/*
Author: Pate Williams (c) January 22, 1995
The following program is a translation of the FORTRAN
subprogram found in Elementary Numerical Analysis by
S. D. Conte and Carl de Boor pages 343-344. The program
uses Romberg extrapolation to find the integral of a
function.
*/
#include "stdafx.h"
#include <math.h>
#include <stdlib.h>
#include <time.h>
#include <stdio.h>
typedef long double real;
real t[10][10];
real f(real x)
{
return(expl(-x * x));
}
real Li(real x)
{
return(1.0 / logl(x));
}
real romberg(
real a, real b, int start, int row,
real(*f)(real))
{
int i, k, m;
real h, ratio, sum;
m = start;
h = (b - a) / m;
sum = 0.5 * (f(a) + f(b));
if (m > 1)
for (i = 1; i < m; i++)
sum += f(a + i * h);
t[1][1] = h * sum;
if (row < 2) return(t[1][1]);
for (k = 2; k <= row; k++)
{
h = 0.5 * h;
m *= 2;
sum = 0.0;
for (i = 1; i <= m; i += 2)
sum += f(a + h * i);
t[k][1] = 0.5 * t[k - 1][1] + sum * h;
for (i = 1; i < k; i++)
{
t[k - 1][i] = t[k][i] - t[k - 1][i];
t[k][i + 1] = t[k][i] - t[k - 1][i] /
(powl(4.0, (real)i) - 1.0);
}
}
if (row < 3) return(t[2][2]);
for (k = 1; k <= row - 2; k++)
for (i = 1; i <= k; i++)
{
if (t[k + 1][i] == 0.0)
ratio = 0.0;
else
ratio = t[k][i] / t[k + 1][i];
t[k][i] = ratio;
}
return(t[row][row - 1]);
}
int main(void)
{
long double experimental = 0.7468241328124271;
int row = 8;
printf("first test\n");
printf("Romberg integration of f(x) = exp(- x * x)\n");
printf("Internet value = 0.7468241328124271\n");
printf("from website = https://www.integral-calculator.com/\n");
printf("Approximation\t\tPercent Difference\tSteps\tRuntime (s)\n");
for (long steps = 100000; steps <= 900000; steps += 100000)
{
clock_t start = clock();
long double trial = romberg(0.0, 1.0, steps, row, f);
clock_t endTm = clock();
long double runtime = ((long double)endTm - start) /
CLOCKS_PER_SEC;
long double pd = 100.0 * fabsl(experimental - trial) /
(0.5 *(experimental + trial));
printf("%16.15lf\t%16.15le\t%6d\t%4.3lf\n",
trial, pd, steps, runtime);
}
printf("second test\n");
printf("Romberg integration of Li(x) = 1 / ln x between 2 and some n\n");
printf("approximates the number of prime numbers between 2 and n\n");
printf("n\t\tLi(n)\t\tSteps\tTime (s)\n");
for (long n = 10; n <= 1000000000; n *= 10)
{
long steps = 1000000;
clock_t start = clock();
long double trial = romberg(2.0, n, steps, row, Li);
clock_t endTm = clock();
long double runtime = ((long double)endTm - start) /
CLOCKS_PER_SEC;
printf("%12ld\t%12.0lf\t%7ld\t%4.3lf\n",
n, trial, steps, runtime);
}
return(0);
}
/*
Author: Pate Williams (c) January 20, 1995
The following is a translation of the Pascal program
sieve found in Pascalgorithms by Edwin D. Reilly and
Francis D. Federighi page 652. This program uses sets
to represent the sieve (see C Programming Language An
Applied Perspective by Lawrence Miller and Alec Qui-
lici pages 160 - 162).
*/
#include "stdafx.h"
#include <math.h>
#include <stdio.h>
#include <time.h>
#define _WORD_SIZE 32
#define _VECT_SIZE 31250000
#define SET_MIN 0
#define SET_MAX 1000000000
typedef long LONG;
typedef long SET[_VECT_SIZE];
typedef LONG ELEMENT;
SET set;
static LONG get_bit_pos(LONG *long_ptr, LONG *bit_ptr,
ELEMENT element)
{
*long_ptr = element / _WORD_SIZE;
*bit_ptr = element % _WORD_SIZE;
return(element >= SET_MIN && element <= SET_MAX);
}
static void set_bit(ELEMENT element, LONG inset)
{
LONG bit, word;
if (get_bit_pos(&word, &bit, element))
inset ? set[word] |= (01 << bit) :
set[word] &= ~(01 << bit);
}
static int get_bit(ELEMENT element)
{
LONG bit, word;
return(get_bit_pos(&word, &bit, element) ?
