Homework # 4 Due March 3, 1999 Numerical Linear Algebra Math 4267 Spring

Use pencil and paper solutions throughout. I want DETAILS. No computer software is allowed except in problem 7a. Of course, you can check your results using such software. Write on one side of the paper only. Place in order assigned and staple results. Put all papers in a manila folder. 10 points per day excluding weekends for late penalty. 10 points penalty over the weekend.

(1) Use the system of equations from p 375, 5a which is viewed as Ax = b,

(a) Solve for x1, x2, and x3 using Gaussian elimination with 3 digit chopping with no pivoting. Call this solution x. This is exactly the problem you worked in Homework # 3,

(b) In order to illustrate iterative refinement, proceed as follows. Use 6 digit chopping and compute r = b - Ax, where x is the solution from a.

(c) Solve Ay = r using Gaussian elimination with 3 digit chopping with no pivoting.

(d) Is x + y using 3 digit chopping closer to the exact solution than x alone?

(2) Consider the matrices in problem 1e and 1f from p 415.

(a) Use a pencil and paper solution to determine if the two matrices are positive definite using theorem 6.24

(b) Use a pencil and paper solution to determine if the two matrices are strictly diagonally dominant.

(3) p 434; 1a, 1c, 4a, 6a, 9b applied to 4a

(4) p 434; 2a, 2b applied to 1a

(5) p 442; 1e

(6) p 442; 2e

(7) p 495, 1b. You will need to apply theorem 8.7 to find 2 functions which are needed to find a₀ and a₁ as mentioned in theorem 8.6, etc,

(a) Use PC-Matlab or similar software to plot both f(x) and linear least square polynomial P(x) on the same axes.

(b) Perform the following integration by hand.

(c) Redo (b) but change the constant term of P(x) by adding 0.1. Contrast the values of the 2 integrals in b and c and explain.