MASTER’S COMPREHENSIVE EXAM–NUMERICAL ANALYSIS AUGUST 1994 INSTRUCTIONS: Calculator required for some problems. Show at least eight (8) figures where numerical results are required.

1. This problem concerns the relationships between mth–order divided differences f[x_{k}, x_{k+1}, …, x_{k+m}], and mth differences Δ^{m}f(x_{k}).

a) State a suitable definition of the mth–order divided differences f[x_{k}, x_{k+1}, …, x_{k+m}].

b) State the recurrence formula expressing an mth–order divided difference f[x_{k}, x_{k+1}, …, x_{k+m}] in terms of (m − 1)st–order divided differences.

c) For the case of equidistant data, i.e., x_{k} = x_{0} + k h, h = Δx_{k}, k = 0, 1, 2, …, state the recurrence formula expressing the mth–order difference Δ^{m}f(x_{k}) in terms of (m − 1)st–order differences.

d) For the case of equidistant data, state an equation expressing the divided difference f[x_{k}, x_{k+1}, …, x_{k+m}] in terms of the difference Δ^{m}f(x_{k}).

e) Prove your equation in part d) above from the facts in parts a)–d).

M O R E  O N  B A C K  • • •

2. Consider the iteration

(∗),  x_{n+1} = g(x_{n}), n = 0, 1, …,

where g(x) is a suitable twice continuously differentiable real function defined on a real interval I = [a, b], a < b.

a) State a definition for the iteration (∗) to converge quadratically to a fixed point x_{∗} = g(x_{∗}) ∈ I. Here, we always assume that |x_{0} − x_{∗}| si sufficiently small.

b) State a sufficient condition for quadratic convergence of the iteration (∗) in terms of properties of the values g(x_{∗}), g′(x_{∗}), g′′(x_{∗}), ….

c) To solve the equation f(x) = 0, where f(x) is a suitable twice continuously differentiable real function with f′(x) ≠ 0 on a real interval I = [a, b], define the iteration function

g(x) = x − a ∗ f(x),

where a is a suitable real numerical constant. For what value of the constant a is the convergence quadratic? Prove your answer!

3. In applying the Composite Trapezoidal Rule T_{N} to approximate the integral

∫_{0}^{1} sin(x^{2}) dx, estimate as accurately as possible the number N of subintervals that are

required to achieve four (4) decimal accuracy.