(set[word] >> bit) & 01 : 0);
}
void set_Add(ELEMENT element)
{
set_bit(element, 1);
}
void set_Del(ELEMENT element)
{
set_bit(element, 0);
}
int set_Mem(ELEMENT element)
{
return get_bit(element);
}
void primes(LONG n)
{
LONG c, i, inc, k;
double x;
clock_t now = clock();
set_Add(2);
for (i = 3; i <= n; i++)
if ((i + 1) % 2 == 0)
set_Add(i);
else
set_Del(i);
c = 3;
do
{
i = c * c;
inc = c + c;
while (i <= n)
{
set_Del(i);
i = i + inc;
}
c += 2;
while (set_Mem(c) == 0) c += 1;
} while (c * c <= n);
k = 0;
for (i = 2; i <= n; i++)
if (set_Mem(i) == 1) k++;
x = n / log(n) - 5.0;
x = x + exp(1.0 + 0.15 * log(n) * sqrt(log(n)));
clock_t later = clock();
double runtime = (double)(later - now) / CLOCKS_PER_SEC;
printf("%10ld\t%10ld\t%10.0lf\t%6.4lf\n",
n, k, x, runtime);
}
int main(void)
{
LONG n = 10L;
printf("--------------------------------------------------------\n");
printf("n\t\tprimes\t\ttheory\t\ttime (s)\n");
printf("--------------------------------------------------------\n");
do
{
primes(n);
clock_t later = clock();
n = 10L * n;
} while (n < 1000000000);
printf("--------------------------------------------------------\n");
return(0);
}
/*
Author: Pate Williams (c) January 22, 1995
The following program is a translation of the FORTRAN
subprogram found in Elementary Numerical Analysis by
S. D. Conte and Carl de Boor pages 343-344. The program
uses Romberg extrapolation to find the integral of a
function.
*/
#include "stdafx.h"
#include <math.h>
#include <stdlib.h>
#include <time.h>
#include <stdio.h>
typedef long double real;
real t[10][10];
real f(real x)
{
return(expl(-x * x));
}
real romberg(real a, real b, int start, int row)
{
int i, k, m;
real h, ratio, sum;
m = start;
h = (b - a) / m;
sum = 0.5 * (f(a) + f(b));
if (m > 1)
for (i = 1; i < m; i++)
sum += f(a + i * h);
t[1][1] = h * sum;
if (row < 2) return(t[1][1]);
for (k = 2; k <= row; k++)
{
h = 0.5 * h;
m *= 2;
sum = 0.0;
for (i = 1; i <= m; i += 2)
sum += f(a + h * i);
t[k][1] = 0.5 * t[k - 1][1] + sum * h;
for (i = 1; i < k; i++)
{
t[k - 1][i] = t[k][i] - t[k - 1][i];
t[k][i + 1] = t[k][i] - t[k - 1][i] /
(powl(4.0, (real)i) - 1.0);
}
}
if (row < 3) return(t[2][2]);
for (k = 1; k <= row - 2; k++)
for (i = 1; i <= k; i++)
{
if (t[k + 1][i] == 0.0)
ratio = 0.0;
else
ratio = t[k][i] / t[k + 1][i];
t[k][i] = ratio;
}
return(t[row][row - 1]);
}
int main(void)
{
long double experimental = 0.7468241328124271;
int row = 8;
printf("Romberg integration of f(x) = exp(- x * x)\n");
printf("Internet value = 0.7468241328124271\n");
printf("from website = https://www.integral-calculator.com/\n");
printf("Approximation\t\tPercent Difference\tSteps\tRuntime (s)\n");
for (long steps = 100000; steps <= 900000; steps += 100000)
{
clock_t start = clock();
long double trial = romberg(0.0, 1.0, steps, row);
clock_t endTm = clock();
long double runtime = ((long double)endTm - start) /
CLOCKS_PER_SEC;
long double pd = 100.0 * fabsl(experimental - trial) /
(0.5 *(experimental + trial));
printf("%16.15lf\t%16.15le\t%6d\t%4.3lf\n",
trial, pd, steps, runtime);
}
return(0);
}
We use two numerical analysis algorithms and two artificial intelligent methods. The first two techniques are from the excellent tome “A Numerical Library in C for Scientists and Engineers” by H.T. Lau, PhD. One does not utilize any derivatives and the other requires first partial derivatives. My homegrown algorithms are my implementation of the particle swarm optimization and an evolutionary hill-climber. The output from my C# application is below:
The two numerical algorithms PRAXIS and FLEMIN perform fairly well and do not require that many function evaluations which is better metric than wall-clock time.
