/// <summary>
/// Computes the reduced row echelon form of the matrix.
/// </summary>
/// <param name="matrix"></param>
/// <returns></returns>
public static AugmentedMatrix ReducedRowEchelonForm(this AugmentedMatrix augMatrix)
{
augMatrix = augMatrix.GaussianElimination();
var gaussElimResult = augMatrix[0];
int iMax = Math.Min(gaussElimResult.Columns, gaussElimResult.Rows);
for (int i = 0, currentCol = 0; i < iMax && currentCol < gaussElimResult.Columns; i++, currentCol++)
{
var currentStartElement = gaussElimResult[i, currentCol];
if (Math.Abs(currentStartElement) > TOLERANCE)
{
// If the current start element does not equal 1 then fix it
if (Math.Abs(currentStartElement - 1) > TOLERANCE)
{
augMatrix = augMatrix.MultiplyRowByScalar(i, 1.0 / currentStartElement);
}
for (int j = 0; j <= i - 1; j++)
{
var ratio = -1.0 * gaussElimResult[j, currentCol] * gaussElimResult[i, currentCol];
augMatrix = augMatrix.AddRows(i, j, ratio);
}
}
else
{
i--; // The test element was zero, do not increment the row
}
}
return(augMatrix);
}
/// <summary>
/// Performs gaussian elimination on the given matrix.
/// </summary>
/// <param name="matrix"></param>
/// <returns></returns>
public static AugmentedMatrix GaussianElimination(this AugmentedMatrix augMatrix)
{
// Derived from ref [1] Theorem 3.5
// identify the row with the first non zero element
int nonZeroRowIndex = 0;
bool run = true;
int nonZeroColumnIndex = 0;
while (run && nonZeroColumnIndex < augMatrix[0].Columns)
{
// Check if columns has non-zero entry
var col = augMatrix[0].GetColumn(nonZeroColumnIndex);
for (int j = 0; j < col.Length; j++)
{
if (Math.Abs(col[j]) > TOLERANCE)
{
nonZeroRowIndex = j;
run = false;
break;
}
}
if (run)
{
nonZeroColumnIndex++;
}
}
// Move this row into the top position
if (nonZeroRowIndex != 0)
{
augMatrix = augMatrix.SwapRows(nonZeroRowIndex, 0);
}
// If the first non zero row does not equal 1 then normalize the
if (Math.Abs(augMatrix[0][0, nonZeroColumnIndex] - 1.0) > TOLERANCE)
{
augMatrix = augMatrix.MultiplyRowByScalar(0, 1.0 / augMatrix[0][0, nonZeroColumnIndex]);
}
int iMax = Math.Min(augMatrix[0].Columns, augMatrix[0].Rows);
// Zero-wise columns
for (int i = 0, currentCol = 0; i < iMax && currentCol < augMatrix[0].Columns; i++, currentCol++)
{
var currentStartElement = augMatrix[0][i, currentCol];
if (Math.Abs(currentStartElement) > TOLERANCE)
{
for (int j = i + 1; j < iMax; j++)
{
var ratio = -1.0 * (augMatrix[0][j, currentCol] / currentStartElement);
augMatrix = augMatrix.AddRows(i, j, ratio);
}
}
else
{
i--; // The test element was zero, do not increment the row
}
}
return(augMatrix);
}
