# Numerical analysis questions

1.

Construct explicitly the interpolating polynomial of degree 4 or less which interpolates the data (0, 0),(1, 1),(−1, 5),(4, 0),(2, 2)? (b)

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How many polynomials of degree exactly 5 interpolate the above data? If none, explain. If there is only one, construct it explicitly. If there are more than one, construct explicitly at least a couple of them. Be sure to explain your answer.

2.

Propose a fixed point iteration to solve 1 − x ^ 2/4 = 0. Do you expect fixed point iteration to calculate the root? Explain your answer.

3.

Recall that the inverse of a 2 × 2 matrix A = a b c d is given by A−1 = 1 ad−bc d -b -c a . Choose an initial guess for solving the system f1(u, v) = u 2 − 2u − v + 0.5 = 0 f2(u, v) = u 2 + 4v 2 − 4 = 0, and give the next step of Newton’s method.

4.

Show that f 0 (x) = 1 /12h − f(x + 2h) + 8f(x + h) − 8f(x − h) + f(x − 2h) + O(h 4 ). HINT: Use a Taylor series expansion to a sufficiently high order

5.

Find the degree of precision of the rule integral from x3 to x0 f(x)dx ≈ 3h 8 f(x0) + 3f(x1) + 3f(x2) + f(x3) , where h = x1 − x0 = x2 − x1 = x3 − x2. Be sure to explain your answer. HINT: Compute R 3 0 x kdx, k = 0, 1, . .

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