Reference no: EM132293881

Question -

Let M ∈ R^mxm be a negative semi-definite matrix, x^TMx ≤ 0 for x ∈ R^m. Consider the initial value problem

(i) Show that the Euclidean norm ||y(·)|| is a decreasing function.

(ii) With the usual notation, consider the approximations y_n, n ≥ 0, produced by the implicit Euler method with a fixed time-step h > 0. Show that the implicit Euler method mimics the continuous problem, in the sense that ||y_n+1|| ≤ ||y_n||, n ∈ N.

(iii) Consider the approximations y_n, n ≥ 0, produced by the trapezoidal method with a fixed time-step h > 0. Show that the method mimics the continuous problem, in the sense that ||y_n+1|| ≤ ||y_n||, n ∈ N.

Attachment:- Assignment File.rar