Blog Entry © Saturday – Tuesday, May 30 – June 2, 2026, by James Pate Williams, Jr., theMicrosoft Bing Copilot, the M365 Copilot Partial Reproduction of Figures 8 and 9 fromChapter 2 of Quantum Mechanics Third Edition by Leonard I. Schiff

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class Figure
{
public:
	static void ComputeFigure(
		bool eight,
		double& xi2, double& eta2,
		std::vector<double>& radius,
		std::vector<double>& xi,
		std::vector<double>& eta,
		std::vector<double>& vix,
		std::vector<double>& viy,
		std::vector<double>& energy1,
		double (*f)(double),
		double (*g)(double));
};


#include "Figure.h"
#include <algorithm>
#include <cmath>
#include <iterator>
#include <set>
#include <vector>

// Function to compute intersection of two sorted containers of doubles with tolerance
template <typename InputIt1, typename InputIt2, typename OutputIt>
void set_intersection_with_tolerance(InputIt1 first1, InputIt1 last1,
	InputIt2 first2, InputIt2 last2,
	OutputIt d_first, double tolerance) {
	while (first1 != last1 && first2 != last2) {
		double a = *first1;
		double b = *first2;

		if (std::fabs(a - b) <= tolerance) {
			// Values are considered equal within tolerance
			*d_first++ = /*a; // or */ (a + b) / 2.0; //if you want averaged value
			++first1;
			++first2;
		}
		else if (a < b - tolerance) {
			++first1;
		}
		else {
			++first2;
		}
	}
}

static double f(double xi)
{
	return xi * tan(xi);
}

static double g(double xi)
{
	return -xi * 1.0 / tan(xi);
}

static void Intersection(
	bool eight, std::vector<double>& intersection)
{
	double del = 0.0, radius2 = 0.0;
	double tolerance = 0.0;
	double (*h)(double);
	int ilimit = 0;
	std::set<double> eta, rad;

	if (eight)
	{
		ilimit = 10000;
		radius2 = 1.0;
		tolerance = 1.0e-1;
		h = f;
	}

	else
	{
		ilimit = 10000;
		radius2 = 4.0;
		tolerance = 1.0e-1;
		h = g;
	}

	del = radius2 / ilimit;
	intersection.clear();

	for (size_t i = 0; i < ilimit; i++)
	{
		double xi0 = i * del;

		if (eight && xi0 >= 0.65)
		{
			double et = h(xi0);
			double r2 = xi0 * xi0 + et * et;
			
			if (fabs(r2 - radius2) < tolerance)
			{
				eta.insert(et);
				rad.insert(r2);
			}
		}

		else if (!eight && xi0 >= 1.8125)
		{
			double et = h(xi0);
			double r2 = xi0 * xi0 + et * et;

			if (fabs(r2 - radius2) < tolerance)
			{
				eta.insert(et);
				rad.insert(r2);
			}
		}
	}

	if (eight)
	{
		size_t count = 0, index = 0;

		for (double val : eta)
		{
			if (index == eta.size() - 1)
			{
				intersection.push_back(val);
				break;
			}

			count++;
			index++;
		}
	}

	else
	{
		int count = 24, index = 0;

		for (double val : eta)
		{
			if (index == eta.size() - count)
			{
				intersection.push_back(val);
				break;
			}

			index++;
		}
	}

	/*set_intersection_with_tolerance(
		eta.begin(), eta.end(),
		rad.begin(), rad.end(),
		std::back_inserter(intersection),
		tolerance);*/
}

void Figure::ComputeFigure(
	bool eight,
	double& xi2, double& eta2,
	std::vector<double>& radius,
	std::vector<double>& xi,
	std::vector<double>& eta,
	std::vector<double>& vix,
	std::vector<double>& viy,
	std::vector<double>& intersection,
	double (*f)(double),
	double (*g)(double))
{
	double xi0 = 0.0;
	double xi1 = 3.5;
	double eta0 = 0.0, eta1 = 0.0;
	double (*h)(double);

	xi.clear();
	eta.clear();

	if (eight)
		h = f;
	else
	{
		xi0 = 1.5;
		h = g;
	}

	eta0 = h(xi0);
	eta1 = h(xi1);

	double deltaXi = (xi1 - xi0) / 10000.0;
	int count = 0;

	vix.clear();
	viy.clear();

	for (int j = 0; j <= 10000; j++)
	{
		double x = j * deltaXi;
		double hx = h(x);
		double vx = x * x + hx * hx;

		if (eight && x >= xi0 && x <= xi1)
		{
			if (count == 0 && x >= xi0)
			{
				xi.push_back(x);
				eta.push_back(hx);
				count = 1;
			}

			else
			{
				xi.push_back(x);
				eta.push_back(hx);
			}

			for (int k = 0; k < 2; k++)
			{
				double r = radius[k];

				vix.push_back(xi[xi.size() - 1]);
				viy.push_back(r * r);
			}
		}

		else if (!eight && x >= xi0 && x <= xi1)
		{
			if (count == 0 && x >= xi0)
			{
				xi.push_back(x);
				eta.push_back(hx);
				count = 1;
			}

			else if (count == 1)
			{
				xi.push_back(x);
				eta.push_back(hx);
			}

			if (xi.size() > 0)
			{
				for (int k = 0; k < 1; k++)
				{
					double r = radius[k];

					vix.push_back(xi[xi.size() - 1]);
					viy.push_back(r * r);
				}
			}
		}

		if (eight && xi[j] >= 0.0 && eta[j] >= 3.5)
			break;
		
		else if (!eight &&
			xi.size() > 0 &&
			xi[xi.size() - 1] > 1.5 &&
			eta.size() > 0 &&
			eta[eta.size() - 1] >= 3.5)
			break;
	}
	
	Intersection(eight, intersection);
}


// SchiffChapter2Fig8and9.cpp : Defines the entry point for the application.
//

#include "framework.h"
#include "SchiffChapter2Fig8and9.h"
#include <time.h>
#include <algorithm>
#include <vector>
#include "Figure.h"

