C++ - Solving Linear Equation (3 X 3) Using Matrix
Assume that you have the following 3 equations and you have to find the value of X, Y and Z using Matrices.
Problem:
X - Y + 2Z = 2
2X - 3Y - Z = 5
3X + .5Y + Z = 3
Answer:
X = 1.16
Y = -0.89
Z = -0.02
This problem has in the matrix form - 3 X 3 square matrix.
Lets form an equation:
ax = b
where a
| 1.00 -1.00 2.00 |
| 2.00 -3.00 -1.00 |
| 3.00 0.50 1.00 |
and b
| 2.00 |
| 5.00 |
| 3.00 |
If we need to find out the X in equation ax = b, then apply the formula:
x = InverseMatrix of A multiplied by B.
Given the matrix A, we have to find out its inverse form. Then do the multiplication with B to get the answer B.
The following program does the work for you. Look at the program and output.
Source Code
#if !defined(MATRIX_H)
#define MATRIX_H
#include <stdio.h>
#include <iostream>
#include <tchar.h>
#include <math.h>
class CMatrix
{
private:
int m_rows;
int m_cols;
char m_name[128];
CMatrix();
public:
double **m_pData;
CMatrix(const char *name, int rows, int cols) : m_rows(rows), m_cols(cols)
{
strcpy(m_name, name);
m_pData = new double*[m_rows];
for(int i = 0; i < m_rows; i++)
m_pData[i] = new double[m_cols];
for(int i = 0; i < m_rows; i++)
{
for(int j = 0; j < m_cols; j++)
{
m_pData[i][j] = 0.0;
}
}
}
CMatrix(const CMatrix &other)
{
strcpy(m_name, other.m_name);
m_rows = other.m_rows;
m_cols = other.m_cols;
m_pData = new double*[m_rows];
for(int i = 0; i < m_rows; i++)
m_pData[i] = new double[m_cols];
for(int i = 0; i < m_rows; i++)
{
for(int j = 0; j < m_cols; j++)
{
m_pData[i][j] = other.m_pData[i][j];
}
}
}
~CMatrix()
{
for(int i = 0; i < m_rows; i++)
delete [] m_pData[i];
delete [] m_pData;
m_rows = m_cols = 0;
}
void SetName(const char *name) { strcpy(m_name, name); }
const char* GetName() const { return m_name; }
void GetInput()
{
std::cin >> *this;
}
void FillSimulatedInput()
{
static int factor1 = 1, factor2 = 2;
std::cout << "\n\nEnter Input For Matrix : " << m_name << " Rows: " << m_rows << " Cols: " << m_cols << "\n";
for(int i = 0; i < m_rows; i++)
{
for(int j = 0; j < m_cols; j++)
{
std::cout << "Input For Row: " << i + 1 << " Col: " << j + 1 << " = ";
int data = ((i + 1) * factor1) + (j + 1) * factor2;
m_pData[i][j] = data / 10.2;
std::cout << m_pData[i][j] << "\n";
factor1 += (rand() % 4);
factor2 += (rand() % 3);
}
std::cout << "\n";
}
std::cout << "\n";
}
double Determinant()
{
double det = 0;
double **pd = m_pData;
switch(m_rows)
{
case 2:
{
det = pd[0][0] * pd[1][1] - pd[0][1] * pd[1][0];
return det;
}
break;
case 3:
{
/***
a b c
d e f
g h i
a b c a b c
d e f d e f
g h i g h i
// det (A) = aei + bfg + cdh - afh - bdi - ceg.
