Author: Don Warrington

Our goal is to demonstrate that, for the Hermite polynomial $latex H_{2n+1}(x)=\sum_{j=0}^{n}f(x_{j})H_{j}(x)+\sum_{j=0}^{n}f'(x_{j})\hat{H}_{j}(x) $ where $latex H_{j}(x)=[1-2(x-x_{j})L'_{j}(x_{j})]L_{j}^{2}(x) $ $latex \hat{H}_{j}(x)=(x-x_{j})L_{j}^{2}(x) $ the error function is given by the equation $latex f(x)-H_{2n+1}(x)=\frac{f^{\left(2n+2\right)}\left(\eta\left(x\right)\right)}{\left(2n+2\right)!}\overset{n}{\underset{j=0}{\prod}}\left(x-x_{j}\right)^{2} $ where $latex f\in C^{2n+2}[a,b] $ Let us begin by considering a point $latex \hat{x}\in[a,b]$ where $latex \hat{x}\neq x_{j}$, i.e., it is not equal to …

Continue reading "Error Function for an Hermite Polynomial"

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"