Master’s Comprehensive Exam - NUMERICAL ANALYSIS - Aug. 1989 INSTRUCTIONS: You may use a calculator. Show eight digits in all numerical values that you are required to compute.

1. For a certain computer with 54–bit binary arithmetic the floating point conversion operator fl satisfies

fl(a) = a(1 + δ_{a}), |δ_{a}| ≤ ϵ, ϵ = 2^{−54}.

Given floating point numbers x, y, z define E = xy + z, an exact mathematical expression. Define F to be the result of computing E in floating point arithmetic.

a) State an expression for F in terms of the data x, y, z and the floating point operator fl.

b) Prove the rounding error may be estimated by |E − F | ≤ 1.01ϵ(2|xy| + |z|).

2. Determine the Hermite cubic polynomial p_{3}(x) for f(x) = sin x at the two points x = 0, \frac{π}{2} , i.e., the cubic polynomial p_{3}(x) is to satisfy the conditions p_{3}(x) = f(x) and p′_{3}(x) = f′(x) for x = 0 and x = \frac{π}{2} .

3. a) State a definition of the composite Simpson rule S_{2n} for approximating ∫_{a}^{b} f(x)dx based on evaluation of f(x) at the 2n + 1 equally spaced nodes x_{k}, k = 0, …, 2n, where x_{0} = a, x_{2n} = b.

b) State an estimate for the error E_{2n}^{S} = | ∫_{a}^{b} f(x)dx − S_{2n}| in terms of the stepsize h = x_{k} − x_{k−1}. [Choose from several different theorems, all involving derivatives of f(x).]

c) Apply the above estimate of E_{2n}^{S} to ∫_{0}^{π} sin xdx when n = 100.

4. Let the matrix A = ( \begin{matrix}1 & .99 \\ .98 & .97 \end{matrix} ). Solving Ax = b with b = ( \begin{matrix}1.99 \\ 1.95 \end{matrix} ) we obtain the

solution vector x = ( \begin{matrix}1 \\ 1 \end{matrix} ), while the right–hand side vector b = ( \begin{matrix}1.98 \\ 1.96 \end{matrix} ) gives a very different result.

a) Explain the dicrepancy between the two solution vectors by means of a suitable inequality that involves a condition number K of A.

b) How large is the condition number K for this matrix? Is its magnitude sufficient to explain the above discrepancy? Your answer should be in quantitative terms.