Numerical Analysis I, Fall 2010 (http://www.math.nthu.edu.tw/˜ wangwc/)

Quiz 06

Dec 31, 2010.

1. Find constants a, b, c and d so that the quadrature ∫_{−1}^{1} f(x)dx = af(−1) + bf(1) + cf′(−1) + df′(1) reaches maximal degree of precision.

2. Derive Gaussian quadrature for ∫_{−1}^{1} f(x)dx for the case n = 2 (two nodes).

3. Find order of convergence of Midpoint rule applied directly (no subtraction of singular part, change of variable, etc) to the integral ∫_{0}^{1} x^{\frac{−1}{3}} dx. You can find the order either numerically or analytically. Give details.

4. Find 10 correct digits of ∫_{0}^{1} \frac{e^{x}}{\sqrt{x}}dx.

5. Estimate number of multiplications/divisions needed for the elimination part of Gauss elimination for an n × n matrix. Give leading order with correct coefficient (i.e. Cn^{p}, find C and p).

Numerical Analysis I, Fall 2010 (http://www.math.nthu.edu.tw/˜ wangwc/)

Quiz 06

Dec 31, 2010.

1. Find constants a, b, c and d so that the quadrature ∫_{−1}^{1} f(x)dx = af(−1) + bf(1) + cf′(−1) + df′(1) reaches maximal degree of precision.

2. Derive Gaussian quadrature for ∫_{−1}^{1} f(x)dx for the case n = 2 (two nodes).

3. Find order of convergence of Midpoint rule applied directly (no subtraction of singular part, change of variable, etc) to the integral ∫_{0}^{1} x^{\frac{−1}{3}} dx. You can find the order either numerically or analytically. Give details.

4. Find 10 correct digits of ∫_{0}^{1} \frac{e^{x}}{\sqrt{x}}dx.

5. Estimate number of multiplications/divisions needed for the elimination part of Gauss elimination for an n × n matrix. Give leading order with correct coefficient (i.e. Cn^{p}, find C and p).