***/
double a = pd[0][0];
double b = pd[0][1];
double c = pd[0][2];
double d = pd[1][0];
double e = pd[1][1];
double f = pd[1][2];
double g = pd[2][0];
double h = pd[2][1];
double i = pd[2][2];
double det = (a*e*i + b*f*g + c*d*h); // - a*f*h - b*d*i - c*e*g);
det = det - a*f*h;
det = det - b*d*i;
det = det - c*e*g;
//std::cout << *this;
//std::cout << "deter: " << det << " \n";
return det;
}
break;
case 4:
{
CMatrix *temp[4];
for(int i = 0; i < 4; i++)
temp[i] = new CMatrix("", 3,3);
for(int k = 0; k < 4; k++)
{
for(int i = 1; i < 4; i++)
{
int j1 = 0;
for(int j = 0; j < 4; j++)
{
if(k == j)
continue;
temp[k]->m_pData[i-1][j1++] = this->m_pData[i][j];
}
}
}
double det = this->m_pData[0][0] * temp[0]->Determinant() -
this->m_pData[0][1] * temp[1]->Determinant() +
this->m_pData[0][2] * temp[2]->Determinant() -
this->m_pData[0][3] * temp[3]->Determinant();
return det;
}
break;
case 5:
{
CMatrix *temp[5];
for(int i = 0; i < 5; i++)
temp[i] = new CMatrix("", 4,4);
for(int k = 0; k < 5; k++)
{
for(int i = 1; i < 5; i++)
{
int j1 = 0;
for(int j = 0; j < 5; j++)
{
if(k == j)
continue;
temp[k]->m_pData[i-1][j1++] = this->m_pData[i][j];
}
}
}
double det = this->m_pData[0][0] * temp[0]->Determinant() -
this->m_pData[0][1] * temp[1]->Determinant() +
this->m_pData[0][2] * temp[2]->Determinant() -
this->m_pData[0][3] * temp[3]->Determinant() +
this->m_pData[0][4] * temp[4]->Determinant();
return det;
}
case 6:
case 7:
case 8:
case 9:
case 10:
case 11:
case 12:
default:
{
int DIM = m_rows;
CMatrix **temp = new CMatrix*[DIM];
for(int i = 0; i < DIM; i++)
temp[i] = new CMatrix("", DIM - 1,DIM - 1);
for(int k = 0; k < DIM; k++)
{
for(int i = 1; i < DIM; i++)
{
int j1 = 0;
for(int j = 0; j < DIM; j++)
{
if(k == j)
continue;
temp[k]->m_pData[i-1][j1++] = this->m_pData[i][j];
}
}
}
double det = 0;
for(int k = 0; k < DIM; k++)
{
if( (k %2) == 0)
det = det + (this->m_pData[0][k] * temp[k]->Determinant());
else
det = det - (this->m_pData[0][k] * temp[k]->Determinant());
}
for(int i = 0; i < DIM; i++)
delete temp[i];
delete [] temp;
return det;
}
break;
}
}
CMatrix& operator = (const CMatrix &other)
{
if( this->m_rows != other.m_rows ||
this->m_cols != other.m_cols)
{
std::cout << "WARNING: Assignment is taking place with by changing the number of rows and columns of the matrix";
}
for(int i = 0; i < m_rows; i++)
delete [] m_pData[i];
delete [] m_pData;
m_rows = m_cols = 0;
strcpy(m_name, other.m_name);
m_rows = other.m_rows;
m_cols = other.m_cols;
m_pData = new double*[m_rows];
for(int i = 0; i < m_rows; i++)
m_pData[i] = new double[m_cols];
for(int i = 0; i < m_rows; i++)
{
for(int j = 0; j < m_cols; j++)
{
m_pData[i][j] = other.m_pData[i][j];
}
}
return *this;
}
CMatrix CoFactor()
{
CMatrix cofactor("COF", m_rows, m_cols);
if(m_rows != m_cols)
return cofactor;
if(m_rows < 2)
return cofactor;
else if(m_rows == 2)
{
cofactor.m_pData[0][0] = m_pData[1][1];
cofactor.m_pData[0][1] = -m_pData[1][0];
cofactor.m_pData[1][0] = -m_pData[0][1];
cofactor.m_pData[1][1] = m_pData[0][0];
return cofactor;
}
else if(m_rows >= 3)
{
