#pragma once
#include <complex>
#include <vector>
class Transform
{
public:
static void VandermondeDFT(
int n,
std::vector<std::complex<double>>& a,
std::vector<std::complex<double>>& y);
static void InverseVandermondeDFT(
int n,
std::vector<std::complex<double>>& a,
std::vector<std::complex<double>>& y);
static std::vector<std::complex<double>> DFT(
std::vector<double>& x, std::vector<double>& f);
static std::vector<double> InverseDFT(
std::vector<double>& f,
std::vector<std::complex<double>>& X);
/*
* Reference: "Elementary Numerical Analysis:
* An Algorithmic Approach Third Edition" (c)
* 1980 by S. D. Conte and Carl de Boor
* Section 6.5 pages 268 - 277 and Section 6.6
* pages 277 - 283
* Input to FFT
* Z1, Z2 complex n-vectors
* n the length of the vectors
* inzee
* = 1 transform in Z1
* = 2 transform in Z2
* Constructs the discrete Fourier transform in the Cooley-
* Tukey way, but with a twist.
*/
static void FFT(
std::vector<std::complex<double>>& Z1,
int& after, int& now, int& before, int& inzee,
std::vector<std::complex<double>>& Z2);
/*
* This computes an in - place complex - to - complex FFT
* x and y are the real and imaginary arrays of 2^m points.
* dir = 1 gives forward transform
* dir = -1 gives reverse transform
* see http://astronomy.swin.edu.au/~pbourke/analysis/dft/
*/
static void FFT(short dir, int m,
std::vector<double>& x, std::vector<double>& y);
/*
* Reference: "Introduction to Algorithms" by
* Thomas H. Cormen, Charles E. Leiserson, and
* Ronald L. Rivest, pages 794 - 795
*/
static void IterativeFFT(
std::vector<std::complex<double>>& a,
std::vector<std::complex<double>>& A);
/*
* Reference: "Introduction to Algorithms" by
* Thomas H. Cormen, Charles E. Leiserson, and
* Ronald L. Rivest, page 788
*/
static std::vector<std::complex<double>> RecursiveFFT(
std::vector<std::complex<double>>& a);
};
#include "Transform.h"
void Transform::VandermondeDFT(
int n,
std::vector<std::complex<double>>& a,
std::vector<std::complex<double>>& y)
{
double pi = 4.0 * atan(1.0);
std::complex<double> z(0.0, 2.0 * pi / n);
std::complex<double> omegaN = exp(z);
std::vector<std::vector<std::complex<double>>> V(n);
for (int k = 0; k < n; k++)
{
V[k].resize(n);
for (int j = 0; j < n; j++)
{
V[k][j] = std::pow(omegaN, k * j);
}
}
for (int k = 0; k < n; k++)
{
std::complex<double> sum = 0.0;
for (int j = 0; j < n; j++)
{
sum += V[k][j] * a[j];
}
y[k] = sum;
}
}
void Transform::InverseVandermondeDFT(
int n,
std::vector<std::complex<double>>& a,
std::vector<std::complex<double>>& y)
{
double pi = 4.0 * atan(1.0);
std::complex<double> nc = { static_cast<double>(n), 0.0 };
std::complex<double> z(0.0, 2.0 * pi / n);
std::complex<double> omegaN = exp(z);
std::vector<std::vector<std::complex<double>>> invV(n);
for (int k = 0; k < n; k++)
{
invV[k].resize(n);
for (int j = 0; j < n; j++)
{
invV[k][j] = std::pow(omegaN, -k * j);
}
}
for (int k = 0; k < n; k++)
{
std::complex<double> sum = 0.0;
for (int j = 0; j < n; j++)
{
sum += invV[k][j] * y[j];
}
a[k] = sum / nc;
}
}
std::vector<std::complex<double>> Transform::DFT(
std::vector<double>& x, std::vector<double>& f)
{