#define MAX_LOADSTRING 100

typedef struct tagPoint2d
{
    double x, y;
} Point2d, * PPoint2d;

// Global Variables:
HINSTANCE hInst;                                // current instance
WCHAR szTitle[MAX_LOADSTRING];                  // The title bar text
WCHAR szWindowClass[MAX_LOADSTRING];            // the main window class name
WCHAR line[8192], text[8192];                   // wide character buffers
WCHAR title[65536];                             // window title
bool eight;                                     // true = plot figure 8
double xi2, eta2;                               // energy cordinates
std::vector<Point2d> points;                    // plotting 2d points
std::vector<double> radius;                     // radius vector { 1.0, 2.0 }
std::vector<double> xi;                         // Greek x-coordinate
std::vector<double> eta;                        // Greek y-coordinate
std::vector<double> V0;                         // potential of the well
std::vector<double> vix;                        // potential x-coordinate
std::vector<double> viy;                        // potential y-coordinate
std::vector<double> energy1;                    // energy eigenvalues

// Forward declarations of functions included in this code module:
ATOM                MyRegisterClass(HINSTANCE hInstance);
BOOL                InitInstance(HINSTANCE, int);
LRESULT CALLBACK    WndProc(HWND, UINT, WPARAM, LPARAM);
INT_PTR CALLBACK    About(HWND, UINT, WPARAM, LPARAM);
INT_PTR CALLBACK    DrawEtaDialog(HWND, UINT, WPARAM, LPARAM);

int APIENTRY wWinMain(_In_ HINSTANCE hInstance,
                     _In_opt_ HINSTANCE hPrevInstance,
                     _In_ LPWSTR    lpCmdLine,
                     _In_ int       nCmdShow)
{
    UNREFERENCED_PARAMETER(hPrevInstance);
    UNREFERENCED_PARAMETER(lpCmdLine);

    // TODO: Place code here.

    // Initialize global strings
    LoadStringW(hInstance, IDS_APP_TITLE, szTitle, MAX_LOADSTRING);
    LoadStringW(hInstance, IDC_SCHIFFCHAPTER2FIG8AND9, szWindowClass, MAX_LOADSTRING);
    MyRegisterClass(hInstance);

    // Perform application initialization:
    if (!InitInstance (hInstance, nCmdShow))
    {
        return FALSE;
    }

    HACCEL hAccelTable = LoadAccelerators(hInstance, MAKEINTRESOURCE(IDC_SCHIFFCHAPTER2FIG8AND9));

    MSG msg;

    // Main message loop:
    while (GetMessage(&msg, nullptr, 0, 0))
    {
        if (!TranslateAccelerator(msg.hwnd, hAccelTable, &msg))
        {
            TranslateMessage(&msg);
            DispatchMessage(&msg);
        }
    }

    return (int) msg.wParam;
}

//
//  FUNCTION: MyRegisterClass()
//
//  PURPOSE: Registers the window class.
//
ATOM MyRegisterClass(HINSTANCE hInstance)
{
    WNDCLASSEXW wcex = { 0 };

    wcex.cbSize = sizeof(WNDCLASSEX);

    wcex.style          = CS_HREDRAW | CS_VREDRAW;
    wcex.lpfnWndProc    = WndProc;
    wcex.cbClsExtra     = 0;
    wcex.cbWndExtra     = 0;
    wcex.hInstance      = hInstance;
    wcex.hIcon          = LoadIcon(hInstance, MAKEINTRESOURCE(IDI_SCHIFFCHAPTER2FIG8AND9));
    wcex.hCursor        = LoadCursor(nullptr, IDC_ARROW);
    wcex.hbrBackground  = (HBRUSH)(COLOR_WINDOW+1);
    wcex.lpszMenuName   = MAKEINTRESOURCEW(IDC_SCHIFFCHAPTER2FIG8AND9);
    wcex.lpszClassName  = szWindowClass;
    wcex.hIconSm        = LoadIcon(wcex.hInstance, MAKEINTRESOURCE(IDI_SMALL));

    return RegisterClassExW(&wcex);
}

//
//   FUNCTION: InitInstance(HINSTANCE, int)
//
//   PURPOSE: Saves instance handle and creates main window
//
//   COMMENTS:
//
//        In this function, we save the instance handle in a global variable and
//        create and display the main program window.
//
BOOL InitInstance(HINSTANCE hInstance, int nCmdShow)
{
   hInst = hInstance; // Store instance handle in our global variable

   HWND hWnd = CreateWindowW(szWindowClass, szTitle, WS_OVERLAPPEDWINDOW,
      CW_USEDEFAULT, 0, CW_USEDEFAULT, 0, nullptr, nullptr, hInstance, nullptr);

   if (!hWnd)
   {
      return FALSE;
   }

   ShowWindow(hWnd, nCmdShow);
   UpdateWindow(hWnd);

   return TRUE;
}

static double fx(double x)
{
    return x * tan(x);
}

static double gx(double x)
{
    return -x * 1.0 / tan(x);
}

//
//  FUNCTION: WndProc(HWND, UINT, WPARAM, LPARAM)
//
//  PURPOSE: Processes messages for the main window.
//
//  WM_COMMAND  - process the application menu
//  WM_PAINT    - Paint the main window
//  WM_DESTROY  - post a quit message and return
//
//
LRESULT CALLBACK WndProc(HWND hWnd, UINT message, WPARAM wParam, LPARAM lParam)
{
    switch (message)
    {
    case WM_COMMAND:
        {
            int wmId = LOWORD(wParam);
            // Parse the menu selections:
            switch (wmId)
            {
            case IDM_ABOUT:
                DialogBox(hInst, MAKEINTRESOURCE(IDD_ABOUTBOX), hWnd, About);
                break;
            case IDM_FIGURE8:
            {
                eight = true;
                text[0] = L'\0';

                clock_t clock0 = clock();
                radius.push_back(1.0);
                radius.push_back(2.0);
                Figure::ComputeFigure(
                    eight, xi2, eta2, radius, xi, eta, vix, viy, energy1, fx, gx);
                clock_t clock1 = clock() - clock0;
                double runtime = (double)clock1 / CLOCKS_PER_SEC;
                swprintf_s(line, 8192, L"Runtime in Seconds = %lf\r\n", runtime);
                wcscat_s(text, 8192, line);

                points.clear();
               
                for (size_t j = 0; j < xi.size(); j++)
                {
                    Point2d pt = { 0 };
                    pt.x = xi[j];
                    pt.y = eta[j];
                    points.push_back(pt);
                }
                
                for (size_t i = 0; i < energy1.size(); i++)
                {
                    swprintf_s(line, 8192, 
                        L"E[%zu] = %lf\r\n", i, energy1[i]);
                    wcscat_s(text, 8192, line);
                }

