x = 0.25
N = 5
Series Cosine(x) = 0.968912421711
C++ cos(x) = 0.968912421711
Series sine(x) = 0.247403959255
C++ sin(x) = 0.247403959255
Series Tangent(x) = 0.255341921221
C++ tan(x) = 0.255341921221
Series sin(2x) = 0.479425538604
C++ sin(2x) = 0.479425538604
C++ 2sin(x)cos(x) = 0.479425538604
End app ? y = yes = n
x = 0.5
N = 5
Series Cosine(x) = 0.877582561890
C++ cos(x) = 0.877582561890
Series sine(x) = 0.479425538604
C++ sin(x) = 0.479425538604
Series Tangent(x) = 0.546302489844
C++ tan(x) = 0.546302489844
Series sin(2x) = 0.841470984807
C++ sin(2x) = 0.841470984808
C++ 2sin(x)cos(x) = 0.841470984808
End app ? y = yes = y
x = 0.75
N = 5
Series Cosine(x) = 0.731688868808
C++ cos(x) = 0.731688868874
Series sine(x) = 0.681638760020
C++ sin(x) = 0.681638760023
Series Tangent(x) = 0.931596460023
C++ tan(x) = 0.931596459944
Series sin(2x) = 0.997494986509
C++ sin(2x) = 0.997494986604
C++ 2sin(x)cos(x) = 0.997494986604
End app ? y = yes = n
x = 1.00
N = 5
Series Cosine(x) = 0.540302303792
C++ cos(x) = 0.540302305868
Series sine(x) = 0.841470984648
C++ sin(x) = 0.841470984808
Series Tangent(x) = 1.557407730344
C++ tan(x) = 1.557407724655
Series sin(2x) = 0.909297423159
C++ sin(2x) = 0.909297426826
C++ 2sin(x)cos(x) = 0.909297426826
End app ? y = yes = n
x = 1.25
N = 5
Series Cosine(x) = 0.315322332275
C++ cos(x) = 0.315322362395
Series sine(x) = 0.948984616456
C++ sin(x) = 0.948984619356
Series Tangent(x) = 3.009569952151
C++ tan(x) = 3.009569673863
Series sin(2x) = 0.598472085108
C++ sin(2x) = 0.598472144104
C++ 2sin(x)cos(x) = 0.598472144104
End app ? y = yes = n
x = 1.50
N = 5
Series Cosine(x) = 0.070736934117
C++ cos(x) = 0.070737201668
Series sine(x) = 0.997494955682
C++ sin(x) = 0.997494986604
Series Tangent(x) = 14.101472846329
C++ tan(x) = 14.101419947172
Series sin(2x) = 0.141119469924
C++ sin(2x) = 0.141120008060
C++ 2sin(x)cos(x) = 0.141120008060
End app ? y = yes = y
// DivMulPowerSeries.cpp (c) Tuesday, April 8, 2025
// by James Pate Williams, Jr.
// https://math.libretexts.org/Bookshelves/Calculus/Calculus_(OpenStax)/10%3A_Power_Series/10.02%3A_Properties_of_Power_Series
// https://en.wikipedia.org/wiki/Formal_power_series
#include <iostream>
#include <iomanip>
#include <string>
#include <vector>
std::vector<double> Multiplication(
std::vector<double> c,
std::vector<double> d,
int N) {
std::vector<double> e(N + 1);
for (int n = 0; n <= N; n++) {
double sum = 0.0;
for (int k = 0; k <= n; k++) {
sum += c[k] * d[n - k];
}
e[n] = sum;
}
return e;
}
std::vector<double> Division(
std::vector<double> a,
std::vector<double> b,
int N) {
double a0 = a[0];
std::vector<double> c(N + 1);
for (int n = 0; n <= N; n++) {
double sum = 0.0;
for (int k = 1; k <= n; k++) {
sum += a[k] * c[n - k];
}
c[n] = (b[n] - sum) / a0;
}
return c;
}
double Factorial(int n) {
double nf = 1.0;
for (int i = 2; i <= n; i++) {
nf *= i;
}
return nf;
}
std::vector<double> Cosine(double x, int N, double& fx) {
std::vector<double> series(N + 1);
fx = 0.0;
for (int n = 0; n <= N; n++) {
int argument = 2 * n;
double coeff = pow(-1, n) / Factorial(argument);
series[n] = coeff;
fx += coeff * pow(x, argument);
}
return series;
}
std::vector<double> Sine(double x, int N, double& fx) {
std::vector<double> series(N + 1);
fx = 0.0;
for (int n = 0; n <= N; n++) {
int argument = 2 * n + 1;
double coeff = pow(-1, n) / Factorial(argument);
series[n] = coeff;
fx += coeff * pow(x, argument);
}
return series;
}
std::vector<double> Tangent(double x, int N, double& fx) {
double fc = 0.0, fs = 0.0;
std::vector<double> seriesC = Cosine(x, N, fc);
std::vector<double> seriesS = Sine(x, N, fs);
std::vector<double> seriesT = Division(seriesS, seriesC, N);
fx = fs / fc;
return seriesT;
}
std::vector<double> Sine2x(double x, int N, double& fx) {
double fc = 0.0, fs = 0.0;
std::vector<double> seriesC = Cosine(x, N, fc);
std::vector<double> seriesS = Sine(x, N, fs);
std::vector<double> series2 = Multiplication(seriesS, seriesC, N);
fx = 2.0 * fs * fc;
return series2;
}
int main()
{
while (true) {
char line[128] = { };
std::cout << "x = ";
std::cin.getline(line, 127);
std::string str1(line);
double x = std::stod(str1);
std::cout << "N = ";
std::cin.getline(line, 127);
std::string str2(line);
int N = std::stoi(str2);
double cx = 0.0, sx = 0.0, tx = 0.0, xx = 0.0;
std::vector<double> cSeries = Cosine(x, N, cx);
std::vector<double> sSeries = Sine(x, N, sx);
