1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120 121 122 123 124 125 126 127 128 129 130 131 132 133 134 135 136 137 138 139 140 141 142 143 144 145 146 147 148 149 150 151 152 153 154 155 156 157 158 159 160 161 162 163 164 165 166 167 168 169 170 171 172 173 174 175 176 177 178 179 180 181 182 183 184 185 186 187 188 189 190 191 192 193 194 195 196 197 198 199 200 201 202 203 204 205 206 207 208 209 210 211 212 213 214 215 216 217 218 219 220 221 222 223 224 225 226 227 228 229 230 231 232
|
// 5.11_Exercises_5.9.cpp : This file contains the 'main' function. Program execution begins and ends there.
/*5.9 Write a module for solving the 3 × 3 linear system Au = b where A is nonsingular*/
/* 1. Make matrix
2. Make vector
3. Evaluate determinant
4. Form cofactor matrix
5. Transpose cofactor matrix to obtain adjoint
6. divide throught the adjoint by the determinant
7. Multiply A^-1 matrix x b column vector = new column vector for x values*/
#include <iostream>
#include <cmath>
using namespace std;
//Function prototypes for formation and deletion of matrix and vector
double** MakeMatrix(int);
double* MakeVector(int);
void EnterMatrixElements(double**, int);
void DeleteMatrix(double**, int);
void DeleteVector(double*);
void MakeDummyVector(double*);
//Function prototypes for display functions for matrix and vector
void DisplayMatrix(double**, int, int);
void DisplayVector(double*, int);
//Function prototypes to obtain inverse of square matrix functions
double ThreebyThreedeterminant(double**); // Evaluate determinant of matrix
double TwobyTwodeterminant(double, double, double, double);
double** InverseA(double**);
//Using x = A^-1 . b (b = column vector/matrix formed from linear equation numbers)
double* Mutiply(double**, double*, double*, int, int, int);
double** MakeMatrix(int size)
{
double** matrix;
matrix = new double* [size];
for (int i = 0; i < size; i++)
{
matrix[i] = new double[size];
}
return matrix;
}
void MakeDummyVector(double* vector)
{
for (int i = 0; i < 3; i++)
{
vector[i] = 0;
}
}
void EnterMatrixElements(double** matrix, int size)
{
cout << "What data would you like to enter for the matrix?" << endl;
cout << "The matrix has a size of " << size * size << " elements" << endl;
double temp;
for (int i = 0; i < size; i++)
{
for (int j = 0; j < size; j++)
{
cout << "What is number you would like to enter?" << endl;
cin >> temp;
matrix[i][j] = temp;
}
}
DisplayMatrix(matrix, size, size);
}
double* MakeVector(int size)
{
//Dynamically allocate memory for vector array
double* vector = new double[size];
return vector;
}
void EnterVectorElements(double* vector)
{
cout << "Enter the values for each element in your column vector" << endl;
//Enter values for vector array elements
for (int i = 0; i < 3; i++)
{
cin >> vector[i];
}
}
void DeleteMatrix(double** matrix, int size)
{
for (int i = 0; i < size; i++)
{
delete[] matrix[i];
}
delete[] matrix;
}
void DeleteVector(double* vector)
{
delete[] vector;
}
void DisplayMatrix(double** matrix, int rows, int columns)
{
cout << "{";
for (int i = 0; i < rows; i++)
{
for (int j = 0; j < columns; j++)
{
cout << matrix[i][j] << ", ";
}
cout << endl;
}
cout << "}";
}
void DisplayVector(double* vector, int size)
{
cout << "The elements of the vector are {";