                MessageBox(hWnd, text, L"Energy Information",
                    MB_OK | MB_ICONINFORMATION);

                DialogBox(hInst, MAKEINTRESOURCE(IDD_DRAW_ETA_DIALOG), hWnd, DrawEtaDialog);
                break;
            }
            case IDM_FIGURE9:
            {
                eight = false;
                text[0] = L'\0';

                clock_t clock0 = clock();
                radius.push_back(2.0);
                Figure::ComputeFigure(
                    eight, xi2, eta2, radius, xi, eta, vix, viy, energy1, fx, gx);
                clock_t clock1 = clock() - clock0;
                double runtime = (double)clock1 / CLOCKS_PER_SEC;
                swprintf_s(line, 8192, L"Runtime in Seconds = %lf\r\n", runtime);
                wcscat_s(text, 8192, line);

                points.clear();
                
                for (size_t j = 0; j < xi.size(); j++)
                {
                    Point2d pt = { 0 };
                    pt.x = xi[j];
                    pt.y = eta[j];
                    points.push_back(pt);
                }

                for (size_t i = 0; i < energy1.size(); i++)
                {
                    swprintf_s(line, 8192,
                        L"E[%zu] = %lf\r\n", i, energy1[i]);
                    wcscat_s(text, 8192, line);
                }

                MessageBox(hWnd, text, L"Energy Information",
                    MB_OK | MB_ICONINFORMATION);

                DialogBox(hInst, MAKEINTRESOURCE(IDD_DRAW_ETA_DIALOG), hWnd, DrawEtaDialog);
                break;
            }
            case IDM_EXIT:
                DestroyWindow(hWnd);
                break;
            default:
                return DefWindowProc(hWnd, message, wParam, lParam);
            }
        }
        break;
    case WM_PAINT:
        {
            PAINTSTRUCT ps;
            HDC hdc = BeginPaint(hWnd, &ps);
            // TODO: Add any drawing code that uses hdc here...
            EndPaint(hWnd, &ps);
        }
        break;
    case WM_DESTROY:
        PostQuitMessage(0);
        break;
    default:
        return DefWindowProc(hWnd, message, wParam, lParam);
    }
    return 0;
}

// Message handler for about box.
INT_PTR CALLBACK About(HWND hDlg, UINT message, WPARAM wParam, LPARAM lParam)
{
    UNREFERENCED_PARAMETER(lParam);
    switch (message)
    {
    case WM_INITDIALOG:
        return (INT_PTR)TRUE;

    case WM_COMMAND:
        if (LOWORD(wParam) == IDOK || LOWORD(wParam) == IDCANCEL)
        {
            EndDialog(hDlg, LOWORD(wParam));
            return (INT_PTR)TRUE;
        }
        break;
    }
    return (INT_PTR)FALSE;
}

static void FindMinMax(
    std::vector<Point2d>& points,
    double& xMin, double& xMax,
    double& yMin, double& yMax)
{
    // uses global 2D double points structure

    xMin = yMin = DBL_MAX;
    xMax = yMax = DBL_MIN;

    for (size_t i = 0; i < points.size(); i++)
    {
        Point2d pt = points[i];
        double x = pt.x;
        double y = pt.y;

        if (x < xMin)
            xMin = x;
        if (x > xMax)
            xMax = x;
        if (y < yMin)
            yMin = y;
        if (y > yMax)
            yMax = y;
    }
}

static void DrawFormattedText(HDC hdc, char text[], RECT rect)
{
    // Draw the text with formatting options
    DrawTextA(hdc, text, -1, &rect, DT_SINGLELINE | DT_NOCLIP);
}

static void DrawQuarterCircleArc(
    HDC hdc,
    float xSlope, float ySlope,
    float xInter, float yInter,
    float radius, bool topToRight)
{
    auto mapX = [&](float x)
        {
            return (int)lroundf(xSlope * x + xInter);
        };

    auto mapY = [&](float y)
        {
            return (int)lroundf(ySlope * y + yInter);
        };

    int x1 = mapX(-radius);
    int y1 = mapY(+radius);
    int x2 = mapX(+radius);
    int y2 = mapY(-radius);

    int left = min(x1, x2);
    int right = max(x1, x2);
    int top = min(y1, y2);
    int bottom = max(y1, y2);

    int xTop = mapX(0.0f);
    int yTop = mapY(radius);

    int xRight = mapX(radius);
    int yRight = mapY(0.0f);

    if (topToRight)
    {
        Arc(hdc, left, top, right, bottom,
            xTop, yTop, xRight, yRight);
    }

    else
    {
        Arc(hdc, left, top, right, bottom,
            xRight, yRight, xTop, yTop);
    }
}