int length = static_cast<int>(x.size());
double pi = 4.0 * atan(1.0);
double pi2oN = 2.0 * pi / length;
int k, n;
std::vector<double> X(length);
std::vector<double> Y(length);
std::vector<std::complex<double>> Z(length);
f.resize(length);
for (k = 0; k < length; k++)
{
X[k] = Y[k] = 0;
for (n = 0; n < length; n++)
{
X[k] += x[n] * cos(pi2oN * k * n);
Y[k] -= x[n] * sin(pi2oN * k * n);
}
f[k] = pi2oN * k;
X[k] /= length;
Y[k] /= length;
Z[k] = { X[k], Y[k] };
}
return Z;
}
std::vector<double> Transform::InverseDFT(
std::vector<double>& f,
std::vector<std::complex<double>>& X)
{
double imag = 0.0;
int length = static_cast<int>(X.size());
std::vector<double> x(length);
for (int n = 0; n < length; n++)
{
imag = x[n] = 0.0;
for (int k = 0; k < length; k++)
{
x[n] += X[k]._Val[0] * cos(f[k] * n)
- X[k]._Val[1] * sin(f[k] * n);
imag += X[k]._Val[0] * sin(f[k] * n)
+ X[k]._Val[1] * cos(f[k] * n);
}
}
return x;
}
static void FFTStep(
std::vector<std::complex<double>>& Zinp,
int after, int now, int before,
std::vector<std::complex<double>>& Zout)
{
double angle = 0.0, ratio = 0.0;
double twoPi = 2.0 * 4.0 * atan(1.0);
int ia = 0, ib = 0, inp = 0, j = 0;
std::complex<double> arg = 1.0, omega = 0, value = 0;
angle = twoPi / ((now + 1) * (after + 1));
omega = std::complex<double>(cos(angle), -sin(angle));
int address = 1;
for (int i = 1; i <= now; i++)
{
for (int j = 1; j <= after; j++)
{
for (int k = 1; k <= before; k++)
{
address = i * j * k;
if (address < Zout.size())
Zout[address] = { 0.0, 0.0 };
}
}
}
address = 1;
for (int j = 1; j <= now; j++)
{
for (ia = 1; ia <= after; ia++)
{
for (ib = 1; ib <= before; ib++)
{
int address = j * ia * ib;
if (address < Zinp.size())
value = Zinp[address];
for (inp = now - 1; inp >= 1; inp--)
{
address = ia * ib * inp;
if (address < Zinp.size())
value = value * arg + Zinp[address];
}
address = ia * j * ib;
if (address < Zout.size())
Zout[address] = value;
}
arg *= omega;
}
}
}
void Transform::FFT(
std::vector<std::complex<double>>& Z1,
int& after, int& now, int& before, int& inzee,
std::vector<std::complex<double>>& Z2)
{
std::vector<int> prime =
{ 0, 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43,
47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 97 };
int next = 1, nextmx = 25;
after = 1;
before = (int)Z1.size();
now = 1;
Label10:
if (before / prime[next] * prime[next] < before)
{
next++;
if (next <= nextmx)
goto Label10;
else
{
now = before;
before = 1;
}
}
else
{
now = prime[next];
before /= prime[next];
}
if (inzee == 1)
FFTStep(Z1, after, now, before, Z2);
else
FFTStep(Z2, after, now, before, Z1);
inzee = 3 - inzee;
if (before == 1)
return;
after *= now;
goto Label10;
}
void Transform::FFT(short dir, int m,
std::vector<double>& x, std::vector<double>& y)
{
int n, i, i1, j, k, i2, l, l1, l2;
double c1, c2, tx, ty, t1, t2, u1, u2, z;
// Calculate the number of points
n = 1;
for (i = 0; i < m; i++)
n *= 2;
// Do the bit reversal
i2 = n >> 1;
j = 0;
for (i = 0; i < n - 1; i++)
{
if (i < j)
{
tx = x[i];
ty = y[i];
x[i] = x[j];
y[i] = y[j];
x[j] = tx;
y[j] = ty;
}
k = i2;
while (k <= j)
{
j -= k;
k >>= 1;
}
j += k;
}
// Compute the FFT
c1 = -1.0;
c2 = 0.0;
