[SOLVED] Numerical Analysis-Homework 2

Description

1. For f ∈ C²[x₀,x₁] and x ∈ (x₀,x₁), linear interpolation of f at x₀ and x₁ yields

.

Consider the case

• Determine ξ(x) explicitly.
• For x ∈ [x₀,x₁], find maxξ(x), minξ(x), and maxf⁰⁰(ξ(x)).

1. Let ^P_m⁺ be the set of all polynomials of degree ≤ m that are non-negative on the real line,

P⁺_m = {p : p ∈ Pm, ∀x ∈ R,p(x) ≥ 0}.

Find such that p(x_i) = f_i for i = 0,1,…,n where f_i ≥ 0 and x_i are distinct points on R.

• Consider f(x) = e^x.
  • Prove by induction that

.

• From Corollary 3.17 we know

ξ ∈ (0,n) s.t..

Determine ξ from the above two equations. Is ξ located to the left or to the right of the midpoint n/2?

12. Consider f(0) = 5, f(1) = 3, f(3) = 5, f(4) = 12.
    • Use the Newton formula to obtain p₃(f;x);
    • The data suggest that f has a minimum in x ∈ (1,3). Find an approximate value for the location x_min of the minimum. V. Consider f(x) = x⁷.
    • Compute f[0,1,1,1,2,2].
    • We know that this divided difference is expressible in terms of the 5th derivative of f evaluated at some ξ ∈ (0,2). Determine ξ.
13. f is a function on [0,3] for which one knows that f(0) = 1,f(1) = 2,f⁰(1) = −1,f(3) = f⁰(3) = 0.
    • Estimate f(2) using Hermite interpolation.
    • Estimate the maximum possible error of the above answer if one knows, in addition, that f ∈ C⁵[0,3] and |f⁽⁵⁾(x)| ≤ M on [0,3]. Express the answer in terms of M.

• Define forward difference by

∆f(x) = f(x + h) − f(x),

∆^k+1f(x) = ∆∆^kf(x) = ∆^kf(x + h) − ∆^kf(x)

and backward difference by

∇f(x) = f(x) − f(x − h),

∇^k+1f(x) = ∇∇^kf(x) = ∇^kf(x) − ∇^kf(x − h).

Prove

∆^kf(x) = k!h^kf[x₀,x₁,…,x_k],

∇^kf(x) = k!h^kf[x₀,x₋₁,…,x_−k], where x_j = x + jh.

• ∗ Assume f is differentiable at x₀. Prove

.