(a) Let F: D + RN for some open subset D CRN and let the standard assumptions hold, then there exists some 8 >0 such that given x) e B:(x*), the inexact Newton iteration x(x+1) = x() - ] (x(K)-F (x(K)) where J is computed columnwise using the first order finite difference approximation J.;(x)) ~ {F(x(X) + he;) F(x())} , with e; the jth canonical basis vector in RN and h sufficiently small, satisfies x(k) B:(x*) (for all k = 1,2...) and converges to x*. You are asked to prove this theorem by answering the following questions. All assumptions made should be clearly stated. (i) Prove that ||J(x(k)) - J(x(k)|| = O(h). (ii) Define error vector elk) = x(k) x*. Show that if x(k) e B:(x*), then ||$(+1) || = (K1 || 8 || + Kah) || (a) || where K1, K2 > 0. You may assume there is v> 0 such that ||J(x())-' || h that there exists some 8 >0 such that given x) e Bs(x*) then x(k) Bs(x*) for all k = 1,2,... and the inexact Newton iteration converges (near) quadratically to x*. (b) Consider the following nonlinear boundary value problem (BVP): -y' = g(y) + f(x), 0 0 such that given x) e B:(x*), the inexact Newton iteration x(x+1) = x() - ] (x(K)-F (x(K)) where J is computed columnwise using the first order finite difference approximation J.;(x)) ~ {F(x(X) + he;) F(x())} , with e; the jth canonical basis vector in RN and h sufficiently small, satisfies x(k) B:(x*) (for all k = 1,2...) and converges to x*. You are asked to prove this theorem by answering the following questions. All assumptions made should be clearly stated. (i) Prove that ||J(x(k)) - J(x(k)|| = O(h). (ii) Define error vector elk) = x(k) x*. Show that if x(k) e B:(x*), then ||$(+1) || = (K1 || 8 || + Kah) || (a) || where K1, K2 > 0. You may assume there is v> 0 such that ||J(x())-' || h that there exists some 8 >0 such that given x) e Bs(x*) then x(k) Bs(x*) for all k = 1,2,... and the inexact Newton iteration converges (near) quadratically to x*. (b) Consider the following nonlinear boundary value problem (BVP): -y' = g(y) + f(x), 0