l2 = 1;
for (l = 0; l < m; l++)
{
l1 = l2;
l2 <<= 1;
u1 = 1.0;
u2 = 0.0;
for (j = 0; j < l1; j++)
{
for (i = j; i < n; i += l2)
{
i1 = i + l1;
t1 = u1 * x[i1] - u2 * y[i1];
t2 = u1 * y[i1] + u2 * x[i1];
x[i1] = x[i] - t1;
y[i1] = y[i] - t2;
x[i] += t1;
y[i] += t2;
}
z = u1 * c1 - u2 * c2;
u2 = u1 * c2 + u2 * c1;
u1 = z;
}
c2 = sqrt((1.0 - c1) / 2.0);
if (dir == 1)
c2 = -c2;
c1 = sqrt((1.0 + c1) / 2.0);
}
// Scaling for forward transform
if (dir == 1)
{
for (i = 0; i < n; i++)
{
x[i] /= n;
y[i] /= n;
}
}
}
static void FFTBase(
std::vector<std::complex<double>> a,
std::vector<std::complex<double>> A)
{
double pi = 4.0 * atan(1.0);
int n = static_cast<int>(a.size());
for (int s = 1; s <= log2(n); s++)
{
int m = static_cast<int>(pow(2, s));
std::complex<double> z(0.0, 2.0 * pi / m);
std::complex<double> omegaM = exp(z);
for (int k = 0; k <= n - 1; k += m)
{
std::complex<double> omega = { 1.0, 0.0 };
for (int j = 0; j <= m / 2 - 1; j++)
{
std::complex<double> t = omega * A[k + j + m / 2];
std::complex<double> u = A[k + j];
std::complex<double> jc = { static_cast<double>(j), 0.0 };
A[k + j] = u + jc;
A[k + j + m / 2] = u - t;
omega *= omegaM;
}
}
}
}
static int Reverse(int k)
{
int digits[32] = { 0 }, i = 0;
while (k > 0)
{
int digit = k & 1;
k >>= 1;
digits[i++] = digit;
}
int result = digits[0];
for (int j = 1; j < i; j++)
result = result * 2 + digits[j];
return result;
}
static void BitReverseCopy(
std::vector<std::complex<double>>& a,
std::vector<std::complex<double>>& A)
{
int n = static_cast<int>(a.size());
for (int k = 0; k <= n - 1; k++)
A[Reverse(k)] = a[k];
}
void Transform::IterativeFFT(
std::vector<std::complex<double>>& a,
std::vector<std::complex<double>>& A)
{
BitReverseCopy(a, A);
double pi = 4.0 * atan(1.0);
int n = static_cast<int>(a.size());
for (int s = 1; s <= static_cast<int>(log2(n)); s++)
{
int m = static_cast<int>(pow(2.0, s));
std::complex<double> z(0.0, 2.0 * pi / m);
std::complex<double> omegaM = exp(z);
std::complex<double> omega = { 1.0, 0.0 };
for (int j = 0; j <= m / 2 - 1; j++)
{
for (int k = j; k <= n - 1; k += m)
{
std::complex<double> t = omega * A[k + m / 2];
std::complex<double> u = A[k];
A[k] = u + t;
A[k + m / 2] = u - t;
omega *= omegaM;
}
}
}
}
std::vector<std::complex<double>> Transform::RecursiveFFT(
std::vector<std::complex<double>>& a)
{
int n = static_cast<int>(a.size());
if (n == 1)
return a;
std::vector<std::complex<double>> a0;
std::vector<std::complex<double>> a1;
std::vector<std::complex<double>> y0;
std::vector<std::complex<double>> y1;
for (int i = 0; i <= n - 2; i++)
a0.push_back(a[i]);
for (int i = 1; i <= n - 1; i++)
a1.push_back(a[i]);
y0 = RecursiveFFT(a0);
y1 = RecursiveFFT(a1);
double pi = 4.0 * atan(1.0);
std::complex<double> z(0.0, 2.0 * pi / n);
std::complex<double> omegaN = exp(z);
std::complex<double> omega(1.0, 0.0);
std::vector<std::complex<double>> y(n, 0.0);
for (int k = 0; k <= n / 2 - 1; k++)
{
y[k] = y0[k] + omega * y1[k];
y[(long long)k + n / 2] = y0[k] - omega * y1[k];
omega *= omegaN;
}
return y;
}
// CooleyTukeyConsole.cpp : This file contains the 'main' function. Program execution begins and ends there.
