// Algorithms from "A Course in Computational
// Algebraic Number Theory" by Henri Cohen
// Implemented by James Pate Williams, Jr.
// Copyright (c) 2023 All Rights Reserved
#pragma once
#include "pch.h"
template<class T> class Matrix
{
public:
size_t m, n;
T** data;
Matrix() { m = 0; n = 0; data = NULL; };
Matrix(size_t m, size_t n)
{
this->m = m;
this->n = n;
data = new T*[m];
if (data == NULL)
exit(-300);
for (size_t i = 0; i < m; i++)
{
data[i] = new T[n];
if (data[i] == NULL)
exit(-301);
}
};
void OutputMatrix(
fstream& outs, char fill, int precision, int width)
{
for (size_t i = 0; i < m; i++)
{
for (size_t j = 0; j < n; j++)
{
outs << setfill(fill) << setprecision(precision);
outs << setw(width) << data[i][j] << '\t';
}
outs << endl;
}
};
};
template<class T> class Vector
{
public:
size_t n;
T* data;
Vector() { n = 0; data = NULL; };
Vector(size_t n)
{
this->n = n;
data = new T[n];
};
void OutputVector(
fstream& outs, char fill, int precision, int width)
{
for (size_t i = 0; i < n; i++)
{
outs << setfill(fill) << setprecision(precision);
outs << setw(width) << data[i] << '\t';
}
outs << endl;
};
};
class LinearAlgebra
{
public:
bool initialized;
size_t m, n;
Matrix<double> M;
Vector<double> B;
LinearAlgebra() {
initialized = false;
m = 0; n = 0;
M.data = NULL;
B.data = NULL;
};
LinearAlgebra(size_t m, size_t n) {
initialized = false;
this->m = m;
this->n = n;
this->M.m = m;
this->M.n = n;
this->B.n = n;
this->M.data = new double*[m];
this->B.data = new double[n];
if (M.data != NULL)
{
for (size_t i = 0; i < m; i++)
{
this->M.data[i] = new double[n];
for (size_t j = 0; j < n; j++)
this->M.data[i][j] = 0;
}
}
if (B.data != NULL)
{
this->B.data = new double[n];
for (size_t i = 0; i < n; i++)
this->B.data[i] = 0;
}
initialized = this->B.data != NULL && this->M.data != NULL;
};
LinearAlgebra(
size_t m, size_t n,
double** M,
double* B)
{
this->m = m;
this->n = n;
this->M.m = m;
this->M.n = n;
this->M.data = new double*[m];
if (M != NULL)
{
for (size_t i = 0; i < m; i++)
{
this->M.data[i] = new double[n];
for (size_t j = 0; j < n; j++)
this->M.data[i][j] = M[i][j];
}
}
if (B != NULL)
{
this->B.data = new double[n];
for (size_t i = 0; i < m; i++)
this->B.data[i] = B[i];
}
initialized = this->B.data != NULL && this->M.data != NULL;
}
~LinearAlgebra()
{
M.m = m;
M.n = n;
B.n = n;
if (B.data != NULL)
delete[] B.data;
for (size_t i = 0; i < m; i++)
if (M.data[i] != NULL)
delete[] M.data[i];
if (M.data != NULL)
delete[] M.data;
}
double DblDeterminant(size_t n, bool failure);
Vector<double> DblGaussianElimination(
bool& failure);
// The following three methods are from the
// textbook "Elementary Numerical Analysis
// An Algorithmic Approach" by S. D. Conte
// and Carl de Boor Translated from the
// original FORTRAN by James Pate Williams, Jr.
