Question

# For a smooth function, : ℝ↦ℝ, consider the finite difference approximation to the second derivative,

′′( )≈ ( +ℎ)+ ( −ℎ)−2 ( )ℎ2

a. Using Taylor's Theorem, determine the bound on the truncation error of this approximation in terms of h and a bound on the function derivative M of appropriate order (to be determined).

b. Assuming the error in function evaluation is bounded by ε, determine the rounding error in evaluating the finite difference approximation formula.

c. Determine the optimum choice of h for which the total error is minimized. What is the value of the minimum error? First, express your answers in terms of M and ε, and then obtain numerical values given that M = 10 and ε = 10-15.

Image transcriptions

2. For a smooth function, f: 1R I—&gt; 1R, consider the finite difference approximation to the second derivative, ,, f(x + h) + f(x — h) — 2f(x) f (X) z —2 h a. Using Taylor's Theorem, determine the bound on the truncation error of this approximation in terms of h and a bound on the function derivative M of appropriate order (to be determined). b. Assuming the error in function evaluation is bounded by 8, determine the rounding error in evaluating the finite difference approximation formula. c. Determine the optimum choice of h for which the total error is minimized. What is the value of the minimum error? First, express your answers in terms of M and a, and then obtain numerical values given that M = 10 and a = 10—15.