Programming Assignment Five

$\def\R{{\bf R}} \hangindent\parindent \item{7a.} Find a standard subroutine for solving tridiagonal systems of equations. Use your mouse to connect to the National Institute of Standards and Technology web page at{\tt\ http://math.nist.gov/}. From this page select the following links in order: \medskip \itemitem{1.} Guide to Available Mathematical Software \itemitem{2.} What Problem it Solves \itemitem{3.} Taxonomy as a Decision Tree \itemitem{4.} D Linear Algebra \itemitem{5.} D2 Solution of Systems of Linear Equations \itemitem{6.} D2a Real Nonsymmetric Matrics \itemitem{7.} D2a2 Banded \itemitem{8.} D2a2a Tridiagonal \medskip \hangindent\parindent How many routines solve the system of linear equations $Ax=b$ where $A$ is a tridiagonal matrix? Download one of them and print it out. \medskip \item{7b.} Consider the two-point linear boundary problem $$ \cases{ y''+\cos(x)y'-(x^2+1)y=2\cr y(-2)=y(2)=-1.\cr } $$ Find conditions on the grid size $h$ such that the corresponding central difference method is guaranteed to have a unique numerical solution. \medskip \item{7c.} Use the central difference method to solve the above boundary problem for grid sizes of $h=4/n$ where $n=4, 8, 16, 32, \ldots, 256$. Plot your solutions. Do they converge? \medskip \item{7d.} Repeat for the homogeneous equation $$ \cases{ y''+\cos(x)y'-(x^2+1)y=0\cr y(-2)=y(2)=-1.\cr } $$ Do solutions to this equation have the weak min-max property?$

Last Updated: Fri Nov 22 10:02:41 PST 2002