for (int i = 0; i < size; i++)
{
cout << vector[i] << endl;
}
cout << "}";
}
double TwobyTwodeterminant(double a, double b, double c, double d)
{
return ((a * d) - (b * c));
}
double ThreebyThreedeterminant(double** A)
{
double determinant = A[0][0] * ((A[1][1] * A[2][2]) - (A[1][2] * A[2][1]))
- A[0][1] * ((A[1][0] * A[2][2]) - (A[1][2] * A[2][0]))
+ A[0][2] * ((A[1][0] * A[2][1]) - (A[1][1] * A[2][0]));
return determinant;
}
double** InverseA(double** A)
{
double** cofactor = MakeMatrix(3);
double** cofactor_transposed = MakeMatrix(3);
double** A_minus_1 = MakeMatrix(3);
double** tempmatrix = MakeMatrix(3);
//Form co factor matrix by recursion using the 2 x 2 determinant function using the A matrix
cofactor[0][0] = TwobyTwodeterminant(A[1][1], A[1][2], A[2][1], A[2][2]);
cofactor[0][1] = -1 * TwobyTwodeterminant(A[1][0], A[1][2], A[2][0], A[2][2]);
cofactor[0][2] = TwobyTwodeterminant(A[1][0], A[1][1], A[2][0], A[2][1]);
cofactor[1][0] = -1 * TwobyTwodeterminant(A[0][1], A[0][2], A[2][1], A[2][2]);
cofactor[1][1] = TwobyTwodeterminant(A[0][0], A[0][2], A[2][0], A[2][2]);
cofactor[1][2] = -1 * TwobyTwodeterminant(A[0][0], A[0][1], A[2][0], A[2][1]);
cofactor[2][0] = TwobyTwodeterminant(A[0][1], A[0][2], A[1][1], A[1][2]);
cofactor[2][1] = -1 * TwobyTwodeterminant(A[0][0], A[0][2], A[1][0], A[1][2]);
cofactor[2][2] = TwobyTwodeterminant(A[0][0], A[0][1], A[1][0], A[1][1]);
cout << " The co factor matrix is {" << endl;
DisplayMatrix(cofactor, 3, 3);
//Transpose the cofactor matrix
for (int i = 0; i < 3; i++)
{
for (int j = 0; j < 3; j++)
{
//Swap columns of cofactors with rows in the traspose
cofactor_transposed[i][j] = cofactor[j][i];
}
}
cout << " The co factor matrix transposed is " << endl;
DisplayMatrix(cofactor_transposed, 3, 3);
//Divide each adjoint element by the determinant of A
for (int i = 0; i < 3; i++)
{
for (int j = 0; j < 3; j++)
{
tempmatrix[i][j] = cofactor_transposed[i][j] / ThreebyThreedeterminant(A);
cout << tempmatrix[i][j] << " = " << cofactor_transposed[i][j] << " / " << ThreebyThreedeterminant(A) << endl;;
}
}
//Assign values to inverse matrix
for (int i = 0; i < 3; i++)
{
for (int j = 0; j < 3; j++)
{
A_minus_1[i][j] = tempmatrix[i][j];
}
}
DisplayMatrix(A_minus_1, 3, 3);
return A_minus_1;
}
double* Mutiply(double** matrix, double* vector1, double* vector2, int rows, int columns, int vector_size)
{
rows, columns = 3;
cout << "Matrix x Vector " << endl;
for (int i = 0; i < rows; i++)
{
for (int j = 0; j < columns; j++)
{
vector2[i] += matrix[i][j] * vector1[j];
}
}
DisplayVector(vector2, 3);
return vector2;
}
int main()
{
cout << "This script will solve a system of linear equations in matrix form " << endl;
double** A = MakeMatrix(3);
EnterMatrixElements(A, 3);
double* b = MakeVector(3);
EnterVectorElements(b);
double* c = MakeVector(3);
MakeDummyVector(c);
cout << ThreebyThreedeterminant(A);
double** A_Minus_1 = MakeMatrix(3);
A_Minus_1 = InverseA(A);
Mutiply(A_Minus_1, b, c, 3, 3, 3);
DeleteMatrix(A, 3);
DeleteVector(b);
DeleteVector(c);
return 0;
}
/* 1. Make matrix
2. Make vector
3. Evaluate determinant
4. Form cofactor matrix
5. Transpose cofactor matrix to obtain adjoint
6. divide throught the adjoint by the determinant
7. Multiply A^-1 matrix x b column vector = new column vector for x values*/
|