Author: Don Warrington

The objective is to show that back substitution is backward stable.  Consider the system $latex Rx=c $ where $latex R$ is an upper triangular $latex m\times m$ matrix and $latex x,c$ are $latex m$ column vectors. For a $latex 3\times3$ matrix system, this would look like: $latex \left[\begin{array}{ccc} r_{1,1} & r_{1,2} & r_{1,3}\\ 0 & …

Continue reading "Stability of Back Substitution for Matrix Systems"

The purpose of this piece is to document the numerical integration of a function in a two-dimensional space using the conjugate gradient method. Basic Problem Statement and Closed Form Solution The boundary value problem is as follows: $latex {\frac{\partial^{2}}{\partial{x}^{2}}}u(x,y)+{\frac{\partial^{2}}{\partial{y}^{2}}}u(x,y)=-5\, y\sin(x)\sin(2\, y)+4\,\sin(x)\cos(2\, y) $ where $latex u(x,0)=0 $ $latex u(x,\pi)=0 $ $latex u(0,y)=0 $ $latex u(\pi,y)=0 …

Continue reading "Numerical Integration of a Function in a Two-Dimensional Space, and Its Solution with Conjugate Gradient"

The object of this is to solve the differential equation $latex {\frac{d^{3}}{d{x}^{3}}}y(x)-\mu\,\left(1-\left(y(x)\right)^{2}\right){\frac{d^{2}}{d{x}^{2}}}y(x)+2\,\mu\, y(x)\left({\frac{d}{dx}}y(x)\right)^{2}+{\frac{d}{dx}}y(x)=0 $ for the following boundary conditions and parameters: $latex \mu=\frac{1}{2}$ $latex y\left(0\right)=0$ $latex {\frac{d}{dx}}y(0)=\frac{1}{2}$ $latex y\left(2\right)=1$ Conventional wisdom would indicate that, because of the high order of the derivatives, this problem cannot be solved using a scalar implementation of simple shooting. However, …

Continue reading "Solving a Third-Order Differential Equation Using Simple Shooting and Regula Falsi"