# hw3 - Homework 3 Math 104B, Winter 2011 Due on Thursday,...

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Homework 3 – Math 104B, Winter 2011 Due on Thursday, February 10th, 2011 Section 7.1: 3.d, 4.d, 5.a, 7, 9, and 11. Section 7.2: 4.d, 8.a, 9, 10, 13, 14, 16, and 17. Additional Problem 1: Given a matrix A R n × n , prove that b A b 1 = max 1 j n n s i =1 | a ij | . Additional Problem 2: Consider the tridiagonal matrix A = ( a i,j ) 1 i,j n given by a i,j = 1 h 2 | i j | = 1 2 h 2 i = j 0 Otherwise (1) obtained when the following ODE, u ′′ ( x ) = f ( x ) , x [0 , 1] u (0) = u (1) = 0 , (2) is discretized using second order centered diFerences: u i +1 2 u i + u i 1 h 2 = f ( x i ) i = 1 , 2 , . . . , n, (3) where h = 1 / ( n + 1). 1. ±or each k = 1 , 2 , . . . , n , show that the vector u ( k ) given by u ( k ) i = sin p πki n + 1 P , i = 1 , 2 , . . . , n (4) is an eigenvector of the matrix A , and determine the corresponding eigenvalue λ k . 2. Set up the Jacobi iteration for system (3), and show that the vectors

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## This note was uploaded on 12/27/2011 for the course MATH 104b taught by Professor Ceniceros,h during the Fall '08 term at UCSB.

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hw3 - Homework 3 Math 104B, Winter 2011 Due on Thursday,...

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