commit f53b8768f80568d06f122ee6717637c197cdc8d6 Author: Safak Date: Sun Jan 4 15:35:46 2026 +0100 init diff --git a/CMakeLists.txt b/CMakeLists.txt new file mode 100755 index 0000000..86fda4a --- /dev/null +++ b/CMakeLists.txt @@ -0,0 +1,15 @@ +# Set the minimum required version of CMake +cmake_minimum_required(VERSION 3.19) + +# Aktiviert die Generierung der compile_commands.json +set(CMAKE_EXPORT_COMPILE_COMMANDS ON) + +# Set the project name +project(HM2_Praktika) + +# Add the subdirectories for each praktikum +add_subdirectory(P1) +add_subdirectory(P2) +add_subdirectory(P3) +add_subdirectory(P4) +add_subdirectory(P5) \ No newline at end of file diff --git a/P1/CMakeLists.txt b/P1/CMakeLists.txt new file mode 100755 index 0000000..0819f77 --- /dev/null +++ b/P1/CMakeLists.txt @@ -0,0 +1,12 @@ +# Set the minimum required version of CMake +cmake_minimum_required(VERSION 3.19) + +# Set the project name +project(HM_P1) + +# Specify the C++ standard +set(CMAKE_CXX_STANDARD 17) +set(CMAKE_CXX_STANDARD_REQUIRED ON) + +# Add all source files +add_executable(HM_P1 main.cpp CMyVektor.cpp) diff --git a/P1/CMyVektor.cpp b/P1/CMyVektor.cpp new file mode 100755 index 0000000..38e0d3d --- /dev/null +++ b/P1/CMyVektor.cpp @@ -0,0 +1,199 @@ +#include "CMyVektor.h" +#include +#include + +int CMyVektor::getDimension() { + return dimension; +} + +// Vektor Element bei index i +double& CMyVektor::operator[](int i) { + return werte.at(i); +} + +// Betrag des Vektors +double CMyVektor::length() { + double l = 0.0; + for (int i = 0; i < werte.size(); i++) + l += pow(werte.at(i), 2); + + return(sqrt(l)); +} + +// Ausgabe im Vektorformat +std::ostream& operator<< (std::ostream& os, CMyVektor& vektor) { + os << "("; + for (int i = 0; i < vektor.getDimension(); i++) { + os << vektor[i]; + if (i < vektor.getDimension() - 1) os << "|"; + } + os << ")"; + + return os; +} + +/* Vektor Addition + a1 + b1 = a1+b1 + a2 + b2 = a2+b2 + : : : + ai + bi = ai+bi +*/ +CMyVektor operator+(CMyVektor a, CMyVektor b) { + // Vektoren addierbar? + if (a.getDimension() != b.getDimension()) + return a; + + CMyVektor sum(a.getDimension()); + + for (int i = 0; i < a.getDimension(); i++) + sum[i] = a[i] + b[i]; + + return sum; +} + +/* Skalare Multiplikation + a1 * s = v1 + a2 * s = v2 + : : : + ai * s = vi +*/ +CMyVektor operator*(double s, CMyVektor a) { + CMyVektor vektor(a.getDimension()); + + for (int i = 0; i < a.getDimension(); i++) + vektor[i] = s * a[i]; + + return vektor; +} + +CMyVektor gradient(double f(CMyVektor x), CMyVektor x) { + CMyVektor grad(x.getDimension()); + CMyVektor xh(x.getDimension()); // x mit "Wackelei" an x_i + xh = x; + + double fx = f(x); + double h = 1e-8; + + // Berechnung der partiellen Ableitung nach x_i und Zusammenfassung in grad + for (int i = 0; i < x.getDimension(); i++) { + // xh[0] xh[1] ... xh[n] + // x_1+h,x_2,...,x_n x_1,x_2+h,...,x_n x_1,x_2,...,x_n+1 + xh[i] += h; + + // grad[i] = df/dx_i => numerische Ableitung nach dem jeweilige x_i + grad[i] = (f(xh) - fx) / h; + + // Zurücksetzen der "Wackelei" + xh[i] -= h; + } + return grad; +} + +CMyVektor gradientenverfahren(double f(CMyVektor x), CMyVektor x, double lambda) { + int schritt_zaehler = 0; + CMyVektor gradfx(x.getDimension()); + CMyVektor x_neu(x.getDimension()); + gradfx = gradient(f, x); + + while (gradient(f, x).length() >= 1e-5 && schritt_zaehler < 25) { + x_neu = x + lambda * gradfx; + + std::cout + << "Schritt " << schritt_zaehler << ": " << std::endl + << " x = " << x << std::endl + << " λ = " << lambda << std::endl + << " f(x) = " << f(x) << std::endl + << " grad f(x) = " << gradfx << std::endl + << " ||grad f(x)|| = " << gradfx.length() << std::endl << std::endl + << " x_neu = " << x_neu << std::endl + << " f(x_neu) = " << f(x_neu) << std::endl << std::endl; + + // Halbierung + if (f(x_neu) <= f(x)) { + double lamda_test = lambda * 0.5; + CMyVektor x_test = x + lamda_test * gradfx; + + std::cout + << " ↯ f(x) = " << f(x) << " ≥ f(x_neu) = " << f(x_neu) << std::endl << std::endl + << " ? Test mit halbierter Schrittweite (λ = " << lamda_test << "): " << std::endl + << " x_test = " << x_test << std::endl + << " f(x_test) = " << f(x_test) << std::endl << std::endl; + + if (f(x) < f(x_test)) + std::cout + << " ✓ f(x) = " << f(x) << " < f(x_test) = " << f(x_test) << std::endl + << " ! Übernehme Schrittweite." << std::endl << std::endl; + + // weiter Halbieren wenn f(x_test) <= f(x) + while (f(x_test) <= f(x)) { + std::cout + << " ↯ f(x) = " << f(x) << " ≥ f(x_test) = " << f(x_test) << std::endl << std::endl; + + lamda_test *= 0.5; + x_test = x + lamda_test * gradfx; + + std::cout + << " ? Test mit halbierter Schrittweite (λ = " << lamda_test << "): " << std::endl + << " x_test = " << x_test << std::endl + << " f(x_test) = " << f(x_test) << std::endl << std::endl; + + // Ausgabe beim letzten Durchlauf der Schleife + if (!(f(x_test) <= f(x))) + std::cout + << " ✓ f(x) = " << f(x) << " < f(x_test) = " << f(x_test) << std::endl << std::endl + << " ! Übernehme Schrittweite λ = " << lamda_test << std::endl << std::endl; + } + + lambda = lamda_test; + x_neu = x_test; + } + + // Verdopplung + else { + double lamda_test = lambda * 2; + CMyVektor x_test = x + lamda_test * gradfx; + + std::cout + << " ? Test mit doppelter Schrittweite (λ = " << lamda_test << "): " << std::endl + << " x_test = " << x_test << std::endl + << " f(x_test) = " << f(x_test) << std::endl << std::endl; + + if (f(x_test) > f(x_neu)) { + x_neu = x_test; + lambda = lamda_test; + std::cout + << " ! Übernehme verdoppelte Schrittweite λ = " << lamda_test << std::endl + << " ✓ f(x_neu) = " << f(x_neu) << " < f(x_test) = " << f(x_test) << std::endl << std::endl; + } + else { + std::cout + << " ↯ f(x_neu) = " << f(x_neu) << " ≥ f(x_test) = " << f(x_test) << std::endl + << " ! behalte aktuelle Schrittweite λ = " << lambda << std::endl << std::endl; + } + } + std::cout << std::endl; + + x = x_neu; + gradfx = gradient(f, x); + schritt_zaehler++; + } + + // Zusammenfassung des Endes + if (gradfx.length() < 1e-5) + std::cout + << "Ende wegen ||grad f(x)|| < 10^-5 bei" << std::endl; + + else + std::cout + << "Ende wegen 25. Schritt" << std::endl; + + std::cout + << " x = " << x << std::endl + << " λ = " << lambda << std::endl + << " f(x) = " << f(x) << std::endl + << " grad f(x) = " << gradfx << std::endl + << " ||grad f(x)|| = " << gradfx.length() << std::endl + << "-----------------------------------------------------" << std::endl << std::endl; + + return x_neu; +} diff --git a/P1/CMyVektor.h b/P1/CMyVektor.h new file mode 100755 index 0000000..aa08f32 --- /dev/null +++ b/P1/CMyVektor.h @@ -0,0 +1,34 @@ +#pragma once +#include +#include + +class CMyVektor { +private: + int dimension; + std::vector werte; +public: + // Vektor mit Dimension anlegen + CMyVektor(int n) : dimension { n } { + werte.resize(n); + } + + CMyVektor(std::vector x) : dimension { (int) x.size() } { + werte.resize(x.size()); + for (int i = 0; i < x.size(); i++) + werte[i] = x[i]; + } + + int getDimension(); + double length(); + + // Komponente von Vektor übergeben + double& operator[](int index); +}; + +CMyVektor operator+(CMyVektor a, CMyVektor b); +CMyVektor operator*(double lambda, CMyVektor a); + +std::ostream& operator<< (std::ostream& os, CMyVektor& vektor); + +CMyVektor gradient(double f(CMyVektor x), CMyVektor x); +CMyVektor gradientenverfahren(double f(CMyVektor x), CMyVektor x, double lambda = 1.0); diff --git a/P1/main.cpp b/P1/main.cpp new file mode 100755 index 0000000..7e382d8 --- /dev/null +++ b/P1/main.cpp @@ -0,0 +1,36 @@ +#include +#include "CMyVektor.h" + +using namespace std; + +// f( [x,y] ) = sin(x * y) + sin(x) + cos(y) +double f(CMyVektor X) { + double x = X[0]; + double y = X[1]; + + return (sin(x * y) + sin(x) + cos(y)); +} + +// g( [x,y,z] ) = -(2x^2 - 2xy + y^2 + z^2 - 2x - 4z) +double g(CMyVektor X) { + double x = X[0]; + double y = X[1]; + double z = X[2]; + + return -(2 * pow(x,2) - 2 * x * y + pow(y,2) + pow(z,2) - 2 * x - 4 * z); +} + +int main() { + CMyVektor X0(std::vector{1,3,5}); + CMyVektor gradX0 = gradient(f, X0); + cout << "grad f" << X0 << "=" << gradX0 << std::endl; + + CMyVektor X1(std::vector{0.2,-2.1}); + gradientenverfahren(f, X1); + + CMyVektor X2(std::vector{0,0,0}); + gradientenverfahren(g, X2, 0.1); + + system("pause"); + return 0; +} \ No newline at end of file diff --git a/P2/CMakeLists.txt b/P2/CMakeLists.txt new file mode 100755 index 0000000..2368bc5 --- /dev/null +++ b/P2/CMakeLists.txt @@ -0,0 +1,12 @@ +# Set the minimum required version of CMake +cmake_minimum_required(VERSION 3.10) + +# Set the project name and specify the C++ as the project language +project(HM_P2) + +# Specify the C++ standard +set(CMAKE_CXX_STANDARD 17) +set(CMAKE_CXX_STANDARD_REQUIRED ON) + +# Add an executable with the above sources +add_executable(HM_P2 main.cpp CMyVektor.cpp CMyMatrix.cpp) \ No newline at end of file diff --git a/P2/CMyMatrix.cpp b/P2/CMyMatrix.cpp new file mode 100755 index 0000000..b3166b0 --- /dev/null +++ b/P2/CMyMatrix.cpp @@ -0,0 +1,145 @@ +#include "CMyMatrix.h" +#include "CMyVektor.h" +#include +#include +#include + +// Konstruktor +CMyMatrix::CMyMatrix(int z, int s) { + zeilen = z; + spalten = s; + matrix.resize(zeilen); + for (int i = 0; i < zeilen; ++i) + matrix[i].resize(spalten); +} + +double CMyMatrix::getZeilen() { + return zeilen; +} + +double CMyMatrix::getSpalten() { + return spalten; +} + +void CMyMatrix::setWert(double wert, int zeile, int spalte) { + matrix[zeile][spalte] = wert; +} + +double CMyMatrix::getWert(int zeile, int spalte) { + return matrix[zeile][spalte]; +} + +double& CMyMatrix::operator()(int zeile, int spalte) { + return matrix[zeile][spalte]; +} + +CMyMatrix CMyMatrix::invers() { + if (this->getZeilen() != 2 || this->getSpalten() != 2) + throw std::runtime_error("Keine 2x2 Matrix"); + + double a = this->getWert(0, 0); + double b = this->getWert(1, 0); + double c = this->getWert(0, 1); + double d = this->getWert(1, 1); + + double determinate = a * d - b * c; + + if (determinate == 0) + throw std::runtime_error("detA == 0"); + + double kehrwertDeterminante = 1 / determinate; + CMyMatrix inverse(2,2); + + inverse.setWert(kehrwertDeterminante * d, 0, 0); + inverse.setWert(kehrwertDeterminante * -c, 0, 1); + inverse.setWert(kehrwertDeterminante * -b, 1, 0); + inverse.setWert(kehrwertDeterminante * a, 1, 1); + + return inverse; +} + +// Matrix-Vektor Multiplikation +CMyVektor operator*(CMyMatrix A, CMyVektor x) { + // M Spalten == V Zeilen + if (A.getSpalten() != x.getDimension()) + throw std::runtime_error("Matrix(m x n) * Vektor(i) n != i"); + + CMyVektor ergebnis(A.getZeilen()); + + for (int i = 0; i < A.getZeilen(); i++) { // Zeile + for (int j = 0; j < A.getSpalten(); j++) // Spalte + ergebnis[i] += A(i, j) * x[j]; // Aufsummierung + } + return ergebnis; +} + +std::ostream& operator<<(std::ostream& os, CMyMatrix x) { + + for (int i = 0; i < x.getZeilen(); ++i) { + os << "\t\t\t("; + for (int j = 0; j < x.getSpalten(); ++j) + j == x.getSpalten() - 1 ? os << std::fixed << std::setprecision(10) << x(i, j) : os << std::fixed << std::setprecision(10) << x(i, j) << "|"; + + os << ")\n"; + } + return os; +} + +CMyMatrix jacobi(CMyVektor x, CMyVektor(*f)(CMyVektor)) { + double const h = pow(10, -4); + CMyVektor f_x = f(x); + CMyVektor tmp{ x }; + + CMyMatrix jacobiMatrix(f_x.getDimension(), x.getDimension()); + + for (int i = 0; i < f_x.getDimension(); ++i) { // Zeile bzw. Teilfunktion f_i + for (int j = 0; j < x.getDimension(); ++j) { // Spalte bzw. Variable x_j + tmp[j] += h; + jacobiMatrix(i, j) = (f(tmp)[i] - f_x[i]) / h; + tmp[j] -= h; + } + } + return jacobiMatrix; +}; + +void newtonverfahren(CMyVektor x, CMyVektor(*f)(CMyVektor)) { + int schritt = 0; + CMyVektor f_x = f(x); + CMyMatrix jacobi_f_x = jacobi(x, f); + CMyMatrix jacobiInvers = jacobi_f_x.invers(); + + double funktionLaenge = f_x.length(); + + CMyVektor dx = -1 * (jacobiInvers * f_x); + + while (schritt != 50 && funktionLaenge >= 1e-5) { + std::cout << "Schritt " << schritt << ":" + << "\n\t 𝐱 = " << x + << "\n\t 𝑓(𝐱) = " << f_x << "\n" + << "\n\t ∂𝑓(𝐱) = \n " << jacobi_f_x + << "\n\t ∂𝑓(𝐱)⁻¹ = \n" << jacobiInvers + << "\n\t Δ𝐱 = " << dx + << "\n\t ||𝑓(𝐱)|| = " << funktionLaenge << std::endl << std::endl; + + schritt++; + x = x + dx; + f_x = f(x); + jacobi_f_x = jacobi(x, f); + jacobiInvers = jacobi_f_x.invers(); + dx = -1 * (jacobiInvers * f_x); + funktionLaenge = f_x.length(); + + if (schritt == 50) { + std::cout << "Ende wegen Schrittanzahl = 50 bei"; + std::cout << "\n\t x = " << x + << "\n\t 𝑓(𝐱) = " << f_x + << "\n\t ||𝑓(𝐱)|| = " << funktionLaenge << std::endl; + } + else if (funktionLaenge < 1e-5) { + std::cout << "Ende wegen ||𝑓(𝐱)||<1e-5 bei"; + std::cout << "\n\t x = " << x + << "\n\t 𝑓(𝐱) = " << f_x + << "\n\t ||𝑓(𝐱)|| = " << funktionLaenge << std::endl; + } + } +}; \ No newline at end of file diff --git a/P2/CMyMatrix.h b/P2/CMyMatrix.h new file mode 100755 index 0000000..d1128fb --- /dev/null +++ b/P2/CMyMatrix.h @@ -0,0 +1,31 @@ +#include +#include +#include "CMyVektor.h" + +class CMyMatrix { +private: + std::vector> matrix; + int zeilen = 0; + int spalten = 0; + +public: + CMyMatrix(int zeilen, int spalten); + + double getZeilen(); + double getSpalten(); + + void setWert(double wert, int zeile, int spalte); + + double getWert(int zeile, int spalte); + + double& operator()(int zeile, int spalte); + + CMyMatrix invers(); + + friend CMyVektor operator*(CMyMatrix A, CMyVektor x); + + friend std::ostream& operator<<(std::ostream& os, CMyMatrix x); +}; + +CMyMatrix jacobi(CMyVektor x, CMyVektor(*funktion)(CMyVektor)); //como friend-> error de compiler "C3767" +void newtonverfahren(CMyVektor x, CMyVektor(*funktion)(CMyVektor)); //libro pagina 221 diff --git a/P2/CMyVektor.cpp b/P2/CMyVektor.cpp new file mode 100755 index 0000000..38e0d3d --- /dev/null +++ b/P2/CMyVektor.cpp @@ -0,0 +1,199 @@ +#include "CMyVektor.h" +#include +#include + +int CMyVektor::getDimension() { + return dimension; +} + +// Vektor Element bei index i +double& CMyVektor::operator[](int i) { + return werte.at(i); +} + +// Betrag des Vektors +double CMyVektor::length() { + double l = 0.0; + for (int i = 0; i < werte.size(); i++) + l += pow(werte.at(i), 2); + + return(sqrt(l)); +} + +// Ausgabe im Vektorformat +std::ostream& operator<< (std::ostream& os, CMyVektor& vektor) { + os << "("; + for (int i = 0; i < vektor.getDimension(); i++) { + os << vektor[i]; + if (i < vektor.getDimension() - 1) os << "|"; + } + os << ")"; + + return os; +} + +/* Vektor Addition + a1 + b1 = a1+b1 + a2 + b2 = a2+b2 + : : : + ai + bi = ai+bi +*/ +CMyVektor operator+(CMyVektor a, CMyVektor b) { + // Vektoren addierbar? + if (a.getDimension() != b.getDimension()) + return a; + + CMyVektor sum(a.getDimension()); + + for (int i = 0; i < a.getDimension(); i++) + sum[i] = a[i] + b[i]; + + return sum; +} + +/* Skalare Multiplikation + a1 * s = v1 + a2 * s = v2 + : : : + ai * s = vi +*/ +CMyVektor operator*(double s, CMyVektor a) { + CMyVektor vektor(a.getDimension()); + + for (int i = 0; i < a.getDimension(); i++) + vektor[i] = s * a[i]; + + return vektor; +} + +CMyVektor gradient(double f(CMyVektor x), CMyVektor x) { + CMyVektor grad(x.getDimension()); + CMyVektor xh(x.getDimension()); // x mit "Wackelei" an x_i + xh = x; + + double fx = f(x); + double h = 1e-8; + + // Berechnung der partiellen Ableitung nach x_i und Zusammenfassung in grad + for (int i = 0; i < x.getDimension(); i++) { + // xh[0] xh[1] ... xh[n] + // x_1+h,x_2,...,x_n x_1,x_2+h,...,x_n x_1,x_2,...,x_n+1 + xh[i] += h; + + // grad[i] = df/dx_i => numerische Ableitung nach dem jeweilige x_i + grad[i] = (f(xh) - fx) / h; + + // Zurücksetzen der "Wackelei" + xh[i] -= h; + } + return grad; +} + +CMyVektor gradientenverfahren(double f(CMyVektor x), CMyVektor x, double lambda) { + int schritt_zaehler = 0; + CMyVektor gradfx(x.getDimension()); + CMyVektor x_neu(x.getDimension()); + gradfx = gradient(f, x); + + while (gradient(f, x).length() >= 1e-5 && schritt_zaehler < 25) { + x_neu = x + lambda * gradfx; + + std::cout + << "Schritt " << schritt_zaehler << ": " << std::endl + << " x = " << x << std::endl + << " λ = " << lambda << std::endl + << " f(x) = " << f(x) << std::endl + << " grad f(x) = " << gradfx << std::endl + << " ||grad f(x)|| = " << gradfx.length() << std::endl << std::endl + << " x_neu = " << x_neu << std::endl + << " f(x_neu) = " << f(x_neu) << std::endl << std::endl; + + // Halbierung + if (f(x_neu) <= f(x)) { + double lamda_test = lambda * 0.5; + CMyVektor x_test = x + lamda_test * gradfx; + + std::cout + << " ↯ f(x) = " << f(x) << " ≥ f(x_neu) = " << f(x_neu) << std::endl << std::endl + << " ? Test mit halbierter Schrittweite (λ = " << lamda_test << "): " << std::endl + << " x_test = " << x_test << std::endl + << " f(x_test) = " << f(x_test) << std::endl << std::endl; + + if (f(x) < f(x_test)) + std::cout + << " ✓ f(x) = " << f(x) << " < f(x_test) = " << f(x_test) << std::endl + << " ! Übernehme Schrittweite." << std::endl << std::endl; + + // weiter Halbieren wenn f(x_test) <= f(x) + while (f(x_test) <= f(x)) { + std::cout + << " ↯ f(x) = " << f(x) << " ≥ f(x_test) = " << f(x_test) << std::endl << std::endl; + + lamda_test *= 0.5; + x_test = x + lamda_test * gradfx; + + std::cout + << " ? Test mit halbierter Schrittweite (λ = " << lamda_test << "): " << std::endl + << " x_test = " << x_test << std::endl + << " f(x_test) = " << f(x_test) << std::endl << std::endl; + + // Ausgabe beim letzten Durchlauf der Schleife + if (!(f(x_test) <= f(x))) + std::cout + << " ✓ f(x) = " << f(x) << " < f(x_test) = " << f(x_test) << std::endl << std::endl + << " ! Übernehme Schrittweite λ = " << lamda_test << std::endl << std::endl; + } + + lambda = lamda_test; + x_neu = x_test; + } + + // Verdopplung + else { + double lamda_test = lambda * 2; + CMyVektor x_test = x + lamda_test * gradfx; + + std::cout + << " ? Test mit doppelter Schrittweite (λ = " << lamda_test << "): " << std::endl + << " x_test = " << x_test << std::endl + << " f(x_test) = " << f(x_test) << std::endl << std::endl; + + if (f(x_test) > f(x_neu)) { + x_neu = x_test; + lambda = lamda_test; + std::cout + << " ! Übernehme verdoppelte Schrittweite λ = " << lamda_test << std::endl + << " ✓ f(x_neu) = " << f(x_neu) << " < f(x_test) = " << f(x_test) << std::endl << std::endl; + } + else { + std::cout + << " ↯ f(x_neu) = " << f(x_neu) << " ≥ f(x_test) = " << f(x_test) << std::endl + << " ! behalte aktuelle Schrittweite λ = " << lambda << std::endl << std::endl; + } + } + std::cout << std::endl; + + x = x_neu; + gradfx = gradient(f, x); + schritt_zaehler++; + } + + // Zusammenfassung des Endes + if (gradfx.length() < 1e-5) + std::cout + << "Ende wegen ||grad f(x)|| < 10^-5 bei" << std::endl; + + else + std::cout + << "Ende wegen 25. Schritt" << std::endl; + + std::cout + << " x = " << x << std::endl + << " λ = " << lambda << std::endl + << " f(x) = " << f(x) << std::endl + << " grad f(x) = " << gradfx << std::endl + << " ||grad f(x)|| = " << gradfx.length() << std::endl + << "-----------------------------------------------------" << std::endl << std::endl; + + return x_neu; +} diff --git a/P2/CMyVektor.h b/P2/CMyVektor.h new file mode 100755 index 0000000..e9fa19e --- /dev/null +++ b/P2/CMyVektor.h @@ -0,0 +1,34 @@ +#pragma once +#include +#include + +class CMyVektor { +private: + int dimension; + std::vector werte; +public: + // Vektor mit Dimension anlegen + CMyVektor(int n) : dimension{ n } { + werte.resize(n); + } + + CMyVektor(std::vector x) : dimension{ (int)x.size() } { + werte.resize(x.size()); + for (int i = 0; i < x.size(); i++) + werte[i] = x[i]; + } + + int getDimension(); + double length(); + + // Komponente von Vektor übergeben + double& operator[](int index); +}; + +CMyVektor operator+(CMyVektor a, CMyVektor b); +CMyVektor operator*(double lambda, CMyVektor a); + +std::ostream& operator<< (std::ostream& os, CMyVektor& vektor); + +CMyVektor gradient(double f(CMyVektor x), CMyVektor x); +CMyVektor gradientenverfahren(double f(CMyVektor x), CMyVektor x, double lambda = 1.0); diff --git a/P2/Praktikum2.pdf b/P2/Praktikum2.pdf new file mode 100755 index 0000000..1a9d385 Binary files /dev/null and b/P2/Praktikum2.pdf differ diff --git a/P2/main.cpp b/P2/main.cpp new file mode 100755 index 0000000..95916b1 --- /dev/null +++ b/P2/main.cpp @@ -0,0 +1,41 @@ +#include +#include "CMyVektor.h" +#include "CMyMatrix.h" + +using namespace std; + +// f( [x,y] ) = sin(x * y) + sin(x) + cos(y) +double ff(CMyVektor X) { + double x = X[0]; + double y = X[1]; + + return (sin(x * y) + sin(x) + cos(y)); +} + +// g( [x,y,z] ) = -(2x^2 - 2xy + y^2 + z^2 - 2x - 4z) +double gg(CMyVektor X) { + double x = X[0]; + double y = X[1]; + double z = X[2]; + + return -(2 * pow(x, 2) - 2 * x * y + pow(y, 2) + pow(z, 2) - 2 * x - 4 * z); +} + +CMyVektor f(CMyVektor x) { + CMyVektor result(3); // da f: R^4 -> R^3 + result[0] = x[0] * x[1] * exp(x[2]); + result[1] = x[1] * x[2] * x[3]; + result[2] = x[3]; + return result; +} + +int main() { + + cout << "Aufgabe_2 :\n" << jacobi(CMyVektor{ {1, 2, 0, 3} }, + [](CMyVektor x) -> CMyVektor { return { {x[0] * x[1] * exp(x[2]), x[1] * x[2] * x[3], x[3]} }; }) + << endl; + + newtonverfahren({ {1, 1} }, + [](CMyVektor x) -> CMyVektor { return { {pow(x[0], 3) * pow(x[1], 3) - 2 * x[1], x[0] - 2} }; }); + return 0; +} \ No newline at end of file diff --git a/P3/CMakeLists.txt b/P3/CMakeLists.txt new file mode 100755 index 0000000..2f4aa19 --- /dev/null +++ b/P3/CMakeLists.txt @@ -0,0 +1,12 @@ +# Set the minimum required version of CMake +cmake_minimum_required(VERSION 3.11) + +# Set the project name and specify the C++ as the project language +project(HM_P3) + +# Specify the C++ standard +set(CMAKE_CXX_STANDARD 17) +set(CMAKE_CXX_STANDARD_REQUIRED ON) + +# Add an executable with the above sources +add_executable(HM_P3 main.cpp CMyVektor.cpp CMyMatrix.cpp C_DGLSolver.cpp) diff --git a/P3/CMyMatrix.cpp b/P3/CMyMatrix.cpp new file mode 100755 index 0000000..b3166b0 --- /dev/null +++ b/P3/CMyMatrix.cpp @@ -0,0 +1,145 @@ +#include "CMyMatrix.h" +#include "CMyVektor.h" +#include +#include +#include + +// Konstruktor +CMyMatrix::CMyMatrix(int z, int s) { + zeilen = z; + spalten = s; + matrix.resize(zeilen); + for (int i = 0; i < zeilen; ++i) + matrix[i].resize(spalten); +} + +double CMyMatrix::getZeilen() { + return zeilen; +} + +double CMyMatrix::getSpalten() { + return spalten; +} + +void CMyMatrix::setWert(double wert, int zeile, int spalte) { + matrix[zeile][spalte] = wert; +} + +double CMyMatrix::getWert(int zeile, int spalte) { + return matrix[zeile][spalte]; +} + +double& CMyMatrix::operator()(int zeile, int spalte) { + return matrix[zeile][spalte]; +} + +CMyMatrix CMyMatrix::invers() { + if (this->getZeilen() != 2 || this->getSpalten() != 2) + throw std::runtime_error("Keine 2x2 Matrix"); + + double a = this->getWert(0, 0); + double b = this->getWert(1, 0); + double c = this->getWert(0, 1); + double d = this->getWert(1, 1); + + double determinate = a * d - b * c; + + if (determinate == 0) + throw std::runtime_error("detA == 0"); + + double kehrwertDeterminante = 1 / determinate; + CMyMatrix inverse(2,2); + + inverse.setWert(kehrwertDeterminante * d, 0, 0); + inverse.setWert(kehrwertDeterminante * -c, 0, 1); + inverse.setWert(kehrwertDeterminante * -b, 1, 0); + inverse.setWert(kehrwertDeterminante * a, 1, 1); + + return inverse; +} + +// Matrix-Vektor Multiplikation +CMyVektor operator*(CMyMatrix A, CMyVektor x) { + // M Spalten == V Zeilen + if (A.getSpalten() != x.getDimension()) + throw std::runtime_error("Matrix(m x n) * Vektor(i) n != i"); + + CMyVektor ergebnis(A.getZeilen()); + + for (int i = 0; i < A.getZeilen(); i++) { // Zeile + for (int j = 0; j < A.getSpalten(); j++) // Spalte + ergebnis[i] += A(i, j) * x[j]; // Aufsummierung + } + return ergebnis; +} + +std::ostream& operator<<(std::ostream& os, CMyMatrix x) { + + for (int i = 0; i < x.getZeilen(); ++i) { + os << "\t\t\t("; + for (int j = 0; j < x.getSpalten(); ++j) + j == x.getSpalten() - 1 ? os << std::fixed << std::setprecision(10) << x(i, j) : os << std::fixed << std::setprecision(10) << x(i, j) << "|"; + + os << ")\n"; + } + return os; +} + +CMyMatrix jacobi(CMyVektor x, CMyVektor(*f)(CMyVektor)) { + double const h = pow(10, -4); + CMyVektor f_x = f(x); + CMyVektor tmp{ x }; + + CMyMatrix jacobiMatrix(f_x.getDimension(), x.getDimension()); + + for (int i = 0; i < f_x.getDimension(); ++i) { // Zeile bzw. Teilfunktion f_i + for (int j = 0; j < x.getDimension(); ++j) { // Spalte bzw. Variable x_j + tmp[j] += h; + jacobiMatrix(i, j) = (f(tmp)[i] - f_x[i]) / h; + tmp[j] -= h; + } + } + return jacobiMatrix; +}; + +void newtonverfahren(CMyVektor x, CMyVektor(*f)(CMyVektor)) { + int schritt = 0; + CMyVektor f_x = f(x); + CMyMatrix jacobi_f_x = jacobi(x, f); + CMyMatrix jacobiInvers = jacobi_f_x.invers(); + + double funktionLaenge = f_x.length(); + + CMyVektor dx = -1 * (jacobiInvers * f_x); + + while (schritt != 50 && funktionLaenge >= 1e-5) { + std::cout << "Schritt " << schritt << ":" + << "\n\t 𝐱 = " << x + << "\n\t 𝑓(𝐱) = " << f_x << "\n" + << "\n\t ∂𝑓(𝐱) = \n " << jacobi_f_x + << "\n\t ∂𝑓(𝐱)⁻¹ = \n" << jacobiInvers + << "\n\t Δ𝐱 = " << dx + << "\n\t ||𝑓(𝐱)|| = " << funktionLaenge << std::endl << std::endl; + + schritt++; + x = x + dx; + f_x = f(x); + jacobi_f_x = jacobi(x, f); + jacobiInvers = jacobi_f_x.invers(); + dx = -1 * (jacobiInvers * f_x); + funktionLaenge = f_x.length(); + + if (schritt == 50) { + std::cout << "Ende wegen Schrittanzahl = 50 bei"; + std::cout << "\n\t x = " << x + << "\n\t 𝑓(𝐱) = " << f_x + << "\n\t ||𝑓(𝐱)|| = " << funktionLaenge << std::endl; + } + else