//
#include <algorithm>
#include <complex>
#include <iomanip>
#include <iostream>
#include <string>
#include <vector>
#include "Transform.h"
int M = 0, N = 0, Sn = 2048;
static double SimpsonsRule(
double a, double b, int n,
double (*f)(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 += f(x);
x += h2;
}
x = a + h2;
for (int i = 2; i < n; i += 2)
{
t += f(x);
x += h2;
}
return h * (f(a) + 4 * s + 2 * t + f(b)) / 3.0;
}
static double f(double x)
{
return x * x * sin(x);
}
static double AMf(double x)
{
return f(x) * cos(M * x);
}
static double BMf(double x)
{
return f(x) * sin(M * x);
}
static double AM(int m)
{
M = m;
double pi = 4.0 * atan(1.0), twoPi = pi + pi;
double integ = SimpsonsRule(-pi, pi, Sn, AMf);
double value = integ / twoPi;
if (m == 0)
return value;
else
return 2.0 * value;
}
static double BM(int m)
{
M = m;
double pi = 4.0 * atan(1.0), twoPi = pi + pi;
double integ = SimpsonsRule(-pi, pi, Sn, BMf);
return 2.0 * integ / twoPi;
}
static void FormatPrint(double x)
{
if (fabs(x) < 1.0e-12)
x = 0.0;
std::cout << std::setw(13);
std::cout << std::setfill(' ');
std::cout << std::setprecision(10);
std::cout << x << '\t';
}
static void FormatPrint(std::complex<double> z)
{
std::cout << std::setw(13);
std::cout << std::setfill(' ');
std::cout << std::setprecision(10);
std::cout << z << '\t';
}
static void GetCooleyTukeyData(
int n0, int n1, int n2, int n3,
std::vector<double>& a,
std::vector<double>& b,
std::vector<double>& p,
std::vector<double>& x,
std::vector<double>& fx)
{
double pi = 4.0 * atan(1.0), twoPi = pi + pi;
x[0] = fx[0] = 0.0;
for (int i = 1; i <= n0; i++)
{
x[i] = twoPi * i / n0;
fx[i] = f(x[i]);
}
a[0] = AM(0);
for (int m = 1; m <= n0; m++)
{
a[m] = AM(m);
b[m] = BM(m);
}
for (int n = 0; n <= n0; n++)
{
double asum = 0.0;
for (int m = 1; m < n0; m++)
asum += a[m] * cos(m * x[n]);
double bsum = 0.0;
for (int m = 1; m <= n0; m++)
bsum += b[m] * sin(m * x[n]);
p[n] = a[0] / 2.0 + asum + bsum;
}
}
static void GetData(
int n0,
std::vector<double>& a,
std::vector<double>& b,
std::vector<double>& p,
std::vector<double>& x,
std::vector<double>& fx)
{
double pi = 4.0 * atan(1.0), twoPi = pi + pi;
for (int i = 0; i < n0; i++)
{
x[i] = twoPi * i / n0;
fx[i] = f(x[i]);
}
a[0] = AM(0);
for (int m = 1; m < n0; m++)
{
a[m] = AM(m);
b[m] = BM(m);
}
for (int n = 0; n < n0; n++)
{
double asum = 0.0;
for (int m = 1; m < n0; m++)
asum += a[m] * cos(m * x[n]);
double bsum = 0.0;
for (int m = 1; m < n0; m++)
bsum += b[m] * sin(m * x[n]);
p[n] = a[0] / 2.0 + asum + bsum;
}
}
static void TestCooleyTukey(
int n0, int n1, int n2, int n3)
{
int n01 = n0 + 1;
std::vector<double> a(n01), b(n01), p(n01), x(n01), fx(n01);
GetCooleyTukeyData(n0, n1, n2, n3, a, b, p, x, fx);
double pi = 4.0 * atan(1.0), twoPi = pi + pi;
int index1 = 1, inzee = 1;
std::vector<std::complex<double>> pp(n01);
