91 lines
2.1 KiB
Matlab
91 lines
2.1 KiB
Matlab
%% Assignment 4
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% Name: Claudio Maggioni
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%
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% Date: 19/3/2019
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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('assignment3.m', options)
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%
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%% Problem 3
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clear;
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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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figure;
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subplot(1,2,1);
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plot(xe, p_5, xe, p_10, xe, y);
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subplot(1,2,2);
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plot(xe, p_5c, xe, p_10c, xe, y);
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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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