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Matrix Manipulation
239 lines (209 loc) · 6.19 KB
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Matrix Manipulation
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// Fundamental Matrix Operations
i/ LU decomposition
ii/ Matrix inversion
iii/Matrix multiplication
iv/MATRIX INVERSION - LU DECOMPOSITION
v/MATRIX ADDITION
vi/MATRIX SUBTRACTION
vii/MATRIX TRANSPOSE
viii/MATRIX MINOR
iX/DETERMINANT OF A MATRIX
X/MATRIX INVERSION USING MINORS
Xi/TRACE MATRIX
#include "stdafx.h"
#include <iostream>
#include <iomanip>
#include <vector>
#include <math.h>
using namespace std;
//***** LU DECOMPOSITION******
// Structure to store upper and lower matrices
typedef struct{
vector<vector<double> > L;
vector<vector<double> > U;
}LUstruct;
// LU decomposition;
LUstruct LU(vector<vector<double> > A) {
int N = A.size();
vector<vector<double> > B(N,vector<double> (N));
for(int i=0;i<=N-1;i++)
for(int j=0;j<=N-1;j++)
B[i][j] = A[i][j];
for(int k=0;k<=N-2;k++) {
for(int i=k+1;i<=N-1;i++)
B[i][k] = B[i][k] / B[k][k];
for(int j=k+1;j<=N-1;j++)
for(int i=k+1;i<=N-1;i++)
B[i][j] = B[i][j] - B[i][k]*B[k][j];
}
vector<vector<double> > L(N,vector<double> (N));
vector<vector<double> > U(N,vector<double> (N));
for(int i=0;i<=N-1;i++) {
L[i][i] = 1.0;
for(int j=0;j<=N-1;j++) {
if(i>j)
L[i][j] = B[i][j];
else
U[i][j] = B[i][j];
}
}
LUstruct Mats;
Mats.L = L;
Mats.U = U;
return Mats;
}
//*****MATRIX INVERSION******
// Inverse of an upper triangular matrix
vector<vector<double> > MatUpTriangleInv(vector<vector<double> > U) {
int N = U.size();
vector<vector<double> > V(N,vector<double> (N));
for(int j=N-1;j>=0;j--) {
V[j][j] = 1.0 / U[j][j];
for(int i=j-1;i>=0;i--)
for(int k=i+1;k<=j;k++)
V[i][j] -= 1.0 / U[i][i] * U[i][k] * V[k][j];
}
return V;
}
// Inverse of a lower triangular matrix
vector<vector<double> > MatLowTriangleInv(vector<vector<double> > L) {
int N = L.size();
vector<vector<double> > V(N,vector<double> (N));
for(int i=0;i<=N-1;i++) {
V[i][i] = 1.0 / L[i][i];
for(int j=i-1;j>=0;j--)
for(int k=i-1;k>=j;k--)
V[i][j] -= 1.0 / L[i][i] * L[i][k] * V[k][j];
}
return V;
}
//***** MATRIX MULTIPLICATION ******
// Multiply two matrices together
// First matrix is (n x k), Second is (k x m)
// Resulting matrix is (n x m)
vector<vector<double> > MMult(vector<vector<double> >A,vector<vector<double> >B, int n, int k, int m) {
vector<vector<double> > C(n, vector<double> (m));
for (int j=0; j<=m-1; j++)
for (int i=0; i<=n-1; i++) {
C[i][j] = 0;
for (int r=0; r<=k-1; r++)
C[i][j] += A[i][r] * B[r][j];
}
return C;
}
//***MATRIX INVERSION - LU DECOMPOSITION **** //
// Inverse of a matrix through LU decomposition
vector<vector<double> > MInvLU(vector<vector<double> > A) {
LUstruct Mats;
Mats = LU(A);
vector<vector<double> > L = Mats.L;
vector<vector<double> > U = Mats.U;
vector<vector<double> > Linv = MatLowTriangleInv(L);
vector<vector<double> > Uinv = MatUpTriangleInv(U);
int n = Linv.size();
int k = Linv[0].size();
int m = Uinv[0].size();
vector<vector<double> > Ainv = MMult(Uinv,Linv,n,k,m);
return Ainv;
}
//***MATRIX ADDITION **** //
// Add two matrices together