std::vector<std::complex<double>> Z1 = { {0, 0} };
std::vector<std::complex<double>> Z2 = { {0, 0} };
Z1.resize(n01, 0.0);
Z2.resize(n01, 0.0);
for (int i = 1; i <= n0; i++)
Z1[i] = fx[i];
int after = 0, now = 0, before = 0;
Transform::FFT(Z1, after, now, before, inzee, Z2);
std::cout << "Forward Transform" << std::endl;
for (int i = 1; index1 < n01 && i <= after; i++)
{
for (int j = 1; index1 < n01 && j <= now; j++)
{
for (int k = 1; index1 < n01 && k <= before; k++)
{
std::cout << std::setw(5);
std::cout << std::setfill(' ');
std::cout << index1 << '\t';
FormatPrint(x[index1]);
FormatPrint(fx[index1]);
FormatPrint(Z1[index1]);
FormatPrint(Z2[index1]);
std::cout << std::endl;
index1++;
}
}
}
}
static void TestFFT(int n0)
{
int m = static_cast<int>(log2(n0));
std::vector<double> a(n0), b(n0), p(n0), x(n0), fx(n0);
GetData(n0, a, b, p, x, fx);
std::vector<double> xx(n0), yy(n0);
for (int i = 0; i < n0; i++)
xx[i] = fx[i];
Transform::FFT(+1, m, xx, yy);
std::cout << "Forward Transform" << std::endl;
for (int i = 0; i < n0; i++)
{
std::cout << std::setw(5);
std::cout << std::setfill(' ');
std::cout << i << '\t';
FormatPrint(x[i]);
FormatPrint(fx[i]);
FormatPrint(p[i]);
FormatPrint(xx[i]);
FormatPrint(yy[i]);
std::cout << std::endl;
if (i == n0 / 2)
break;
}
Transform::FFT(-1, m, xx, yy);
std::cout << "Backward Transform" << std::endl;
for (int i = 0; i < n0; i++)
{
std::cout << std::setw(5);
std::cout << std::setfill(' ');
std::cout << i << '\t';
FormatPrint(x[i]);
FormatPrint(fx[i]);
FormatPrint(p[i]);
FormatPrint(xx[i]);
FormatPrint(yy[i]);
std::cout << std::endl;
if (i == n0 / 2)
break;
}
}
static void TestIterativeFFT(int n0)
{
std::vector<double> a(n0), b(n0), p(n0), x(n0), fx(n0);
GetData(n0, a, b, p, x, fx);
double pi = 4.0 * atan(1.0), twoPi = pi + pi;
std::vector<std::complex<double>> AA(n0);
std::vector<std::complex<double>> aa(n0);
std::vector<std::complex<double>> yy(n0);
for (int i = 0; i < n0; i++)
aa[i] = fx[i];
Transform::IterativeFFT(aa, AA);
std::cout << "Forward Transform" << std::endl;
for (int i = 0; i < n0; i++)
{
std::cout << std::setw(5);
std::cout << std::setfill(' ');
std::cout << i << '\t';
FormatPrint(x[i]);
FormatPrint(fx[i]);
FormatPrint(AA[i]);
std::cout << std::endl;
}
}
static void TestRecursiveFFT(int n0)
{
std::vector<double> a(n0), b(n0), p(n0), x(n0), fx(n0);
std::vector<double> xx(n0), yy(n0);
GetData(n0, a, b, p, x, fx);
double pi = 4.0 * atan(1.0), twoPi = pi + pi;
std::vector<std::complex<double>> A = { {0, 0} };
std::vector<std::complex<double>> Y = { {0, 0} };
A.resize(n0);
for (int i = 0; i < n0; i++)
A[i] = fx[i];
Y = Transform::RecursiveFFT(A);
std::cout << "Forward Transform" << std::endl;
for (int i = 0; i < n0; i++)
{
std::cout << std::setw(5);
std::cout << std::setfill(' ');
std::cout << i << '\t';
FormatPrint(x[i]);
