeric_odeint/old_concepts/dense_ouput_stepper.html index 125b75c4..92bc7e8f 100644 --- a/doc/boost_sandbox_numeric_odeint/old_concepts/dense_ouput_stepper.html +++ b/doc/boost_sandbox_numeric_odeint/old_concepts/dense_ouput_stepper.html @@ -3,8 +3,8 @@ Dense ouput stepper - - + + @@ -15,13 +15,14 @@
                              PrevUpHomeNext
                              -

                              +
                              -
                              Copyright © 2009 -2011 Karsten Ahnert and Mario Mulansky

                              +

                              Copyright © 2009-2011 Karsten Ahnert + and Mario Mulansky

                              Distributed under 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)

                              diff --git a/doc/boost_sandbox_numeric_odeint/old_concepts/error_stepper.html b/doc/boost_sandbox_numeric_odeint/old_concepts/error_stepper.html index bce3b512..d669dc9a 100644 --- a/doc/boost_sandbox_numeric_odeint/old_concepts/error_stepper.html +++ b/doc/boost_sandbox_numeric_odeint/old_concepts/error_stepper.html @@ -3,8 +3,8 @@ Error stepper - - + + @@ -15,7 +15,7 @@
                              PrevUpHomeNext
                              -
                              +

                              Error stepper @@ -40,18 +40,18 @@

                              Methods

                              -
                                -
                              • +
                                  +
                                • Error_Stepper() Constructor.
                                  • -
                                • +
                                • Error_Stepper( container_type &x ) Constructor that allocates internal memory to store intermediate results of the same size as x.
                                  • -
                                • +
                                • void do_step( DynamicalSystem &system , container_type @@ -183,7 +183,7 @@ Note, that the time t is not automatically increased by this method.

                                  -
                                  • +
                                    • void do_step( DynamicalSystem &system , container_type @@ -199,18 +199,18 @@ that represents the derivative x'(t) = f(x,t) at the time t.

                                      -
                                        -
                                      • +
                                          +
                                        • void adjust_size( const container_type &x ) Adjusts the internal memory to store intermediate results of the same size as x. This function must be called whenever the system size changes during the integration.
                                          • -
                                        • +
                                        • order_type order_error_step() Returns the order of the result x(t+dt) of the algorithm.
                                          • -
                                        • +
                                        • order_type order_error() Returns the order of the error estimation of the algorithm.
                                          • @@ -218,21 +218,22 @@

                                          Stepper that model this concept

                                          -
                                            -
                                          • +
                                              +
                                            • stepper_rk5_ck
                                              • -
                                            • +
                                            • stepper_rk78_fehlberg
                                              • -
                                            • +
                                            • stepper_half_step
                                          -
                                          Copyright © 2009 -2011 Karsten Ahnert and Mario Mulansky

                                          +

                                          Copyright © 2009-2011 Karsten Ahnert + and Mario Mulansky

                                          Distributed under 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)

                                          diff --git a/doc/boost_sandbox_numeric_odeint/old_concepts/size_adjusting_stepper.html b/doc/boost_sandbox_numeric_odeint/old_concepts/size_adjusting_stepper.html index c54bc218..8b51376c 100644 --- a/doc/boost_sandbox_numeric_odeint/old_concepts/size_adjusting_stepper.html +++ b/doc/boost_sandbox_numeric_odeint/old_concepts/size_adjusting_stepper.html @@ -3,8 +3,8 @@ Size adjusting stepper - - + + @@ -15,13 +15,14 @@
                                          PrevUpHomeNext
                                          -

                                          +
                                          -
                                          Copyright © 2009 -2011 Karsten Ahnert and Mario Mulansky

                                          +

                                          Copyright © 2009-2011 Karsten Ahnert + and Mario Mulansky

                                          Distributed under 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)

                                          diff --git a/doc/boost_sandbox_numeric_odeint/old_reference.html b/doc/boost_sandbox_numeric_odeint/old_reference.html index bc0874da..9f32316b 100644 --- a/doc/boost_sandbox_numeric_odeint/old_reference.html +++ b/doc/boost_sandbox_numeric_odeint/old_reference.html @@ -3,8 +3,8 @@ Old Reference - - + + @@ -15,7 +15,7 @@
                                          PrevUpHomeNext
                                          -
                                          +
                                          @@ -31,7 +31,8 @@
                                          -
                                          Copyright © 2009 -2011 Karsten Ahnert and Mario Mulansky

                                          +

                                          Copyright © 2009-2011 Karsten Ahnert + and Mario Mulansky

                                          Distributed under 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)

                                          diff --git a/doc/boost_sandbox_numeric_odeint/old_reference/algebras.html b/doc/boost_sandbox_numeric_odeint/old_reference/algebras.html index 33abc8dc..35ae5a85 100644 --- a/doc/boost_sandbox_numeric_odeint/old_reference/algebras.html +++ b/doc/boost_sandbox_numeric_odeint/old_reference/algebras.html @@ -3,8 +3,8 @@ Algebras - - + + @@ -15,12 +15,13 @@
                                          PrevUpHomeNext
                                          -

                                          +
                                          -
                                          Copyright © 2009 -2011 Karsten Ahnert and Mario Mulansky

                                          +

                                          Copyright © 2009-2011 Karsten Ahnert + and Mario Mulansky

                                          Distributed under 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)

                                          diff --git a/doc/boost_sandbox_numeric_odeint/old_reference/integration_functions.html b/doc/boost_sandbox_numeric_odeint/old_reference/integration_functions.html index 6c4b3705..7cc7b3bc 100644 --- a/doc/boost_sandbox_numeric_odeint/old_reference/integration_functions.html +++ b/doc/boost_sandbox_numeric_odeint/old_reference/integration_functions.html @@ -3,8 +3,8 @@ Integration functions - - + + @@ -15,7 +15,7 @@
                                          PrevUpHomeNext
                                          -
                                          +
                                          -

