MASTER’S COMPREHENSIVE EXAMS NUMERICAL ANALYSIS JANUARY 1999

INSTRUCTIONS: Calculator required. Show eight or more decimal figures. Two problems completely solved will be sufficient for a passing grade.

1. Let ϵ and K be positive numbers with 0 < K < 1 and L = \frac{1}{1−K}. Prove the following Lemma: Given (real or complex) numbers δ_{j}, j = 1, …, n, with |δ_{j}| ≤ ϵ, j = 1, …, n, define α_{n} by the equation 1 + α_{n} = ∏_{j=1}^{n}(1 + δ_{j}). If nϵ ≤ K then |α_{n}| ≤ L nϵ. Hint: Use mathematical induction. The Lemma, which is used for bounding roundoff errors, is true in the stronger version with L = 1 + K but the proof is slightly more difficult.

2. Solve approximately the equation x = g(x) for the function

g(x) = ln x − \frac{x}{3} + 3

in the interval 2 ≤ x ≤ 4 by means of the iteration x_{n+1} = g(x_{n}), n = 0, 1, …, with x_{0} = 3.0

a) Compute explicitly x_{1}, x_{2}, x_{3}.

b) Apply the contraction mapping theorem (or, its proof and corollaries) to obtain an accurate upper bound for the error |x_{3} − x_{∗}|, where x_{∗} is the exact solution of the equation x = g(x) in the interval.

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3. Let f(x) be a sufficiently differentiable function on [a, b].

a) State the composite trapezoidal rule for ∫_{a}^{b} f(x) dx.

b) Give a formula for the error. Explain.

c) Use the composite trapezoidal rule to estimate ∫_{0}^{1} e^{−x^{2}}dx to within 0.005.

d) For the value of h found in part c) above, calculate an asymptotic error estimate. Use this to give a corrected value (this is sometimes referred to as the corrected trapezoidal rule).