Applied Matrix Theory - Math Summer 09 MATLAB Project 3 Created by Prof. Diego Maldonado and Prof. Virginia Naibo

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1 Applied Matrix Theory - Math Summer 09 MATLAB Project Created by Prof. Diego Maldonado and Prof. Virginia Naibo In this project you will be introduced to MATLAB programming. What you have to submit: Submit the required MATLAB codes from TASK III and TASK IV (M-files) electronically by to your lab instructor. Please use the names YourLastnameFirstnameIII.m and YourLastnameFirstnameIV.m, respectively, and write Lab, your first name and last name in the subject of your . Important notice: Lab assignments must be submitted using your KSU address. Lab assignments submitted from a non-ksu address will not be considered. Files with an incorrect extension will not be considered. TASK I: Introduction to M-files and MATLAB flow control statements. This task consists mostly of reading material and a couple of exercises, so that you get familiarized with some of MATLAB programming language. Keywords: M-file, if, else, elseif, for, while MATLAB M-files can be scripts that execute a series of MATLAB statements or can be functions that can accept arguments and can produce one or more outputs. M-files are created using a text editor and are stored with a name of the form filename.m. You can use the MATLAB editor to write your M-files: open MATLAB and type edit in the prompt. Example 1 (M-file script): lines.m % lines.m gives the graphs of the lines % 2y+x=2 and y+x= for x between -2 and 5 on the same window. % This is what you did in TASK III of Lab 2, line by line. x=-2:0.1:5; y1=1-0.5 x; y2=-x; plot(x,y1,x,y2) grid on; Remark: Use the symbol % to make comments in your script. Everything in a line following the symbol % is ignored by MATLAB when the file is executed. Exercise. Download lines.m. Then type

2 >> help lines What do you get? To run the file type >> lines To edit the file type >> edit lines The MATLAB editor will pop up and you will be able to see the code. Example 2 (M-file function): subim.m function[s] = subim(a,m,n,rx,cy) % SUBIM Extracts a subimage from a given image. % The function s syntax is subim(a,m,n,rx,cy). % The original image is represented by the matrix A. % The subimage is of size m-by-n, and the coordinates % of its top, left corner are (rx,cy). s=a(rx:rx+m-1,cy:cy+n-1); imshow(s); Remark: Note that the name of the file before.m coincides with the name of the function appearing in the first line of the code. Exercise. Download the file subim.m. Type >> help subim What do you get? Download goldhill.jpg and apply the function subim to this image. Some flow control statements that we will be using: if statement. if, together with else and elseif, executes a group of statements based on a specified logical condition. Syntax: if expression statements or if expression1 statements1 elseif expression2 statements2 else statements 2

3 for statement. for executes a group of statements a specified number of times. Syntax: for index=start:increment: statements while statement. while executes a group of statements an indefinite number of times, based on a specified logical condition. Syntax: while expression statements TASK II. Analyzing an M-file. Keywords: ceil, disp, while, input, error, if, else, elseif Relational operators: <, <=, >, >=, ==, = Logical operators: &,, Download guess.m. Find out what guess.m does, then run guess.m, and finally, edit guess.m. Study this file. Try to understand the language code. For your convenience, the code of guess.m is in the appix of these instructions. TASK III. Write an M-file function, based on Theorem 1., page 27, of the textbook, that tells whether a system of linear equations has no solution, unique solution, or infinitely many solutions. The code should be such that you enter the coefficient matrix A and the corresponding column vector b. If the number of rows of A is different from the number of entries in b, the output should be an error message and execution stops. Otherwise, the output is a message indicating whether the system has no solution, unique solution, or infinitely many solutions. In this last case, it also tells the number of free variables of the system. In the case of unique solution, the code should also give the solution. Name your file YourLastnameFirstnameIII.m To check what your code should do, download the pseudo-code labtask.p into MATLAB. Type >> labtask(a,b) where A and b are the pairs of matrices given in the appix. Try all the pairs of matrices given in the appix, to see all the different outputs. Your code should work exactly as labtask.p. Remark: You will not be able to edit labtask.p because it is pseudo-code. Hint: Use Theorem 1., to establish the three possible cases. error, disp, rank, rref. Use if, elseif, else,

4 TASK IV. Warm up for task IV: (a) Use MATLAB to find the equation of the cubic y = ax + bx 2 + cx + d that passes through the points (2, 2), ( 1, 9), ( 2, 2), and (1, 2). This is, you have to solve for a, b, c and d in the following linear system: 2 a b + 2 c + d = 2 ( 1) a + ( 1) 2 b + ( 1) c + d = 9 ( 2) a + ( 2) 2 b + ( 2) c + d = 2 1 a b + 1 c + d = 2 (b) Use MATLAB to plot the cubic obtained in (a) using a range of x that contains the x-components of the given points. Syntax: x=-:0.1:; plot(x, a*x. +b*x. 2+c*x+d) (c) Let (x 1, y 1 ), (x 2, y 2 ), (x, y ), (x, y ) be points in the plane. Let y = ax + bx 2 + cx + d be a cubic curve passing through all points (assuming there is one). What system of linear equations must be solved in order to find the coefficients a, b, c, and d? Identify the corresponding augmented matrix. (d) Let (x 1, y 1 ), (x 2, y 2 ), (x, y ), (x, y ) be points in the plane. Identify the cases in which there is only one cubic curve going through all points, infinitely many cubic curves going through all points, and no cubic curve going through all points. What you have to do in task IV: Write an M-file function that takes the coordinates of four points in the plane, and tells whether there is exactly one, infinitely many, or no cubic curve passing through all points. When there is exactly one curve, the program solves the required linear system for the coefficients a, b, c and d of the cubic polynomial passing through the points, and plots the cubic curve y = ax + bx 2 + cx + d using a range of x that contains the x-components of the given points. The input should be a 2 matrix, where each row corresponds to the coordinates of a point (first column corresponding to x-components and second column corresponding to the y-components). Save your code under the name YourLastnameFirstnameIV.m To check how your code should work apply the pseudo-code labtask.p to the following groups of points: (2, 2), ( 1, 9), ( 2, 2), (1, 2). (2, 2), (2, 9), ( 2, 2), (1, 2).

5 (2, 2), ( 1, 9), ( 2, 2), ( 2, 2). The syntax is (using the first group of points): >> labtask([2 2; -1 9; -2 2; 1 2]) Hint: Adapt the code that you wrote in TASK III. In the case of unique cubic use the commands from the warm up to plot the curve. Use the following range for x: Calling C the matrix of given points x=min(c(:,1))-1:0.1:max(c(:,1))+1; 5

6 APPENDIX. The code of guess.m is in the next page. Matrices to use with labtask.p A1= b1= A2= A= A= A5= b2= b= b= [ ] b5= 6