int DIM = m_rows;
CMatrix ***temp = new CMatrix**[DIM];
for(int i = 0; i < DIM; i++)
temp[i] = new CMatrix*[DIM];
for(int i = 0; i < DIM; i++)
for(int j = 0; j < DIM; j++)
temp[i][j] = new CMatrix("", DIM - 1,DIM - 1);
for(int k1 = 0; k1 < DIM; k1++)
{
for(int k2 = 0; k2 < DIM; k2++)
{
int i1 = 0;
for(int i = 0; i < DIM; i++)
{
int j1 = 0;
for(int j = 0; j < DIM; j++)
{
if(k1 == i || k2 == j)
continue;
temp[k1][k2]->m_pData[i1][j1++] = this->m_pData[i][j];
}
if(k1 != i)
i1++;
}
}
}
bool flagPositive = true;
for(int k1 = 0; k1 < DIM; k1++)
{
flagPositive = ( (k1 % 2) == 0);
for(int k2 = 0; k2 < DIM; k2++)
{
if(flagPositive == true)
{
cofactor.m_pData[k1][k2] = temp[k1][k2]->Determinant();
flagPositive = false;
}
else
{
cofactor.m_pData[k1][k2] = -temp[k1][k2]->Determinant();
flagPositive = true;
}
}
}
for(int i = 0; i < DIM; i++)
for(int j = 0; j < DIM; j++)
delete temp[i][j];
for(int i = 0; i < DIM; i++)
delete [] temp[i];
delete [] temp;
}
return cofactor;
}
CMatrix Adjoint()
{
CMatrix cofactor("COF", m_rows, m_cols);
CMatrix adj("ADJ", m_rows, m_cols);
if(m_rows != m_cols)
return adj;
cofactor = this->CoFactor();
// adjoint is transpose of a cofactor of a matrix
for(int i = 0; i < m_rows; i++)
{
for(int j = 0; j < m_cols; j++)
{
adj.m_pData[j][i] = cofactor.m_pData[i][j];
}
}
return adj;
}
CMatrix Transpose()
{
CMatrix trans("TR", m_cols, m_rows);
for(int i = 0; i < m_rows; i++)
{
for(int j = 0; j < m_cols; j++)
{
trans.m_pData[j][i] = m_pData[i][j];
}
}
return trans;
}
CMatrix Inverse()
{
CMatrix cofactor("COF", m_rows, m_cols);
CMatrix inv("INV", m_rows, m_cols);
if(m_rows != m_cols)
return inv;
// to find out Determinant
double det = Determinant();
cofactor = this->CoFactor();
// inv = transpose of cofactor / Determinant
for(int i = 0; i < m_rows; i++)
{
for(int j = 0; j < m_cols; j++)
{
inv.m_pData[j][i] = cofactor.m_pData[i][j] / det;
}
}
return inv;
}
CMatrix operator + (const CMatrix &other)
{
if( this->m_rows != other.m_rows ||
this->m_cols != other.m_cols)
{
std::cout << "Addition could not take place because number of rows and columns are different between the two matrices";
return *this;
}
CMatrix result("", m_rows, m_cols);
for(int i = 0; i < m_rows; i++)
{
for(int j = 0; j < m_cols; j++)
{
result.m_pData[i][j] = this->m_pData[i][j] + other.m_pData[i][j];
}
}
return result;
}
CMatrix operator - (const CMatrix &other)
{
if( this->m_rows != other.m_rows ||
this->m_cols != other.m_cols)
{
std::cout << "Subtraction could not take place because number of rows and columns are different between the two matrices";
return *this;
}
CMatrix result("", m_rows, m_cols);
for(int i = 0; i < m_rows; i++)
{
for(int j = 0; j < m_cols; j++)
{
result.m_pData[i][j] = this->m_pData[i][j] - other.m_pData[i][j];
}
}
return result;
}
CMatrix operator * (const CMatrix &other)
{
if( this->m_cols != other.m_rows)
{
std::cout << "Multiplication could not take place because number of columns of 1st Matrix and number of rows in 2nd Matrix are different";
return *this;
}
CMatrix result("", this->m_rows, other.m_cols);