“The Greek forces included 300 Spartans and their helots with 2,120 Arcadians, 1,000 Lokrians, 1,000 Phokians, 700 Thespians, 400 Corinthians, 400 Thebans, 200 men from Phleious, and 80 Mycenaeans.” The Persians led by Xerxes I were myriad.
There have been at least two large scale motion pictures made about the battle namely “The 300 Spartans (1962)” and the “300 (2006)”. I have seen both movies, but I prefer the 1962 version due to attempted adherence to Herodotus’ version of the history. The ending of the 1962 movie has a small remaining contingent of Spartans led by King Leonidas I dropping their shields in a mass suicide by a barrage of Persian arrows (“the sky was black with arrows”). “Oh stranger, tell the Spartans that we lie here obedient to their word.” The mantra of the Spartans was to come home to Sparta carrying their shields or to come home being carried dead on their shields. The 300 Spartans – Wikipedia
The Gibb’s overshoot phenomenon is very well defined by the Fourier series expansion of the Heaviside step function. The following images were produced by a new C/C++ Win32 application that I created in the period February 10, 2023, to February 14, 2023.
The code was adapted from “Fourier Series and Boundary Value Problems” by Ruel V. Churchill and James Ward Brown. The integration algorithm was translated from C source code found in the tome “A Numerical Library in C for Scientists and Engineers” by H.T. Lau, PhD pages 299-303.
I use a very slowly converging infinite series and Fourier series to compute Pi.
// https://en.wikipedia.org/wiki/Rexx#NUMERIC\
// Translated from Rexx code
// by James Pate Williams, Jr.
// Added a lot of functionality
// Copyrighted February 7 - 10, 2023
#include "FourierSeries.h"
#include <stdlib.h>
#include <iomanip>
#include <iostream>
#include <chrono>
using namespace std;
typedef chrono::high_resolution_clock Clock;
void Option3(int digits, int N)
{
long double i = 0;
long double sgn = 1;
long double sum = 0;
long double pi0 = Pi;
long double pi1 = 0.0;
while (i <= N)
{
sum += sgn / (2.0 * i + 1);
sgn *= -1;
i = i + 1.0;
}
pi1 = 4.0 * sum;
cout << setw(static_cast<std::streamsize>((int)digits) + 2);
cout << setprecision(digits) << pi1 << '\t';
cout << setprecision(digits) << (fabsl(pi0 - pi1)) << endl;
}
void Option4(int digits, int N)
{
long double* a = new long double[N + 1];
long double* b = new long double[N + 1];
memset(a, 0, (N + 1) * sizeof(long double));
memset(b, 0, (N + 1) * sizeof(long double));
FourierSeries::CreateCosSeries(N, a);
FourierSeries::CreateSinSeries(N, b);
long double pi = 4.0 * FourierSeries::Series(N, 1.0, a, b);
cout << setw(static_cast<std::streamsize>((int)digits) + 2);
cout << setprecision(digits) << pi << '\t';
cout << setprecision(digits) << (fabsl(pi - Pi)) << endl;
delete[] a;
delete[] b;
}
int main()
{
int choice, digits, N = 0, N1 = 0;
while (1)
{
cout << "==Menu==" << endl;
cout << "1 Compute Sqrt(2)" << endl;
cout << "2 Compute Exp(1)" << endl;
cout << "3 Compute Pi" << endl;
cout << "4 Compute Pi (Fourier Series)" << endl;
cout << "5 Generate Option 3 Table" << endl;
cout << "6 Generate Option 4 Table" << endl;
cout << "7 Exit" << endl;
cin >> choice;
if (choice == 7)
break;
cout << " Digits = ";
cin >> digits;
if (choice >= 3)
{
cout << " Terms = ";