// Copyright (c) 2023 All Rights Reserved
bool DblGaussianFactor(
size_t n,
Vector<int>& pivot);
bool DblGaussianSolution(
int n,
Vector<double>& x,
Vector<int>& pivot);
bool DblSubstitution(
size_t n,
Vector<double>& x,
Vector<int>& pivot);
bool DblInverse(
size_t n,
Matrix<double>& A,
Vector<int>& pivot);
};
#include "pch.h"
#include "LinearAlgebra.h"
double LinearAlgebra::DblDeterminant(
size_t n, bool failure)
{
double deter = 1;
Vector<int> pivot(n);
if (!initialized || m != n)
{
failure = true;
return 0.0;
}
if (!DblGaussianFactor(n, pivot))
{
failure = true;
return 0.0;
}
for (size_t i = 0; i < n; i++)
deter *= M.data[i][i];
return deter;
}
Vector<double> LinearAlgebra::DblGaussianElimination(
bool& failure)
{
double* C = new double[m];
Vector<double> X(n);
X.data = new double[n];
if (X.data == NULL)
exit(-200);
if (!initialized)
{
failure = true;
delete[] C;
return X;
}
for (size_t i = 0; i < m; i++)
C[i] = -1;
failure = false;
for (size_t j = 0; j < n; j++)
{
bool found = false;
size_t i = j;
while (i < n && !found)
{
if (M.data[i][j] != 0)
found = true;
else
i++;
}
if (!found)
{
failure = true;
break;
}
if (i > j)
{
for (size_t l = j; l < n; l++)
{
double t = M.data[i][l];
M.data[i][l] = M.data[j][l];
M.data[j][l] = t;
t = B.data[i];
B.data[i] = B.data[j];
B.data[j] = t;
}
}
double d = 1.0 / M.data[j][j];
for (size_t k = j + 1; k < n; k++)
C[k] = d * M.data[k][j];
for (size_t k = j + 1; k < n; k++)
{
for (size_t l = j + 1; l < n; l++)
M.data[k][l] = M.data[k][l] - C[k] * M.data[j][l];
B.data[k] = B.data[k] - C[k] * B.data[j];
}
}
for (int i = (int)n - 1; i >= 0; i--)
{
double sum = 0;
for (size_t j = i + 1; j < n; j++)
sum += M.data[i][j] * X.data[j];
X.data[i] = (B.data[i] - sum) / M.data[i][i];
}
delete[] C;
return X;
}
bool LinearAlgebra::DblGaussianFactor(
size_t n,
Vector<int>& pivot)
// returns false if matrix is singular
{
Vector<double> d(n);
double awikod, col_max, ratio, row_max, temp;
int flag = 1;
size_t i_star, itemp;
for (size_t i = 0; i < n; i++)
{
pivot.data[i] = i;
row_max = 0;
for (size_t j = 0; j < n; j++)
row_max = max(row_max, abs(M.data[i][j]));
if (row_max == 0)
{
flag = 0;
row_max = 1;
}
d.data[i] = row_max;
}
if (n <= 1) return flag != 0;
// factorization
for (size_t k = 0; k < n - 1; k++)
{
// determine pivot row the row i_star
col_max = abs(M.data[k][k]) / d.data[k];
i_star = k;
for (size_t i = k + 1; i < n; i++)
{
awikod = abs(M.data[i][k]) / d.data[i];
if (awikod > col_max)
{
col_max = awikod;
i_star = i;
}
}
if (col_max == 0)
flag = 0;
else
{
if (i_star > k)
{
// make k the pivot row by
// interchanging with i_star
flag *= -1;
itemp = pivot.data[i_star];
pivot.data[i_star] = pivot.data[k];
pivot.data[k] = itemp;
temp = d.data[i_star];
d.data[i_star] = d.data[k];
d.data[k] = temp;
for (size_t j = 0; j < n; j++)
{
temp = M.data[i_star][j];
M.data[i_star][j] = M.data[k][j];
M.data[k][j] = temp;
}
}
// eliminate x[k]
for (size_t i = k + 1; i < n; i++)
{
M.data[i][k] /= M.data[k][k];
ratio = M.data[i][k];
for (size_t j = k + 1; j < n; j++)
M.data[i][j] -= ratio * M.data[k][j];
}
}
if (M.data[n - 1][n - 1] == 0) flag = 0;
}
if (flag == 0)
return false;
return true;
}
bool LinearAlgebra::DblGaussianSolution(
int n,
Vector<double>& x,
Vector<int>& pivot)
{
if (!DblGaussianFactor(n, pivot))
return false;
return DblSubstitution(n, x, pivot);
}
bool LinearAlgebra::DblSubstitution(
size_t n, Vector<double>& x,
Vector<int>& pivot)
{
double sum;
size_t j, n1 = n - 1;
if (n == 1)
{
x.data[0] = B.data[0] / M.data[0][0];
return true;
}
// forward substitution
x.data[0] = B.data[pivot.data[0]];
for (int i = 1; i < (int)n; i++)
{
for (j = 0, sum = 0; j < (size_t)i; j++)
sum += M.data[i][j] * x.data[j];
x.data[i] = B.data[pivot.data[i]] - sum;
}
// backward substitution
x.data[n1] /= M.data[n1][n1];
for (int i = n - 2; i >= 0; i--)
{
double sum = 0.0;
for (j = i + 1; j < n; j++)
sum += M.data[i][j] * x.data[j];
x.data[i] = (x.data[i] - sum) / M.data[i][i];
}
return true;
}
bool LinearAlgebra::DblInverse(
size_t n,
Matrix<double>& A,
Vector<int>& pivot)
{
Vector<double> x(n);
if (!DblGaussianFactor(n, pivot))
return false;
for (size_t i = 0; i < n; i++)
{
for (size_t j = 0; j < n; j++)
B.data[j] = 0;
}
for (size_t i = 0; i < n; i++)
{
B.data[i] = 1;
if (!DblSubstitution(n, x, pivot))
return false;
B.data[i] = 0;
for (size_t j = 0; j < n; j++)
A.data[i][j] = x.data[pivot.data[j]];
}
return true;