if (funktionLaenge < 1e-5) { + std::cout << "Ende wegen ||𝑓(𝐱)||<1e-5 bei"; + std::cout << "\n\t x = " << x + << "\n\t 𝑓(𝐱) = " << f_x + << "\n\t ||𝑓(𝐱)|| = " << funktionLaenge << std::endl; + } + } +}; \ No newline at end of file diff --git a/P3/CMyMatrix.h b/P3/CMyMatrix.h new file mode 100755 index 0000000..d1128fb --- /dev/null +++ b/P3/CMyMatrix.h @@ -0,0 +1,31 @@ +#include +#include +#include "CMyVektor.h" + +class CMyMatrix { +private: + std::vector> matrix; + int zeilen = 0; + int spalten = 0; + +public: + CMyMatrix(int zeilen, int spalten); + + double getZeilen(); + double getSpalten(); + + void setWert(double wert, int zeile, int spalte); + + double getWert(int zeile, int spalte); + + double& operator()(int zeile, int spalte); + + CMyMatrix invers(); + + friend CMyVektor operator*(CMyMatrix A, CMyVektor x); + + friend std::ostream& operator<<(std::ostream& os, CMyMatrix x); +}; + +CMyMatrix jacobi(CMyVektor x, CMyVektor(*funktion)(CMyVektor)); //como friend-> error de compiler "C3767" +void newtonverfahren(CMyVektor x, CMyVektor(*funktion)(CMyVektor)); //libro pagina 221 diff --git a/P3/CMyVektor.cpp b/P3/CMyVektor.cpp new file mode 100755 index 0000000..6eb4509 --- /dev/null +++ b/P3/CMyVektor.cpp @@ -0,0 +1,212 @@ +#include "CMyVektor.h" +#include +#include + +int CMyVektor::getDimension() { + return dimension; +} + +// Vektor Element bei index i +double& CMyVektor::operator[](int i) { + return werte.at(i); +} + +double CMyVektor::getElement(int i) { + return werte.at(i - 1); +} + +void CMyVektor::setElement(int i, double j) { + werte.at(i - 1) = j; +} + +void CMyVektor::setElements(std::vector x) { + for (int i = 0; i < x.size(); i++) + werte[i] = x[i]; +} + +// Betrag des Vektors +double CMyVektor::length() { + double l = 0.0; + for (int i = 0; i < werte.size(); i++) + l += pow(werte.at(i), 2); + + return(sqrt(l)); +} + +// Ausgabe im Vektorformat +std::ostream& operator<< (std::ostream& os, CMyVektor& vektor) { + os << "("; + for (int i = 0; i < vektor.getDimension(); i++) { + os << vektor[i]; + if (i < vektor.getDimension() - 1) os << "|"; + } + os << ")"; + + return os; +} + +/* Vektor Addition + a1 + b1 = a1+b1 + a2 + b2 = a2+b2 + : : : + ai + bi = ai+bi +*/ +CMyVektor operator+(CMyVektor a, CMyVektor b) { + // Vektoren addierbar? + if (a.getDimension() != b.getDimension()) + return a; + + CMyVektor sum(a.getDimension()); + + for (int i = 0; i < a.getDimension(); i++) + sum[i] = a[i] + b[i]; + + return sum; +} + +/* Skalare Multiplikation + a1 * s = v1 + a2 * s = v2 + : : : + ai * s = vi +*/ +CMyVektor operator*(double s, CMyVektor a) { + CMyVektor vektor(a.getDimension()); + + for (int i = 0; i < a.getDimension(); i++) + vektor[i] = s * a[i]; + + return vektor; +} + +CMyVektor gradient(double f(CMyVektor x), CMyVektor x) { + CMyVektor grad(x.getDimension()); + CMyVektor xh(x.getDimension()); // x mit "Wackelei" an x_i + xh = x; + + double fx = f(x); + double h = 1e-8; + + // Berechnung der partiellen Ableitung nach x_i und Zusammenfassung in grad + for (int i = 0; i < x.getDimension(); i++) { + // xh[0] xh[1] ... xh[n] + // x_1+h,x_2,...,x_n x_1,x_2+h,...,x_n x_1,x_2,...,x_n+1 + xh[i] += h; + + // grad[i] = df/dx_i => numerische Ableitung nach dem jeweilige x_i + grad[i] = (f(xh) - fx) / h; + + // Zurücksetzen der "Wackelei" + xh[i] -= h; + } + return grad; +} + +CMyVektor gradientenverfahren(double f(CMyVektor x), CMyVektor x, double lambda) { + int schritt_zaehler = 0; + CMyVektor gradfx(x.getDimension()); + CMyVektor x_neu(x.getDimension()); + gradfx = gradient(f, x); + + while (gradient(f, x).length() >= 1e-5 && schritt_zaehler < 25) { + x_neu = x + lambda * gradfx; + + std::cout + << "Schritt " << schritt_zaehler << ": " << std::endl + << " x = " << x << std::endl + << " λ = " << lambda << std::endl + << " f(x) = " << f(x) << std::endl + << " grad f(x) = " << gradfx << std::endl + << " ||grad f(x)|| = " << gradfx.length() << std::endl << std::endl + << " x_neu = " << x_neu << std::endl + << " f(x_neu) = " << f(x_neu) << std::endl << std::endl; + + // Halbierung + if (f(x_neu) <= f(x)) { + double lamda_test = lambda * 0.5; + CMyVektor x_test = x + lamda_test * gradfx; + + std::cout + << " ↯ f(x) = " << f(x) << " ≥ f(x_neu) = " << f(x_neu) << std::endl << std::endl + << " ? Test mit halbierter Schrittweite (λ = " << lamda_test << "): " << std::endl + << " x_test = " << x_test << std::endl + << " f(x_test) = " << f(x_test) << std::endl << std::endl; + + if (f(x) < f(x_test)) + std::cout + << " ✓ f(x) = " << f(x) << " < f(x_test) = " << f(x_test) << std::endl + << " ! Übernehme Schrittweite." << std::endl << std::endl; + + // weiter Halbieren wenn f(x_test) <= f(x) + while (f(x_test) <= f(x)) { + std::cout + << " ↯ f(x) = " << f(x) << " ≥ f(x_test) = " << f(x_test) << std::endl << std::endl; + + lamda_test *= 0.5; + x_test = x + lamda_test * gradfx; + + std::cout + << " ? Test mit halbierter Schrittweite (λ = " << lamda_test << "): " << std::endl + << " x_test = " << x_test << std::endl + << " f(x_test) = " << f(x_test) << std::endl << std::endl; + + // Ausgabe beim letzten Durchlauf der Schleife + if (!(f(x_test) <= f(x))) + std::cout + << " ✓ f(x) = " << f(x) << " < f(x_test) = " << f(x_test) << std::endl << std::endl + << " ! Übernehme Schrittweite λ = " << lamda_test << std::endl << std::endl; + } + + lambda = lamda_test; + x_neu = x_test; + } + + // Verdopplung + else { + double lamda_test = lambda * 2; + CMyVektor x_test = x + lamda_test * gradfx; + + std::cout + << " ? Test mit doppelter Schrittweite (λ = " << lamda_test << "): " << std::endl + << " x_test = " << x_test << std::endl + << " f(x_test) = " << f(x_test) << std::endl << std::endl; + + if (f(x_test) > f(x_neu)) { + x_neu = x_test; + lambda = lamda_test; + std::cout + << " ! Übernehme verdoppelte Schrittweite λ = " << lamda_test << std::endl + << " ✓ f(x_neu) = " << f(x_neu) << " < f(x_test) = " << f(x_test) << std::endl << std::endl; + } + else { + std::cout + << " ↯ f(x_neu) = " << f(x_neu) << " ≥ f(x_test) = " << f(x_test) << std::endl + << " ! behalte aktuelle Schrittweite λ = " << lambda << std::endl << std::endl; + } + } + std::cout << std::endl; + + x = x_neu; + gradfx = gradient(f, x); + schritt_zaehler++; + } + + // Zusammenfassung des Endes + if (gradfx.length() < 1e-5) + std::cout + << "Ende wegen ||grad f(x)|| < 10^-5 bei" << std::endl; + + else + std::cout + << "Ende wegen 