FormatPrint(fx[i]);
FormatPrint(A[i]);
FormatPrint(Y[i]);
std::cout << std::endl;
}
}
static void TestDFT(int n0)
{
std::vector<double> a(n0), b(n0), p(n0), x(n0), fx(n0), ff(n0);
GetData(n0, a, b, p, x, fx);
std::vector<std::complex<double>> zz = Transform::DFT(fx, ff);
std::cout << "Forward Transform" << std::endl;
for (int i = 0; i < n0; i++)
{
std::cout << std::setw(5);
std::cout << std::setfill(' ');
std::cout << i << '\t';
FormatPrint(x[i]);
FormatPrint(fx[i]);
FormatPrint(ff[i]);
FormatPrint(zz[i]);
std::cout << std::endl;
}
std::cout << "Inverse Transform" << std::endl;
std::vector<double> inv = Transform::InverseDFT(ff, zz);
for (int i = 0; i < n0; i++)
{
std::cout << std::setw(5);
std::cout << std::setfill(' ');
std::cout << i << '\t';
FormatPrint(x[i]);
FormatPrint(fx[i]);
FormatPrint(ff[i]);
FormatPrint(inv[i]);
std::cout << std::endl;
}
}
static void TestVandermondeDFT(int n0)
{
std::vector<double> a(n0), b(n0), p(n0), x(n0), fx(n0);
GetData(n0, a, b, p, x, fx);
std::vector<std::complex<double>> aa(n0);
std::vector<std::complex<double>> yy(n0);
for (int i = 0; i < n0; i++)
aa[i] = fx[i];
Transform::VandermondeDFT(n0, aa, yy);
std::cout << "Vandermonde DFT" << std::endl;
for (int i = 0; i < n0; i++)
{
std::cout << std::setw(5);
std::cout << std::setfill(' ');
std::cout << i << '\t';
FormatPrint(x[i]);
FormatPrint(fx[i]);
FormatPrint(p[i]);
FormatPrint(yy[i]);
std::cout << std::endl;
}
Transform::InverseVandermondeDFT(n0, aa, yy);
std::cout << "Inverse Vandermonde DFT" << std::endl;
for (int i = 0; i < n0; i++)
{
std::cout << std::setw(5);
std::cout << std::setfill(' ');
std::cout << i << '\t';
FormatPrint(x[i]);
FormatPrint(fx[i]);
FormatPrint(p[i]);
FormatPrint(aa[i]);
std::cout << std::endl;
}
}
static int Horner(char line[])
{
int length = static_cast<int>(strlen(line));
int sum = line[0] - '0';
for (int i = 1; i < length; i++)
sum = sum * 10 + line[i] - '0';
return sum;
}
int main()
{
char line[128];
while (true)
{
std::cout << "== Menu ==" << std::endl;
std::cout << "1 Cooley-Tukey" << std::endl;
std::cout << "2 FFT" << std::endl;
std::cout << "3 Iterative FFT" << std::endl;
std::cout << "4 Recursive FFT" << std::endl;
std::cout << "5 DFT" << std::endl;
std::cout << "6 Vandermonde DFT" << std::endl;
std::cout << "7 Exit" << std::endl;
std::cout << "Option 1 - 7 = ";
std::cin.getline(line, 128);
int option = Horner(line);
if (option == 7)
break;
if (option < 1 || option > 7)
{
std::cout << "Unknown Option Number" << std::endl;
continue;
}
if (option == 1)
{
int n0 = 0, n1 = 0, n2 = 0, n3 = 0;
std::cout << "n1 = ";
std::cin.getline(line, 128);
n1 = Horner(line);
std::cout << "n2 = ";
std::cin.getline(line, 128);
n2 = Horner(line);
std::cout << "n3 = ";
std::cin.getline(line, 128);
n3 = Horner(line);
n0 = n1 * n2 * n3;
TestCooleyTukey(n0, n1, n2, n3);
