Numerical Analysis I, Fall 2017 (http://www.math.nthu.edu.tw/˜ wangwc/)

Quiz 05

Dec 22, 2017.

1. Use any desingularization method to evaluate the improper integral

∫_{0}^{1} x^{\frac{−1}{3}} e^{x} dx

with a composite quadrature rule of 4th order accuracy. Demonstrate numerically that the result is indeed 4th order accurate.

2. Let A be an N^{2} × N^{2} matrix with a_{ij} = 0 except for −N ≤ i − j ≤ 2N (only 3N + 1 diagonals have nonzero entries). Suppose that Ax = b can be solved using Gaussian elimination without pivoting. Find the leading order operation count (multiplication/division only) CN^{p} for the elimination and backward substitution, respectively. Give all details.

3. Write a pseudo-code for Gaussian elimination, assuming no pivoting is needed. Then another pseudo-code for backward substitution. Partial credits is nearly impossible for incorrect algorithms. Check carefully.

4. Perform the required row interchanges for the following linear system using scaled partial pivoting:

\begin{aligned}& x_{2}+ & x_{3} = & 6 \\ x_{1}− & 2x_{2}− & x_{3} = & 4 \\ x_{1}− & x_{2}+ & x_{3} = & 5 \end{aligned} ,  (1)

Give all details and the final linear system. Need not solve it.

5. Find P, L, U in the factorization P A = LU for the matrix

A = ( \begin{matrix}0 & 0 & −1 & 1 \\ 1 & 1 & −1 & 2 \\ −1 & −1 & 2 & 0 \\ 1 & 2 & 0 & 2 \end{matrix} )  (2)

where P is the permutation matrix corresponding to partial pivoting.