Programming Assignment Three

Your work should be presented in the form of a typed report using clear and properly punctuated English. Where appropriate include full program listings and output. If you choose to work in a group of two, please turn in independently prepared reports.

$\hangindent\parindent \item{3a.} Let $A$ be the $n$ by $n$ tridiagonal matrix with entries $a_{ii}=-2$ for $i=1,2,\ldots n$ and $a_{j,j+1}=a_{j+1,j}=1$ for $j=1,2,\ldots n-1$. Write a program that uses the power method to find $\rho(A)=|\lambda|$ where $\lambda$ is the eigenvalue of largest absolute value that solves the eigenvalue problem $A\xi=\lambda\xi$. Use your program to approximate $\rho(A)$ to three significant decimal digits for matrices $A$ with sizes $n=1, 2,\ldots, 45$. \bigskip \item{b.} Make a conjecture as to how $\rho(A)$ depends on $n$. Can you prove your conjecture analytically? \bigskip \item{c.} Consider the linear model $$ F(x)=\sum_{i=0}^3 c_i x^i. $$ Use the least squares method to find values of $c_i$ such that $F(n)\approx \rho(A)$ for $n=1,2,\ldots, 45$. \bigskip \item{d.} Plot $\rho(A)$ and $F(n)$ on the same graph. What can be said about the goodness of fit.$

Last updated: Thu Sep 26 14:35:24 PDT 2002