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# include <cppad/utility/runge_45.hpp>
xf = Runge45(F, M, ti, tf, xi)
xf = Runge45(F, M, ti, tf, xi, e)
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:
@[@
\begin{array}{rcl}
X(ti) & = & xi \\
X'(t) & = & F[t , X(t)]
\end{array}
@]@
If your set of ordinary differential equations
are stiff, an implicit method may be better
(perhaps Rosen34
.)
Runge
does not depend on any of its
Scalar
input values provided that
the operation sequence for
F.Ode(t, x, f)
Scalar
inputs (see below).
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
has the prototype
Vector xf
xf
is equal to
n
(see description of Vector
below).
@[@
X(tf) = xf + O( h^6 )
@]@
where @(@
h = (tf - ti) / M
@)@ is the step size.
If
xf
contains not a number nan
,
see the discussion for f
.
Fun
and the object
F
satisfy the prototype
Fun &F
F
(and the class
Fun
)
must have a member function named Ode
that supports the syntax
F.Ode(t, x, f)
t
to
F.Ode
has prototype
const Scalar &t
x
to
F.Ode
has prototype
const Vector &x
n
(see description of Vector
below).
f
to
F.Ode
has prototype
Vector &f
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.
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
has prototype
size_t M
M
, the more accurate the
return value
xf
is as an approximation
for @(@
X(tf)
@)@.
ti
has prototype
const Scalar &ti
t
in the
differential equation; i.e.,
the time corresponding to the value
xi
.
tf
has prototype
const Scalar &tf
t
in the
differential equation; i.e.,
the time corresponding to the value
xf
.
xi
has the prototype
const Vector &xi
xi
is equal to
n
.
It specifies the value of @(@
X(ti)
@)@
e
is optional and has the prototype
Vector &e
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
@[@
e = O( h^5 )
@]@
where @(@
h = (tf - ti) / M
@)@ is the step size.
If on output,
e
contains not a number nan,
see the discussion for f
.
Scalar
must satisfy the conditions
for a NumericType
type.
The routine CheckNumericType
will generate an error message
if this is not the case.
Scalar
objects
a
and
b
a = fabs(b)
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
must be a SimpleVector
class with
elements of type Scalar
.
The routine CheckSimpleVector
will generate an error message
if this is not the case.
Runge45
must not be parallel
execution mode.
Runge45.
Runge45
in the context of algorithmic differentiation.
It also returns true if it succeeds and false otherwise.
cppad/runge_45.hpp.