for(int i = 0; i < this->m_rows; i++)
{
for(int j = 0; j < other.m_cols; j++)
{
for(int k = 0; k < this->m_cols; k++)
{
result.m_pData[i][j] += this->m_pData[i][k] * other.m_pData[k][j];
}
}
}
return result;
}
bool operator == (const CMatrix &other)
{
if( this->m_rows != other.m_rows ||
this->m_cols != other.m_cols)
{
std::cout << "Comparision could not take place because number of rows and columns are different between the two matrices";
return false;
}
CMatrix result("", m_rows, m_cols);
bool bEqual = true;
for(int i = 0; i < m_rows; i++)
{
for(int j = 0; j < m_cols; j++)
{
if(this->m_pData[i][j] != other.m_pData[i][j])
bEqual = false;
}
}
return bEqual;
}
friend std::istream& operator >> (std::istream &is, CMatrix &m);
friend std::ostream& operator << (std::ostream &os, const CMatrix &m);
};
std::istream& operator >> (std::istream &is, CMatrix &m)
{
std::cout << "\n\nEnter Input For Matrix : " << m.m_name << " Rows: " << m.m_rows << " Cols: " << m.m_cols << "\n";
for(int i = 0; i < m.m_rows; i++)
{
for(int j = 0; j < m.m_cols; j++)
{
std::cout << "Input For Row: " << i + 1 << " Col: " << j + 1 << " = ";
is >> m.m_pData[i][j];
}
std::cout << "\n";
}
std::cout << "\n";
return is;
}
std::ostream& operator << (std::ostream &os,const CMatrix &m)
{
os << "\n\nMatrix : " << m.m_name << " Rows: " << m.m_rows << " Cols: " << m.m_cols << "\n\n";
for(int i = 0; i < m.m_rows; i++)
{
os << " | ";
for(int j = 0; j < m.m_cols; j++)
{
char buf[32];
double data = m.m_pData[i][j];
if( m.m_pData[i][j] > -0.00001 &&
m.m_pData[i][j] < 0.00001)
data = 0;
sprintf(buf, "%10.2lf ", data);
os << buf;
}
os << "|\n";
}
os << "\n\n";
return os;
}
#endif
int main()
{
CMatrix a("A", 6,6);
CMatrix b("B", 6,1);
a.FillSimulatedInput();
b.FillSimulatedInput();
std::cout << a << "\n Determinant : ";
std::cout << a.Determinant() << "\n";
std::cout << b << "\n Determinant : ";
std::cout << b.Determinant() << "\n";
CMatrix ainv = a.Inverse();
CMatrix q = a * ainv;
q.SetName("A * A'");
std::cout << q << "\n";
CMatrix x = ainv * b;
x.SetName("X");
std::cout << x << "\n";
CMatrix y = a * x; // we will get B
y.SetName("Y");
std::cout << y << "\n";
return 0;
}
Output
Enter Input For Matrix : A Rows: 3 Cols: 3
Input For Row: 1 Col: 1 = 1
Input For Row: 1 Col: 2 = -1
Input For Row: 1 Col: 3 = 2
Input For Row: 2 Col: 1 = 2
Input For Row: 2 Col: 2 = -3
Input For Row: 2 Col: 3 = -1
Input For Row: 3 Col: 1 = 3
Input For Row: 3 Col: 2 = 0.5
Input For Row: 3 Col: 3 = 1
Enter Input For Matrix : B Rows: 3 Cols: 1
Input For Row: 1 Col: 1 = 2
Input For Row: 2 Col: 1 = 5
Input For Row: 3 Col: 1 = 3
Matrix : A Rows: 3 Cols: 3
| 1.00 -1.00 2.00 |
| 2.00 -3.00 -1.00 |
| 3.00 0.50 1.00 |
Determinant : 22.5
Matrix : B Rows: 3 Cols: 1
| 2.00 |
| 5.00 |
| 3.00 |
Determinant : 7.29768e-164
Matrix : A * A' Rows: 3 Cols: 3
| 1.00 0.00 0.00 |
| 0.00 1.00 0.00 |
| 0.00 0.00 1.00 |
Matrix : X Rows: 3 Cols: 1
| 1.16 |
| -0.89 |
| -0.02 |
Matrix : Y Rows: 3 Cols: 1
| 2.00 |
| 5.00 |
| 3.00 |
Press any key to continue . . .
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