                                          +
                                          -
                                          -
                                          Copyright © 2009 -2011 Karsten Ahnert and Mario Mulansky

                                          +

                                          Copyright © 2009-2011 Karsten Ahnert + and Mario Mulansky

                                          Distributed under 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)

                                          diff --git a/doc/boost_sandbox_numeric_odeint/old_reference/operations.html b/doc/boost_sandbox_numeric_odeint/old_reference/operations.html index d9d8d5ca..ef8a45a3 100644 --- a/doc/boost_sandbox_numeric_odeint/old_reference/operations.html +++ b/doc/boost_sandbox_numeric_odeint/old_reference/operations.html @@ -3,8 +3,8 @@ Operations - - + + @@ -15,12 +15,13 @@
                                          PrevUpHomeNext
                                          -

                                          +
                                          -
                                          Copyright © 2009 -2011 Karsten Ahnert and Mario Mulansky

                                          +

                                          Copyright © 2009-2011 Karsten Ahnert + and Mario Mulansky

                                          Distributed under 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)

                                          diff --git a/doc/boost_sandbox_numeric_odeint/old_reference/resizing.html b/doc/boost_sandbox_numeric_odeint/old_reference/resizing.html index 9a88f763..14749420 100644 --- a/doc/boost_sandbox_numeric_odeint/old_reference/resizing.html +++ b/doc/boost_sandbox_numeric_odeint/old_reference/resizing.html @@ -3,8 +3,8 @@ Resizing - - + + @@ -14,12 +14,13 @@
                                          PrevUpHome
                                          -

                                          +
                                          -
                                          Copyright © 2009 -2011 Karsten Ahnert and Mario Mulansky

                                          +

                                          Copyright © 2009-2011 Karsten Ahnert + and Mario Mulansky

                                          Distributed under 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)

                                          diff --git a/doc/boost_sandbox_numeric_odeint/old_reference/stepper_classes.html b/doc/boost_sandbox_numeric_odeint/old_reference/stepper_classes.html index 0775a70a..5f52e624 100644 --- a/doc/boost_sandbox_numeric_odeint/old_reference/stepper_classes.html +++ b/doc/boost_sandbox_numeric_odeint/old_reference/stepper_classes.html @@ -3,8 +3,8 @@ Stepper classes - - + + @@ -15,7 +15,7 @@
                                          PrevUpHomeNext
                                          -
                                          +

                                          Stepper classes @@ -191,7 +191,8 @@

                                          -
                                          Copyright © 2009 -2011 Karsten Ahnert and Mario Mulansky

                                          +

                                          Copyright © 2009-2011 Karsten Ahnert + and Mario Mulansky

                                          Distributed under 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)

                                          diff --git a/doc/boost_sandbox_numeric_odeint/tutorial.html b/doc/boost_sandbox_numeric_odeint/tutorial.html index 0ea093c8..0ba767e6 100644 --- a/doc/boost_sandbox_numeric_odeint/tutorial.html +++ b/doc/boost_sandbox_numeric_odeint/tutorial.html @@ -3,8 +3,8 @@ Tutorial - - + + @@ -15,7 +15,7 @@
                                          PrevUpHomeNext
                                          -
                                          +
                                          @@ -39,7 +39,8 @@
                                          -
                                          Copyright © 2009 -2011 Karsten Ahnert and Mario Mulansky

                                          +

                                          Copyright © 2009-2011 Karsten Ahnert + and Mario Mulansky

                                          Distributed under 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)

                                          diff --git a/doc/boost_sandbox_numeric_odeint/tutorial/all_examples.html b/doc/boost_sandbox_numeric_odeint/tutorial/all_examples.html index 4ab2eadd..7705102e 100644 --- a/doc/boost_sandbox_numeric_odeint/tutorial/all_examples.html +++ b/doc/boost_sandbox_numeric_odeint/tutorial/all_examples.html @@ -3,8 +3,8 @@ All examples - - + + @@ -15,7 +15,7 @@
                                          PrevUpHomeNext
                                          -
                                          +

                                          All examples @@ -236,7 +236,8 @@

                                          -
                                          Copyright © 2009 -2011 Karsten Ahnert and Mario Mulansky

                                          +

                                          Copyright © 2009-2011 Karsten Ahnert + and Mario Mulansky

                                          Distributed under 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)

                                          diff --git a/doc/boost_sandbox_numeric_odeint/tutorial/chaotic_systems_and_lyapunov_exponents.html b/doc/boost_sandbox_numeric_odeint/tutorial/chaotic_systems_and_lyapunov_exponents.html index 8494961e..f576b099 100644 --- a/doc/boost_sandbox_numeric_odeint/tutorial/chaotic_systems_and_lyapunov_exponents.html +++ b/doc/boost_sandbox_numeric_odeint/tutorial/chaotic_systems_and_lyapunov_exponents.html @@ -3,8 +3,8 @@ Chaotic systems and Lyapunov exponents - - + + @@ -15,7 +15,7 @@
                                          PrevUpHomeNext
                                          -
                                          +

                                          Chaotic systems and Lyapunov exponents @@ -141,8 +141,8 @@ can be used for example via 

                                          state_type x;
-// initialize x
-explicit_rk4< state_type > rk4;
+// initialize x
+explicit_rk4< state_type > rk4;
 integrate_n_steps( rk4 , lorenz_with_lyap , x , 0.0 , 0.01 , 1000 );

                                          @@ -157,8 +157,8 @@ 

                                          state_type x;
-// initialize x
-explicit_rk4< state_type > rk4;
+// initialize x
+explicit_rk4< state_type > rk4;
 integrate_n_steps( rk4 , lorenz , x , 0.0 , 0.01 , 1000 );

                                          @@ -213,8 +213,8 @@

                                          -
                                          // perform 10000 transient steps
-integrate_n_steps( rk4 , lorenz() , std::make_pair( x.begin() , x.begin() + n ) , 0.0 , dt , 10000 );
+
                                          // perform 10000 transient steps
+integrate_n_steps( rk4 , lorenz() , std::make_pair( x.begin() , x.begin() + n ) , 0.0 , dt , 10000 );
 
                                          
 
                                          
       
                                          
@@ -222,22 +222,22 @@
         Having integrated a sufficient number of transients steps we are now able
         to calculate the Lyapunov exponents:
       
                                          
-
                                            
-
                                          1. 
+
                                              
+
                                            1. 
             We initialize the perturbations. They are stored linearly behind the
             state of the Lorenz system. The perturbation are initialized such that
             they fullfill < p ​i , p​j > = δ ​ij  where <,>
             is the classical scalar product and δ ​ij is the Kronecker
             symbol.
           
