MATH 417: Numerical Analysis
Instructor: Prof. Wolfgang Bangerth
Teaching Assistants: Dukjin Nam
Homework assignment 6 – due 3/22/2007
Problem 1 (Jacobi iteration). Let A, b be the 100 ×100 matrix and 100-
dimensional vector defined by
Aij =
2.01 if i=j,
−1 if i=j±1,
0 otherwise,
bi=1
100 sin 2πi
50 .
Apply Jacobi’s method to solving Ax =b. Write a program that implements
the Jacobi method and start with a vector x0with randomly chosen elements in
the range −1≤(x0)i≤1 (i.e. with elements generated from what the rand()
function or a similar replacement returns).
(Hint: It is not necessary to actually store the complete matrix just to
multiply with it. Rather, use that the i-th component of the vector Ay is
(Ay)i=Pn
j=1 Aijyj= 2.01yi−yi−1−yi+1 at least for 2 ≤i≤n−1, and
obvious modifications for j= 1 and j=n.)
Run 200 Jacobi iterations and plot the values of (x(k))iagainst ifor every
few iterations, for example k= 0,2,5,10,20,50,100,200. What do you observe?
(5 points)
Problem 2 (Alternative vector norms). Let Ab e a symmetric and posi-
tive definite n×nmatrix. Show that
kxkA=√xTAx
is a norm for vectors x∈Rn. (Hint: Use the eigenvalue and eigenvector decom-
position of symmetric positive definite matrices.)
(3 points)
Problem 3 (Jacobi iteration). Solve problems 7.3.1 a) and b) of the book
(using paper and pencil). (3 points)
Problem 4 (Gauss-Seidel iteration). Solve problems 7.3.3 of the book
(using paper and pencil) for parts a) and b). (3 points)
Happy spring break!
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