25. Schritt" << std::endl; + + std::cout + << " x = " << x << std::endl + << " λ = " << lambda << std::endl + << " f(x) = " << f(x) << std::endl + << " grad f(x) = " << gradfx << std::endl + << " ||grad f(x)|| = " << gradfx.length() << std::endl + << "-----------------------------------------------------" << std::endl << std::endl; + + return x_neu; +} diff --git a/P3/CMyVektor.h b/P3/CMyVektor.h new file mode 100755 index 0000000..8a00f60 --- /dev/null +++ b/P3/CMyVektor.h @@ -0,0 +1,38 @@ +#pragma once +#include +#include + +class CMyVektor { +private: + int dimension; + std::vector werte; +public: + // Vektor mit Dimension anlegen + CMyVektor(int n) : dimension{ n } { + werte.resize(n); + } + + CMyVektor(std::vector x) : dimension{ (int)x.size() } { + werte.resize(x.size()); + for (int i = 0; i < x.size(); i++) + werte[i] = x[i]; + } + + double getElement(int i); + void setElement(int i, double j); + void setElements(std::vector x); + + int getDimension(); + double length(); + + // Komponente von Vektor übergeben + double& operator[](int index); +}; + +CMyVektor operator+(CMyVektor a, CMyVektor b); +CMyVektor operator*(double lambda, CMyVektor a); + +std::ostream& operator<< (std::ostream& os, CMyVektor& vektor); + +CMyVektor gradient(double f(CMyVektor x), CMyVektor x); +CMyVektor gradientenverfahren(double f(CMyVektor x), CMyVektor x, double lambda = 1.0); diff --git a/P3/C_DGLSolver.cpp b/P3/C_DGLSolver.cpp new file mode 100755 index 0000000..c9e2561 --- /dev/null +++ b/P3/C_DGLSolver.cpp @@ -0,0 +1,110 @@ +#include "C_DGLSolver.h" +using namespace std; + +C_DGLSolver::C_DGLSolver() { +} + +// Konstruktor für DGL 1. Ordnung +C_DGLSolver::C_DGLSolver(CMyVektor(*f_DGL_System)(CMyVektor y, double x)) { + isDGLSystem = true; + fDglSystemZeiger = f_DGL_System; +} + +// Konstruktor für DGL n-ter Ordnung +C_DGLSolver::C_DGLSolver(double(*f_DGL_nterOrdnung)(CMyVektor y, double x)) { + isDGLSystem = false; + fDglNterOrdnungZeiger = f_DGL_nterOrdnung; +} + +C_DGLSolver::~C_DGLSolver() { +} + +CMyVektor C_DGLSolver::euler(CMyVektor y, double xStart, double XEnd, double schritte) { + double schrittweite = (XEnd - xStart) / schritte; + CMyVektor yCopy(y.getDimension()); + CMyVektor yAbl(y.getDimension()); + yCopy = y; + + double x = xStart; + int s = 0; + cout << "h = " << schrittweite << endl; + + for (; s < schritte; s++, x += schrittweite) { + yAbl = ableitungen(yCopy, x); + + cout << "Schritt " << s << ": " << endl + << "\t" << " 𝑥 = " << x << endl + << "\t" << " 𝑦 = " << yCopy << endl + << "\t" << "𝑦' = " << yAbl << endl << endl; + + for (int i = 1; i <= yCopy.getDimension(); i++) + yCopy.setElement(i, yCopy.getElement(i) + schrittweite * yAbl.getElement(i)); + } + + cout << "Ende bei " << endl; + cout << "\t" << " 𝑥 = " << x << endl; + cout << "\t" << " 𝑦 = " << yCopy << endl; + return yCopy; +} + +CMyVektor C_DGLSolver::heun(CMyVektor y, double xStart, double XEnd, double schritte) { + double schrittweite = (XEnd - xStart) / schritte; + CMyVektor yCopy(y.getDimension()); + CMyVektor yCopyTest(y.getDimension()); + yCopy = y; + + CMyVektor yAbl(y.getDimension()); + CMyVektor yAblTest(y.getDimension()); + CMyVektor fx_mStg(y.getDimension()); + + double x = xStart; + int s = 0; + cout << "h = " << schrittweite << endl; + + for (; s < schritte; x += schrittweite, s++) { + yAbl = ableitungen(yCopy, x); + + cout << "Schritt " << s << ": " << endl; + cout << "\t" << " 𝑥 = " << x << endl; + cout << "\t" << " 𝑦 = " << yCopy << endl; + + cout << "\t" << "𝑦ₒ' = " << yAbl << endl << endl; + + + for (int i = 1; i <= yCopy.getDimension(); i++) + yCopyTest.setElement(i, yCopy.getElement(i) + schrittweite * yAbl.getElement(i)); + + yAblTest = ableitungen(yCopyTest, x + schrittweite); + + for (int i = 1; i <= fx_mStg.getDimension(); i++) + fx_mStg.setElement(i, 0.5 * (yAbl.getElement(i) + yAblTest.getElement(i))); + + for (int i = 1; i <= yCopy.getDimension(); i++) + yCopy.setElement(i, yCopy.getElement(i) + schrittweite * fx_mStg.getElement(i)); + + cout << "\t" << " 𝑦ₜ = " << yCopyTest << endl; + cout << "\t" << "𝑦ₜ' = " << yAblTest << endl << endl; + cout << "\t" << "𝑦ₘ' = " << fx_mStg << endl << endl; + } + + cout << "Ende bei " << endl + << "\t" << " 𝑥 = " << x << endl + << "\t" << " 𝑦 = " << yCopy << endl; + + return yCopy; +} + +CMyVektor C_DGLSolver::ableitungen(CMyVektor y, double x) { + CMyVektor result(y.getDimension()); + + if (isDGLSystem) { + result = fDglSystemZeiger(y, x); + } + else { + for (int i = 1; i < y.getDimension(); i++) + result.setElement(i, y.getElement(i + 1)); + + result.setElement(y.getDimension(), fDglNterOrdnungZeiger(y, x)); + } + return result; +} diff --git a/P3/C_DGLSolver.h b/P3/C_DGLSolver.h new file mode 100755 index 0000000..54a42c0 --- /dev/null +++ b/P3/C_DGLSolver.h @@ -0,0 +1,26 @@ +#pragma once +#ifndef C_DGLSolver_h +#include +#include +#include +#include "CMyVektor.h" + +class C_DGLSolver +{ +public: + C_DGLSolver(); + C_DGLSolver(CMyVektor(*f_DGL_System)(CMyVektor y, double x)); + C_DGLSolver(double(*f_DGL_nterOrdnung)(CMyVektor y, double x)); + ~C_DGLSolver(); + CMyVektor euler(CMyVektor y, double xStart, double XEnd, double h); + CMyVektor heun(CMyVektor y, double xStart, double XEnd, double h); + + +private: + CMyVektor(*fDglSystemZeiger)(CMyVektor y, double x); + double(*fDglNterOrdnungZeiger)(CMyVektor y, double x); + bool isDGLSystem; + CMyVektor ableitungen(CMyVektor y, double x); +}; + +#endif // !C_DGLSolver_h diff --git a/P3/Heun Verfahren.ods b/P3/Heun Verfahren.ods new file mode 100755 index 0000000..7dc35b0 Binary files /dev/null and b/P3/Heun Verfahren.ods differ diff --git a/P3/Praktikum3.pdf b/P3/Praktikum3.pdf new file mode 100755 index 0000000..2022ec0 Binary files /dev/null and b/P3/Praktikum3.pdf differ diff --git a/P3/main.cpp b/P3/main.cpp new file mode 100755 index 0000000..2005a39 --- /dev/null +++ b/P3/main.cpp @@ -0,0 +1,99 @@ +#include +#include "CMyVektor.h" +#include "CMyMatrix.h" +#include "C_DGLSolver.h" +#include +using namespace std; + +CMyVektor DGLSystem(CMyVektor y, double x) { + CMyVektor result(2); + result.setElement(1, 2 * y.getElement(2) - x * y.getElement(1)); + result.setElement(2, y.getElement(1) * y.getElement(2) - 2 * pow(x, 3)); + return result; +} + +double DGL_dritter_Ordnung(CMyVektor y, double x) { + return 2 * x * y.getElement(2) * y.getElement(3) + 2 * pow(y.getElement(1), 2) * y.getElement(2); +} + +int main() { + int