                                              2. 
-
                                            2. 
+
                                            3. 
             Integrate 100 steps of the full system with perturbations
           
                                              4. 
-
                                            4. 
+
                                            5. 
             Orthonormalize the perturbation using Gram-Schmidt orthonormalization
             algorithm.
           
                                              6. 
-
                                            6. 
+
                                            7. 
             Repeat step 2 and 3. Every 10000 steps write the current Lyapunov exponent.
           
                                              8. 
 
                                            
@@ -273,7 +273,8 @@
 
                                          
 
-
                                          
 
                                          Copyright © 2009 -2011 Karsten Ahnert and Mario Mulansky
                                          
+
                                          Copyright © 2009-2011 Karsten Ahnert
+      and Mario Mulansky
                                          
         Distributed under 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)
       
                                          
diff --git a/doc/boost_sandbox_numeric_odeint/tutorial/harmonic_oscillator.html b/doc/boost_sandbox_numeric_odeint/tutorial/harmonic_oscillator.html
index 26036355..c07d5718 100644
--- a/doc/boost_sandbox_numeric_odeint/tutorial/harmonic_oscillator.html
+++ b/doc/boost_sandbox_numeric_odeint/tutorial/harmonic_oscillator.html
@@ -3,8 +3,8 @@
 
 Harmonic oscillator
 
-
-
+
+
 
 
 
@@ -15,7 +15,7 @@
 
                                          
 PrevUpHomeNext
 
                                          
-
                                          
+
                                          
 
                                          
-
                                          
+
                                          
 
                                          
 Define
         the ODE
@@ -103,7 +103,7 @@
           which allows for cleaner code.
         
                                          
 
                                          
-
                                          
+
                                          
 
                                          
 Stepper
         Types
@@ -1120,7 +1120,7 @@
           which kind of steppers should be applied.
         
                                          
 
                                          
-
                                          
+
                                          
 
                                          
 Integration
         with Constant Step Size
@@ -1164,7 +1164,7 @@
 
                                          
         
                                          
 
                                          
-
                                          
+
                                          
 
                                          
 Integration
         with Adaptive Step Size
@@ -1453,7 +1453,8 @@
 
                                          
 
-
                                          
 
                                          Copyright © 2009 -2011 Karsten Ahnert and Mario Mulansky
                                          
+
                                          Copyright © 2009-2011 Karsten Ahnert
+      and Mario Mulansky
                                          
         Distributed under 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)
       
                                          
diff --git a/doc/boost_sandbox_numeric_odeint/tutorial/references.html b/doc/boost_sandbox_numeric_odeint/tutorial/references.html
index 03ce8a67..86871b10 100644
--- a/doc/boost_sandbox_numeric_odeint/tutorial/references.html
+++ b/doc/boost_sandbox_numeric_odeint/tutorial/references.html
@@ -3,8 +3,8 @@
 
 References
 
-
-
+
+
 
 
 
@@ -15,7 +15,7 @@
 
                                          
 PrevUpHomeNext
 
                                          
-
                                          
+
                                          
 
                                          
@@ -74,7 +74,8 @@
 
                                          
 
-
                                          
 
                                          Copyright © 2009 -2011 Karsten Ahnert and Mario Mulansky
                                          
+
                                          Copyright © 2009-2011 Karsten Ahnert
+      and Mario Mulansky
                                          
         Distributed under 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)
       
                                          
diff --git a/doc/boost_sandbox_numeric_odeint/tutorial/solar_system.html b/doc/boost_sandbox_numeric_odeint/tutorial/solar_system.html
index d7e89781..05d75810 100644
--- a/doc/boost_sandbox_numeric_odeint/tutorial/solar_system.html
+++ b/doc/boost_sandbox_numeric_odeint/tutorial/solar_system.html
@@ -3,8 +3,8 @@
 
 Solar system
 
-
-
+
+
 
 
 
@@ -15,7 +15,7 @@
 
                                          
 PrevUpHomeNext
 
                                          
-
                                          
+
                                          
 
                                          
-
                                          
+
                                          
 
                                          
 Gravitation
         and energy conservation
@@ -88,7 +88,7 @@
           f( q​i ).
         
                                          
 
                                          
-
                                          
+
                                          
 
                                          
 Define
         the system function
@@ -114,20 +114,20 @@
         typedef T value_type;
         typedef point< value_type , dim > point_type;
 
-        // ...
-        // constructors
-
-        // ...
-        // operators
-
+        // ...
+        // constructors
+
+        // ...
+        // operators
+
     private:
 
         T m_val[dim];
     };
 
-    //...
-    // more operators
-    
+    //...
+    // more operators
+    
 
                                          
         
                                          
 
                                          
@@ -138,8 +138,8 @@
 
                                          
           
 
                                          
-
                                          // we simulate 5 planets and the sun
-const size_t n = 6;
+
                                          // we simulate 5 planets and the sun
+const size_t n = 6;
 
 typedef point< double , 3 > point_type;
 typedef boost::array< point_type , n > container_type;
@@ -273,7 +273,8 @@
 
                                          
 
