MASTER’S COMPREHENSIVE EXAM–NUMERICAL ANALYSIS AUGUST 1992

Instructions: Show all work in detail and numerical values to eight (8) figures at least, or exactly. Calculator required (minimum display eight (8) figures).

1. In an equally spaced table of a function f(x), with stepsize Δx_{k} = h, cubic interpolation in the central interval is to be applied at four succesive tabular points (x_{i}, y_{i}), with ≤ x_{i} = x_{0} + ih, i = 0, 1, 2, 3 . This means that we obtain a cubic interpolating polynomial p_{3}(x), that is to be evaluated for x_{1} < x < x_{2} only, hence the reference to the central interval. For the special case of f(x) = x^{4} determine exactly the largest possible error |e_{3}(x)| = |f(x) − p_{3}(x)| when x lies in the central interval.

HINT: Begin by stating an appropriate formula for the error term. Your result must be a function of the stepsize h.

2. To within 8 figure accuracy at least, compute a real root of the equation f(x) = x^{5} − 4x^{4} + 9x^{3} − 11x^{2} + 6x − 2 = 0

by any method.

3. Data (x_{i}, y_{i}), i = 0, 1, …, 5, tabulated below (at right) are equally spaced abscissae and corresponding y-values of a certain function y = f(x).

a) Compute the table of first and second differences Δy_{i}, Δ^{2}y_{i}, i = 0, …, 4 (resp. 3) .

b) From the 2nd differences that you have computed,

prove the inequality

0.021 ≤ max_{1≤x≤2} |f′′(x)|.

Here, f′′(x) is the 2nd derivative of the function y = f(x), that is tabulated at right.

x  y = f(x)_{0.0  0.23607}_{0.2  0.28035}_{0.4  0.32379}_{0.6  0.36643}_{0.8  0.40832}_{1.0  0.44949}

4. The three-point open Newton-Cotes quadrature formula is given by the equation

∫_{−1}^{1} f(x) dx = A_{1}f(−\frac{1}{2}) + A_{2}f(0) + A_{3}f(\frac{1}{2}) + E_{3}^{NC^{op}}.

The error term is known to satisfy E_{3}^{NC^{op}} = c_{4}f^{(4)}(x̃), −1 < x̃ < 1.

a) Determine the weights A_{k}, k = 1, 2, 3.

HINT: Substitute for f(x) several functions such that f^{(4)}(x) ≡ 0, to obtain three equations in three unknowns, noting that

f^{(4)}(x) ≡ 0 ⇒ E_{3}^{NC^{op}} = 0.

b) Compute the error constant c_{4} from the fact that f(x) = x^{4} ⇒ f^{(4)}(x) ≡ 24.