%% Vandermonde Matrix System for polynomial interpolation %% Introduction % % In this lab, we will learn how to solve the Vandermonde matrix system to % find the coefficients of an interpolating polynomial. We will also see % how the matlab command polyfit can be used to % obtain the same set of coefficients. % %% Generate the data clear all; close all; N = 10; % We will try to fit a tenth degree polynomial x = linspace(-1,1,N+1)'; a = 3; b = 8; y = a + (b-a)*rand(N+1,1); % Generate random numbers in [a,b] plot(x,y,'k.','markersize',30); snapnow; %% Solving the Vandermonde system % % Solve the Vandermonde system using vander % V = vander(x); a = V\y; %% % Plot the results of the Vandermonde system xs = linspace(-1,1,100); ys = polyval(a,xs); plot(xs,ys,'r','linewidth',1); hold on; plot(x,y,'k.','markersize',30); title('Using vander','fontsize',18); snapnow; %% Using the polynomial interpolation available in Matlab. % % Use the Matlab function polyfit % % p = polyfit(x,y,N); % Supply the degree of the polynomial ys = polyval(p,xs); plot(xs,ys,'ko','linewidth',1); % Plot using the 'circle' symbol title('Using polyfit','fontsize',18); snapnow; %% Exercises % %
% For each of the following problems, use the vander % function to answer the questions. Plot the data points and the % curve you find. %
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  1. Find the slope and the y-intercept of the line that passes through % the two points $(-3.2, 1.2)$ and $(5.4,-4.3)$.
  2. %
  3. Find the maximum value of the parabola that passes through the three % points % $$(-3.2, 4.5), \quad (1.2,6.1), \quad % (6.1,-3.4)$$ %
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