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Numerical Toolbox

Welcome to the Numerical Toolbox, a collection of tools designed to provide numerical solutions to various mathematical problems. Whether you're working on linear algebra, solving systems of equations, finding eigenvalues, fitting curves, or tackling differential equations, this toolbox has you covered.

Basic Vector and Matrix Operations

CVector: Basic Vector Operations

CVector<double> v1{ 1, 2, 3, 4, 5 };
std::cout << v1 << "\n";
CVector<double> v2{ 1, 2, 3, 4, 5 };
std::cout << v1 + v2 << "\n";
std::cout << v1 - v2 << "\n";
std::cout << v1 * v2 << "\n";

Matrix: Basic Matrix Operations

Matrix<double> m1 = { 3,3, 1, 2, 3, 4, 5, 6, 7, 8, 9 }; // First two numbers are the dimensions of the matrix
std::cout << m1 << "\n";
Matrix<double> m2 = { 3,3, 1, 2, 3, 4, 5, 6, 7, 8, 9 };
std::cout << m1 + m2 << "\n";
std::cout << m1 - m2 << "\n";
std::cout << m1 * m2 << "\n";

CVector<double> v3 = { 1, 2, 3 };
std::cout << m1 * v3 << "\n";

You can assign a vector to a row of a matrix. Must be the same size as the row

m1(2) = v3;
std::cout << m1 << "\n";

MATRIXTOOLBOX: More Matrix Tricks

Generate a random matrix

Matrix<double> rand_mat = MATRIXTOOLBOX::random_matrix(4, 4, -500, 500, MATRIXTOOLBOX::RandMode::POSITIVEDEF, 10);
std::cout << rand_mat << "\n";

Generate an identity matrix

Matrix<double> identity_mat = MATRIXTOOLBOX::identity(4);

Transpose a matrix

Matrix<double> test = {3,3,3,2,1,
                            4,2,2,
                            5,6,2};
std::cout << MATRIXTOOLBOX::Transpose(test) << "\n";

Calculate the determinant of a matrix

double det{};
MATRIXTOOLBOX::Determinant(test, det) ;
//-6
std::cout << det<< "\n";

Linear Algebra

System of Linear Equations

Solve the system of linear equations:

$$ \begin{align*} x + y + z &= 3 \\ 2x + 3y + 4z &= 20 \\ x + y - z &= 0 \end{align*} $$

Matrix<double> A { 3, 3, 1, 1, 1, 
                        2, 3, 4, 
                        1, 1, -1 };
CVector<double> b { 3, 20, 0 };
CVector<double> x;
MATRIXTOOLBOX::AxEqb(A, x, b);
std::cout << x << "\n";

x = {0, 0, 0};
MATRIXTOOLBOX::LUSolve(A, x, b);
std::cout << x << "\n";

x = {0, 0, 0};
MATRIXTOOLBOX::LDLTSolve(A, x, b);
std::cout << x << "\n";

Use Inverse Iteration to find the eigenvalue of a matrix nearest to an initial guess.

A = { 3, 3, 2, 1, 1,
            1, 3, 1, 
            1, 1, 4};
CVector<double> eigenvector = { 1, 1, 1 };
eigenvector/=sqrt(3);
double eigenvalue{5.0};
MATRIXTOOLBOX::EIGEN::vector_iter_inverse(A, eigenvector, 5.0, eigenvalue, 100);
// 5.214319743184
std::cout << "Eigenvalue: " << eigenvalue << "\n";
std::cout << "Eigenvector: " << eigenvector << "\n";

Solve an eigenvalue problem of the form $$BX = \lambda{}X$$

Matrix <double> B = {2, 2 , 3,-1,
                            -1, 3};
Matrix <double> LAM;
// add true as the last argument to print the eigenvalues and eigenvectors
MATRIXTOOLBOX::EIGEN::eigen_Jacobi(A, LAM, 100, true);
//Diagonals are the eigenvalues
std::cout << LAM << "\n";

Solve a eigenvalue problem of the form $$KX = \lambda{}MX$$

Matrix<double> K = {3,3,3,2,1,
                        2,2,1,
                        1,1,1};
Matrix<double> M(MATRIXTOOLBOX::identity(K.get_rows()));
Matrix<double> X;
Matrix<double> LAMDBA;
std::cout << "Generalized Eigenvalue Problem\n";
MATRIXTOOLBOX::EIGEN::generalized_eigen_Jacobi(K, X, LAMDBA, M, 100, true);

Curve Fitting

Least Squares Fit

Performing a least squares fit to a function $$x^2$$, we utilize the LSF class to obtain coefficients, residuals, and the coefficient of determination $$r^2$$.

