% 16.01 / 16.02 Unified Engineering, Fall 2006
% Introduction to MATLAB: Matrix Computations
% Instructor: Violeta Ivanova, MIT Academic Computing, violeta@mit.edu

% EXERCISE 2 System of Linear Equations
% MATLAB vectors, matrices, matrix operations, curve fitting, eigenvalues, plotting. 

% The coordinates (x1,y1), (x2,y2), (x3,y3) and (x4,y4) of four points are
% given: (1, 54), (5, 82), (8, 40), and (10, 10).
% Find a cubic equation y(x) that passes through the four points i.e.:
% y = c1*x^3 + c2*x^2 + c3*x + c4

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% A. SOLVE A SYSTEM OF LINEAR EQUATIONS FOR C1, C2, C3, AND C4.

% A1. Define column vectors X and Y from the points' coordinates 
% X = (x1, x2, x3, x4) and Y = (y1, y2, y3, y4)




% A2. Define matrix A from X
% If C = (c1, c2, c3, c4) is the vector we want to solve for, 
% we need to define a matrix from the vector X, so that Y = AC.
% Hint: use the .^ operator and function ONES to simplify the code.




% A3. Solve Y = AC for C



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% B. COMPUTE A QUADRATIC AND A STRAIGHT LINE FIT TO THE POINTS
% Hint: use function POLYFIT

% B1. Quadratic fit


% B2 Straight line fit


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% C. PLOT THE CUBIC, QUADRATIC, STRAIGHT LINE AND POINTS

% C1. Compute a cubic curve, a quadratic curve, and a straight line for an
% interval of x from 0 to 10.
% Hint: use built-in function POLYVAL



% C2. Plot the cubic curve, quadratic curve, straight line, and points on
% on the same figure.




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% D. EIGENVVALUES AND EIGENVECTORS

% An eigenvalue and eingenvector of a square matrix A are a scalar lambda
% and a nonzero vector V such that A*lambda = A*V.
 
% D1. Compute the eigenvalues and eigenvectors of A
% Hint: use built-in function EIG.



% D2. Check solution (optional)