145 lines
3.2 KiB
Matlab
145 lines
3.2 KiB
Matlab
%% Assignment 4
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% Name: Claudio Maggioni
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%
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% Date: 2020-05-17
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%
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% This is a template file for the first assignment to get started with running
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% and publishing code in Matlab. Each problem has its own section (delineated
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% by |%%|) and can be run in isolation by clicking into the particular section
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% and pressing |Ctrl| + |Enter| (evaluate current section).
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%
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% To generate a pdf for submission in your current directory, use the following
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% three lines of code at the command window:
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%
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% >> options.format = 'pdf'; options.outputDir = pwd; publish('assignment4.m', options)
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%
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%% Problem 3
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clear all;
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clc;
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clf reset;
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n=10;
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f = @(x) (exp(-(x^2)/2)/sqrt(2*pi));
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x_5 = computeEquidistantXs(5);
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x_10 = computeEquidistantXs(10);
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x_5c = computeChebyshevXs(5);
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x_10c = computeChebyshevXs(10);
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xe = (-1:0.01:1)';
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y = zeros(size(xe));
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for i = 1:size(xe, 1)
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y(i) = f(xe(i));
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end
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p_5 = computePolypoints(f, xe, x_5, 5);
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p_10 = computePolypoints(f, xe, x_10, 10);
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p_5c = computePolypoints(f, xe, x_5c, 5);
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p_10c = computePolypoints(f, xe, x_10c, 10);
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plot(xe, p_5, xe, p_10, xe, y);
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figure;
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plot(xe, p_5c, xe, p_10c, xe, y);
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%% Question 6
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y = [-0.0044; -0.0771; -0.2001; -0.3521; -0.3520; 0; 0.5741; 0.8673; ...
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0.5741; 0; 0.3520; -0.3521; 0.2001; -0.0771; -0.0213; -0.0044];
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figure;
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alpha = b3interpolate(y);
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x = (0:0.01:16)';
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y_c = spline_curve(alpha, x);
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plot(x, y_c);
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%% Question 3 (continued)
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function x = computeEquidistantXs(n)
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x = zeros(2*n+1,1);
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for i = 1:2*n+1
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x(i) = (i-n-1)/n;
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end
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end
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function x = computeChebyshevXs(n)
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x = zeros(2*n+1,1);
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for i = 1:2*n+1
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x(i) = cos((2*i - 1) *pi / (4*n + 2));
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end
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end
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function p = computePolypoints(f, xe, x, n)
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p = zeros(size(xe));
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for i = 1:(2*n+1)
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e_i = zeros(2*n+1, 1);
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e_i(i) = 1;
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N = NewtonInterpolation(x, e_i);
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p = p + f(x(i)) * HornerNewton(N, x, xe);
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end
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end
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% Assuming x and y are column vectors with the same length
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function N = NewtonInterpolation (x,y)
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n = size(x, 1);
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N = y;
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for i = 1:n
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N(n:-1:i+1) = (N(n:-1:i+1) - N(n-1:-1:i)) ./ (x(n:-1:i+1) - x(n-i:-1:1));
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end
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end
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% N is the array of coefficients
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%
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% xi evaluation points
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function p = HornerNewton(N, x, xe)
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n = size(x, 1);
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p = ones(size(xe, 1), 1) * N(n);
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for i = n-1:-1:1
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p = p .* (xe - x(i)) + N(i);
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end
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end
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%% Problem 6 (continued)
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% assuming x_i = i - 1
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function [alpha] = b3interpolate(y)
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n = size(y, 1);
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A = zeros(n+2);
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B = [y; 0; 0];
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for x_i = 0:n-1
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for j = -1:n
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%fprintf("A(%d, %d) = B3(%d - %d)\n", x_i + 1, j+2, x_i, j);
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A(x_i + 1, j + 2) = B3(x_i - j);
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end
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end
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A(n+1, 1:3) = [1 -2 1];
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A(n+2, n-1:n+1) = [1 -2 1];
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alpha = A \ B;
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end
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function [v] = spline_curve(alpha, x)
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n = size(alpha, 1) - 2;
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v = zeros(size(x));
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for i = 1:size(x,1)
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for j = -1:n
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v(i) = v(i) + alpha(j + 2) * B3(x(i) - j);
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end
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end
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end
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function [y] = B3(x)
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y = zeros (size (x));
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i1 = find (-2 < x & x < -1);
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i2 = find (-1 <= x & x < 1);
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i3 = find (1 <= x & x < 2);
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y(i1) = 0.5 * (x(i1) + 2) .^ 3;
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y(i2) = 0.5 * (3 * abs (x(i2)) .^ 3 - 6 * x(i2) .^ 2 + 4);
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y(i3) = 0.5 * (2 - x(i3)) .^ 3;
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y = y / 3;
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end
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