Homework # 2 Due Feb 3 in class

Note: You must turn in

if you are using PC-Matlab

(1) a labeled, virus free disk with directory hw# containing 2 m-files for parts (1) and (2). I will run these m-files and read your output.

(2) a hard copy of the source code files and and all graphs.

If you use C++,

(1) turn in all your files on a disk and

(2) a hard copy of the source code files and and all graphs.

From class, we have discussed how to solve a system of linear equations AX = B where the matrix A is a tridiagonal matrix. That is, all entries are zero except the main diagonal as well as the super diagonal and subdiagonal. We used both C++ and PC-Matlab to solve such a system.

Use C++ or PC-Matlab to do the following.

(1) Find the solution of a system EX = B where E is a tridiagonal matrix where the diagonal D has all 3's on it, all 1's on the superdiagonal, and all 2's on the subdiagonal and zeroes elsewhere where E is 30x30. Also B is 30x1 and consists of 1, 2, 3, ..., 30. If you use MatLab, you can write N = 30; D= 3*ones(N,1); etc.

(2) We wish to approximate the solution to the differential equation

y'' + y = sin(x), y(0) = 1, y(pi/2) = 0. This requires the solving of AX = B where A is a tridiagonal matrix.

Due to our earlier work, we can use from ftp://www.etsu.edu/kerleyl/nla/ trisys.m in PC-Matlab plus another m-file that you need to construct

use a modification of the C++ programs tri.h, trimain.cpp, and tri.cpp.

First we need to determine what A and B are. Then from this, we can determine Sup, D, Sub, and B as was discussed in class and also mentioned in trisys.m in PC-Matlab. Study p . 641 to determine what A and B are. DO NOT TRY TO WRITE THE ALGORITHM 11.3 discussed on pages 641-642. Do not reinvent the wheel. Use the programs from the Web that I just mentioned. Note h = (b-a)/(N+1) where a = 0, b = pi/2, and N = number of interior points of [a,b]. Use N = 500. Then x_i = a + ih, i = 0, 1, ..., N + 1. We will solve for w_i where w_i = approximation at x_i where i = 1,...,N. Observe that we know x₁ and x_N+1.

Write 3 columns for your output. Column 1 contains x_i, column 2 contains the solution to the system AX = B and column 3 contains the exact solution which is (1-x/2) cos(x). In PC-Matlab, Out=[X1 X E] will produce a matrix of 3 columns as needed where X1 describes x_i and X is approximate solution, and E is exact solution. However don't write out 500 rows. Find a way to print out say every 10th row. Hint: Short = Out(1:10:500,:) constructs a matrix which extracts row 1, row 11, row 21, etc from Out. Also the following code is helpful for displaying attractive output.

disp(' x Approx Exact')

disp(' ')

disp(Short)

Caution: C = A*B will find the matrix product of 2 matrices whereas E= A.*B finds pointwise product of the elements of the matrices A and B.

Now plot columns 1 and 2 on one graph and columns 1 and 3 on a second graph. This is easy in PC-Matlab. If you use C++, use a spreadsheet or similar software to produce a plot.