-
                                          
 
                                          Copyright © 2009 -2011 Karsten Ahnert and Mario Mulansky
                                          
+
                                          Copyright © 2009-2011 Karsten Ahnert
+      and Mario Mulansky
                                          
         Distributed under 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)
       
                                          
diff --git a/doc/boost_sandbox_numeric_odeint/tutorial/special_topics.html b/doc/boost_sandbox_numeric_odeint/tutorial/special_topics.html
index 2ec48199..c48d35a9 100644
--- a/doc/boost_sandbox_numeric_odeint/tutorial/special_topics.html
+++ b/doc/boost_sandbox_numeric_odeint/tutorial/special_topics.html
@@ -3,8 +3,8 @@
 
 Special topics
 
-
-
+
+
 
 
 
@@ -15,7 +15,7 @@
 
                                          
 PrevUpHomeNext
 
                                          
-
                                          
+
                                          
 
                                          
 Special
       topics
@@ -43,7 +43,7 @@
 
                                          
         explain boost::ref
       
                                          
-
                                          
+
                                          
 
                                          
 Complex
         state types
@@ -127,7 +127,7 @@
           The full cpp file for the Stuart Landau example can be found here stuart_landau.cpp
         
                                          
 
                                          
-
                                          
+
                                          
 
                                          
 Lattice
         systems
@@ -166,8 +166,8 @@
 
     fpu( const double beta = 1.0 ) : m_beta( beta ) { }
 
-    // system function defining the ODE
-    void operator()( const container_type &q , container_type &dpdt ) const
+    // system function defining the ODE
+    void operator()( const container_type &q , container_type &dpdt ) const
     {
         size_t n = q.size();
         double tmp = q[0] - 0.0;
@@ -185,17 +185,17 @@
         dpdt[n-1] += tmp2;
     }
 
-    // calculates the energy of the system
-    double energy( const container_type &q , const container_type &p ) const
+    // calculates the energy of the system
+    double energy( const container_type &q , const container_type &p ) const
     {
-        // ...
-    }
+        // ...
+    }
 
-    // calculates the local energy of the system
-    void local_energy( const container_type &q , const container_type &p , container_type &e ) const
+    // calculates the local energy of the system
+    void local_energy( const container_type &q , const container_type &p , container_type &e ) const
     {
-        // ...
-    }
+        // ...
+    }
 };
 
 
                                          
@@ -273,7 +273,7 @@
           The full cpp file for this FPU example can be found here fpu.cpp
         
                                          
 
                                          
-
                                          
+
                                          
 
                                          
 Ensembles
         of oscillators
@@ -411,8 +411,8 @@
 boost::uniform_real<> unif( 0.0 , 2.0 * M_PI );
 boost::variate_generator< boost::mt19937&, boost::uniform_real<> > gen( rng , unif );
 
-// gamma = 1, the phase transition occurs at epsilon = 2
-phase_ensemble ensemble( n , 1.0 );
+// gamma = 1, the phase transition occurs at epsilon = 2
+phase_ensemble ensemble( n , 1.0 );
 statistics_observer obs;
 
 for( double epsilon = 0.0 ; epsilon < 5.0 ; epsilon += 0.1 )
@@ -420,14 +420,14 @@
     ensemble.set_epsilon( epsilon );
     obs.reset();
 
-    // start with random initial conditions
-    generate( x.begin() , x.end() , gen );
+    // start with random initial conditions
+    generate( x.begin() , x.end() , gen );
 
-    // calculate some transients steps
-    integrate_const( runge_kutta4< container_type >() , boost::ref( ensemble ) , x , 0.0 , 10.0 , dt );
+    // calculate some transients steps
+    integrate_const( runge_kutta4< container_type >() , boost::ref( ensemble ) , x , 0.0 , 10.0 , dt );
 
-    // integrate and compute the statistics
-    integrate_const( runge_kutta4< container_type >() , boost::ref( ensemble ) , x , 0.0 , 100.0 , dt , boost::ref( obs ) );
+    // integrate and compute the statistics
+    integrate_const( runge_kutta4< container_type >() , boost::ref( ensemble ) , x , 0.0 , 100.0 , dt , boost::ref( obs ) );
     cout << epsilon << "\t" << obs.get_K_mean() << endl;
 }
 
@@ -439,7 +439,7 @@
           The full cpp file for this example can be found here phase_oscillator_ensemble.cpp
         
                                          
 
                                          
-
                                          
+
                                          
 
                                          
 Using
         boost::units
@@ -582,7 +582,7 @@
           The full cpp file for this example can be found here harmonic_oscillator_units.cpp.
         
                                          
 
                                          
-
                                          
+
                                          
 
                                          
 Using
         matrices as state types
@@ -659,7 +659,7 @@
           The full cpp for this example can be found here two_dimensional_phase_lattice.cpp.
         
                                          
 
                                          
-
                                          
+
                                          
 
                                          
 Partial
         differential equations
@@ -674,11 +674,11 @@
           Ginzburg-Landau
         
                                          
 
                                          
-
                                          
+
                                          
-
                                          
+
                                          
 
                                          
 Using
         arbitrary precision floating point types
@@ -763,7 +763,7 @@
           The full example can be found at lorenz_gmpxx.cpp.
         