std::vector<double> tSeries = Tangent(x, N, tx);
std::vector<double> xSeries = Sine2x(x, N, xx);
std::cout << std::fixed << std::setprecision(12);
std::cout << "Series Cosine(x) = " << cx << std::endl;
std::cout << "C++ cos(x) = " << cos(x) << std::endl;
std::cout << "Series sine(x) = " << sx << std::endl;
std::cout << "C++ sin(x) = " << sin(x) << std::endl;
std::cout << "Series Tangent(x) = " << tx << std::endl;
std::cout << "C++ tan(x) = " << tan(x) << std::endl;
std::cout << "Series sin(2x) = " << xx << std::endl;
std::cout << "C++ sin(2x) = " << sin(x + x) << std::endl;
std::cout << "C++ 2sin(x)cos(x) = " << 2.0 * sin(x) * cos(x);
std::cout << std::endl;
std::cout << "End app ? y = yes = ";
std::cin.getline(line, 127);
if (line[0] == 'Y' || line[0] == 'y') {
break;
}
}
return 0;
}
I originally implemented this algorithm in FORTRAN IV in the Summer Quarter of 1982 at the Georgia Institute of Technology. I was taking a course named “Scientific Computing I” taught by Professor Gunter Meyer. I made a B in the class. Later in 2015 I re-implemented the recipe in C# using Visual Studio 2008 Professional. VS 2015 did not have support for complex numbers nor large integers. In December of 2015 I upgraded to Visual Studio 2015 Professional which has support for big integers and complex numbers. I used Visual Studio 2019 Community version for this project. Root below should be function.
I became fascinated with secret key cryptography as a child. Later, as an adult, in around 1979, I started creating crude symmetric cryptographic algorithms. I became further enthralled with cryptography and number theory in 1996 upon reading Applied Cryptography, Second Edition: Protocols, Algorithms, and Source Code inC by Bruce Schneier and later the Handbook of Applied Cryptography by Alfred J. Menezes, Paul C. van Oorschot, and Scott A. Vanstone. After implementing many of the algorithms in both tomes, I communicated my results to two of the authors namely Bruce Schneier and Professor Alfred J. Menezes. In 1997 I developed a website devoted to constraint satisfaction problems and their solutions, cryptography, and number theory. I posted legal C and C++ source code. Professor Menezes advertised my website along with his treatise. See the following blurb:
In the spirit of my twin scientific infatuations, I offer yet another C integer factoring implementation utilizing the Free Large Integer Package (known more widely as lip) which was created by Arjen K. Lenstra (now a Professor Emeritus). This implementation includes Henri Cohen’s Trial Division algorithm, the Brent-Cohen-Pollard rho method, the Cohen-Pollard p – 1 stage 1 method, and the Lenstra lip Elliptic Curve Method. If I can get the proper authorization, I will later post the source code.
total time required for initialization: 0.056000 seconds
enter number below:
2^111+2
== Menu ==
1 Trial Division
2 Pollard-Brent-Cohen rho
3 p - 1 Pollard-Cohen
4 Lenstra's Elliptic Curve Method
5 Pollard-Lenstra rho
1
2596148429267413814265248164610050
number is composite
factors:
total time required factoring: 0.014000 seconds:
2
5 ^ 2
41
397
2113
enter number below:
0
total time required for initialization: 0.056000 seconds
enter number below:
2^111+2
== Menu ==
1 Trial Division
2 Pollard-Brent-Cohen rho
3 p - 1 Pollard-Cohen
4 Lenstra's Elliptic Curve Method
5 Pollard-Lenstra rho
2
2596148429267413814265248164610050
number is composite
factors:
total time required factoring: 1.531000 seconds:
2
5 ^ 2
41
397
2113
415878438361
3630105520141
enter number below:
0
total time required for initialization: 0.055000 seconds
enter number below:
2^111+2
== Menu ==
1 Trial Division
2 Pollard-Brent-Cohen rho
3 p - 1 Pollard-Cohen
4 Lenstra's Elliptic Curve Method
5 Pollard-Lenstra rho
3
2596148429267413814265248164610050
number is composite
factors:
total time required factoring: 0.066000 seconds:
2
5 ^ 2
41
838861
415878438361
3630105520141
enter number below:
0
total time required for initialization: 0.056000 seconds
enter number below:
2^111+2
== Menu ==
1 Trial Division
2 Pollard-Brent-Cohen rho
3 p - 1 Pollard-Cohen
4 Lenstra's Elliptic Curve Method
5 Pollard-Lenstra rho
4
2596148429267413814265248164610050
number is composite
factors:
total time required factoring: 0.013000 seconds:
2
5
205
838861
415878438361
3630105520141
enter number below:
0
Numerical Explorations 