# hw7 - Math 471 Fall 2007 Homework 7 Assigned Friday Due...

This preview shows pages 1–2. Sign up to view the full content.

Math 471, Fall 2007 Homework 7 Assigned: Friday, October 26, 2007. Due: Friday, November 2, 2007 . 1. (Positive Definite Matrices, page 221, #9) . Let A be an n × n symmetric positive definite matrix. (a) Show that a ii > 0 for each i = 1 , 2 , 3 , · · · , n . (b) Show that a 2 ij < a ii a jj for each i 6 = j . Some hints: Establish the 2 × 2 case. Then reduce the general problem to this special case using the definition of positive definiteness. 2. (Poisson BVP, page 741, #7) Consider the Poisson problem 2 u ∂x 2 + 2 u ∂y 2 = 0 on R = { ( x, y ) | 0 < x < 1 , 0 < y < 1 } u ( x, 0) = 0 , u ( x, 1) = 1 (1 + x 2 ) + 1 , u (0 , y ) = y 1 + y 2 , u (1 , y ) = y 4 + y 2 , whose exact solution is u ( x, y ) = y (1 + x ) 2 + y 2 . (a) Taking N = M = 4, set up and solve the corresponding system of finite differ- ence equations. (b) Numerically verify the second-order accuracy of the numerical method. 3. (Iterative Methods: Analysis). Consider a linear system with matrix A = 2 1 1 4 . (a) Write down the iteration matrices T jac and T gs for Jacobi’s Method and Gauss– Seidel.

This preview has intentionally blurred sections. Sign up to view the full version.

View Full Document
This is the end of the preview. Sign up to access the rest of the document.

{[ snackBarMessage ]}

### What students are saying

• As a current student on this bumpy collegiate pathway, I stumbled upon Course Hero, where I can find study resources for nearly all my courses, get online help from tutors 24/7, and even share my old projects, papers, and lecture notes with other students.

Kiran Temple University Fox School of Business ‘17, Course Hero Intern

• I cannot even describe how much Course Hero helped me this summer. It’s truly become something I can always rely on and help me. In the end, I was not only able to survive summer classes, but I was able to thrive thanks to Course Hero.

Dana University of Pennsylvania ‘17, Course Hero Intern

• The ability to access any university’s resources through Course Hero proved invaluable in my case. I was behind on Tulane coursework and actually used UCLA’s materials to help me move forward and get everything together on time.

Jill Tulane University ‘16, Course Hero Intern