                                          
 
                                          
-
                                          
+
                                          
 
                                          
 Self
         expanding lattices
@@ -804,8 +804,8 @@
         : m_max_N( max_N ) , m_beta( beta ) , m_pot_start_index( pot_start_index ) , m_pot( max_N )
     {
         srand( time( NULL ) );
-        // fill random potential with iid values from [0,1]
-        boost::mt19937 rng;
+        // fill random potential with iid values from [0,1]
+        boost::mt19937 rng;
         boost::uniform_real<> unif( 0.0 , 1.0 );
         boost::variate_generator< boost::mt19937&, boost::uniform_real<> > gen( rng , unif );
         generate( m_pot.begin() , m_pot.end() , gen );
@@ -813,9 +813,9 @@
 
     void operator()( const coord_type &q , coord_type &dpdt )
     {
-        // calculate dpdt = -dH/dq of this hamiltonian system
-        // dp_i/dt = - V_i * q_i^3 - beta*(q_i - q_{i-1})^3 + beta*(q_{i+1} - q_i)^3
-        const int N = q.size();
+        // calculate dpdt = -dH/dq of this hamiltonian system
+        // dp_i/dt = - V_i * q_i^3 - beta*(q_i - q_{i-1})^3 + beta*(q_{i+1} - q_i)^3
+        const int N = q.size();
         double diff = q[0] - q[N-1];
         for( int i=0 ; i<N ; ++i )
         {
@@ -828,8 +828,8 @@
 
     void energy_distribution( const coord_type &q , const coord_type &p , coord_type &energies )
     {
-        // computes the energy per lattice site normalized by total energy
-        const size_t N = q.size();
+        // computes the energy per lattice site normalized by total energy
+        const size_t N = q.size();
         double en = 0.0;
         for( size_t i=0 ; i<N ; i++ )
         {
@@ -848,8 +848,8 @@
 
     double energy( const coord_type &q , const coord_type &p )
     {
-        // calculates the total energy of the excitation
-        const size_t N = q.size();
+        // calculates the total energy of the excitation
+        const size_t N = q.size();
         double en = 0.0;
         for( size_t i=0 ; i<N ; i++ )
         {
@@ -876,23 +876,23 @@
 
                                          
           
 
                                          
-
                                          //start with 60 sites
-const int N_start = 60;
+
                                          //start with 60 sites
+const int N_start = 60;
 coord_type q( N_start , 0.0 );
 q.reserve( max_N );
 coord_type p( N_start , 0.0 );
 p.reserve( max_N );
-// start with uniform momentum distribution over 20 sites
-fill( p.begin()+20 , p.end()-20 , 1.0/sqrt(20) );
+// start with uniform momentum distribution over 20 sites
+fill( p.begin()+20 , p.end()-20 , 1.0/sqrt(20) );
 
 coord_type distr( N_start , 0.0 );
 distr.reserve( max_N );
 
-// create the system
-compacton_lattice lattice( max_N , beta , (max_N-N_start)/2 );
+// create the system
+compacton_lattice lattice( max_N , beta , (max_N-N_start)/2 );
 
-//create the stepper, note that we use an always_resizer because state size might change during steps
-typedef symplectic_rkn_sb3a_mclachlan< coord_type , coord_type , double , coord_type , coord_type , double ,
+//create the stepper, note that we use an always_resizer because state size might change during steps
+typedef symplectic_rkn_sb3a_mclachlan< coord_type , coord_type , double , coord_type , coord_type , double ,
         range_algebra , default_operations , always_resizer > hamiltonian_stepper;
 hamiltonian_stepper stepper;
 hamiltonian_stepper::state_type state = make_pair( q , p );
@@ -961,7 +961,8 @@
 
                                          
 
-
                                          
 
                                          Copyright © 2009 -2011 Karsten Ahnert and Mario Mulansky
                                          
+
                                          Copyright © 2009-2011 Karsten Ahnert
+      and Mario Mulansky
                                          
         Distributed under 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)
       
                                          
diff --git a/doc/boost_sandbox_numeric_odeint/tutorial/stiff_systems.html b/doc/boost_sandbox_numeric_odeint/tutorial/stiff_systems.html
index 6732f950..511dcd5d 100644
--- a/doc/boost_sandbox_numeric_odeint/tutorial/stiff_systems.html
+++ b/doc/boost_sandbox_numeric_odeint/tutorial/stiff_systems.html
@@ -3,8 +3,8 @@
 
 Stiff systems
 
-
-
+
+
 
 
 
@@ -15,7 +15,7 @@
 
                                          
 PrevUpHomeNext
 
                                          
-
                                          
+
                                          
 
                                          
 Stiff
       systems
@@ -155,7 +155,8 @@
 
                                          
 
-
                                          
 
                                          Copyright © 2009 -2011 Karsten Ahnert and Mario Mulansky
                                          
+
                                          Copyright © 2009-2011 Karsten Ahnert
+      and Mario Mulansky
                                          
         Distributed under 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)
       
                                          
diff --git a/doc/boost_sandbox_numeric_odeint/tutorial/using_cuda_and_thrust.html b/doc/boost_sandbox_numeric_odeint/tutorial/using_cuda_and_thrust.html
index 57a56968..0221e4d1 100644
--- a/doc/boost_sandbox_numeric_odeint/tutorial/using_cuda_and_thrust.html
+++ b/doc/boost_sandbox_numeric_odeint/tutorial/using_cuda_and_thrust.html
@@ -3,8 +3,8 @@
 
 Using Cuda and Thrust
 
-
-
+
+
 
 
 
@@ -15,7 +15,7 @@
 
                                          
 PrevUpHomeNext
 
                                          
-
                                          
+
                                          
 
                                          
 Using
       Cuda and Thrust
@@ -66,7 +66,7 @@
           in the last example.
         
                                          
 
                                          
-
                                          
+
                                          
 
                                          
 Phase
         oscillator ensemble
@@ -89,13 +89,13 @@
 
                                          
           
 
                                          
-
                                          //change this to float if your device does not support double computation
-typedef double value_type;
+
                                          //change this to float if your device does not support double computation
+typedef double value_type;
 
-//change this to host_vector< ... > of you want to run on CPU
-typedef thrust::device_vector< value_type > state_type;
-// typedef thrust::host_vector< value_type > state_type;
-
                                          
+//change this to host_vector< ... > of you want to run on CPU
+typedef thrust::device_vector< value_type > state_type;
+// typedef thrust::host_vector< value_type > state_type;
+
                                          
 
                                          
         
                                          
 
                                          
@@ -115,8 +115,8 @@
     }
 };
 
-// ...
-
+// ...
+
 thrust::for_each( x.begin() , x.end() , add_two() );
 
 
                                          
@@ -125,11 +125,11 @@
 
                                          
           For the purpose of integrating the phase oscillator ensemble we need
         
                                          
-
                                            
-
                                          • 
+
                                              
+
                                            • 
               to calculate the system function, hence the r.h.s. of the ODE.
             
