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Numerical Analysis

David Gleich

Purdue University

Spring 2016

Course number CS-51400, MATH-51400

Tuesday and Thursday, noon-1:15pm

Location Lawson B155

Homework 4

Please answer the following questions in complete sentences in a typed manuscript and submit the solution on blackboard by on March 7th at noon. (These will be back before the midterm.)

Problem 0: Homework checklist

Problem 1: Gautschi Exercise 3.13

  1. Construct a trapezoidal-like formula which is exact for and .

  2. Does this formula integrate constants exactly?

  3. (Ignore part (b) if you are looking at the book.)

Problem 2: Gautschi Exercise 3.36

The Gaussian quadrature rule for the Chebyshev weight function is known to be: where Use this fact to show that the unit disk has area .

Problem 3: Monte Carlo, Gaussian-Legendre, and Clenshaw-Curtis Quadrature

In this problem, we'll study how different quadrature methods converge on a variety of problems. For a technical paper on this idea, see Trefethen, Is Gaussian Quadrature better than Clenshaw-Curtis? SIAM Review, 2008. In this problem, we'll be studying and reproducing Fig. 2 from that paper.

The functions are all defined on the region and are:

The quadrature methods are: