Repository files navigation

Implementations of various algorithms used in Numerical Analysis, from root-finding up to gradient descent and numerically solving PDEs.

Information:

• ab2.m: Implementation of the 2-step Adams-Bashforth method for solving ODEs.
  • Current configuration is for the ODE$y' = 1 + \sin(t)-y$,$y(2)=1$
• adamspc.m: Implementation of an Adams 2nd order Predictor-Corrector Method, using the explicit 2-step Adams-Bashforth method and the implicit 2-step Adams-Moulton method
  • Current configuration is for the ODE$y' = 1 + (t-y)^2$,$y(2) = 1$.
• aitken.m: Implementation of Aitken's Delta-Squared Process for Series Acceleration.
  • Current configuration is for the sequence$\left(p_n\right)$ given by$p_n = \sqrt{\frac{1}{3}e^{p_{n-1}}}$.
• bisection.m: Implementation of Bisection method for finding roots
  • Current configuration is for$f(x) = x + \cos(x)$ in the interval$[-5,5]$ with tolerance$10^{-10}$.
• broyden.m: Implementation of Broyden's Method for finding solutions to nonlinear systems of equations.
  • Current configuration is for the system$\vec{F}(x,y) = (x^2 + y - 11, x + y^2 - 7)$
• euler.m: Implementation of Euler's method for solving ODes
  • Current configuration is for the ODE$y' = 1 + (t-y)^2$,$y(2) = 1$. Implementations on multiple step sizes are given.
• euler_system.m: Implementation of Euler's Method for solving a system of ODEs.
  • Current configuration is for the ODE$y'' = -\sin(y)$,$y(0) = -1$,$y'(0) = 0$.
• fdm.m: Implementation of the Finite Difference method for solving BVPs of the form$y'' = p(x)y' + q(x)y + r(x)$ with$y\in [a,b]$,$y(a) = \alpha$,$y(b) = \beta$.
  • Current configuration is for the system:$p(x) = 2$,$q(x) = -1$,$r(x) = xe^x - x$, with$a = 0$,$b = 2$,$y(a) = 0$,$y(b) = -4$.
• fdmlaplace.m: Implementation of the Finite Difference Method for solving PDEs that satisfy Laplace's Equation$u_{xx} + u_{yy} = 0$
  • Current configuration is for the boundary conditions$(x,y)\in [0,1]\times[0,1]$ given by$u(0,y) = \sin(2\pi y)$,$u(1,y) = u(x,0) = u(x,1) = 0$.
• fixed_point_gauss_seidel.m: Fixed point iteration for a system of nonlinear equations but implemented with the Gauss-Seidel Method.
  • Current configuration is for the system of equations$\vec{G}(x,y) = (\sqrt{1-y^2}, \sqrt{\frac{9-5x^2}{21}})$.
• fixed_point_system.m: Fixed point iteration for a system of nonlinear equations.
  • Current configuration is for the system of equations$\vec{G}(x,y) = (\sqrt{1-y^2}, \sqrt{\frac{9-5x^2}{21}})$.
• fixedptiteration.m: Fixed point iteration method for finding roots
  • Current configuration is for the functions:$g(x) = \pi + \frac{1}{2}\sin(\frac{x}{2})$,$g_1(x) = (-2x^2+x+3)^\frac{1}{4}$,$g_2(x) = \sqrt{\frac{1}{2}(-x^4+x+3)}$,$g_3(x) = \sqrt{\frac{x+3}{x^2+2}}$, and$g_4(x) = \frac{3x^4+2x^2+3}{4x^3+4x-1}$.
• leapfrog: Implementation of the leapfrog method for solving ODEs
  • Current configuration is for the ODE$y' = 1 + \sin(t)-y$,$y(2)=1$.
• lshooting.m: Implementation of the Linear Shooting Method for solving BVPs of the form$y'' = p(x)y' + q(x)y + r(x)$ with$y\in [a,b]$,$y(a) = \alpha$,$y(b) = \beta$.
  • Current configuration is for the system:$p(x) = 2$,$q(x) = -1$,$r(x) = xe^x - x$, with$a = 0$,$b = 2$,$y(a) = 0$,$y(b) = -4$.
• meuler.m: Implementation of the Modified Euler's Method for solving ODEs
  • Current configuration is for the ODE$y' = 1 + (t-y)^2$,$y(2) = 1$. Implementations on multiple step sizes are given.
• modnewton.m: Implementation of a Modified Newton-Raphson method to avoid multiple roots. This is similar to Halley's Method, but instead of using$\frac{f(x)}{\sqrt{|f'(x)|}}$ like in Halley's method, instead using$\frac{f(x)}{f'(x)}$.
  • Current configuration is on the equation$f(x) = e^x - 1 - x - \frac{x^2}{2}$.
• newton.m: Implementation of the Newton-Raphson method for finding roots.
  • Current configuration is on the equation$f(x) = e^x - 1 - x - \frac{x^2}{2}$, with tolerance of$10^{-10}$ and initial guess of$p_0 = 1$.
• newtonsys.m: Implementation of the Newton-Raphson method for a system of nonlinear equations.
  • Current configuration is for the system$\vec{F}(x,y) = (x^2 + y - 11, x + y^2 - 7)$, with an initial guess of$(-0.164, 1)$.
• nlshooting.m: Implementation of the Nonlinear Shooting Method for solving BVPs of the form$y'' = p(x)y' + q(x)y + r(x)$ with$y\in [a,b]$,$y(a) = \alpha$,$y(b) = \beta$.
  • Current configuration is for the system:$p(x) = 2$,$q(x) = -1$,$r(x) = xe^x - x$, with$a = 0$,$b = 2$,$y(a) = 0$,$y(b) = -4$.
• rk4_system.m: Implementation of the 4th order Runge-Kutta method for solving a system of ODEs.
  • Current configuration is for the ODE$y'' = -y$,$y(0) = 1$,$y'(0) = 0$.
• secant.m: Implementation of the Secant method for finding roots
  • Current configuration is for the function$f(x) = -x^3-\cos(x)$ with tolerance$10^{-20}$ and a maximum of$100$ iterations with the initial guesses$p_0 =-1$,$p_1 = 0$.
• spline.m: Building splines with natural boundary conditions.
  • Current configuration is for building a cubic spline for the points$(0.1, -0.62049958), (0.2,-0.28398668), (0.3, 0.00660095), (0.4,0.24842440)$.
• steepdesc.m: Implementation of Steepest Gradient Descent for solving systems of nonlinear equations.
  • Current configuration is for the system$\vec{F}(x,y) = (x^2 + y - 11, x + y^2 - 7)$, with an initial guess of$(-0.164, 1)$.