                                              • 
-
                                            • 
+
                                            • 
               this involves computing the mean field of the oscillator example, i.e.
               the values of R and θ
             
                                              • 
@@ -230,8 +230,8 @@
         }
     };
 
-    // ...
-
+    // ...
+
     void operator() ( const state_type &x , state_type &dxdt , const value_type dt ) const
     {
         std::pair< value_type , value_type > mean_field = mean_field_calculator::get_mean( x );
@@ -243,8 +243,8 @@
                 );
     }
 
-    // ...
-};
+    // ...
+};
 
 
                                              
         
                                              
@@ -309,7 +309,7 @@
           The full example can be found at phase_oscillator_example.cu.
         
                                              
 
                                            
-
                                            
+
                                            
 
                                            
 Large
         oscillator chains
@@ -336,12 +336,12 @@
 
                                            
           
 
                                            
-
                                            //change this to host_vector< ... > of you want to run on CPU
-typedef thrust::device_vector< value_type > state_type;
+
                                            //change this to host_vector< ... > of you want to run on CPU
+typedef thrust::device_vector< value_type > state_type;
 typedef thrust::device_vector< size_t > index_vector_type;
-//typedef thrust::host_vector< value_type > state_type;
-//typedef thrust::host_vector< size_t > index_vector_type;
-
+//typedef thrust::host_vector< value_type > state_type;
+//typedef thrust::host_vector< size_t > index_vector_type;
+
 class phase_oscillators
 {
 
@@ -351,29 +351,29 @@
     {
         template< class Tuple >
         __host__ __device__
-        void operator()( Tuple t )  // this functor works on tuples of values
-        {
-            // first, unpack the tuple into value, neighbors and omega
-            const value_type phi = thrust::get<0>(t);
-            const value_type phi_left = thrust::get<1>(t);  // left neighbor
-            const value_type phi_right = thrust::get<2>(t); // right neighbor
-            const value_type omega = thrust::get<3>(t);
-            // the dynamical equation
-            thrust::get<4>(t) = omega + sin( phi_right - phi ) + sin( phi - phi_left );
+        void operator()( Tuple t )  // this functor works on tuples of values
+        {
+            // first, unpack the tuple into value, neighbors and omega
+            const value_type phi = thrust::get<0>(t);
+            const value_type phi_left = thrust::get<1>(t);  // left neighbor
+            const value_type phi_right = thrust::get<2>(t); // right neighbor
+            const value_type omega = thrust::get<3>(t);
+            // the dynamical equation
+            thrust::get<4>(t) = omega + sin( phi_right - phi ) + sin( phi - phi_left );
         }
     };
 
     phase_oscillators( const state_type &omega )
         : m_omega( omega ) , m_N( omega.size() ) , m_prev( omega.size() ) , m_next( omega.size() )
     {
-        // build indices pointing to left and right neighbours
-        thrust::counting_iterator<size_t> c( 0 );
+        // build indices pointing to left and right neighbours
+        thrust::counting_iterator<size_t> c( 0 );
         thrust::copy( c , c+m_N-1 , m_prev.begin()+1 );
-        m_prev[0] = 0; // m_prev = { 0 , 0 , 1 , 2 , 3 , ... , N-1 }
-
+        m_prev[0] = 0; // m_prev = { 0 , 0 , 1 , 2 , 3 , ... , N-1 }
+
         thrust::copy( c+1 , c+m_N , m_next.begin() );
-        m_next[m_N-1] = m_N-1; // m_next = { 1 , 2 , 3 , ... , N-1 , N-1 }
-    }
+        m_next[m_N-1] = m_N-1; // m_next = { 1 , 2 , 3 , ... , N-1 , N-1 }
+    }
 
     void operator() ( const state_type &x , state_type &dxdt , const value_type dt )
     {
@@ -425,32 +425,32 @@
 
                                            
           
 
                                            
-
                                            // create random number generators
-boost::mt19937 rng;
+
                                            // create random number generators
+boost::mt19937 rng;
 boost::uniform_real< value_type > unif( 0.0 , 2.0 * pi );
 boost::variate_generator< boost::mt19937&, boost::uniform_real< value_type > > gen( rng , unif );
 
-// create initial conditions and omegas on host:
-vector< value_type > x_host( N );
+// create initial conditions and omegas on host:
+vector< value_type > x_host( N );
 vector< value_type > omega_host( N );
 for( size_t i=0 ; i<N ; ++i )
 {
     x_host[i] = gen();
-    omega_host[i] = ( N - i ) * epsilon; // decreasing frequencies
-}
+    omega_host[i] = ( N - i ) * epsilon; // decreasing frequencies
+}
 
-// copy to device
-state_type x = x_host;
+// copy to device
+state_type x = x_host;
 state_type omega = omega_host;
 
-// create stepper
-runge_kutta4< state_type , value_type , state_type , value_type , thrust_algebra , thrust_operations > stepper;
+// create stepper
+runge_kutta4< state_type , value_type , state_type , value_type , thrust_algebra , thrust_operations > stepper;
 
-// create phase oscillator system function
-phase_oscillators sys( omega );
+// create phase oscillator system function
+phase_oscillators sys( omega );
 
-// integrate
-integrate_const( stepper , sys , x , 0.0 , 10.0 , dt );
+// integrate
+integrate_const( stepper , sys , x , 0.0 , 10.0 , dt );
 
 thrust::copy( x.begin() , x.end() , std::ostream_iterator< value_type >( std::cout , "\n" ) );
 std::cout << std::endl;
@@ -461,7 +461,7 @@
           The full example can be found at phase_oscillator_chain.cu.
         
