Numerical Analysis – Page 28 – Numerical Explorations

Win32 C Real Double Precision and Integer Matrix Add and Subtract by James Pate Williams, Jr.

void DblAdd(int l, double* a, double* b, double* c)
{
	_asm
	{
		mov eax, a; load a matrix address
		mov ebx, b; load a matrix address
		mov edi, c; load result address
		mov edx, l; load matrix length
		finit;
	ll:	fld qword ptr[eax];
		fld qword ptr[ebx];
		fadd;
		fstp qword ptr[edi];
		fwait;
		add edi, 8;
		add eax, 8;
		add ebx, 8;
		dec edx;
		jnz ll;
	}
}

void DblSub(int l, double* a, double* b, double* c)
{
	_asm
	{
		mov eax, a; load a matrix address
		mov ebx, b; load a matrix address
		mov edi, c; load result address
		mov edx, l; load matrix length
		finit;
	ll:	fld qword ptr[eax];
		fld qword ptr[ebx];
		fsub;
		fstp qword ptr[edi];
		fwait;
		add edi, 8;
		add eax, 8;
		add ebx, 8;
		dec edx;
		jnz ll;
	}
}

C Matrix Operations: Add, Subtract, Multiply, and Determinant Calculation by James Pate Williams, Jr.

Eigenfunctions of a Nonlinear Second Order Ordinary Differential Equation Initial Value Eigenvalue Problem by James Pate Williams, Jr.

Solutions of a nonlinear second order ordinary differential equation initial value eigenvalue problem:

y”(x) = x * x * y(x) + n * n * y(x) * y(x) for all x in [0, 1]
y(0) = 0
y'(0) = 1
n in [0, 1, 2, …]

I graphed the first five eigenfunctions.

Blog Entry Sunday April 17, 2022

radioactive-decay-chain-example Download

Blog Entry Friday April 15, 2022 – The Gamma Function and Related Functions

gamma-functions-blog-entry Download

Blog Entry of January 27, 2022 by James Pate Williams, Jr

blog-entry-01-27-2022-1 Download

Python Code to Implement the Linear Algebraic Rule by Cramer for a 3 by 3 Set of Linear Equations by James Pate Williams Jr

The algebraic example comes from the website:

Cramers Rule Calculator (ncalculators.com)

pythoncramersrule3x3code Download

C++ Orthogonal Polynomials Static Library by James Pate Williams, Jr

Three Term Recurrence Relations for the Legendre Functions by James Pate Williams, Jr.

bachelorthesis.dvi (uni-ulm.de)

My implementations and additional graphs for the Legendre functions mentioned in the thesis cited in the preceding line and PDF. The Legendre polynomials, functions, and associated functions have many applications in quantum mechanics and other branches of applied and theoretical physics.

Graph of P(cos x, 8) Real Valued Function

Graph of Q(cos x, 8) Real Valued Function

First Order Perturbation Calculation for the Helium Atom Ground State by James Pate Williams, Jr.

The first order perturbation calculation for the helium atom ground state is treated in detail in the textbook “Quantum Mechanics Third Edition” by Leonard I. Schiff pages 257 to 259. I offer a numerical algorithm for computing the electron-electron repulsion interaction which is analytically determined by Schiff and other scientists. Next is the graphical user interface for the application and its output.

The Ep text box is the ground state energy as found by a first order perturbation computation. The Ee text box is the experimental ground state energy. The IA text box is the analytic electron-electron repulsion interaction determined by Schiff and other quantum mechanics researchers. The IC text box is my numerical contribution. All the energies are in electron volts.