From 631cd8d9934a3814cfb12000ebe27b02db6af277 Mon Sep 17 00:00:00 2001 From: Claudio Maggioni Date: Sun, 17 May 2020 12:19:11 +0200 Subject: [PATCH] hw4: almost done --- hw4/assignment4.m | 66 ++++++++++++++++++++++++++++++++++++++++---- hw4/assignment4.pdf | Bin 0 -> 44485 bytes hw4/hw4.pdf | Bin 117732 -> 117779 bytes hw4/hw4.tex | 2 +- 4 files changed, 61 insertions(+), 7 deletions(-) create mode 100644 hw4/assignment4.pdf diff --git a/hw4/assignment4.m b/hw4/assignment4.m index c1e8c9c..0072451 100644 --- a/hw4/assignment4.m +++ b/hw4/assignment4.m @@ -1,7 +1,7 @@ %% Assignment 4 % Name: Claudio Maggioni % -% Date: 19/3/2019 +% Date: 2020-05-17 % % This is a template file for the first assignment to get started with running % and publishing code in Matlab. Each problem has its own section (delineated @@ -11,11 +11,13 @@ % To generate a pdf for submission in your current directory, use the following % three lines of code at the command window: % -% >> options.format = 'pdf'; options.outputDir = pwd; publish('assignment3.m', options) +% >> options.format = 'pdf'; options.outputDir = pwd; publish('assignment4.m', options) % %% Problem 3 -clear; +clear all; +clc; +clf reset; n=10; f = @(x) (exp(-(x^2)/2)/sqrt(2*pi)); @@ -36,12 +38,21 @@ p_10 = computePolypoints(f, xe, x_10, 10); p_5c = computePolypoints(f, xe, x_5c, 5); p_10c = computePolypoints(f, xe, x_10c, 10); -figure; -subplot(1,2,1); plot(xe, p_5, xe, p_10, xe, y); -subplot(1,2,2); +figure; plot(xe, p_5c, xe, p_10c, xe, y); +%% Question 6 + +y = [-0.0044; -0.0771; -0.2001; -0.3521; -0.3520; 0; 0.5741; 0.8673; ... + 0.5741; 0; 0.3520; -0.3521; 0.2001; -0.0771; -0.0213; -0.0044]; +figure; +alpha = b3interpolate(y); +x = (0:0.01:16)'; +y_c = spline_curve(alpha, x); +plot(x, y_c); + +%% Question 3 (continued) function x = computeEquidistantXs(n) x = zeros(2*n+1,1); for i = 1:2*n+1 @@ -88,3 +99,46 @@ function p = HornerNewton(N, x, xe) p = p .* (xe - x(i)) + N(i); end end + +%% Problem 6 (continued) + +% assuming x_i = i - 1 +function [alpha] = b3interpolate(y) + n = size(y, 1); + A = zeros(n+2); + B = [y; 0; 0]; + for x_i = 0:n-1 + for j = -1:n + %fprintf("A(%d, %d) = B3(%d - %d)\n", x_i + 1, j+2, x_i, j); + A(x_i + 1, j + 2) = B3(x_i - j); + end + end + A(n+1, 1:3) = [1 -2 1]; + A(n+2, n-1:n+1) = [1 -2 1]; + alpha = A \ B; +end + +function [v] = spline_curve(alpha, x) + n = size(alpha, 1) - 2; + v = zeros(size(x)); + for i = 1:size(x,1) + for j = -1:n + v(i) = v(i) + alpha(j + 2) * B3(x(i) - j); + end + end +end + +function [y] = B3(x) +y = zeros (size (x)); +i1 = find (-2 < x & x < -1); +i2 = find (-1 <= x & x < 1); +i3 = find (1 <= x & x < 2); + +y(i1) = 0.5 * (x(i1) + 2) .^ 3; +y(i2) = 0.5 * (3 * abs (x(i2)) .^ 3 - 6 * x(i2) .^ 2 + 4); +y(i3) = 0.5 * (2 - x(i3)) 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a/hw4/hw4.tex b/hw4/hw4.tex index a2af726..d82a35c 100644 --- a/hw4/hw4.tex +++ b/hw4/hw4.tex @@ -184,7 +184,7 @@ a_0\\a_1\\a_2\\a_{-1}\\ y_0\\y_1\\y_2\\y_{3}\\ \end{bmatrix}\] -\subsection*{Question 5} +\section*{Question 5} $$1 = y_0 = s(x_0) = 0 = a_{-1}B_3(1) +a_0B_3(0) +a_1B_3(-1)