Solutions for FCUP's Numerical Analysis Assignments
This commit is contained in:
Diogo Cordeiro
6
3 - Splines/source/input.txt
Normal file
6
3 - Splines/source/input.txt
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@@ -0,0 +1,6 @@
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0 1.4
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1 0.6
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2 1.0
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2.5 0.6
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3 0.6
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4 1.0
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51
3 - Splines/source/matriz.txt
Normal file
51
3 - Splines/source/matriz.txt
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@@ -0,0 +1,51 @@
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mat:
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1,0,0,0,0,0,
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0.166667,0.666667,0.166667,0,0,0,
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0,0.166667,0.5,0.0833333,0,0,
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0,0,0.0833333,0.333333,0.0833333,0,
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0,0,0,0.0833333,0.5,0.166667,
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0,0,0,0,0,1,
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b:
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0
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1.2
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-1.2
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0.8
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0.4
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0
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x:
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0
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2.76846
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-3.87386
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3.30622
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0.248963
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0
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mat:
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1,0,0,0,0,0,0,0,0,
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0.0416667,0.166667,0.0416667,0,0,0,0,0,0,
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0,0.0416667,0.166667,0.0416667,0,0,0,0,0,
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0,0,0.0416667,0.166667,0.0416667,0,0,0,0,
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0,0,0,0.0416667,0.166667,0.0416667,0,0,0,
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0,0,0,0,0.0416667,0.166667,0.0416667,0,0,
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0,0,0,0,0,0.0416667,0.166667,0.0416667,0,
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0,0,0,0,0,0,0.0416667,0.166667,0.0416667,
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0,0,0,0,0,0,0,0,1,
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b:
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0
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7.862
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-10.7327
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12.1347
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2
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-8.13471
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14.7327
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-3.862
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0
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x:
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0
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73.9996
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-107.31
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97.6564
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7.91753
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-81.3265
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122.156
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-53.7109
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0
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0
3 - Splines/source/new.foo
Normal file
0
3 - Splines/source/new.foo
Normal file
211
3 - Splines/source/newton.cpp
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211
3 - Splines/source/newton.cpp
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@@ -0,0 +1,211 @@
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#include <algorithm>
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#include <cmath>
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#include <cstdlib>
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#include <cstring>
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#include <fstream>
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#include <iomanip>
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#include <ios>
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#include <iostream>
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#include <numeric>
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#include <string>
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#include <utility>
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#include <vector>
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using namespace std::literals;
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using points_t = std::pair<std::vector<double>, std::vector<double>>;
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using matrix_t = std::vector<std::vector<double>>;
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std::vector<double> newton_differences(points_t points) {
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std::vector<double> factors{};
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auto &[x, fx] = points;
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int n = points.first.size();
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for (int i = 1; i <= n; ++i) {
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factors.push_back(fx[0]);
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for (int j = 0; j < (n - i); ++j) {
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fx[j] = (fx[j + 1] - fx[j]) / (x[j + i] - x[j]);
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}
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}
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return factors;
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}
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double newton_polynomial(points_t points, std::vector<double> factors, double x) {
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auto xs = points.first;
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int n = points.second.size() - 1;
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double val = 0;
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for (int k = 0; k <= n; ++k) {
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double acc = 1;
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for(int i = 0; i < k; ++i) {
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acc *= (x - xs[i]);
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}
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val += acc * factors[k];
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}
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return val;
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}
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void calculate_natural_cubic_spline_matrix(points_t points, matrix_t &mat) {
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auto &[xs, fx] = points;
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int n = xs.size();
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// Construção da matriz
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for (int i = 1; i < n - 1; ++i) {
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mat[i][i - 1] = (xs[i] - xs[i - 1])/6;
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mat[i][i] = (xs[i + 1] - xs[i - 1])/3;
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mat[i][i + 1] = (xs[i + 1] - xs[i])/6;
