\(\newcommand{\W}[1]{ \; #1 \; }\) \(\newcommand{\R}[1]{ {\rm #1} }\) \(\newcommand{\B}[1]{ {\bf #1} }\) \(\newcommand{\D}[2]{ \frac{\partial #1}{\partial #2} }\) \(\newcommand{\DD}[3]{ \frac{\partial^2 #1}{\partial #2 \partial #3} }\) \(\newcommand{\Dpow}[2]{ \frac{\partial^{#1}}{\partial {#2}^{#1}} }\) \(\newcommand{\dpow}[2]{ \frac{ {\rm d}^{#1}}{{\rm d}\, {#2}^{#1}} }\)
Runge45¶
View page sourceAn Embedded 4th and 5th Order Runge-Kutta ODE Solver¶
Syntax¶
include <cppad/utility/runge_45.hpp>
Runge45
( F , M , ti , tf , xi )Runge45
( F , M , ti , tf , xi , e )Purpose¶
This is an implementation of the Cash-Karp embedded 4th and 5th order Runge-Kutta ODE solver described in Section 16.2 of Numerical Recipes . We use \(n\) for the size of the vector xi . Let \(\B{R}\) denote the real numbers and let \(F : \B{R} \times \B{R}^n \rightarrow \B{R}^n\) be a smooth function. The return value xf contains a 5th order approximation for the value \(X(tf)\) where \(X : [ti , tf] \rightarrow \B{R}^n\) is defined by the following initial value problem:
If your set of ordinary differential equations are stiff, an implicit method may be better (perhaps Rosen34 .)
Operation Sequence¶
The operation sequence for Runge does not depend on any of its Scalar input values provided that the operation sequence for
F .
Ode
( t , x , f )
does not on any of its Scalar inputs (see below).
Include¶
The file cppad/utility/runge_45.hpp
is included by cppad/cppad.hpp
but it can also be included separately with out the rest of
the CppAD
routines.
xf¶
The return value xf has the prototype
Vector xf
and the size of xf is equal to n (see description of Vector below).
where \(h = (tf - ti) / M\) is the step size. If xf contains not a number nan , see the discussion for f .
Fun¶
The class Fun and the object F satisfy the prototype
Fun & F
The object F (and the class Fun )
must have a member function named Ode
that supports the syntax
F .
Ode
( t , x , f )
t¶
The argument t to F . Ode
has prototype
const
Scalar & t
(see description of Scalar below).
x¶
The argument x to F . Ode
has prototype
const
Vector & x
and has size n (see description of Vector below).
f¶
The argument f to F . Ode
has prototype
Vector & f
On input and output, f is a vector of size n
and the input values of the elements of f do not matter.
On output,
f is set equal to \(F(t, x)\) in the differential equation.
If any of the elements of f have the value not a number nan
the routine Runge45
returns with all the
elements of xf and e equal to nan
.
Warning¶
The argument f to F . Ode
must have a call by reference in its prototype; i.e.,
do not forget the &
in the prototype for f .
M¶
The argument M has prototype
size_t
M
It specifies the number of steps to use when solving the differential equation. This must be greater than or equal one. The step size is given by \(h = (tf - ti) / M\), thus the larger M , the more accurate the return value xf is as an approximation for \(X(tf)\).
ti¶
The argument ti has prototype
const
Scalar & ti
(see description of Scalar below). It specifies the initial time for t in the differential equation; i.e., the time corresponding to the value xi .
tf¶
The argument tf has prototype
const
Scalar & tf
It specifies the final time for t in the differential equation; i.e., the time corresponding to the value xf .
xi¶
The argument xi has the prototype
const
Vector & xi
and the size of xi is equal to n . It specifies the value of \(X(ti)\)
e¶
The argument e is optional and has the prototype
Vector & e
If e is present, the size of e must be equal to n . The input value of the elements of e does not matter. On output it contains an element by element estimated bound for the absolute value of the error in xf
where \(h = (tf - ti) / M\) is the step size.
If on output, e contains not a number nan
,
see the discussion for f .
Scalar¶
The type Scalar must satisfy the conditions for a NumericType . The routine CheckNumericType will generate an error message if this is not the case.
fabs¶
In addition, the following function must be defined for Scalar objects a and b
a =
fabs
( b )
Note that this operation is only used for computing e ; hence
the operation sequence for xf can still be independent of
the arguments to Runge45
even if
fabs
( b ) =std::max
(-
b , b )
.
Vector¶
The type Vector must be a SimpleVector class with elements of type Scalar . The routine CheckSimpleVector will generate an error message if this is not the case.
Parallel Mode¶
For each set of types
Scalar ,
Vector , and
Fun ,
the first call to Runge45
must not be parallel execution mode.
Example¶
The file
runge45_1.cpp
contains a simple example and test of Runge45
.
The file
runge_45.cpp contains an example using Runge45
in the context of algorithmic differentiation.
It also returns true if it succeeds and false otherwise.
Source Code¶
The source code for this routine is in the file
cppad/runge_45.hpp
.