/// <summary>
/// Invert the matrix by using Gauss-Jordan using GF16 operations
/// </summary>
/// <remarks>
/// The inversion is done by adding the identity matrix to the right
/// and using Gauss-Jordan algorithm (elementary
/// operations on rows) until we get the identity matrix
/// on the left. Then we strip the identity on the left
/// in order to get the reverted matrix.
/// </remarks>
/// <param name="mtxIn_GF16">The matrix to invert</param>
/// <returns>The inverted matrix</returns>
public static GF16[,] Invert(GF16[,] mtxIn_GF16)
{
// Add the ID to the right to be able to apply Gauss-Jordan to revert the matrix
GF16[,] mtxWithID_GF16 = FEC_Matrix.AddIdentityRight(mtxIn_GF16);
// Apply Gauss Jordan algo to revert the matrix
GF16[,] mtxInvert_GF16 = FEC_Matrix.GaussJordan(mtxWithID_GF16);
// Remove the ID on the left
GF16[,] mtxInvertStripped_GF16 = FEC_Matrix.StripIdentity(mtxInvert_GF16);
// Return the inverted matrix
return mtxInvertStripped_GF16;
}
/// <summary>
/// Remove the identity at the front of the matrix
/// </summary>
/// <param name="matrixIn">The input matrix</param>
/// <returns>Stripped Matrix without the identity at the front</returns>
private static GF16[,] StripIdentity(GF16[,] matrixIn)
{
int rowLength = matrixIn.GetLength(1);
int columnLength = matrixIn.GetLength(0);
GF16[,] matrixOut = new GF16[columnLength - rowLength, rowLength];
for (int row = 0; row < rowLength; row++)
for (int column = rowLength; column < columnLength; column++)
matrixOut[column - rowLength, row] = matrixIn[column, row];
return matrixOut;
}
/// <summary>
/// Copy a matrix to a destination matrix
/// </summary>
/// <param name="matrixIn">The matrix to copy</param>
/// <param name="matrixDestination">The destination matrix</param>
private static void MatrixCopyTo(GF16[,] matrixIn, GF16[,] matrixDestination)
{
int rowLength = matrixIn.GetLength(1);
int columnLength = matrixIn.GetLength(0);
for (int row = 0; row < rowLength; row++)
for (int column = 0; column < columnLength; column++)
matrixDestination[column, row] = matrixIn[column, row];
}
/// <summary>
/// Gauss - Jordan elimination.
/// This algorithm takes a matrix and use elementary
/// row operations to put the matrix in a row echelon form. This
/// means that the result matrix will have the identity matrix at the
/// begining.
/// This method can be used in 2 specific scenarios:
/// - To solve a linear system numerically
/// - To invert a matrix
/// </summary>
/// <example>
/// For instance the matrix:
/// 0 1 1 1 0 0
/// 0 1 2 0 1 0
/// 1 1 4 0 0 1
///
/// Will become (over int):
/// [ ID ] [ rest ]
/// 1 0 0 2 -3 1
/// 0 1 0 2 -1 0
/// 0 0 1 -1 1 0
/// </example>
/// <remarks>
/// For the Reed Solomon encoding, Gauss - Jordan elimination
/// is used to invert the decoding matrix. This is needed to
/// recover from packet loss.
/// Important note: The input is modified
/// </remarks>
/// <param name="matrixIn">The matrix to transform in a row echelon form</param>
/// <returns>The matrix in a row echelon from</returns>
private static GF16[,] GaussJordan(GF16[,] matrixIn)
{
// Matrix size variables
int rowLength = matrixIn.GetLength(1);
int columnLength = matrixIn.GetLength(0);
// The number of pivots needed is the min between row/column
// For instance the matrix:
// 0 1 1 1 0 0
// 0 1 2 0 1 0
// 1 1 4 0 0 1
// has 3 pivots
int pivots = Math.Min(rowLength, columnLength);
// Loop through all the pivot columns
for (int pivotColumn = 0; pivotColumn < pivots; pivotColumn++)
{
// TODO: Place Step 1 in a separate method
// Step 1: Find the Pivot Row and reduce the pivot to have a factor of 1
int pivotRow = -1;
// Loop through all the rows to find a pivot for the pivotColumn
for (int row = 0; row < rowLength; row++)
{
// A pivot must be non-zero...