cin >> N;
N1 = N + 1;
}
auto start_time = Clock::now();
if (choice == 1)
{
long double n = 2;
long double r = 1;
while (1)
{
long double rr = (n / r + r) / 2;
if (rr == r)
break;
r = rr;
}
cout << setprecision(digits) << r << endl;
}
else if (choice == 2)
{
long double e = 2.5;
long double f = 0.5;
long double n = 3;
do
{
f /= n;
long double ee = e + f;
if (ee == e)
break;
e = ee;
n += 1;
} while (1);
cout << setprecision(digits) << e << endl;
}
else if (choice == 3)
Option3(digits, N);
else if (choice == 4)
Option4(digits, N);
else if (choice == 5)
{
for (int n = 8; n <= N; n *= 2)
Option3(digits, n);
}
else if (choice == 6)
{
for (int n = 8; n <= N; n *= 2)
Option4(digits, n);
}
auto end_time = Clock::now();
cout << "runtime in milliseconds = ";
cout << std::chrono::duration_cast<std::chrono::milliseconds>
(end_time - start_time).count();
cout << endl;
cout << "runtime in nanoseconds = ";
cout << std::chrono::duration_cast<std::chrono::nanoseconds>
(end_time - start_time).count();
cout << endl;
}
return 0;
}
#pragma once
const long double c = 2.0e9;
const long double Pi = 3.1415926535897932384626433832795;
class FourierSeries
{
private:
static long double f(long double x);
static long double cosTermFunction(int n, long double x);
static long double sinTermFunction(int n, long double x);
public:
static void CreateCosSeries(int N, long double a[]);
static void CreateSinSeries(int N, long double b[]);
static long double Series(int N, long double x,
long double a[], long double b[]);
};
#include <math.h>
#include "FourierSeries.h"
#include "Integral.h"
long double FourierSeries::f(long double x)
{
return atanl(x);
}
long double FourierSeries::cosTermFunction(int n, long double x)
{
return cosl(n * x * Pi / c) * f(x);
}
long double FourierSeries::sinTermFunction(int n, long double x)
{
return sinl(n * x * Pi / c) * f(x);
}
void FourierSeries::CreateCosSeries(int N, long double a[])
{
long double e[7] = { 0 };
e[1] = e[2] = 1.0e-12;
for (int n = 0; n <= N; n++)
a[n] = Integral::integral(
n, -c, c, cosTermFunction, e, 1, 1) / c;
}
void FourierSeries::CreateSinSeries(int N, long double b[])
{
long double e[7] = { 0 };
e[1] = e[2] = 1.0e-12;
for (int n = 1; n <= N; n++)
b[n] = Integral::integral(
n, -c, c, sinTermFunction, e, 1, 1) / c;
}
long double FourierSeries::Series(int N, long double x,
long double a[], long double b[])
{
long double sum = 0.0;
for (int n = 1; n <= N; n++)
sum += a[n] * cosl(2.0 * n * x) + b[n] * sinl(2.0 * n * x);
return a[0] / 2.0 + sum / 2.0;
}
#pragma once
// Translated from C source code found in the tome
// "A Numerical Library in C for Scientists and
// Engineers" by H.T. Lau, PhD
class Integral
{
private:
static long double integralqad(
int n,
int transf, long double (*fx)(int, long double), long double e[],
long double* x0, long double* x1, long double* x2, long double* f0, long double* f1,
long double* f2, long double re, long double ae, long double b1);
static void integralint(
int n,
int transf, long double (*fx)(int, long double), long double e[],
long double* x0, long double* x1, long double* x2, long double* f0, long double* f1,
long double* f2, long double* sum, long double re, long double ae, long double b1,
long double hmin);
public:
static long double integral(int n, long double a, long double b,
