Another good idea for y" = -y is the trapezoidal method (half forwardlhalf back): This

Chapter 6, Problem 30

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Another good idea for y" = -y is the trapezoidal method (half forwardlhalf back): This may be the best way to keep (Yn , Zn) exactly on a circle. Trapezoidal [ 1 -fl.t /2 ] [ Yn+l ] = [ 1 fl.t /2 ] [ Yn ]. fl.t /2 1 Zn+l -fl.t /2 I Zn (a) Invert the left matrix to write this equation as U n+l = AU n. Show that A is an orthogonal matrix: AT A = I. These points Un never leave the circle. A = (1- B)-I(1 + B) is always an orthogonal matrix if BT = -B. (b) (Optional MATLAB) Take 32 steps from U 0 = {l, 0) to U 32 with fl.t = 2n /32. Is U 32 = U o? I think there is a small error.