                                            
 
                                            
-
                                            
+
                                            
 
                                            
 Parameter
         studies
@@ -523,8 +523,8 @@
         __host__ __device__
         void operator()( T t ) const
         {
-            // unpack the parameter we want to vary and the Lorenz variables
-            value_type R = thrust::get< 3 >( t );
+            // unpack the parameter we want to vary and the Lorenz variables
+            value_type R = thrust::get< 3 >( t );
             value_type x = thrust::get< 0 >( t );
             value_type y = thrust::get< 1 >( t );
             value_type z = thrust::get< 2 >( t );
@@ -640,29 +640,29 @@
 
                                            
 
                                            state_type x( 6 * N );
 
-// initialize x,y,z
-thrust::fill( x.begin() , x.begin() + 3 * N , 10.0 );
+// initialize x,y,z
+thrust::fill( x.begin() , x.begin() + 3 * N , 10.0 );
 
-// initial dx
-thrust::fill( x.begin() + 3 * N , x.begin() + 4 * N , 1.0 );
+// initial dx
+thrust::fill( x.begin() + 3 * N , x.begin() + 4 * N , 1.0 );
 
-// initialize dy,dz
-thrust::fill( x.begin() + 4 * N , x.end() , 0.0 );
+// initialize dy,dz
+thrust::fill( x.begin() + 4 * N , x.end() , 0.0 );
 
 
-// create error stepper, can be used with make_controlled or make_dense_output
-typedef runge_kutta_dopri5< state_type , value_type , state_type , value_type , thrust_algebra , thrust_operations > stepper_type;
+// create error stepper, can be used with make_controlled or make_dense_output
+typedef runge_kutta_dopri5< state_type , value_type , state_type , value_type , thrust_algebra , thrust_operations > stepper_type;
 
 
 lorenz_system lorenz( N , beta );
 lorenz_perturbation_system lorenz_perturbation( N , beta );
 lyap_observer obs( N , 1 );
 
-// calculate transients
-integrate_adaptive( make_controlled( 1.0e-6 , 1.0e-6 , stepper_type() ) , lorenz , std::make_pair( x.begin() , x.begin() + 3 * N ) , 0.0 , 10.0 , dt );
+// calculate transients
+integrate_adaptive( make_controlled( 1.0e-6 , 1.0e-6 , stepper_type() ) , lorenz , std::make_pair( x.begin() , x.begin() + 3 * N ) , 0.0 , 10.0 , dt );
 
-// calculate the Lyapunov exponents -- the main loop
-double t = 0.0;
+// calculate the Lyapunov exponents -- the main loop
+double t = 0.0;
 while( t < 10000.0 )
 {
     integrate_adaptive( make_controlled( 1.0e-6 , 1.0e-6 , stepper_type() ) , lorenz_perturbation , x , t , t + 1.0 , 0.1 );
@@ -685,7 +685,8 @@
 
                                            
 
-
                                            
 
                                            Copyright © 2009 -2011 Karsten Ahnert and Mario Mulansky
                                            
+
                                            Copyright © 2009-2011 Karsten Ahnert
+      and Mario Mulansky
                                            
         Distributed under 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)
       
                                            
diff --git a/doc/index.html b/doc/index.html
index d6455445..681ef265 100644
--- a/doc/index.html
+++ b/doc/index.html
@@ -3,15 +3,15 @@
 
 Chapter 1. boost.sandbox.numeric.odeint
 
-
-
+
+
 
 
 
 
                                            
 
                                            
 
                                            Next
                                            
-
                                            
+
                                            
 
                                            
 
                                            
 Chapter 1. boost.sandbox.numeric.odeint
                                            
@@ -21,9 +21,10 @@
 
                                            
 Mario Mulansky
 
                                            
-
                                            Copyright © 2009 -2011 Karsten Ahnert and Mario Mulansky
                                            
+
                                            Copyright © 2009-2011 Karsten Ahnert
+      and Mario Mulansky
                                            
 
                                            
-
                                            
+
                                            
         Distributed under 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)
       
                                            
@@ -133,7 +134,7 @@
 
                                            
 
                                            
 
-
+
                                            Last revised: November 27, 2011 at 15:44:29 GMT
                                            Last revised: November 29, 2011 at 08:29:31 GMT
                                            		
 
                                            		
 
                                            
 
                                            
diff --git a/doc/odeint/reference.html b/doc/odeint/reference.html
index 06c2dd53..e8a7c0df 100644
--- a/doc/odeint/reference.html
+++ b/doc/odeint/reference.html
@@ -3,8 +3,8 @@
 
 Reference
 
-
-
+
+
 
 
 
@@ -15,11 +15,12 @@
 
                                            
 PrevUpHomeNext
 
                                            
-
                                            
+
                                            
 Reference
                                            
 
-
                                            
 
                                            Copyright © 2009 -2011 Karsten Ahnert and Mario Mulansky
                                            
+
                                            Copyright © 2009-2011 Karsten Ahnert
+      and Mario Mulansky
                                            
         Distributed under 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)
       
                                            
diff --git a/libs/numeric/odeint/doc/extend.qbk b/libs/numeric/odeint/doc/extend.qbk
index ef4e5059..8ae743d6 100644
--- a/libs/numeric/odeint/doc/extend.qbk
+++ b/libs/numeric/odeint/doc/extend.qbk
@@ -319,7 +319,7 @@ If you use this algebra, the following operators have to be defined for the stat
 As an example for the employment of the `vector_space_algebra` we will adopt
 `ublas::vector` from __ublas to work as a state type in odeint.
 This is particularily easy because `ublas::vector` supports vector-vector
-addition and scalar-vector multiplication as well as `boost::size`.
+addition and scalar-vector multiplication described above as well as `boost::size`.
 It also has a resize member function so all that has to be done in this case
 is to declare resizability:
 
@@ -339,6 +339,51 @@ libs/numeric/odeint/examples/ublas/lorenz_ublas.cpp lorenz_ublas.cpp].
 