}
else if (option == 2)
{
std::cout << "n0 = ";
std::cin.getline(line, 128);
int n0 = Horner(line);
if (n0 % 2 != 0)
{
std::cout << "n0 must be a power of 2";
std::cout << std::endl;
continue;
}
TestFFT(n0);
}
else if (option == 3)
{
std::cout << "n0 = ";
std::cin.getline(line, 128);
int n0 = Horner(line);
if (n0 % 2 != 0)
{
std::cout << "n0 must be a power of 2";
std::cout << std::endl;
continue;
}
TestIterativeFFT(n0);
}
else if (option == 4)
{
std::cout << "n0 = ";
std::cin.getline(line, 128);
int n0 = Horner(line);
if (n0 % 2 != 0)
{
std::cout << "n0 must be a power of 2";
std::cout << std::endl;
continue;
}
TestRecursiveFFT(n0);
}
else if (option == 5)
{
std::cout << "n0 = ";
std::cin.getline(line, 128);
int n0 = Horner(line);
if (n0 % 2 != 0)
{
std::cout << "n0 must be a power of 2";
std::cout << std::endl;
continue;
}
TestDFT(n0);
}
else if (option == 6)
{
std::cout << "n0 = ";
std::cin.getline(line, 128);
int n0 = Horner(line);
if (n0 % 2 != 0)
{
std::cout << "n0 must be a power of 2";
std::cout << std::endl;
continue;
}
TestVandermondeDFT(n0);
}
}
return 0;
}
Author: jamespatewilliamsjr
My whole legal name is James Pate Williams, Jr. I was born in LaGrange, Georgia approximately 70 years ago. I barely graduated from LaGrange High School with low marks in June 1971. Later in June 1979, I graduated from LaGrange College with a Bachelor of Arts in Chemistry with a little over a 3 out 4 Grade Point Average (GPA). In the Spring Quarter of 1978, I taught myself how to program a Texas Instruments desktop programmable calculator and in the Summer Quarter of 1978 I taught myself Dayton BASIC (Beginner's All-purpose Symbolic Instruction Code) on LaGrange College's Data General Eclipse minicomputer. I took courses in BASIC in the Fall Quarter of 1978 and FORTRAN IV (Formula Translator IV) in the Winter Quarter of 1979. Professor Kenneth Cooper, a genius poly-scientist taught me a course in the Intel 8085 microprocessor architecture and assembly and machine language. We would hand assemble our programs and insert the resulting machine code into our crude wooden box computer which was designed and built by Professor Cooper. From 1990 to 1994 I earned a Bachelor of Science in Computer Science from LaGrange College. I had a 4 out of 4 GPA in the period 1990 to 1994. I took courses in C, COBOL, and Pascal during my BS work. After graduating from LaGrange College a second time in May 1994, I taught myself C++. In December 1995, I started using the Internet and taught myself client-server programming. I created a website in 1997 which had C and C# implementations of algorithms from the "Handbook of Applied Cryptography" by Alfred J. Menezes, et. al., and some other cryptography and number theory textbooks and treatises.
View all posts by jamespatewilliamsjr
Numerical Explorations 