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# include <cppad/utility/rosen_34.hpp>
xf = Rosen34(F, M, ti, tf, xi)
xf = Rosen34(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 not stiff
an explicit method may be better (perhaps Runge45
.)
cppad/rosen_34.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^5 )
@]@
where @(@
h = (tf - ti) / M
@)@ is the step size.
If
xf
contains not a number nan
,
see the discussion of f
.
Fun
and the object
F
satisfy the prototype
Fun &F
F.Ode(t, x, f)
F.Ode_ind(t, x, f_t)
F.Ode_dep(t, x, f_x)
t
has prototype
const Scalar &t
x
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)
@)@
(see
F(t, x)
in Description
).
f_t
to
F.Ode_ind
has prototype
Vector &f_t
f_t
is a vector of size
n
and the input values of the elements of
f_t
do not matter.
On output, the i-th element of
f_t
is set equal to @(@
\partial_t F_i (t, x)
@)@
(see
F(t, x)
in Description
).
f_x
to
F.Ode_dep
has prototype
Vector &f_x
f_x
is a vector of size
n*n
and the input values of the elements of
f_x
do not matter.
On output, the [
i*n+j
] element of
f_x
is set equal to @(@
\partial_{x(j)} F_i (t, x)
@)@
(see
F(t, x)
in Description
).
f
,
f_t
, or
f_x
have the value not a number nan,
the routine Rosen34 returns with all the
elements of
xf
and
e
equal to nan.
f
,
f_t
, and
f_x
must have a call by reference in their prototypes; i.e.,
do not forget the & in the prototype for
f
,
f_t
and
f_x
.
F.Ode_ind(t, x, f_t)
F.Ode_dep(t, x, f_x)
t
and
x
have not changed between calls.
In many cases it is faster to compute the values of
f_t
and
f_x
together and then pass them back one at a time.
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^4 )
@]@
where @(@
h = (tf - ti) / M
@)@ is the step size.
Scalar
must satisfy the conditions
for a NumericType
type.
The routine CheckNumericType
will generate an error message
if this is not the case.
In addition, the following operations must be defined for
Scalar
objects
a
and
b
:
| Operation | Description |
a < b
|
less than operator (returns a bool object)
|
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.
Rosen34
must not be parallel
execution mode.
cppad/rosen_34.hpp.