 [endsect]
 
+[section Point type]
+
+[import ../examples/lorenz_point.cpp]
+
+Here we show how to implement the required operators on a state type.
+As example we define a new class `point3D` representing a three-dimensional
+vector with components x,y,z and define addition and scalar multiplication
+operators for it.
+We use __boost_operators to reduce the amount of code to be written.
+The class for the point type looks as follows:
+
+[point3D]
+
+By deriving from __boost_operators classes we don't have to define outer class
+operators like `operator+( point3D , point3D )` because that is taken care of
+by the operators library.
+Note that for simple Runge-Kutta schemes (like `runge_kutta4`) only the `+`
+and `*` operators are required.
+If, however, a controlled stepper is used one also needs to specify the
+division operator `/` because calculation of the error term involves an
+element wise division of the state types.
+Additionally, controlled steppers require an `abs` function calculating the
+element-wise absolute value for the state type:
+
+[point3D_abs_div]
+
+Finally, we have to add a specialization for `reduce` implementing a reduction
+over the state type:
+
+[point3D_reduce]
+
+Again, note that the two last steps were only required if you want to use
+controlled steppers.
+For simple steppers definition of the simple `+=` and `*=` operators are
+sufficient.
+Having defined such a point type, we can easily perform the integration on a lorenz
+system by using the `vector_space_algebra` again:
+
+[point3D_main]
+
+The whole example can be found in [github_link
+libs/numeric/odeint/examples/lorenz_point.cpp lorenz_point.cpp]
+
+[endsect]
+
 [endsect]
 
 [endsect]
diff --git a/libs/numeric/odeint/examples/Jamfile b/libs/numeric/odeint/examples/Jamfile
index 734f883c..3c2c2180 100644
--- a/libs/numeric/odeint/examples/Jamfile
+++ b/libs/numeric/odeint/examples/Jamfile
@@ -27,6 +27,7 @@ exe resizing_lattice : resizing_lattice.cpp ;
 exe list_lattice : list_lattice.cpp ;
 exe stepper_details : stepper_details.cpp ;
 exe my_vector : my_vector.cpp ;
+exe lorenz_point : lorenz_point.cpp ;
 
 # build-project mtl ;
 # build-project ublas ;
diff --git a/libs/numeric/odeint/examples/lorenz_point.cpp b/libs/numeric/odeint/examples/lorenz_point.cpp
new file mode 100644
index 00000000..4602eb56
--- /dev/null
+++ b/libs/numeric/odeint/examples/lorenz_point.cpp
@@ -0,0 +1,111 @@
+/* Example for the lorenz system with a 3D point type */
+
+#include 
+
+#include 
+
+#include 
+
+
+//[point3D
+class point3D :     
+    boost::additive1< point3D ,
+    boost::additive2< point3D , double ,
+    boost::multiplicative2< point3D , double > > >
+{
+public:
+
+    double x , y , z;
+
+    point3D()
+        : x( 0.0 ) , y( 0.0 ) , z( 0.0 )
+    { }
+    
+    point3D( const double val )
+        : x( val ) , y( val ) , z( val )
+    { }
+
+    point3D( const double _x , const double _y , const double _z  )
+        : x( _x ) , y( _y ) , z( _z )
+    { }
+
+    point3D& operator+=( const point3D &p )
+    {
+        x += p.x; y += p.y; z += p.z;
+        return *this;
+    }
+
+    point3D& operator*=( const double a )
+    {
+        x *= a; y *= a; z *= a;
+        return *this;
+    }
+
+};
+//]
+
+//[point3D_abs_div
+// only required for steppers with error control
+point3D operator/( const point3D &p1 , const point3D &p2 )
+{
+    return point3D( p1.x/p2.x , p1.y/p2.y , p1.z/p1.z );
+}
+
+point3D abs( const point3D &p )
+{
+    return point3D( abs(p.x) , abs(p.y) , abs(p.z) );
+}
+//]
+
+
+//[point3D_reduce
+namespace boost { namespace numeric { namespace odeint {
+// specialization of vector_space_reduce, only required for steppers with error control
+template<>
+struct vector_space_reduce< point3D >
+{
+    template< class Value , class Op >
+    Value operator() ( const point3D &p , Op op , Value init )
+    {
+        init = op( init , p.x );
+        init = op( init , p.y );
+        init = op( init , p.z );
+        return init;
+    }
+};
+} } } 
+//]
+
+std::ostream& operator<<( std::ostream &out , const point3D &p )
+{
+    out << p.x << " " << p.y << " " << p.z;
+    return out;
+}
+
+//[point3D_main
+const double sigma = 10.0;
+const double R = 28.0;
+const double b = 8.0 / 3.0;
+
+void lorenz( const point3D &x , point3D &dxdt , const double t )
+{
+    dxdt.x = sigma * ( x.y - x.x );
+    dxdt.y = R * x.x - x.y - x.x * x.z;
+    dxdt.z = -b * x.z + x.x * x.y;
+}
+
+using namespace boost::numeric::odeint;
+
+int main()
+{
+
+    point3D x( 10.0 , 5.0 , 5.0 );
+    //typedef runge_kutta4< point3D , double , point3D , 
+    //                      double , vector_space_algebra > stepper;
+    typedef runge_kutta_dopri5< point3D , double , point3D , 
+                                double , vector_space_algebra > stepper;
+    integrate_adaptive( make_controlled( 1E-8 , 1E-8 ) , lorenz , x , 
+                        0.0 , 10.0 , 0.1 );
+    std::cout << x << std::endl;
+}
+//]