/// <summary>
/// This function executes the backward reduction algorithm provided in
/// the assignment text given an augmented matrix in row Echelon form.
/// </summary>
///
/// <param name="a">
/// An N-by-M augmented matrix in row Echelon form.
/// </param>
///
/// <returns>
/// The resulting N-by-M matrix after executing the algorithm.
/// </returns>
public static Matrix BackwardReduction(this Matrix a)
{
Vector pRow = new Vector(0);
int pRowIndex = -1;
int uCol = -1;
// Finds pivot row
for (int i = a.M_Rows - 1; i >= 0; i--)
{
pRow = a.Row(i);
if (BasicExtensions.IsZeroVector(pRow))
{
continue;
}
pRowIndex = i;
uCol = BasicExtensions.FirstValueIndex(pRow);
break;
}
// Scale pivot row to 1
a.ElementaryRowScaling(pRowIndex, 1.0 / pRow[uCol]);
// Using pivot row to reduce entries to zero
for (int i = 0; i < a.M_Rows; i++)
{
if (i == pRowIndex)
{
break;
}
a.ElementaryRowReplacement(i, -a[i, uCol], pRowIndex);
}
// If this was the last row to be scaled end of recursion is reached.
if (a.M_Rows - 1 <= 0)
{
return(a);
}
// step 2 forming submatrix to process other rows. Last row of a is excluded.
Vector[] subMatrixVectors = new Vector[a.M_Rows - 1];
for (int i = 0; i < subMatrixVectors.Length; i++)
{
subMatrixVectors[i] = a.Row(i);
}
Matrix subMatrix = BasicExtensions.VectorCombine(subMatrixVectors);
subMatrix = subMatrix.BackwardReduction();
Vector[] retVectors = new Vector[a.M_Rows];
retVectors[a.M_Rows - 1] = a.Row(a.M_Rows - 1);
for (int i = 0; i < a.M_Rows - 1; i++)
{
retVectors[i] = subMatrix.Row(i);
}
return(BasicExtensions.VectorCombine(retVectors));
}
/// <summary>
/// This function executes the forward reduction algorithm provided in
/// the assignment text to achieve row Echelon form of a given
/// augmented matrix.
/// </summary>
///
/// <param name="a">
/// An N-by-M augmented matrix.
/// </param>
///
/// <returns>
/// An N-by-M matrix that is the row Echelon form.
/// </returns>
public static Matrix ForwardReduction(this Matrix a)
{
// If submatrix has zero rows, end of recursion is reached
if (a.M_Rows - 1 <= 0)
{
return(a);
}
Vector colJ = new Vector(0);
var pRow = -1;
// step 1 locate pivot column
for (var colLoop = 0; colLoop < a.N_Cols; colLoop++)
{
colJ = a.Column(colLoop);
if (BasicExtensions.IsZeroVector(colJ))
{
continue;
}
// step 2 find first non-zero entry
pRow = BasicExtensions.FirstValueIndex(colJ);
break;
}
// if all rows are zero rows end of recursion is reached.
if (pRow == -1)
{
return(a);
}
// step 2 reduction
a.ElementaryRowInterchange(0, pRow);
for (var i = 1; i < colJ.Size; i++)
{
a.ElementaryRowReplacement(i, (-colJ[i] / colJ[0]), 0);
}
// step 3 forming submatrix.
Vector[] subMatrixVectors = new Vector[a.M_Rows - 1];
for (int i = 1; i < a.M_Rows; i++)
{
subMatrixVectors[i - 1] = a.Row(i);
}
Matrix subMatrix = BasicExtensions.VectorCombine(subMatrixVectors);
// applying steps 1 and 2 on submatrix
subMatrix = subMatrix.ForwardReduction();
// building return matrix with vectors
Vector[] retVectors = new Vector[a.M_Rows];
retVectors[0] = a.Row(0);
for (int i = 1; i < a.M_Rows; i++)
{
retVectors[i] = subMatrix.Row(i - 1);
}
return(BasicExtensions.VectorCombine(retVectors));
}