INT_PTR CALLBACK DrawEtaDialog(
    HWND hDlg, UINT message, WPARAM wParam, LPARAM lParam)
{
    UNREFERENCED_PARAMETER(lParam);
    switch (message)
    {
    case WM_INITDIALOG:
        SetWindowText(hDlg, title);
        return (INT_PTR)TRUE;
    case WM_COMMAND:
        if (LOWORD(wParam) == IDOK || LOWORD(wParam) == IDCANCEL)
        {
            EndDialog(hDlg, LOWORD(wParam));
            return (INT_PTR)TRUE;
        }
        break;
    case WM_PAINT:
        double h = 0, pi = 0, plm = 0, theta = 0;
        double xMax = 0, xMin = 0, yMax = 0, yMin = 0;
        FindMinMax(points, xMin, xMax, yMin, yMax);
        float xSpan = (float)(xMax - xMin);
        float ySpan = (float)(yMax - yMin);
        RECT rect = { };
        GetClientRect(hDlg, &rect);
        float width = (float)(rect.right - rect.left + 1);
        float height = (float)(rect.bottom - rect.top - 32 + 1);
        float sx0 = 2.0f * width / 16.0f;
        float sx1 = 14.0f * width / 16.0f;
        float sy0 = 2.0f * height / 16.0f;
        float sy1 = 14.0f * height / 16.0f;
        float deltaX = xSpan / 8.0f;
        float deltaY = ySpan / 8.0f;
        float xSlope, xInter, ySlope, yInter;
        xSlope = (sx1 - sx0) / xSpan;
        xInter = (float)(sx0 - xSlope * xMin);
        ySlope = (sy0 - sy1) / ySpan;
        yInter = (float)(sy0 - ySlope * yMax);
        float px = 0, py = 0, sx = 0, sy = 0;
        float vx = 0, vy = 0;
        PAINTSTRUCT ps;
        POINT wPt = { };
        HDC hdc = BeginPaint(hDlg, &ps);
        int i = 0;
        float x = (float)xMin;
        float y = (float)yMax;
        px = x;
        py = y;
        sx = xSlope * px + xInter;
        sy = ySlope * py + yInter;
        MoveToEx(hdc, (int)sx, (int)sy0, &wPt);
        char buffer[128] = { };

        while (i <= 8)
        {
            sx = xSlope * x + xInter;
            wPt.x = wPt.y = 0;
            MoveToEx(hdc, (int)sx, (int)sy0, &wPt);
            LineTo(hdc, (int)sx, (int)sy1);

            sprintf_s(buffer, "%5.4lf", x);
            SIZE size = { };
            GetTextExtentPoint32A(
                hdc,
                buffer,
                (int)strlen(buffer),
                &size);
            RECT textRect = { };
            textRect.left = (long)(sx - size.cx / 2.0f);
            textRect.right = (long)(sx + size.cx / 2.0f);
            textRect.top = (long)sy1;
            textRect.bottom = (long)(sy1 + size.cy / 2.0f);
            DrawFormattedText(hdc, buffer, textRect);
            x += deltaX;
            i++;
        }

        i = 0;
        y = (float)yMin;

        while (i <= 8)
        {
            sy = ySlope * y + yInter;
            wPt.x = wPt.y = 0;
            MoveToEx(hdc, (int)sx0, (int)sy, &wPt);
            LineTo(hdc, (int)sx, (int)sy);

            if (i != 0)
            {
                sprintf_s(buffer, "%5.3lf", y);
                SIZE size = { };
                GetTextExtentPoint32A(
                    hdc,
                    buffer,
                    (int)strlen(buffer),
                    &size);
                RECT textRect = { };
                textRect.left = (long)(sx0 - size.cx - size.cx / 5.0f);
                textRect.right = (long)(sx0 - size.cx / 2.0f);
                textRect.top = (long)(sy - size.cy / 2.0f);
                textRect.bottom = (long)(sy + size.cy / 2.0f);
                DrawFormattedText(hdc, buffer, textRect);
            }

            y += deltaY;
            i++;
        }

        HGDIOBJ bPenNew = NULL;
        HGDIOBJ hPenOld = NULL;

        bPenNew = CreatePen(PS_SOLID, 2, RGB(0, 0, 255));
        hPenOld = SelectObject(hdc, bPenNew);

        HRGN hrgn = CreateRectRgn((int)sx0, (int)sy0, (int)sx1, (int)sy1);
        
        // Select the clipping region into the DC
        if (SelectClipRgn(hdc, hrgn) == ERROR) {
            MessageBox(hDlg, L"Failed to select clip region", 
                L"Error", MB_ICONERROR);
            return (INT_PTR)FALSE;
        }

        SelectClipRgn(hdc, hrgn);

        px = (float)points[0].x;
        py = (float)points[0].y;
        sx = xSlope * px + xInter;
        sy = ySlope * py + yInter;
        wPt.x = wPt.y = 0;
        MoveToEx(hdc, (int)sx, (int)sy, &wPt);

        for (size_t j = 1; j < points.size(); j++)
        {
            px = (float)points[j].x;
            py = (float)points[j].y;
            sx = xSlope * px + xInter;
            sy = ySlope * py + yInter;
            LineTo(hdc, (int)sx, (int)sy);
        }

        float radius = 0.0f;

        if (eight)
            radius = 1.0f;
        else
            radius = 2.0f;

        DrawQuarterCircleArc(
            hdc, xSlope, ySlope,
            xInter, yInter, radius, false);

        if (eight)
        {
            radius = 2.0f;

            DrawQuarterCircleArc(
                hdc, xSlope, ySlope,
                xInter, yInter, radius, false);
        }

        SelectObject(hdc, hPenOld);
        DeleteObject(bPenNew);

        return (INT_PTR)FALSE;
    }

    return (INT_PTR)FALSE;
}

Blog Entry © Thursday, May 28, 2026, by James Pate Williams, Jr. and Microsoft’s Copilot Solution of the Potential Equation in Rectangle using Fixed Point Iteration in Python