option; + + cout << "Was soll berechnet werden? \n" << + "1 Euler-Verfahren DGL\n" << + "2 Heun-Verfahren DGL\n" << + "3 Abweichungen DGL 3.O\n" << + "0 Beenden\n" + "> "; + cin >> option; + + while (0 < option && option < 4) { + switch (option) { + case 1: { + CMyVektor startwerte(vector{0, 1}); + double xStart = 0; // x0 = 0 + double xEnd = 2; + C_DGLSolver dglSystemTest(DGLSystem); + dglSystemTest.euler(startwerte, xStart, xEnd, 100); + break; + } + case 2: { + CMyVektor startwerte(vector{0, 1}); + double xStart = 0; // x0 = 0 + double xEnd = 2; + C_DGLSolver dglSystemTest(DGLSystem); + dglSystemTest.heun(startwerte, xStart, xEnd, 100); + break; + } + case 3: { + CMyVektor startwerte(3); + CMyVektor eulerResult(3); + CMyVektor heunResult(3); + CMyVektor allResults(8); + C_DGLSolver dglSystemTest(DGL_dritter_Ordnung); + + + double schritte; + double xStart; + double xEnd; + int j = 1; + for (int i = 1; i < 5; i++) { + schritte = pow(10, i); + xStart = 1; + xEnd = 2; + startwerte.setElements(vector{1, -1, 2}); + eulerResult = dglSystemTest.euler(startwerte, xStart, xEnd, schritte); + allResults.setElement(j, eulerResult.getElement(1) - 0.5); + + startwerte.setElements(vector{1, -1, 2}); + heunResult = dglSystemTest.heun(startwerte, xStart, xEnd, schritte); + allResults.setElement(++j, heunResult.getElement(1) - 0.5); + j++; + } + cout << endl; + int k = 1; + cout << setw(10) << "Verfahren" << setw(10) << "Schritte" << setw(10) << "Δ\n"; + for (int j = 1; j <= 4; j++) { + double schritte = pow(10, j); + // E = s+1 , H = E*2 + cout << setw(10) << "Euler" << setw(10) << setprecision(0) << schritte << setw(10) << scientific << setprecision(1) << allResults.getElement(k) << endl; + cout << setw(10) << "Heun" << setw(10) << setprecision(0) << schritte << setw(10) << scientific << setprecision(1) << allResults.getElement(++k) << endl << endl; + + k++; + } + break; + } + } + + cout << "\nHauptmenü: \n" << + "1 Euler-Verfahren DGL\n" << + "2 Heun-Verfahren DGL\n" << + "3 Euler/Heun Vergleich DGL 3.O\n" << + "0 Beenden\n" + "> "; + cin >> option; + + }; + cout << endl << endl; + system("PAUSE"); +} \ No newline at end of file diff --git a/P4/Bilder/1/Daten_original1.txt 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+CKomplex::CKomplex(double a, double b) : real(a), imag(b) { +} + +// Konstruktor mit einem Argument fr e^(j*phi) +CKomplex::CKomplex(double phi) : real(cos(phi)), imag(sin(phi)) { +} + +// Methode zur Rckgabe des Realteils +double CKomplex::re() const { + return real; +} + +// Methode zur Rckgabe des Imaginrteils +double CKomplex::im() const { + return imag; +} + +// berladene Methode fr die Addition von zwei komplexen Zahlen +CKomplex CKomplex::operator+(const CKomplex &other) const { + double a = real + other.real; + double b = imag + other.imag; + return CKomplex(a, b); +} + +// berladene Methode fr die Subtraktion von zwei komplexen Zahlen +CKomplex CKomplex::operator-(const CKomplex &other) const { + double a = real - other.real; + double b = imag - other.imag; + return CKomplex(a, b);} + +// berladene Methode fr die Multiplikation von zwei komplexen Zahlen +CKomplex CKomplex::operator*(const CKomplex &other) const { + double a = real * other.real - imag * other.imag; + double b = real * other.imag + imag * other.real; + return CKomplex(a, b); +} + +// berladene Methode fr die Multiplikation einer double-Zahl mit einer komplexen Zahl +CKomplex CKomplex::operator*(double scalar) const { + double a = real * scalar; + double b = imag * scalar; + return CKomplex(a, b); +} + +// Methode zur Berechnung des Betrags der komplexen Zahl +double CKomplex::abs() const { + return sqrt(real * real + imag * imag); +} diff --git a/P4/CKomplex.h b/P4/CKomplex.h new file mode 100755 index 0000000..616b714 --- /dev/null +++ b/P4/CKomplex.h @@ -0,0 +1,24 @@ +class CKomplex { +private: + double real; + double imag; + +public: + // Konstruktoren + CKomplex(); + CKomplex(double a, double b); + CKomplex(double phi); + + // Methoden zur Rckgabe des Real- und Imaginrteils + double re() const; + double im() const; + + // berladene Operatoren fr Addition und Multiplikation + CKomplex operator+(const CKomplex& other) const; + CKomplex operator-(const CKomplex& other) const; + CKomplex operator*(const CKomplex& other) const; + CKomplex operator*(double scalar) const; + + // Methode zur Berechnung des Betrags + double abs() const; +}; \ No newline at end of file diff --git a/P4/CMakeLists.txt b/P4/CMakeLists.txt new file mode 100755 index 0000000..f445b0e --- /dev/null +++ b/P4/CMakeLists.txt @@ -0,0 +1,12 @@ +# Set the minimum required version of CMake +cmake_minimum_required(VERSION 3.19) + +# Set the project name and specify the C++ as the project language +project(HM_P4) + +# Specify the C++ standard +set(CMAKE_CXX_STANDARD 17) +set(CMAKE_CXX_STANDARD_REQUIRED ON) + +# Add an executable with the above sources +add_executable(HM_P4 main.cpp CKomplex.cpp Fourier.cpp) \ No newline at end of file diff --git a/P4/Daten_original1.txt b/P4/Daten_original1.txt new file mode 100755 index 0000000..f8b1842 --- /dev/null +++ b/P4/Daten_original1.txt @@ -0,0 +1,1001 @@ +1000 +0 11.90000000 0 +1 11.76079970 0 +2 11.62170428 0 +3 11.48272862 0 +4 11.34388760 0 +5 11.20519610 0 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_USE_MATH_DEFINES +#include +#include "Fourier.h" + +using namespace std; + +vector fourier(vector werte, int vz) { + int N = werte.size(); + vector ret; + + for (int n = 0; n < N; n++) { + CKomplex v(0,0); + for (int k = 0; k < N; k++) { + double phi = (vz * -2.0 * M_PI * k * n) / (double) N; + CKomplex e = werte[k] * CKomplex(phi); + v = v + e; + } + v = v * (1 / sqrt(N)); + ret.push_back(v); + } + return ret; +} + +vector Fourier::hin(vector werte) { + return fourier(werte, -1); +} + +vector Fourier::zurueck(vector werte) { + return fourier(werte, 1); +} diff --git a/P4/Fourier.h b/P4/Fourier.h new file mode 100755 index 0000000..015f3ae --- /dev/null +++ b/P4/Fourier.h @@ -0,0 +1,12 @@ +#ifndef __FOURIER_H__ +# define __FOURIER_H__ +#include +#include "CKomplex.h" + +namespace Fourier +{ + extern std::vector hin(std::vector werte); + extern std::vector zurueck(std::vector werte); +} + +#endif diff --git a/P4/main.cpp b/P4/main.cpp new file mode 100755 index 0000000..c497fc5 --- /dev/null +++ b/P4/main.cpp @@ -0,0 +1,106 @@ +#include +#include +#include +#include +#include "Fourier.h" + +using