Numerical Analysis I, Fall 2014 (http://www.math.nthu.edu.tw/˜ wangwc/)

Midterm 01

Oct 31, 2014. Hand in your code for problem 1, 2, 3, 6 for possible partial credit in case your numerical answer in wrong.

1. (20 pts) Use any method to find a solution of (1 + x)^{1/3} − (1 − x)^{1/3} = 10^{−10} to 10 correct digits.

2. (12 pts) Find p_{20} to 10 correct digits if p_{0} = 1, p_{1} = 0.9, p_{n} = 4p_{n−1} − 2.79p_{n−2}.

3. (10+6 pts)

(a) Give a locally convergent fixed point iteration for solving f(x) = x − 3 sin(x) − 0.01 = 0. Give the formula and find a solution to 10 correct digits.

(b) Give an upper bound for the number of steps it takes to reach |x_{n} − x^{∗}| < 10^{−6} with x_{0} = 1.

4. (12 pts) Derive Aitken’s Δ^{2} method from the assumption

\frac{p_{n+1} − p}{p_{n} − p} ≈ \frac{p_{n+2} − p}{p_{n+1} − p}

5. (12+12 pts)

(a) Give a cubically convergent method to solve for e^{x} − 1 = 0. Give the formula and prove that it is cubically convergent (locally). If you cannot do it, do the same for a quadratically convergent method for partial credit.

(b) Find α and λ (the constants in the definition of order of convergence) analytically for your method if it is applied to solve the equation e^{x} − x − 1 = 0 instead. If you cannot do it, do the same for a quadratically convergent method for partial credit (no matter whether you did cubically or quadratically convergent method in (a) ).

6. (20 pts) Find P (0.3) where P (x) is the Lagrange polynomial interpolating the data (0, sin 0), (0.25, sin 0.25), (0.5, sin 0.5), (0.75, sin 0.75), (1, sin 1). Give formula and find the numerical value.

7. (12 pts) Suppose that we are to construct a piecewise polynomial interpolation S(x) on the data (x_{0}, f(x_{0})), (x_{1}, f(x_{1})), ⋯, (x_{n}, f(x_{n})), with additional continuity conditions for S′, S′′ and S′′′ on the interior nodes x_{1}, ⋯, x_{n−1}. If we use polynomials of the same degrees on each of the interval [x_{0}, x_{1}], ⋯, [x_{n−1}, x_{n}], what is the minimal degree needed in each interval? How many additional end conditions are needed? Explain.