# NOTE:
# This implementation prioritizes clarity and correctness over optimization.
# Further performance improvements can be made if needed.
# (c) May 26, 2026 by James Pate Willims, Jr.
# I had some help from the Microsoft Copilot
# to calculate runtimes and define matrices
# Computes the potential in a rectangle
# Reference: "Boundary Value Problems
# Second Edition" by David L. Powers
# See pages 179 to 182 for the analytic
# solution of this Laplace Equation
# Stand alone application using
# Microsoft Visual Studio 2022
# Community Version

import math
import time

xi = yi = 10
u = [[0.0 for _ in range(xi + 2)] for _ in range(yi + 2)]
v = [[0.0 for _ in range(xi + 2)] for _ in range(yi + 2)]

def ComputeBoundaryValues(x, y):
    if x == 0:
        return 0
    if x == 1:
        return 0
    if y == 0 or y == 1:
        if x > 0.0 and x < 0.5:
            return 2.0 * x
        elif x >= 0.5 and x < 1.0:
            return 2.0 - 2.0 * x
                 
    return 0.0

def ComputeParams(its, norm, params):
    params['iterations'] = its
    params['norm'] = norm

def Compute(h, k, xi, yi, maxIts, params):
    # Use a simple fixed-point iteration to
    # compute an approximate solution
    for i in range(0, xi + 1):
        for j in range(0, yi + 1):
            u[i][j] = ComputeBoundaryValues(i * h, j * k)

    for its in range(1, maxIts + 1):
        for i in range(1, xi):
            for j in range(1, yi): 
                u[i][j] = 0.25 * (u[i + 1][j] + u[i - 1][j] + u[i][j + 1] + u[i][j - 1]);

    norm = 0

    for i in range(0, xi + 1):
        for j in range(0, yi + 1):
            norm += math.fabs(u[i][j] * u[i][j])

    norm = math.sqrt(norm)
    params['iterations'] = its
    params['norm'] = norm

def f(x, y):
    # Analytic solution series expansion n = 1 to 100 
    sum = 0.0

    for n in range(1, 101):
        factor1 = math.sin(n * math.pi / 2.0) / (n * n)
        factor2 = math.sinh(n * math.pi * y)
        factor3 = math.sinh(n * math.pi * (1 - y))
        factor4 = math.sin(n * math.pi * x)
        term = (factor2 + factor3) / math.sinh(n * math.pi)
        sum += factor1 * term * factor4
    return 8.0 * sum / (math.pi * math.pi)

avgPE = 0
deltaX = 1.0 / xi
deltaY = 1.0 / yi
maxIts = 50
start_time = time.perf_counter()

for i in range(0, xi + 1):
    for j in range(0, yi + 1):
        v[i][j] = f(i * deltaX, j * deltaY)

minPE = +1000000000
maxPE = -1000000000
params = {}

Compute(deltaX, deltaY, xi, yi, maxIts, params)
print("Approximate\tAnalytic\tPercent Error")

for i in range(0, xi + 1):
    for j in range(0, yi + 1):
        if (math.fabs(u[i][j]) > 1.0e-12 and
            math.fabs(v[i][j]) > 1.0e-12):
            pe = 100.0 * math.fabs((v[i][j] - u[i][j]) / v[i][j])
        else:
            pe = 0.0

        avgPE += pe

        if (pe < minPE):
            minPE = pe

        if (pe > maxPE):
            maxPE = pe

        if math.fabs(pe) != 0.0:
            print("{:10.8f}".format(u[i][j]), "\t", "{:10.8f}".format(v[i][j]), "\t", "{:10.8f}".format(pe))

avgPE /= (xi * yi)
end_time = time.perf_counter()
# Calculate elapsed time in milliseconds
elapsed_ms = (end_time - start_time) * 1000

print("Iterations = ", params['iterations'])
print("Frobenius Norm = ", params['norm'])
print("Minimum Percent Error = ", "{:10.8f}".format(minPE))
print("Average Percent Error = ", "{:10.8f}".format(avgPE))
print("Maximum Percent Error = ", "{:10.8f}".format(maxPE))
print("Elapsed Milliseconds  = ", "{:10.8f}".format(elapsed_ms))