long double (*fx)(int, long double), long double e[],
int ua, int ub);
};
#include <math.h>
#include "Integral.h"
long double Integral::integral(
int n,
long double a, long double b,
long double (*fx)(int, long double), long double e[],
int ua, int ub)
{
long double x0, x1, x2, f0, f1, f2, re, ae, b1 = 0, x;
re = e[1];
if (ub)
ae = e[2] * 180.0 / fabsl(b - a);
else
ae = e[2] * 90.0 / fabsl(b - a);
if (ua) {
e[3] = e[4] = 0.0;
x = x0 = a;
f0 = (*fx)(n, x);
}
else {
x = x0 = a = e[5];
f0 = e[6];
}
e[5] = x = x2 = b;
e[6] = f2 = (*fx)(n, x);
e[4] += integralqad(n, 0, fx, e, &x0, &x1, &x2, &f0, &f1, &f2, re, ae, b1);
if (!ub) {
if (a < b) {
b1 = b - 1.0;
x0 = 1.0;
}
else {
b1 = b + 1.0;
x0 = -1.0;
}
f0 = e[6];
e[5] = x2 = 0.0;
e[6] = f2 = 0.0;
ae = e[2] * 90.0;
e[4] -= integralqad(n, 1, fx, e, &x0, &x1, &x2, &f0, &f1, &f2, re, ae, b1);
}
return e[4];
}
long double Integral::integralqad(
int n,
int transf, long double (*fx)(int, long double), long double e[],
long double* x0, long double* x1, long double* x2, long double* f0, long double* f1,
long double* f2, long double re, long double ae, long double b1)
{
/* this function is internally used by INTEGRAL */
long double sum, hmin, x, z;
hmin = fabs((*x0) - (*x2)) * re;
x = (*x1) = ((*x0) + (*x2)) * 0.5;
if (transf) {
z = 1.0 / x;
x = z + b1;
(*f1) = (*fx)(n, x) * z * z;
}
else
(*f1) = (*fx)(n, x);
sum = 0.0;
integralint(n, transf, fx, e, x0, x1, x2, f0, f1, f2, &sum, re, ae, b1, hmin);
return sum / 180.0;
}
void Integral::integralint(
int n,
int transf, long double (*fx)(int, long double), long double e[],
long double* x0, long double* x1, long double* x2, long double* f0, long double* f1,
long double* f2, long double* sum, long double re, long double ae, long double b1,
long double hmin)
{
/* this function is internally used by INTEGRALQAD of INTEGRAL */
int anew;
long double x3, x4, f3, f4, h, x, z, v, t;
x4 = (*x2);
(*x2) = (*x1);
f4 = (*f2);
(*f2) = (*f1);
anew = 1;
while (anew) {
anew = 0;
x = (*x1) = ((*x0) + (*x2)) * 0.5;
if (transf) {
z = 1.0 / x;
x = z + b1;
(*f1) = (*fx)(n, x) * z * z;
}
else
(*f1) = (*fx)(n, x);
x = x3 = ((*x2) + x4) * 0.5;
if (transf) {
z = 1.0 / x;
x = z + b1;
f3 = (*fx)(n, x) * z * z;
}
else
f3 = (*fx)(n, x);
h = x4 - (*x0);
v = (4.0 * ((*f1) + f3) + 2.0 * (*f2) + (*f0) + f4) * 15.0;
t = 6.0 * (*f2) - 4.0 * ((*f1) + f3) + (*f0) + f4;
if (fabsl(t) < fabsl(v) * re + ae)
(*sum) += (v - t) * h;
else if (fabsl(h) < hmin)
e[3] += 1.0;
else {
integralint(n, transf, fx, e, x0, x1, x2, f0, f1, f2, sum,
re, ae, b1, hmin);
*x2 = x3;
*f2 = f3;
anew = 1;
}
if (!anew) {
*x0 = x4;
*f0 = f4;
}
}
}
I created a C# application to model an ideal gas and real gases. The real gas algorithms were the Van der Waal Equation of State and the Virial Equation of State.
The functions to be calculated are the exclusive or (XOR) function, the eight input and output identity function, and four non-linear three input and one output functions. I used the logistic function that represents 0 as 0.1 and 1 as 0.9. The three input functions use 8 hidden units, a tolerance of 1.0e-9, and maximum number of epochs as 1,000,000. The backpropagation feed-forward neural network algorithm is from Tom M. Mitchell’s classic textbook “Machine Learning” which has a copyright of 1997.
Numerical Explorations 