Determinant of a matrix by Gaussian elimination C++ - c++

Question

I was trying to find the code for finding the determinant of a square matrix , and I came across this code.
int det(vector<vector<int> > mat) {
int n = mat.size();
for(int col = 0; col < n; ++col) {
bool found = false;
for(int row = col; row < n; ++row) {
if(mat[row][col]) {
mat[row].swap(mat[col]);
found = true;
break;
}
}
if(!found) {
return 0;
}
for(int row = col + 1; row < n; ++row) {
while(true) {
int del = mat[row][col] / mat[col][col];
for (int j = col; j < n; ++j) {
mat[row][j] -= del * mat[col][j];
}
if (mat[row][col] == 0)
break;
else
mat[row].swap(mat[col]);
}
}
}
li res = 1;
for(int i = 0; i < n; ++i) {
res *= mat[i][i];
}
return abs(res);
}
But I am having trouble understanding line 20-29 i.e where the subtraction of row from multiple of another row is performed. I mean why the while loop is required here ? as I am
subtracting the quotient*dividend , It should always be 0 , right ? So I think it should be just one iteration. So , why we need to perform this mat[row].swap(mat[col]); operation ?
Thanks in advance.

Answer 1

There is some strange logic in your code to account for the fact that you are performing your calculations using integer arithmetic.
Say you have a 3x3 matrix in which the first two rows are:
4 6 5
1 2 3
When you compute del for col=0 and row=1, you will get:
del = 1/4 = 0
With that, when you compute:
mat[row][j] -= del * mat[col][j];
mat[row][j] doesn't change at all.
To account for that, you swap the rows. Now the first two rows are:
1 2 3
4 6 5
With the rows swapped like that, the value of del is 4/1 = 4. Now the line:
mat[row][j] -= del * mat[col][j];
does make a difference. The value of mat[1][0] ends up being zero, which is what you need. So you break out of the while loop.
Here's an instrumented version of your function that produces a lot of debug output, with a helper function to print the matrix and the main function to test the code.
#include <iostream>
#include <vector>
#include <stdlib.h>
using namespace std;
void printMatrix(vector<vector<int> > const& mat)
{
int n = mat.size();
for(int row = 0; row < n; ++row) {
for(int col = 0; col < n; ++col) {
cout << mat[row][col] << " ";
}
cout << "\n";
}
cout << "\n";
}
int det(vector<vector<int> > mat) {
int n = mat.size();
for(int col = 0; col < n; ++col) {
cout << "Column: " << col << "\n";
printMatrix(mat);
bool found = false;
for(int row = col; row < n; ++row) {
if(mat[row][col]) {
cout << "Got non-zero value for row " << row << " and col " << col << "\n";
if ( row != col )
{
cout << "(1) Swapping rows " << col << " and " << row << "\n";
mat[row].swap(mat[col]);
printMatrix(mat);
}
else
{
cout << "Not swapping rows\n";
}
found = true;
break;
}
}
if(!found) {
cout << "Did not find a non-zero row. Column: " << col << "\n";
return 0;
}
for(int row = col + 1; row < n; ++row) {
while(true) {
int del = mat[row][col] / mat[col][col];
cout << "del: " << del << "\n";
for (int j = col; j < n; ++j) {
mat[row][j] -= del * mat[col][j];
}
if (mat[row][col] == 0)
{
break;
}
else
{
cout << "(2) Swapping rows " << col << " and " << row << "\n";
mat[row].swap(mat[col]);
printMatrix(mat);
}
}
}
}
printMatrix(mat);
long res = 1;
for(int i = 0; i < n; ++i) {
res *= mat[i][i];
}
return abs(res);
}
int main()
{
vector<vector<int> > mat = { {4, 6, 5}, {1, 2, 3}, {8, 10, 9} };
int r = det(mat);
cout << "Determinant: " << r << endl;
return 0;
}