Approximate     Analytic        Percent Error
0.20000000       0.19999972      0.00013831
0.16633455       0.16663592      0.18085704
0.13739427       0.13768928      0.21425509
0.11591159       0.11605132      0.12040154
0.10292732       0.10291871      0.00836375
0.09864668       0.09854114      0.10710198
0.10305612       0.10291871      0.13351975
0.11613103       0.11605132      0.06868445
0.13763395       0.13768928      0.04018144
0.16650309       0.16663592      0.07971563
0.20000000       0.19999972      0.00013831
0.40000000       0.39999927      0.00018169
0.32813882       0.32854798      0.12453472
0.26763492       0.26776105      0.04710489
0.22369849       0.22344522      0.11334607
0.19755216       0.19705465      0.25246951
0.18899096       0.18834090      0.34515051
0.19778524       0.19705465      0.37075402
0.22409557       0.22344522      0.29105301
0.26806863       0.26776105      0.11486989
0.32844379       0.32854798      0.03171204
0.40000000       0.39999927      0.00018169
0.60000000       0.59999811      0.00031532
0.47888974       0.47875999      0.02710181
0.38176991       0.38059768      0.30799710
0.31425594       0.31267225      0.50650231
0.27518847       0.27354681      0.60013876
0.26255192       0.26082096      0.66365837
0.27549367       0.27354681      0.71170992
0.31477587       0.31267225      0.67278721
0.38233779       0.38059768      0.45720485
0.47928905       0.47875999      0.11050679
0.60000000       0.59999811      0.00031532
0.80000000       0.79999202      0.00099701
0.60602379       0.60222488      0.63081180
0.46685956       0.46199176      1.05365615
0.37704317       0.37308607      1.06063957
0.32711029       0.32392135      0.98448025
0.31121899       0.30818168      0.98555933
0.32745161       0.32392135      1.08985035
0.37762462       0.37308607      1.21648870
0.46749464       0.46199176      1.19112076
0.60647034       0.60222488      0.70496220
0.80000000       0.79999202      0.00099701
1.00000000       0.99594729      0.40692036
0.67874673       0.65811281      3.13531720
0.50319768       0.49282441      2.10486107
0.40066316       0.39470092      1.51057096
0.34574699       0.34157202      1.22228048
0.32848272       0.32468552      1.16950228
0.34608839       0.34157202      1.32223096
0.40124475       0.39470092      1.65792212
0.50383291       0.49282441      2.23375660
0.67919339       0.65811281      3.20318626
1.00000000       0.99594729      0.40692036
0.80000000       0.79999202      0.00099701
0.60615279       0.60222488      0.65223206
0.46709299       0.46199176      1.10418350
0.37734885       0.37308607      1.14257161
0.32745218       0.32392135      1.09002599
0.31156100       0.30818168      1.09653477
0.32776105       0.32392135      1.18538026
0.37787503       0.37308607      1.28360546
0.46766769       0.46199176      1.22857864
0.60655688       0.60222488      0.71933113
0.80000000       0.79999202      0.00099701
0.60000000       0.59999811      0.00031532
0.47910949       0.47875999      0.07300025
0.38216756       0.38059768      0.41247566
0.31477665       0.31267225      0.67303766
0.27577086       0.27354681      0.81304189
0.26313452       0.26082096      0.88702848
0.27602079       0.27354681      0.90440981
0.31520243       0.31267225      0.80920943
0.38263258       0.38059768      0.53465917
0.47943646       0.47875999      0.14129609
0.60000000       0.59999811      0.00031532
0.40000000       0.39999927      0.00018169
0.32837881       0.32854798      0.05149022
0.26806920       0.26776105      0.11508237
0.22426717       0.22344522      0.36785094
0.19818820       0.19705465      0.57524397
0.18962723       0.18834090      0.68297865
0.19836093       0.19705465      0.66290001
0.22456142       0.22344522      0.49953914
0.26839057       0.26776105      0.23510691
0.32860478       0.32854798      0.01728752
0.40000000       0.39999927      0.00018169
0.20000000       0.19999972      0.00013831
0.16650327       0.16663592      0.07960469
0.13769959       0.13768928      0.00748883
0.11631140       0.11605132      0.22411146
0.10337449       0.10291871      0.44285504
0.09909401       0.09854114      0.56105764
0.10346087       0.10291871      0.52678322
0.11645855       0.11605132      0.35090580
0.13786030       0.13768928      0.12420922
0.16661627       0.16663592      0.01179315
0.20000000       0.19999972      0.00013831
Iterations =  50
Frobenius Norm =  4.028216200275417
Minimum Percent Error =  0.00000000
Average Percent Error =  0.54286140
Maximum Percent Error =  3.20318626
Elapsed Milliseconds  =  36.23520000
Press any key to continue . . .

Blog Entry © Wednesday, May 27, 2026, by James Pate Williams, Jr. and Microsoft’s Copilot Grade School Arithmetic

Technical Report Grade School ArithmeticDownload
#pragma once
#include <stdint.h>

/* Algorithm due to Microsft's Coilot 
function udiv_restoring(N, D, n) :
    R = 0
    Q = 0

    negD = (~D + 1)

    for i from n - 1 down to 0
    {
        R = (R << 1) | ((N >> i) & 1)

    T = R + negD

    if MSB(T) == 0:
        R = T
        Q = Q | (1 << i)

    return (Q, R) 
 */

class Arithmetic
{
public:
    static bool udiv_restoring(
        uint32_t numer,
        uint32_t denom,
        uint32_t& quo,
        uint32_t& rem,
        int n);
    static bool umul_shift_add(
        uint32_t a,
        uint32_t b,
        uint64_t& product,
        int n);
};


#include <cstdint>
#include "Arithmetic.h"

static inline uint32_t mask_n(int bits) {
    return (bits >= 32) ? 0xFFFFFFFFu : ((1u << bits) - 1u);
}

static inline uint32_t msb(uint32_t x, int bits) {
    // returns top bit of a 'bits'-wide value
    return (x >> (bits - 1)) & 1u;
}

bool Arithmetic::udiv_restoring(
    uint32_t numer,
    uint32_t denom,
    uint32_t& quo,
    uint32_t& rem,
    int n)
{
    if (denom == 0 || n <= 0 || n > 32) return false;
    
    if (numer == 0)
    {
        quo = rem = 0;
        return true;
    }

    quo = 0;
    rem = 0;

    if (n == 32) {
        uint64_t R = 0;
        uint64_t D = (uint64_t)denom;
        uint64_t maskW = (1ull << 33) - 1ull;           // 33-bit mask
        uint64_t negD = ((~D) + 1ull) & maskW;          // 33-bit two's complement

        for (int i = n - 1; i >= 0; --i) {
            R = ((R << 1) | ((numer >> i) & 1u)) & maskW;

            uint64_t T = (R + negD) & maskW;         // R - D

            // Sign bit is bit 32 (the 33rd bit)
            if (((T >> 32) & 1ull) == 0ull) {
                R = T;
                quo |= (1u << i);
            }
        }
        rem = (uint32_t)(R & 0xFFFFFFFFu);
        return true;
    }

    // n < 32 case: we can keep everything in uint32_t using (n+1) bits
    uint32_t maskN = mask_n(n);
    uint32_t maskW = mask_n(n + 1);

    uint32_t N = numer & maskN;
    uint32_t D = denom & maskN;

    // Two's complement of D in (n+1) bits
    uint32_t Dw = D;                  // placed in low bits of (n+1)-wide register
    uint32_t negD = ((~Dw) + 1u) & maskW;

    uint32_t R = 0;

    for (int i = n - 1; i >= 0; --i) {
        R = ((R << 1) | ((N >> i) & 1u)) & maskW;

        uint32_t T = (R + negD) & maskW;            // trial subtract: R - D (in w bits)

        if (msb(T, n + 1) == 0) {                   // non-negative in (n+1) bits
            R = T;
            quo |= (1u << i);
        }
    }

    rem = R & maskN;                                // remainder fits in n bits
    return true;
}

bool Arithmetic::umul_shift_add(
    uint32_t a,
    uint32_t b,
    uint64_t& product,
    int n)
{
    if (n <= 0 || n > 32) return false;

    uint64_t A = a;    // promote to avoid overflow
    uint32_t B = b;

    product = 0;

    for (int i = 0; i < n; ++i) {
        if (B & 1u) {
            product += A;
        }