namespace std; + +vector werte_einlesen(string dateiname) { + int i, N, idx; + double re, im; + vector werte; + ifstream fp; + fp.open(dateiname); + if (fp.is_open()) { + fp >> N; + werte.resize(N); + CKomplex null(0, 0); + for (i = 0; i < N; i++) + werte[i] = null; + while (!fp.eof()) { + fp >> idx >> re >> im; + CKomplex a(re, im); + werte[idx] = a; + } + fp.close(); + } + return werte; +} + +void werte_ausgeben(string dateiname, vector &werte, double epsilon = -1.0) { + + int i; + int N = werte.size(); + ofstream fp; + fp.open(dateiname); + if (fp.is_open()) { + fp << N << endl; + for (i = 0; i < N; i++) + if (werte[i].abs() > epsilon) + fp << i << "\t" << setprecision(10) << werte[i].re() << "\t" << werte[i].im() << endl; + fp.close(); + } +} + +double abweichung(vector &a, vector &b) { + double r; + if (a.size() == b.size()) { + for (size_t i(0U), sz(a.size()); i != sz; i++) { + CKomplex a1(a[i]); + CKomplex b1(b[i]); + CKomplex difC = b1 - a1; + double difAbs = difC.abs(); + if (r < difAbs) + r = difAbs; + } + } + return r; +} + +void abweichungBestimmen(string dateinameTransformiert, vector transformierteWerte, double epsilon, + string dateinameOriginal) { + vector originaleWerte(werte_einlesen(dateinameOriginal)); + werte_ausgeben(dateinameTransformiert, transformierteWerte, epsilon); + + vector transformierteWerteEpsilon = werte_einlesen(dateinameTransformiert); + vector zurueckTransformiertWerte = Fourier::zurueck(transformierteWerteEpsilon); + + string nameZ = "Z" + dateinameTransformiert; + werte_ausgeben(nameZ, zurueckTransformiertWerte, -1); + + cout << "Maximale Abweichung bei epsilon " << epsilon << " = " << abweichung( + originaleWerte, zurueckTransformiertWerte) << endl; +} + +int main() { + string dateinameOriginal = "Daten_original2.txt"; + + vector originalWerte = werte_einlesen(dateinameOriginal); + + vector transformierteWerte = Fourier::hin(originalWerte); + + /* + werte_ausgeben("Daten_original_TEST.txt", originalWerte); + + + werte_ausgeben("Daten_original1_trafo.txt", transformierteWerte, -1); + + vector transformierteWerteA = werte_einlesen("Daten_original1_trafo.txt"); + vector zurueckTransformiertWerte = Fourier::zurueck(transformierteWerteA); + werte_ausgeben("Daten_original1_Rtrafo.txt", zurueckTransformiertWerte); + + cout << "Maximale Abweichung bei epsilon " << -1 << " = " << abweichung( + zurueckTransformiertWerte, originalWerte) << endl; + + */ + + abweichungBestimmen("Daten_transformiert_default.txt", transformierteWerte, -1, dateinameOriginal); + abweichungBestimmen("Daten_transformiert_0_001.txt", transformierteWerte, 0.001, dateinameOriginal); + abweichungBestimmen("Daten_transformiert_0_01.txt", transformierteWerte, 0.01, dateinameOriginal); + abweichungBestimmen("Daten_transformiert_0_1.txt", transformierteWerte, 0.1, dateinameOriginal); + abweichungBestimmen("Daten_transformiert_1_0.txt", transformierteWerte, 1.0, dateinameOriginal); + + + return 0; +} diff --git a/P5/CMakeLists.txt b/P5/CMakeLists.txt new file mode 100755 index 0000000..0e1e1ca --- /dev/null +++ b/P5/CMakeLists.txt @@ -0,0 +1,12 @@ +# Set the minimum required version of CMake +cmake_minimum_required(VERSION 3.19) + +# Set the project name and specify the C++ as the project language +project(HM_P5) + +# Specify the C++ standard +set(CMAKE_CXX_STANDARD 17) +set(CMAKE_CXX_STANDARD_REQUIRED ON) + +# Add an executable with the above sources +add_executable(HM_P5 main.cpp) \ No newline at end of file diff --git a/P5/main.cpp b/P5/main.cpp new file mode 100755 index 0000000..a03e238 --- /dev/null +++ b/P5/main.cpp @@ -0,0 +1,117 @@ +#include +#include +#include +#include +#include +#include + +class CZufall { +public: + void initialisiere(int s) { + srand(s); + } + + int wert(int a, int b) { + return a + rand() % (b - a + 1); + } + + void test(int a, int b, int N) { + std::vector haeufigkeiten(b - a + 1, 0); + for (int i = 0; i < N; ++i) { + int zahl = wert(a, b); + haeufigkeiten[zahl - a]++; + } + for (int i = 0; i < haeufigkeiten.size(); ++i) { + std::cout << a + i << ": " << haeufigkeiten[i] << std::endl; + } + } + + void test_falsch(int a, int b, int N) { + std::vector haeufigkeiten(b - a + 1, 0); + for (int i = 0; i < N; ++i) { + initialisiere(time(NULL)); + int zahl = wert(a, b); + haeufigkeiten[zahl - a]++; + } + for (int i = 0; i < haeufigkeiten.size(); ++i) { + std::cout << a + i << ": " << haeufigkeiten[i] << std::endl; + } + } +}; + +class CLotto { +private: + int k, n; + std::vector tippzettel; + CZufall zufall; + +public: + CLotto(int k, int n, int s) : k(k), n(n) { + if (s < 0) { + zufall.initialisiere(time(NULL)); + } else { + zufall.initialisiere(s); + } + } + + void setzeTippzettel(const std::vector& tipp) { + tippzettel = tipp; + } + + std::vector ziehung() { + std::set unique_numbers; + while (unique_numbers.size() < k) { + int zahl = zufall.wert(1, n); + unique_numbers.insert(zahl); + } + return std::vector(unique_numbers.begin(), unique_numbers.end()); + } + + int anzahlRichtige(const std::vector& ziehung) { + int richtige = 0; + for (int zahl : ziehung) { + if (std::find(tippzettel.begin(), tippzettel.end(), zahl) != tippzettel.end()) { + richtige++; + } + } + return richtige; + } + + double monteCarloSimulation(int r, int N, bool gleicherTippzettel) { + int treffer = 0; + for (int i = 0; i < N; ++i) { + if (!gleicherTippzettel) { + setzeTippzettel(ziehung()); + } + std::vector aktuelleZiehung = ziehung(); + if (anzahlRichtige(aktuelleZiehung) == r) { + treffer++; + } + } + return static_cast(treffer) / N; + } +}; + +int main() { + // Aufgabe 1: Testen der Klasse CZufall + CZufall zufall; + zufall.initialisiere(42); + zufall.test(3, 7, 10000); + zufall.test_falsch(3, 7, 10000); + + // Aufgabe 2: Testen der Klasse CLotto + CLotto lotto(8, 38, -1); + std::vector tippzettel = {1, 2, 3, 4, 5, 6, 7, 8}; + lotto.setzeTippzettel(tippzettel); + std::vector ziehung = lotto.ziehung(); + std::cout << "Anzahl Richtige: " << lotto.anzahlRichtige(ziehung) << std::endl; + + // Monte-Carlo-Simulation + double wahrscheinlichkeit = lotto.monteCarloSimulation(2, 10000, true); + double wahrscheinlichkeit2 = lotto.monteCarloSimulation(2, 10000, false); + std::cout << "Wahrscheinlichkeit fr genau 3 Richtige: " << wahrscheinlichkeit << std::endl; + std::cout << "Wahrscheinlichkeit fr genau 3 Richtige: " << wahrscheinlichkeit2 << std::endl; + + + return 0; +} \ No newline at end of file