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mat[i][n] = (fx[i + 1] - fx[i])/(xs[i + 1] - xs[i]) - (fx[i] - fx[i - 1])/(xs[i] - xs[i - 1]);
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}
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mat[0][0] = 1;
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mat[0][n - 1] = 0;
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mat[n - 1][n - 1] = 1;
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mat[n - 1][n] = 0;
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std::ofstream matriz("matriz.txt", std::ios_base::app);
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matriz << "mat:\n";
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for (auto i : mat){
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for (auto j = i.begin(); j != i.end() - 1; ++ j)
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matriz << *j << ",";
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matriz << "\n";
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}
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matriz << "b:\n";
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for (auto i : mat){
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matriz << i[n] << '\n';
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}
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// Passar para a forma triangular
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for (int k = 0; k < n; ++k) {
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for (int i = k + 1; i < n; ++i) {
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if (mat[k][k] != 0) {
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double mul = mat[i][k]/mat[k][k];
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for (int j = k; j < n; ++j) {
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mat[i][j] -= mul * mat[k][j];
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}
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mat[i][n] -= mul * mat[k][n];
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}
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}
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}
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// Resolução da matriz
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for (int i = n - 1; i > 0; --i) {
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if (mat[i][i] != 0) {
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double mul = mat[i - 1][i]/mat[i][i];
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for (int j = 1; j < n + 1; ++j) {
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mat[i - 1][j] -= mul * mat[i][j];
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}
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mat[i - 1][i] = 0;
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mat[i][n] /= mat[i][i];
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mat[i][i] = 1;
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}
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}
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matriz << "x:\n";
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for (auto i : mat){
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matriz << i[n] << '\n';
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}
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}
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double natural_cubic_spline(points_t points, matrix_t &mat, double x) {
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auto &[xs, fx] = points;
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int n = xs.size();
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int i = 0;
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for (int i_ = 0; i_ < n; ++i_) {
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if (xs[i_] > x){
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i = i_;
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break;
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}
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}
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double hi = xs[i] - xs[i - 1];
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return mat[i - 1][n] * std::pow((xs[i] - x), 3)/(6 * hi) +
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mat[i][n] * std::pow((x - xs[i - 1]), 3)/(6 * hi) +
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(fx[i - 1] - mat[i - 1][n] * (hi * hi)/6)*(xs[i] - x)/hi +
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(fx[i] - mat[i][n] * (hi * hi)/6)*(x - xs[i - 1])/hi;
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}
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void exercise_a(points_t points) {
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auto &[xs, fx] = points;
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auto factors = newton_differences(points);
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std::ofstream poly{"a_polynomial.txt"};
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for(double x = 0; x < 4; x += 0.001)
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poly << x << " " << newton_polynomial(points, factors, x) << '\n';
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std::ofstream spline{"a_spline.txt"};
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unsigned long n = xs.size();
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matrix_t mat(n, std::vector<double>(n + 1));
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calculate_natural_cubic_spline_matrix(points, mat);
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for(double x = 0; x < 4; x += 0.001)
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spline << x << " " << natural_cubic_spline(points, mat, x) << '\n';
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}
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void exercise_b() {
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points_t points;
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auto &[xs, fx] = points;
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auto f = [](double x) { return 4 * std::pow(x, 2) + std::sin(9 * x); };
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for (double x = -1; x <= 1; x += (1 - -1)/8.0) {
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xs.push_back(x);
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fx.push_back(f(x));
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}
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std::ofstream poly{"b_polynomial.txt"};
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auto factors = newton_differences(points);
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for(double x = -1; x < 1; x += 0.001)
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poly << x << " " << newton_polynomial(points, factors, x) << '\n';
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std::ofstream spline{"b_spline.txt"};
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unsigned long n = xs.size();
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matrix_t mat(n, std::vector<double>(n + 1));
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calculate_natural_cubic_spline_matrix(points, mat);
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for(double x = -1; x < 1; x += 0.001)
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spline << x << " " << natural_cubic_spline(points, mat, x) << '\n';
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std::cout << "Erros poly: "
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<< std::abs(std::round((f(0.3) - newton_polynomial(points, factors, 0.3)) * 100.0) / 100.0) << " "
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<< std::abs(std::round((f(0.83) - newton_polynomial(points, factors, 0.83)) * 100.0) / 100.0) << '\n';
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std::cout << "Erros spline: "
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<< std::abs(std::round((f(0.3) - natural_cubic_spline(points, mat, 0.3))*100.0)/100.0) << " "
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<< std::abs(std::round((f(0.83) - natural_cubic_spline(points, mat,0.83))*100.0)/100.0) << '\n';
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std::cout << "p(0.3) = " << newton_polynomial(points, factors, 0.3) << ' '