        A <<= 1;
        B >>= 1;
    }

    return true;
}


#include <chrono>
#include <cstdint>
#include <iostream>
#include <limits>
#include <random>
#include <string>
#include "Arithmetic.h"

namespace {

    constexpr int TESTS_PER_N = 200000;

    uint32_t make_mask(int n) {
        return (n == 32) ? 0xFFFFFFFFu : ((1u << n) - 1u);
    }

    void clear_bad_input() {
        std::cin.clear();
        std::cin.ignore(std::numeric_limits<std::streamsize>::max(), '\n');
    }

    template <typename TrialFn>
    double run_suite(const char* label, TrialFn trial, bool verbose) {
        std::mt19937 rng(12345); // deterministic

        auto t0 = std::chrono::high_resolution_clock::now();

        for (int n = 1; n <= 32; ++n) {
            const uint32_t mask = make_mask(n);

            for (int i = 0; i < TESTS_PER_N; ++i) {
                if (!trial(rng, mask, n)) {
                    std::cout << label << ": FAILED (n=" << n << ", i=" << i << ")\n";
                    return -1.0;
                }
            }

            if (verbose) {
                std::cout << "n=" << n << " passed\n";
            }
        }

        auto t1 = std::chrono::high_resolution_clock::now();
        double secs = std::chrono::duration<double>(t1 - t0).count();

        std::cout << label << " runtime = " << secs << " sec\n";
        return secs;
    }

    bool trial_division(std::mt19937& rng, uint32_t mask, int n) {
        const uint32_t numer = rng() & mask;
        const uint32_t denom = (rng() & mask) | 1u; // ensure non-zero

        uint32_t q = 0, r = 0;
        if (!Arithmetic::udiv_restoring(numer, denom, q, r, n)) {
            std::cout << "Failure numer=" << numer << " denom=" << denom << "\n";
            return false;
        }

        const uint32_t q2 = numer / denom;
        const uint32_t r2 = numer % denom;

        if (q != q2 || r != r2) {
            std::cout << "Mismatch n=" << n
                << " numer=" << numer
                << " denom=" << denom
                << " got q=" << q << " r=" << r
                << " expected q=" << q2 << " r=" << r2 << "\n";
            return false;
        }
        return true;
    }

    bool trial_multiplication(std::mt19937& rng, uint32_t mask, int n) {
        const uint32_t a = rng() & mask;
        const uint32_t b = rng() & mask;

        uint64_t prod = 0;
        if (!Arithmetic::umul_shift_add(a, b, prod, n)) {
            std::cout << "Failure a=" << a << " b=" << b << "\n";
            return false;
        }

        const uint64_t expected = static_cast<uint64_t>(a) * static_cast<uint64_t>(b);
        if (prod != expected) {
            std::cout << "Mismatch n=" << n
                << " a=" << a
                << " b=" << b
                << " got=" << prod
                << " expected=" << expected << "\n";
            return false;
        }
        return true;
    }

} // namespace

int main() {
    bool verbose = true;

    while (true) {
        std::cout << "\nArithmetic Lab\n";
        std::cout << "1. Test Division (restoring)\n";
        std::cout << "2. Test Multiplication (shift-add)\n";
        std::cout << "3. Run ALL tests\n";
        std::cout << "4. Toggle verbose (currently " << (verbose ? "ON" : "OFF") << ")\n";
        std::cout << "5. Exit\n";
        std::cout << "Choice: ";

        int choice = 0;
        if (!(std::cin >> choice)) {
            clear_bad_input();
            std::cout << "Invalid input. Please enter a number.\n";
            continue;
        }

        if (choice == 1) {
            run_suite("Division test", trial_division, verbose);
        }
        else if (choice == 2) {
            run_suite("Multiplication test", trial_multiplication, verbose);
        }
        else if (choice == 3) {
            const double d = run_suite("Division test", trial_division, verbose);
            if (d >= 0.0) run_suite("Multiplication test", trial_multiplication, verbose);
        }
        else if (choice == 4) {
            verbose = !verbose;
        }
        else if (choice == 5) {
            return 0;
        }
        else {
            std::cout << "Invalid choice.\n";
        }
    }
}

Blog Entry (c) Tuesday, May 26, 2026, by James Pate Williams, Jr. and Microsoft’s Copilot Hydrogen-Like Atomic Radial Wave Functions

Included a downloadable PDF and Microsoft Excel Workbook.

Hydrogen Like Radial Wave FunctionsDownload
Hydrogen Like Atomic Wave FunctonsDownload

Blog Entry (c) Monday, May 25, 2026, by James Pate Williams, Jr. Quantum Mechanical Linear Harmonic Oscillator

Python Quantum Mechanical Linear Harmonic Oscillator Wave Functions n0 to n10Download
Python Quantum Mechanical Linear Harmonic Oscillator WorkbookDownload

Blog Entry (c) Friday, May 22, 2024, by James Pate Williams, Jr. and Microsoft’s Copilot (AI Agent)

Cheerwing U12S Mini RC Helicopter with Camera Remote Control Helicopter for Kids and Adults Preflight and Start Checklist found on Amazon.com

Helicopter ChecklistDownload

Blog Entry © Sunday, May 17, 2026, by James Pate Williams, Jr. Derivation of the Time Independent Free Particle Schrödinger Equation in Confocal Parabolic Coordinates

Derivation of the Time Independent Schrödinger Equation in Confocal Parabolic CoordinatesDownload

Blog Entry © Saturday, May 16, 2026, by James Pate Williams, Jr. Some More Linear Algebra Examples

Blog Entry Saturday May 16 2026Download

Blog Entry © Thursday, May 14, 2026, by James Pate Williams, Jr. More Numerical Integration Results

