/// <summary>
/// Does the work of matrix inversion. Assumes that this is an r by 2r matrix.
/// </summary>
private void GaussianElimination()
{
// Clear out the part below the main diagonal and scale the main diagonal to be 1.
for (int r = 0; r < Rows; r++)
{
// If the element on the diagonal is 0, find a row below
// that has a non-zero and swap them.
if (_data[r, r] == (byte)0)
{
for (int rowBelow = r + 1; rowBelow < Rows; rowBelow++)
{
if (_data[rowBelow, r] != 0)
{
SwapRows(r, rowBelow);
break;
}
}
}
// If we couldn't find one, the matrix is singular.
Throw.If(_data[r, r] == 0, "Matrix is singular");
// Scale to 1.
if (_data[r, r] != 1)
{
byte scale = GaloisField.Divide(1, _data[r, r]);
for (int c = 0; c < Columns; c++)
{
_data[r, c] = GaloisField.Multiply(_data[r, c], scale);
}
}
// Make everything below the 1 be a 0 by subtracting a multiple of it.
// Note: subtraction and addition are both exclusive or in the Galois field.
for (int rowBelow = r + 1; rowBelow < Rows; rowBelow++)
{
if (_data[rowBelow, r] != 0)
{
byte scale = _data[rowBelow, r];
for (int c = 0; c < Columns; c++)
{
_data[rowBelow, c] ^= GaloisField.Multiply(scale, _data[r, c]);
}
}
}
}
// Now clear the part above the main diagonal.
for (int d = 0; d < Rows; d++)
{
for (int rowAbove = 0; rowAbove < d; rowAbove++)
{
if (_data[rowAbove, d] != (byte)0)
{
byte scale = _data[rowAbove, d];
for (int c = 0; c < Columns; c++)
{
_data[rowAbove, c] ^= GaloisField.Multiply(scale, _data[d, c]);
}
}
}
}
}