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10.3.2 Shooting method
We assume a value for
and solve the differential equation as an IVP
using the fourth order Runge-Kutta method. Using the first
fundamental form (3.13), given
we can obtain
from
(10.41)
Here we also have to
assume the entire arc length of the geodesic path
to stop the
integration. Thus the unknowns can be considered as
and
. If
we denote the computed value of
as
, the
difference can be given as
. We need
to adjust
and
to make the difference zero. This can be done
by employing Newton's method
(10.42)
where the Jacobian matrix is evaluated numerically. We first change
slightly to
and integrate the ordinary
differential equations as an IVP to evaluate the end point
, from which we can
compute the partial derivatives
and
as
(10.43)
(10.44)
Similarly we change
slightly to
and integrate the ordinary
differential equations as IVP to evaluate the end point
, from which we can
compute the partial derivatives
and
(10.45)
(10.46)
Next: 10.3.3 Relaxation method
Up: 10.3 Two point boundary
Previous: 10.3.1 Introduction
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December 2009