MATLAB Assignment 8

Introduction to Linear Algebra (Weeks 11 and 12) Spring,2018

1. MATLAB Submission Problem 3 ( Due Date : May 24 (Thu) )Referring to the instruction below, you are required to submit thisproblem.

A common problem in experimental work is to find a curve y =f(x) of a specified form corresponding to experimentally determinedvalues of x and y, say

(x1, y1), (x2, y2), · · · , (xn, yn). The followings are thefour important models in applications.

• Linear line model (y = ax + b)
• Exponential model (y = aebx)
• Logarithmic model (y = a + b ln x)

A function file LS_solver.m is to fit given experimental data tothe proper mathematical model using least squares method. Thefunction file given as follows:

   % --------- function file \"LS_solver.m\" --------- %   % input data: x, y and opt

   %  if opt=1, linear model (y=a*x+b)

   %  if opt=2, exponential model (y=a*exp(b*x))

   %  if opt=3, logarithmic model (y=a+b*ln(x))   function [a, b]=LS_solver(x, y, opt)

     [m1, n1]=size(x);  [m2, n2]=size(y); % Size of the input data.     xx=linspace(min(x), max(x), 100); % xx will be used to plot the fitting curve.

     if (m1~=1)||(m2~=1)||(n1~=n2)  % If the input data size is not proper,       fprintf(’Error: Improper input data.\n’);  % error message.

     elseif (opt==1)||(opt==2)||(opt==3) % option = 1, 2, 3.       figure; plot(x, y, ’o’); % Plot the given data points.       hold on; % Ready to draw the next graph.       switch opt

         case 1 % Linear model           fprintf(’Linear model\n’);           % ---- Complete here ---- %           a=sol(1); b=sol(2); % Fitting constants a and b.           plot(xx, a*xx+b); % Plot the fitting curve with a and b.           title(’Linear model (y=a*x+b)’);

         case 2 % Exponential model           fprintf(’Exponential model\n’);           % ---- Complete here ---- %

      case 3 % Logarithmic model        fprintf(’Logarithmic models\n’);        % ---- Complete here ---- %

    end    hold off; % no more graph.

  else % for invalid [opt]    fprintf(’Error: Improper option value.\n’); % error message.    return; % Return the process.

end end

If you execute the following MATLAB commands:

>> x=[2 3 4 5 6 7 8 9]; y=[1.75 1.91 2.03 2.13 2.22 2.30 2.37 2.43];>> [a b]=LS_solver(x, y, 1)

Then, you may obtain the following results with the figure (SeeFigure 1 below):

Linear model a=

0.0948

b=
1.6213

Figure 1: Execution result

(a) Download the function file LS_solver.m in KLMS and completethe missing parts.

(b) Use LS_solver.m to fit an exponential model to the followingdata (Table 1), and

graph the curve and data points in the same figure.
Table 1: Data points of Problem 1-ii (exponential model)

(c) Use LS_solver.m to fit a logarithmic model to the followingdata (Table 2), and graph the curve and data points in the samefigure.

Table 2: Data points of Problem 1-iii (logarithmic model)

You may use the backslash operator in MATLAB (syntax : A \ b fora linear system Ax = b) and refer to the T5 and T7 in Section 7.8of the textbook.

Submission Guidelines.

Save your resulting images as Id_b.fig and Id_c.fig,respectively.

Upload your m-file, resulting images and hardcopy solution (pdfformat is recom-

mended) on the Homework Box for MATLAB Submission Problem 3.

Print out your hardcopy solution and submit it to yourrecitation TA at the beginning

of your recitation class.

Add comments in your m-file using %.

Late submission will not be allowed.

2. Read the attachments “MATLAB Week11.pdf”, “MATLAB Week12.pdf”and practice by yourself.

x	0	1	2	3	4	5	6	7
y	3.9	5.3	7.2	9.6	12	17	23	31

x	2	3	4	5	6	7	8	9
y	4.07	5.30	6.21	6.79	7.32	7.91	8.23	8.51