The computed values of y(B)will not, in general, agree with the corresponding boundary condition at s= B Consequently, we need to adjust the initial values and try again The process is repeated until the computed values at the final point agree with the boundary conditions and referred as shooting method Formulation: Using the first fundamental form, given pa we can obtain gA from FpA±VFmA-G(EnA-1) Thus we assume pA and solve the differential equation as IvP using, say runge- Kutta method. 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 pA and s If we denote the computed value of(uB, UB) as(uB, UB), the difference can be given as (ub -uB, UB We need to adjust pa and s to make the difference zero This can be done by employing the Newtons method B n+1 dub uB(PA+△pA)-ubB(PA) dus uB(s+△s)-uB(s) ap B UR(PA+ ApA)-uR(pa) auk u*(s+As)-uB(s) e Relaxation method The second method is based on a finite difference approximation to a on a mesh of points in the interval [A, B This method starts with an initial guess and improves the solution iteratively and referred as. direct method. relaxation method or finite difference method The shooting method is often very sensitive to the unknown initial angles at point A and unless a good initial guess is provided, the integrated path will never reach the other point B, while the relaxation method starts with two end points fixed and relaxes to the true solution and hence it is much more stable Let us consider a mesh of points satisfying A=S1<S2 <sm=B. We approximate the n first order differential equations by the trapezoidal rule [8 -11 =Gk+Gk-l],k=2,3 (20.24)� � � (�) (�) – The computed values of y(B) will not, in general, agree with the corresponding boundary condition at s = B. – Consequently, we need to adjust the initial values and try again. – The process is repeated until the computed values at the final point agree with the boundary conditions and referred as shooting method. – Formulation: Using the first fundamental form, given pA we can obtain qA from − 2 F pA ± F2pA − G(Ep2 q A − 1) A = . G Thus we assume pA and solve the differential equation as IVP using, say Runge￾Kutta method. 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 pA and s. � B), the difference can be given � If we denote the computed value of (uB, vB) as (uB, v as (u This can be done by employing the Newton’s method � T B − vB) � B − uB, v . We need to adjust pA and s to make the difference zero. ⎣ � �−1 B �s � B ⎤ ⎦ ⎤ ⎦ �u �u ⎤ ⎦ pA pA �pA uB − uB = − � B �s � B �v �v s n+1 s �p vB − vB n A n where � B � B(pA) � B(pA + �pA) − u � B � B(s) � πu u πu uB(s + �s) − u = , = πPA �pA πs �s � B v � B(pA) � B(pA + �pA) − v � B � B(s) � πv πv vB(s + �s) − v = , = πPA �pA πs �s • Relaxation method dy – The second method is based on a finite difference approximation to ds on a mesh of points in the interval [A, B]. – This method starts with an initial guess and improves the solution iteratively and referred as, direct method, relaxation method or finite difference method. – The shooting method is often very sensitive to the unknown initial angles at point A and unless a good initial guess is provided, the integrated path will never reach the other point B, while the relaxation method starts with two end points fixed and relaxes to the true solution and hence it is much more stable. – Let us consider a mesh of points satisfying A = s1 < s2 < . . . < sm = B. We approximate the n first order differential equations by the trapezoidal rule [8]. Yk − Yk−1 1 = [Gk + Gk−1], k = 2, 3, . . . , m (20.24) sk − sk−1 2 6