Tiny numbers in place of zero?

Question

I have been making a matrix class (as a learning exercise) and I have come across and issue whilst testing my inverse function.

I input a arbitrary matrix as such:

2 1 1
1 2 1
1 1 2

And got it to calculate the inverse and I got the correct result:

0.75 -0.25 -0.25
-0.25 0.75 -0.25
-0.25 -0.25 0.75

But when I tried multiplying the two together to make sure I got the identity matrix I get:

1 5.5111512e-017 0
0 1 0
-1.11022302e-0.16 0 1

Why am I getting these results? I would understand if I was multiplying weird numbers where I could understand some rounding errors but the sum it's doing is:

2 * -0.25 + 1 * 0.75 + 1 * -0.25

which is clearly 0, not 5.111512e-017

If I manually get it to do the calculation; eg:

std::cout << (2 * -0.25 + 1 * 0.75 + 1 * -0.25) << "\n";

I get 0 as expected?

All the numbers are represented as doubles. Here's my multiplcation overload:

Matrix operator*(const Matrix& A, const Matrix& B)
{
    if(A.get_cols() == B.get_rows())
    {
        Matrix temp(A.get_rows(), B.get_cols());
        for(unsigned m = 0; m < temp.get_rows(); ++m)
        {
            for(unsigned n = 0; n < temp.get_cols(); ++n)
            {
                for(unsigned i = 0; i < temp.get_cols(); ++i)
                {
                    temp(m, n) += A(m, i) * B(i, n);
                }
            }
        }

        return temp;
    }

    throw std::runtime_error("Bad Matrix Multiplication");
}

and the access functions:

double& Matrix::operator()(unsigned r, unsigned c)
{
    return data[cols * r + c];
}

double Matrix::operator()(unsigned r, unsigned c) const
{
    return data[cols * r + c];
}

Here's the function to find the inverse:

Matrix Inverse(Matrix& M)
{
    if(M.rows != M.cols)
    {
        throw std::runtime_error("Matrix is not square");
    }

    int r = 0;
    int c = 0;
    Matrix augment(M.rows, M.cols*2);
    augment.copy(M);

    for(r = 0; r < M.rows; ++r)
    {
        for(c = M.cols; c < M.cols * 2; ++c)
        {
            augment(r, c) = (r == (c - M.cols) ? 1.0 : 0.0);
        }
    }

    for(int R = 0; R < augment.rows; ++R)
    {
        double n = augment(R, R);
        for(c = 0; c < augment.cols; ++c)
        {
            augment(R, c) /= n;
        }

        for(r = 0; r < augment.rows; ++r)
        {
            if(r == R) { continue; }
            double a = augment(r, R);

            for(c = 0; c < augment.cols; ++c)
            {
                augment(r, c) -= a * augment(R, c);
            }
        }
    }

    Matrix inverse(M.rows, M.cols);
    for(r = 0; r < M.rows; ++r)
    {
        for(c = M.cols; c < M.cols * 2; ++c)
        {
            inverse(r, c - M.cols) = augment(r, c);
        }
    }

    return inverse;
}

Answer 1

Please read this paper: What Every Computer Scientist Should Know About Floating-Point Arithmetic