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10.3.2 Shooting method

We assume a value for $ p_A$ and solve the differential equation as an IVP using the fourth order Runge-Kutta method. Using the first fundamental form (3.13), given $ p_A$ we can obtain $ q_A$ from
$\displaystyle q_A = \frac{-Fp_A \pm \sqrt{F^2 p_A^2 - G(Ep_A^2-1)}}{G}\;.$     (10.41)

Here we also have to assume the entire arc length of the geodesic path $ s$ to stop the integration. Thus the unknowns can be considered as $ p_A$ and $ s$ . If we denote the computed value of $ (u_B, v_B)$ as $ (u_B^*, v_B^*)$ , the difference can be given as $ (u_{B}^*-u_{B}, v_{B}^*-v_{B})$ . We need to adjust $ p_A$ and $ s$ to make the difference zero. This can be done by employing Newton's method
$\displaystyle \left( \begin{array}{c} p_A \ s \end{array} \right)_{i+1} = \lef... ...eft( \begin{array}{c} u_{B}^{*}-u_{B} \ v_{B}^{*}-v_{B} \end{array} \right)\;,$     (10.42)

where the Jacobian matrix is evaluated numerically. We first change $ p_A$ slightly to $ p_A+\Delta p_A$ and integrate the ordinary differential equations as an IVP to evaluate the end point $ (u^*_B(p_A+\Delta p_A,s), v^*_B(p_A+\Delta p_A,s))$ , from which we can compute the partial derivatives $ \frac{\partial u_B^*}{\partial p_A}$ and $ \frac{\partial v_B^*}{\partial p_A}$ as
$\displaystyle \frac{\partial u_B^*}{\partial p_A} = \frac{u_B^*(p_A+\Delta p_A,s)-u_B^*(p_A,s)}{\Delta p_A}\;,$     (10.43)
$\displaystyle \frac{\partial v_B^*}{\partial p_A} = \frac{v_B^*(p_A+\Delta p_A,s)-v_B^*(p_A,s)}{\Delta p_A}\;.$     (10.44)

Similarly we change $ s$ slightly to $ s+\Delta s$ and integrate the ordinary differential equations as IVP to evaluate the end point $ (u^*_B(p_A, s+\Delta s), v^*_B(p_A, s+\Delta s))$ , from which we can compute the partial derivatives $ \frac{\partial u_B^*}{\partial s}$ and $ \frac{\partial v_B^*}{\partial s}$
$\displaystyle \frac{\partial u_B^*}{\partial s} = \frac{u_B^*(p_A, s+\Delta s)-u_B^*(p_A,s)}{\Delta s}\;,$     (10.45)
$\displaystyle \frac{\partial v_B^*}{\partial s} = \frac{v_B^*(p_A, s+\Delta s)- v_B^*(p_A, s)}{\Delta s}\;.$     (10.46)

next up previous contents index
Next: 10.3.3 Relaxation method Up: 10.3 Two point boundary Previous: 10.3.1 Introduction   Contents   Index
December 2009