if (matrixIn[pivotColumn, row].Value != 0)
{
// ...and don't pivot on a row that has previous non-zero column entries
// Loop through columns of the row Check is the row found respect the non-zero column rule
bool isRowWithPreviousZero = true;
for (int column = 0; column < pivotColumn; column++)
{
// Check for non-zero
if (matrixIn[column, row].Value != 0)
{
// The row is not a pivot because it has previous non-zero entries
isRowWithPreviousZero = false;
break;
}
}
// If the row found has previous zero, it is a pivot row
if (isRowWithPreviousZero)
{
// The current row is the pivot
pivotRow = row;
GF16 factorPivot = matrixIn[pivotColumn, row];
// Reduce the pivot row to have a factor of 1
for(int column = pivotColumn; column < columnLength; column++)
{
matrixIn[column, row] /= factorPivot;
}
// We can exit the loop through rows in Step 1
break;
}
}
}
// Ensure that a pivot has been found
if (pivotRow == -1)
{
throw new ApplicationException(Strings.NoPivotRowFound);
}
// TODO: Place Step 2 in a separate method
// Step 2: Do elementary operation to all rows (except the pivot row and
// the rows that have zero on the pivot column)
// The final goal is to have a zero for all rows on the column corresponding to the
// pivot column
// Loop trough the rows to do elementary operation (subtraction)
for (int row = 0; row < rowLength; row++)
{
// Note: We skip the pivot row and the rows that has zero on the pivot
// column (no operation requiered)
if ((row != pivotRow) && (matrixIn[ pivotColumn, row].Value != 0))
{
// Now, determine the factor by which the pivot row must be added to the target
// row to cancel out the pivot value
GF16 factorOperation = matrixIn[ pivotColumn, row ];
// Perform the operation for each column of the current row
for(int column = 0; column < columnLength; column++)
{
// Perform a subtraction. The final goal is to have a zero for all rows
// on the column corresponding to the pivot column
matrixIn[column, row] -= factorOperation * matrixIn[column, pivotRow];
}
}
}
}
// At this point the work manipulation is done but we might get the rows
// out of order.
// For instance, the input matrix:
// 0 1 1 1 0 0
// 0 1 2 0 1 0
// 1 1 4 0 0 1
//
// will become the following (over int operation):
// 0 1 0 2 -1 0
// 0 0 1 -1 1 0
// 1 0 0 2 -3 1
//
// so now, we need to swap the row to place them in order to obtain the following:
// [ ID ] [ rest ]
// 1 0 0 2 -3 1
// 0 1 0 2 -1 0
// 0 0 1 -1 1 0
// TODO: Place Step 3 in a separate method
// Step 3: Swap raw into order to have an identity matrix at the front
for(int row = 0; row < rowLength; row++) // can use row < rowLength -1, because last row should always be in order?
{
// Skip rows that are in proper order
if (matrixIn[row,row].Value != 1)
{
// Not in proper order, search down through the matrix
bool found = false;
for (int searchRow = 0; searchRow < rowLength; searchRow++)
{
if (matrixIn[row, searchRow].Value == 1) // found the row we're looking for
{
found = true;
// Do the swap operation
// Save the target row to temporary swap space first
GF16[] swapRowTemp = new GF16[columnLength];
for(int swapRowColumn = 0; swapRowColumn < columnLength; swapRowColumn++)
{
swapRowTemp[swapRowColumn] = matrixIn[swapRowColumn, row];
}
// Replace the target row contents with the found row contents
for(int swapRowColumn = 0; swapRowColumn < columnLength; swapRowColumn++)
{
matrixIn[swapRowColumn, row] = matrixIn[swapRowColumn,searchRow];
}
// Replace the found row contents with the temporary swap space (previous target row contents)
for(int swapRowColumn = 0; swapRowColumn < columnLength; swapRowColumn++)
{
matrixIn[swapRowColumn,searchRow] = swapRowTemp[swapRowColumn];
}
}
}
if(!found)
throw new ApplicationException(Strings.SwapRowNotFound);
}
}
return matrixIn;
}
/// <summary>
/// Add identity to the right side of a matrix
/// </summary>
/// <remarks>This is use when doing Jordan-Gauss elimination</remarks>
/// <param name="matrixIn">The matrix</param>
/// <returns>The new matrix with the identity</returns>
private static GF16[,] AddIdentityRight (GF16[,] matrixIn)
{
Int32 rowLength = matrixIn.GetLength(1);
Int32 columnLength = matrixIn.GetLength(0);
GF16[,] matrixOut = new GF16[columnLength * 2, rowLength];
// Add in identity portion to the top
for (Int32 column = 0; column < columnLength; column++)
{
matrixOut[column + columnLength, column] = (UInt16)1;
}
// Copy the previous matrix to the bottom
MatrixCopyTo(matrixIn, matrixOut);
return matrixOut;
}
