Blog Entry Tuesday, June 18, 2024 (c) James Pate Williams, Jr. FreeLIP Computation of Euler Numbers and Tangent Numbers

FreeLIP is a free large integer package solely created by Professor Emeritus Arjen K. Lenstra of the Number Field Sieve fame. He developed FreeLIP while he was an employee of AT&T – Lucent in the late 1980s. His copyright notice in the header file, lip.h, states copyright from 1989 to 1997. I have been using this excellent number theoretical package since the late 1990s. See the paper by Donald E. Knuth and Thomas J. Buckholtz for the formula for Tangent Numbers. I can’t remember where I got the Euler Numbers recurrence relation. I wrote a C# application in 2015 for computing Euler Numbers. The code below is in the vanilla C computer language. Excellent resources for the Euler and tangent numbers also known as zag numbers are:

https://oeis.org/A122045

https://oeis.org/A000182

Blog Entry Sunday, June 16, 2024 (c) James Pate Williams, Jr. Chapter 4 Matrices and Systems of Linear Equations from a Textbook by S. D. Conte and Carl de Boor

Blog Entry Friday, June 14, 2024 (c) James Pate Williams, Jr.

For the last week or so I have been working my way through Chapter 3 The Solution of Nonlinear Equations found in the textbook “Numerical Analysis: An Algorithmic Approach” by S. D. Conte and Carl de Boor. I also used some C source code from “A Numerical Library in C for Scientists and Engineers” by H. T. Lau, PhD. I implemented twenty examples and exercises from the previously mentioned chapter.

Blog Entry June 5-7, 2024, (c) James Pate Williams, Jr. All Applicable Rights Reserved Chapter 7 Example and Some Exercises from “Numerical Analysis: An Algorithmic Approach (c) 1980 by S. D. Conte and Carl de Boor (Numerical Differentiation and Numerical Integration)

Blog Entry June 3-4, 2024, (c) James Pate Wiliams, Jr., Solution of Tridiagonal Matrix Problems

The first solution is from the textbook, Elementary Numerical Analysis: An Algorithmic Approach (c) 1980 by S. D. Conte and Carl de Boor. I translated the FORTRAN code to vanilla C using Visual Studio 2019 Community Version. The second solution is from Boundary Value Problems Second Edition (c) 1979 by David L. Powers. It solves a simple second order linear ordinary differential equation using the finite element difference equation method.

Blog Entry for Early Morning Friday, May 31, 2024, Solutions of a Second Order Self-Adjoint Ordinary Differential Equation by Three Methods

Recent Email that I Wrote on May 23, 2024

Does the following thought experiment make sense?

Suppose we have a positively charged quantum mechanical particle in a finite potential energy well. Also suppose there is a free negatively charged quantum mechanical particle outside the potential energy well. There is a measurable probability that the positively charged particle will tunnel through the potential energy well and perhaps be attracted to the negatively charged particle. Likewise, the negatively charged particle has a finite probability of penetrating the potential energy well and hooking up with the positively charged particle should it still be trapped in the well. There is no “spooky action at a distance” to use Albert Einstein’s 1930s definition of quantum entanglement in this example since this electromagnetic attraction is a local phenomenon (?). The positively charged particle cannot exert an attractive force until it tunnels through the energy barrier or otherwise the negatively charged particle winds up breaking into the well. I don’t know exactly how quantum electrodynamics would explain this example. Perhaps the positively charged particle is a positron (antimatter lepton) and the negatively charged particle is a plain vanilla electron. We know that the local result of the interaction of our two matter-antimatter particles is an annihilation event whereby two energetic photons are created, or other products are generated.
https://en.wikipedia.org/wiki/Annihilation#/media/File:Electron_Positron_Annihilation.png