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<< "p(0.83) = " << newton_polynomial(points, factors, 0.83) << '\n';
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std::cout << "p(0.3) = " << natural_cubic_spline(points, mat, 0.3) << ' '
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<< "p(0.83) = " << natural_cubic_spline(points, mat, 0.83) << '\n';
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}
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int main(int argc, char **argv) {
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std::istream *s;
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std::ifstream file;
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if (argc == 2) {
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if (argv[1] == "-"s) {
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s = &std::cin;
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} else {
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file.open(argv[1]);
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s = &file;
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}
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} else{
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std::cerr << "An argument required\n";
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return 1;
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}
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points_t points;
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std::pair<double, double> p;
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while(true) {
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*s >> p.first >> p.second;
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if(s->eof())
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break;
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points.first.push_back(p.first);
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points.second.push_back(p.second);
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}
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exercise_a(points);
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exercise_b();
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}
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17
3 - Splines/source/old.foo
Normal file
17
3 - Splines/source/old.foo
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@@ -0,0 +1,17 @@
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Mat:
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1,0,0,0,0,0,0,
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0,1,-7.14916e-18,2.73077e-18,-4.18062e-19,0,2.76846,
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0,0,1,-7.50963e-18,1.14967e-18,0,-3.87386,
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0,0,-3.57458e-18,1,-4.38965e-18,0,3.30622,
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0,0,0,-7.50963e-18,1,0,0.248963,
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0,0,0,0,0,1,0,
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Mat:
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1,0,0,0,0,0,0,0,0,0,
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0,1,-1.73472e-18,4.64061e-19,2.85733e-19,5.30686e-20,-1.26382e-20,-4.49207e-21,0,73.9996,
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0,0,1,-1.74023e-18,-1.0715e-18,-1.99007e-19,4.73931e-20,1.68453e-20,0,-107.31,
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0,0,-1.73472e-18,1,4.00026e-18,7.4296e-19,-1.76934e-19,-6.2889e-20,0,97.6564,
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0,0,0,-1.74023e-18,1,-2.77283e-18,6.60344e-19,2.34711e-19,0,7.91753,
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0,0,0,0,4.00026e-18,1,-2.46444e-18,-8.75954e-19,0,-81.3265,
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0,0,0,0,0,-2.77283e-18,1,3.26911e-18,0,122.156,
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0,0,0,0,0,0,-2.46444e-18,1,0,-53.7109,
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0,0,0,0,0,0,0,0,1,0,
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39
3 - Splines/source/residue.py
Executable file
39
3 - Splines/source/residue.py
Executable file
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#!/usr/bin/env python3
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import numpy as np
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from scipy.linalg import solve
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from numpy.linalg import norm
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# Matriz construída pelo programa em C++
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A = np.matrix([[1, 0, 0, 0, 0, 0],
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[0.166667, 0.666667, 0.166667, 0, 0, 0],
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[0, 0.166667, 0.5, 0.0833333, 0, 0],
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[0, 0, 0.0833333, 0.333333, 0.0833333, 0],
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[0, 0, 0, 0.0833333, 0.5, 0.166667],
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[0, 0, 0, 0, 0, 1]])
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b = np.array([0, 1.2, -1.2, 0.8, 0.4, 0])
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x2 = np.array([0, 2.76846, -3.87386, 3.30622, 0.248963, 0])
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x = solve(A, b)
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print(norm(x - x2))
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A = np.matrix([[1, 0, 0, 0, 0, 0, 0, 0, 0],
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[0.0416667, 0.166667, 0.0416667, 0, 0, 0, 0, 0, 0],
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[0, 0.0416667, 0.166667, 0.0416667, 0, 0, 0, 0, 0],
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[0, 0, 0.0416667, 0.166667, 0.0416667, 0, 0, 0, 0],
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[0, 0, 0, 0.0416667, 0.166667, 0.0416667, 0, 0, 0],
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[0, 0, 0, 0, 0.0416667, 0.166667, 0.0416667, 0, 0],
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[0, 0, 0, 0, 0, 0.0416667, 0.166667, 0.0416667, 0],
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[0, 0, 0, 0, 0, 0, 0.0416667, 0.166667, 0.0416667],
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[0, 0, 0, 0, 0, 0, 0, 0, 1]])
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b = np.array([0, 7.862, -10.7327, 12.1347, 2, -8.13471, 14.7327, -3.862, 0])
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x2 = np.array([0, 73.9996, -107.31, 97.6564, 7.91753, -81.3265, 122.156, -53.7109, 0])
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x = solve(A, b)
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print(norm(x - x2))
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17
3 - Splines/source/test.foo
Normal file
17
3 - Splines/source/test.foo
Normal file
@@ -0,0 +1,17 @@
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Mat:
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1,0,0,0,0,0,0,
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0,1,0,0,0,0,2.76846,
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0,0,1,0,0,0,-3.87386,
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0,0,-3.57458e-18,1,0,0,3.30622,
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0,0,0,-7.50963e-18,1,0,0.248963,
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0,0,0,0,0,1,0,
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Mat:
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1,0,0,0,0,0,0,0,0,0,
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0,1,0,0,0,0,0,0,0,73.9996,
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0,0,1,0,0,0,0,0,0,-107.31,
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0,0,-1.73472e-18,1,0,0,0,0,0,97.6564,
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0,0,0,-1.74023e-18,1,0,0,0,0,7.91753,
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0,0,0,0,4.00026e-18,1,0,0,0,-81.3265,
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0,0,0,0,0,-2.77283e-18,1,0,0,122.156,
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0,0,0,0,0,0,-2.46444e-18,1,0,-53.7109,
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0,0,0,0,0,0,0,0,1,0,
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