Blog Entry Thursday May 14 2026Download
// NumericalIntegrals.cpp (c) Thursday, May 14, 2026
// by James Pate Williams, Jr., BA, BS, MSwE, PhD

#include <iomanip>
#include <iostream>
#include <vector>
#include <stdlib.h>

static double f(double x) {
    return sin(x);
}

static double MonteCarlo(double a, double b,
    double (*f)(double), int n){ 
    double sum = 0;

    for (int i = 0; i < n; i++) {
        double x = (b - a) * (double)rand() / RAND_MAX;

        sum += f(x);
    }

    return (b - a) * sum / n;
}

static double Factorial(int n) {
    double factorial = 1.0;

    for (int i = 2; i <= n; i++)
        factorial *= i;

    return factorial;
}

static double Series(double a, double b, int n)
{
    double sumA = 0.0, sumB = 0.0;
    int sign = 1;

    for (int i = 0; i <= n; i++) {
        sumA += sign * pow(a, 2 * i + 2) /
            Factorial(2 * i + 2);

        sign *= -1;
    }

    sign = 1;

    for (int i = 0; i <= n; i++) {      
        sumB += sign * pow(b, 2 * i + 2) /
            Factorial(2 * i + 2);

        sign *= -1;
    }

    return sumB - sumA;
}

static double CompositeTrapezoidalRule(
    double a, double b, int n) {
    double pi = 4.0 * atan(1.0);
    double endPts = 0.5 * (f(a) + f(b));
    double sum = 0, xk = 0.0;
    double h = (b - a) / n;

    for (int k = 1; k <= n - 1; k++) {
        xk = a + k * h;
        sum += f(xk);
    }

    return h * (0.5 * endPts + sum);
}

static double SimpsonsRule(
    int n, double a, double b, double(*fx)(double)) {
    double h = (b - a) / n;
    double h2 = 2.0 * h;
    double s = 0.0;
    double t = 0.0;
    double x = a + h;

    for (int i = 1; i < n; i += 2) {
        s += fx(x);
        x += h2;
    }

    x = a + h2;

    for (int i = 2; i < n; i += 2) {
        t += fx(x);
        x += h2;
    }

    return h * (fx(a) + 4 * s + 2 * t + fx(b)) / 3.0;
}

static void Romberg(double a, double b,
    double (*f)(double), int mStart, int nRow,
    std::vector<std::vector<double>>& T) {
    int m = mStart;
    double h = (b - a) / m;
    double sum = 0.5 * (f(a) + f(b));

    if (m > 1) {
        for (int i = 1; i <= m - 1; i++) {
            sum += f(a + i * h);
        }
    }

    T[0][0] = sum * h;

    std::cout << "romberg t-table" << std::endl;
    std::cout << std::fixed;
    std::cout << std::setprecision(5) << T[0][0];
    std::cout << std::endl;

    if (nRow < 2)
        return;

    for (int k = 2; k <= nRow; k++) {
        h /= 2.0;
        m *= 2;
        sum = 0.0;

        for (int i = 1; i <= m; i += 2) {
            sum += f(a + i * h);
        }

        T[k][1] = 0.5 * T[k - 1LL][1] + sum * h;

        for (int j = 1; j <= k - 1; j++) {
            T[k - 1LL][j] = T[k][j] - T[k - 1LL][j];
            T[k][j + 1LL] = T[k][j] + T[k - 1LL][j] /
                (pow(4.0, j) - 1.0);
        }

        for (int j = 1; j <= k; j++) {
            std::cout << std::fixed;
            std::cout << std::setprecision(5);
            std::cout << T[k][j] << '\t';
        }

        std::cout << std::endl;
    }

    if (nRow < 3) {
        return;
    }

    std::cout << "table of ratios" << std::endl;
    double ratio = 0.0;

    for (int k = 1; k <= nRow - 2; k++) {
        for (int j = 1; j <= k; j++) {
            if (T[k + 1LL][j] == 0.0) {
                ratio = 0.0;
            }

            else {
                ratio = T[k][j] / T[k + 1LL][j];
            }

            T[k][j] = ratio;
        }

        for (int j = 1; j <= k; j++) {
            std::cout << std::fixed;
            std::cout << std::setprecision(5);
            std::cout << T[k][j] << '\t';
        }

        std::cout << std::endl;
    }
}

double MonteCarloVolume(double R, int n)
{
    double pi = 4.0 * atan(1.0), pi2 = 2.0 * pi;
    double R2 = R * R, sum = 0;

    for (int i = 0; i < n; i++)
    {
        double r = R2 * (double)rand() / RAND_MAX;
        double t = pi * (double)rand() / RAND_MAX;
        double p = pi2 * (double)rand() / RAND_MAX;
        sum += r * r * sin(t);
    }

    return R * pi * pi2 * sum / n;
}

int main()
{
    srand(1);
    std::vector<std::vector<double>> T;
    T.resize(35);
    for (int i = 0; i < 35; i++) {
        T[i].resize(35);
    }
    Romberg(0.0, 2.0, f, 2, 7, T);
    std::cout << std::setprecision(11);
    std::cout << "analytic integral of sine = " << -cos(2.0) + cos(0.0);
    std::cout << std::endl;
    std::cout << "simpson's rule integral   = " << SimpsonsRule(500, 0, 2.0, f);
    std::cout << std::endl;
    std::cout << "monte carlo integral      = " << MonteCarlo(0.0, 2.0, f, 2130);
    std::cout << std::endl;
    std::cout << "infinite series integral  = " << Series(0.0, 2.0, 16);
    std::cout << std::endl;
    double integral = CompositeTrapezoidalRule(0.0, 2.0, 175000000);
    std::cout << "romberg integral          = " << integral << std::endl;
    std::cout << "actual spherical volume   = " << 4.0 * 4.0 * atan(1.0) / 3.0;
    std::cout << std::endl;
    double volume = MonteCarloVolume(1.0, 1000000);
    std::cout << "approx spherical volume   = " << volume;
    std::cout << std::endl;
}