}
/*
** Cohen's linear algebra test program
** Implemented by James Pate Williams, Jr.
** Copyright (c) 2023 All Rights Reserved
*/
#include "pch.h"
#include "LinearAlgebra.h"
double GetDblNumber(fstream& inps)
{
char ch = inps.get();
string numberStr;
while (ch == ' ' || ch == '\t' || ch == '\r' || ch == '\n')
ch = inps.get();
while (ch == '+' || ch == '-' || ch == '.' ||
ch >= '0' && ch <= '9')
{
numberStr += ch;
ch = inps.get();
}
double x = atof(numberStr.c_str());
return x;
}
int GetIntNumber(fstream& inps)
{
char ch = inps.get();
string numberStr;
while (ch == ' ' || ch == '\t' || ch == '\r' || ch == '\n')
ch = inps.get();
while (ch == '+' || ch == '-' || ch >= '0' && ch <= '9')
{
numberStr += ch;
ch = inps.get();
}
int x = atoi(numberStr.c_str());
return x;
}
int main()
{
fstream inps;
inps.open("CLATestFile.txt", fstream::in);
if (inps.fail())
{
cout << "Input file opening error!" << endl;
return -1;
}
fstream outs;
outs.open("CLAResuFile.txt", fstream::out | fstream::ate);
if (outs.fail())
{
cout << "Output file opening error!" << endl;
return -2;
}
size_t m, n;
while (!inps.eof())
{
m = GetIntNumber(inps);
if (inps.eof())
return 0;
if (m < 1)
{
cout << "The number of rows must be >= 1" << endl;
return -100;
}
n = GetIntNumber(inps);
if (n < 1)
{
cout << "The number of rows must be >= 1" << endl;
return -101;
}
LinearAlgebra la(m, n);
Matrix<double> copyM(m, n);
Vector<double> copyB(n);
for (size_t i = 0; i < m; i++)
{
for (size_t j = 0; j < n; j++)
{
double x = GetDblNumber(inps);
la.M.data[i][j] = x;
copyM.data[i][j] = x;
}
}
for (size_t i = 0; i < n; i++)
{
la.B.data[i] = GetDblNumber(inps);
copyB.data[i] = la.B.data[i];
}
bool failure = false;
Vector<double> X = la.DblGaussianElimination(failure);
if (!failure)
X.OutputVector(outs, ' ', 5, 8);
else
{
cout << "Cohen Gaussian elimination failure!" << endl;
exit(-102);
}
for (size_t i = 0; i < m; i++)
{
la.B.data[i] = copyB.data[i];
for (size_t j = 0; j < n; j++)
{
la.M.data[i][j] = copyM.data[i][j];
}
}
Matrix<double> A(n, n);
Vector<int> pivot(n);
if (!la.DblGaussianSolution(n, X, pivot))
exit(-103);
X.OutputVector(outs, ' ', 5, 8);
for (size_t i = 0; i < m; i++)
{
la.B.data[i] = copyB.data[i];
for (size_t j = 0; j < n; j++)
{
la.M.data[i][j] = copyM.data[i][j];
}
}
double deter = la.DblDeterminant(n, failure);
outs << deter << endl;
for (size_t i = 0; i < m; i++)
{
la.B.data[i] = copyB.data[i];
for (size_t j = 0; j < n; j++)
{
la.M.data[i][j] = copyM.data[i][j];
}
}
if (!la.DblInverse(n, A, pivot))
{
cout << "Conte Gaussian inverse matrix failure!" << endl;
exit(-104);
}
else
A.OutputMatrix(outs, ' ', 5, 8);
}
inps.close();
outs.close();
}
2
2
1 1
1 2
7 11
2
2
1 1
1 3
7 11
2
2
6 3
4 8
5 6
2
2
5 3
10 4
8 6
3
3
2 1 -1
-3 -1 2
-2 1 2
8 -11 -3
3 4
3 4
1
2 -1
-1 1
5 2
5 2
2
1.5 -0.5
-0.5 0.5
0.61111 0.44444
0.61111 0.44444
36
0.22222 -0.11111
-0.083333 0.16667
-1.4 5
-1.4 5
-10
-0.4 1
0.3 -0.5
2 3 -1
2 3 -1
1
4 5 -2
3 4 -2
-1 -1 1
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 