LSF lsf;
int terms = 3;
x = CVector<double> {1, 2, 3, 4, 5, 6, 7};
CVector<double> y {1, 4, 9, 16, 25, 36, 49};

CVector<double> coeff(terms);
double residuals;
double r2;
lsf.fit(terms, x, y, coeff, residuals, r2);
std::cout << coeff << "\n";
std::cout << "residuals = " << residuals << "\n";
std::cout << "r^2 = " << r2 << "\n";

Modifying it a bit to introduce noise

x = CVector<double> {1.01, 1.99, 3.01, 3.99, 5.01, 5.99, 7.01};
y = CVector<double> {1.01, 3.98, 9.02, 15.99, 25.01, 35.99, 49.01};

lsf.fit(terms, x, y, coeff, residuals, r2);
std::cout << coeff << "\n";
std::cout << "residuals = " << residuals << "\n";
std::cout << "r^2 = " << r2 << "\n";

Ordinary Differential Equations (ODEs)

Solving Linear ODEs

Solving the differential equation $$Y''' + Y'' + Y' + Y = 0$$ with initial conditions, where $$Y(0) = 0$$, $$Y'(0) = 0$$, $$Y''(0) = 0$$, and $$Y'''(0) = 10$$.

Decouple into 3 first-order ODEs:

$$ \begin{align*} Y_1' &= Y_2 \\ Y_2' &= Y_3 \\ Y_3' &= -Y_1 - Y_2 - Y_3 \end{align*} $$

LINODESOLVER::FunctionContainer functions;
functions.addFunction([](CVector<double> points) {return points(3); });
functions.addFunction([](CVector<double> points) {return points(4); });
functions.addFunction([](CVector<double> points) {return -points(2) - points(3) - points(4); });
Matrix<double> solution;
solution = LINODESOLVER::ode45wrapperLinear(functions, 0, 10, { 0, 0, 0, 10 }, 0.1);
std::cout << std::setprecision(15) << solution(solution.get_rows()) << "\n";
std::cout << "Number of points: " << solution.get_rows() << "\n";

Integration and Differentiation

Integration

auto integral_func = [](int FUNCNO, double x) {return x*x*x; };
auto solution_func = [](double x) {return x*x*x*x/4; };

const int FUNC_NO = 1;
const double upper_limit = 79;
const double lower_limit = 10;

print(std::setprecision(15), "Solution: ", solution_func(upper_limit) - solution_func(lower_limit), "\n");
print("Trapezodial: ", Trapezodial(FUNC_NO, 16356, upper_limit, lower_limit, integral_func), "\n");
print("Simpson: ", Simpson(FUNC_NO, 16356, upper_limit, lower_limit, integral_func), "\n");
print("Gauss-Legendre: ", quadrature(FUNC_NO, 5, upper_limit, lower_limit, integral_func), "\n");

Differentiation

auto diff_func = [](int FUNCNO, double x) {return 1000*atan(x); };
auto diff_solution_func = [](double x) {return 1000/(1+x*x); };

double central_point = 100;
double h = 0.01;
print(std::setprecision(15), "Solution: ", diff_solution_func(central_point), "\n");
print("Forward Difference: ", ForwardMethod(FUNC_NO, central_point, h, diff_func), "\n");
print("Backward Difference: ", BackwardMethod(FUNC_NO, central_point, h, diff_func), "\n");
print("Central Difference: ", CentralMethod(FUNC_NO, central_point, h, diff_func), "\n");
print("Fourth Order Central Difference: ", FourthOrderCentralMethod(FUNC_NO, central_point, h, diff_func), "\n");