vector<vector<double> > MAdd(vector<vector<double> >A, vector<vector<double> >B, int n, int m) {
vector<vector<double> > C(n, vector<double> (m));
for (int j=0; j<=m-1; j++)
for (int i=0; i<=n-1; i++)
C[i][j] = A[i][j] + B[i][j];
return C;
}
//***MATRIX SUBTRACTION **** //
// Subtract two matrices
vector<vector<double> > MSub(vector<vector<double> >A,vector<vector<double> >B, int n, int m) {
vector<vector<double> > C(n, vector<double> (m));
for (int j=0; j<=m-1; j++)
for (int i=0; i<=n-1; i++)
C[i][j] = A[i][j] - B[i][j];
return C;
}
//***MATRIX TRANSPOSE **** //
// Transpose of a matrix
vector<vector<double> > MTrans(vector<vector<double> >A, int n, int m) {
vector<vector<double> > C(m, vector<double> (n));
for (int j=0; j<=m-1; j++)
for (int i=0; i<=n-1; i++)
C[j][i] = A[i][j];
return C;
}
//***MATRIX MINOR **** //
// Minor of matrix A, eliminating row RowExclude and column ColExclude
// RowExclude and ColExclude run from 0 to (n-1)
vector<vector<double> > MMinor(vector<vector<double> > A, int RowExclude, int ColExclude) {
int n = A.size();
vector<vector<double> > M((n-1), vector<double> (n-1));
int ColCount=0;
for (int j=0; j<=n-1; j++) {
if (j!=ColExclude) {
int RowCount = 0;
for (int i=0; i<=n-1; i++)
if (i!=RowExclude) {
M[RowCount][ColCount] = A[i][j];
RowCount++;
}
ColCount++;
}
}
return M;
}
//***DETERMINANT OF A MATRIX**** //
// Determinant of a square matrix
// Expand along the first row
double MDet(vector<vector<double> > A) {
int n = A.size();
double det = 0;
if (n==1) det = A[0][0];
else {
if (n==2) det = A[0][0]*A[1][1] - A[1][0]*A[0][1];
else
for (int j=0; j<=n-1; j++)
det += A[0][j]*pow(-1.0,j)*MDet(MMinor(A,0,j));
}
return det;
}
//***MATRIX INVERSION USING MINORS **** //
// Inverse of a Matrix using minors
vector<vector<double> > MInv(vector<vector<double> >A) {
int n = A.size();
vector<vector<double> > C(n, vector<double> (n));
double det = MDet(A);
for (int j=0; j<=n-1; j++)
for (int i=0; i<=n-1; i++)
C[i][j] = pow(-1.0,(i+j))*MDet(MMinor(A,i,j)) / det;
return MTrans(C, n, n);
}
//***TRACE MATRIX**** //
// Trace of a Matrix
double MTrace(vector<vector<double> > A) {
double sum = 0;
double n = A.size();
for (int i=0; i<=n-1; i++)
sum += A[i][i];
return sum;
}
int main() {
int cols = 5;
int rows = 5;
// Initialize the matrices and read in the data
vector<vector<double> > X(rows, vector<double> (cols));
vector<vector<double> > Xinv1(rows, vector<double> (cols));
vector<vector<double> > Xinv2(rows, vector<double> (cols));
X[0][0] = 1.2; X[0][1] = 0.1; X[0][2] = 0.4; X[0][3] = 0.5; X[0][4] = 0.3;
X[1][0] = 9.1; X[1][1] = 3.1; X[1][2] = 1.7; X[1][3] = 0.1; X[1][4] = 1.3;
X[2][0] = 1.0; X[2][1] = 1.2; X[2][2] = 2.6; X[2][3] = 0.9; X[2][4] = 0.1;
X[3][0] = 3.6; X[3][1] = 8.0; X[3][2] = 0.4; X[3][3] = 0.6; X[3][4] = 4.6;
X[4][0] = 3.1; X[4][1] = 4.1; X[4][2] = 0.4; X[4][3] = 0.9; X[4][4] = 0.1;
// X inverse using minor expansion
Xinv1 = MInv(X);
// X inverse using LU decomposition
Xinv2 = MInvLU(X);
cout << "The trace of X-inverse using minors is " << MTrace(Xinv1) << endl;
cout << "The trace of X-inverse using LU-decomp is " << MTrace(Xinv2) << endl;
}