/// <summary>
/// Initializes the preconditioner and loads the internal data structures.
/// </summary>
/// <param name="matrix">
/// The <see cref="Matrix"/> upon which this preconditioner is based. Note that the
/// method takes a general matrix type. However internally the data is stored
/// as a sparse matrix. Therefore it is not recommended to pass a dense matrix.
/// </param>
/// <exception cref="ArgumentNullException"> If <paramref name="matrix"/> is <see langword="null" />.</exception>
/// <exception cref="ArgumentException">If <paramref name="matrix"/> is not a square matrix.</exception>
public void Initialize(Matrix matrix)
{
if (matrix == null)
{
throw new ArgumentNullException("matrix");
}
if (matrix.RowCount != matrix.ColumnCount)
{
throw new ArgumentException(Resources.ArgumentMatrixSquare, "matrix");
}
var sparseMatrix = (matrix is SparseMatrix) ? matrix as SparseMatrix : SparseMatrix.OfMatrix(matrix);
// The creation of the preconditioner follows the following algorithm.
// spaceLeft = lfilNnz * nnz(A)
// for i = 1, .. , n
// {
// w = a(i,*)
// for j = 1, .. , i - 1
// {
// if (w(j) != 0)
// {
// w(j) = w(j) / a(j,j)
// if (w(j) < dropTol)
// {
// w(j) = 0;
// }
// if (w(j) != 0)
// {
// w = w - w(j) * U(j,*)
// }
// }
// }
//
// for j = i, .. ,n
// {
// if w(j) <= dropTol * ||A(i,*)||
// {
// w(j) = 0
// }
// }
//
// spaceRow = spaceLeft / (n - i + 1) // Determine the space for this row
// lfil = spaceRow / 2 // space for this row of L
// l(i,j) = w(j) for j = 1, .. , i -1 // only the largest lfil elements
//
// lfil = spaceRow - nnz(L(i,:)) // space for this row of U
// u(i,j) = w(j) for j = i, .. , n // only the largest lfil - 1 elements
// w = 0
//
// if max(U(i,i + 1: n)) > U(i,i) / pivTol then // pivot if necessary
// {
// pivot by swapping the max and the diagonal entries
// Update L, U
// Update P
// }
// spaceLeft = spaceLeft - nnz(L(i,:)) - nnz(U(i,:))
// }
// Create the lower triangular matrix
_lower = new SparseMatrix(sparseMatrix.RowCount);
// Create the upper triangular matrix and copy the values
_upper = new SparseMatrix(sparseMatrix.RowCount);
// Create the pivot array
_pivots = new int[sparseMatrix.RowCount];
for (var i = 0; i < _pivots.Length; i++)
{
_pivots[i] = i;
}
Vector workVector = new DenseVector(sparseMatrix.RowCount);
Vector rowVector = new DenseVector(sparseMatrix.ColumnCount);
var indexSorting = new int[sparseMatrix.RowCount];
// spaceLeft = lfilNnz * nnz(A)
var spaceLeft = (int)_fillLevel * sparseMatrix.NonZerosCount;
// for i = 1, .. , n
for (var i = 0; i < sparseMatrix.RowCount; i++)
{
// w = a(i,*)
sparseMatrix.Row(i, workVector);
// pivot the row
PivotRow(workVector);
var vectorNorm = workVector.InfinityNorm();
// for j = 1, .. , i - 1)
for (var j = 0; j < i; j++)
{
// if (w(j) != 0)
// {
// w(j) = w(j) / a(j,j)
// if (w(j) < dropTol)
// {
// w(j) = 0;
// }
// if (w(j) != 0)
// {
// w = w - w(j) * U(j,*)
// }
if (workVector[j] != 0.0)
{
// Calculate the multiplication factors that go into the L matrix
workVector[j] = workVector[j] / _upper[j, j];
if (Math.Abs(workVector[j]) < _dropTolerance)
{
workVector[j] = 0.0f;
}
// Calculate the addition factor
if (workVector[j] != 0.0)
{
// vector update all in one go
_upper.Row(j, rowVector);
// zero out columnVector[k] because we don't need that
// one anymore for k = 0 to k = j
for (var k = 0; k <= j; k++)
{
rowVector[k] = 0.0f;
}
rowVector.Multiply(workVector[j], rowVector);
workVector.Subtract(rowVector, workVector);
}
}
}
// for j = i, .. ,n
for (var j = i; j < sparseMatrix.RowCount; j++)
{
// if w(j) <= dropTol * ||A(i,*)||
// {
// w(j) = 0
// }
if (Math.Abs(workVector[j]) <= _dropTolerance * vectorNorm)
{
workVector[j] = 0.0f;
}
}
// spaceRow = spaceLeft / (n - i + 1) // Determine the space for this row
var spaceRow = spaceLeft / (sparseMatrix.RowCount - i + 1);
// lfil = spaceRow / 2 // space for this row of L
var fillLevel = spaceRow / 2;
FindLargestItems(0, i - 1, indexSorting, workVector);
// l(i,j) = w(j) for j = 1, .. , i -1 // only the largest lfil elements
var lowerNonZeroCount = 0;
var count = 0;
for (var j = 0; j < i; j++)
{
if ((count > fillLevel) || (indexSorting[j] == -1))
{
break;
}
_lower[i, indexSorting[j]] = workVector[indexSorting[j]];
count += 1;
lowerNonZeroCount += 1;
}
FindLargestItems(i + 1, sparseMatrix.RowCount - 1, indexSorting, workVector);
// lfil = spaceRow - nnz(L(i,:)) // space for this row of U
fillLevel = spaceRow - lowerNonZeroCount;
// u(i,j) = w(j) for j = i + 1, .. , n // only the largest lfil - 1 elements
var upperNonZeroCount = 0;
count = 0;
for (var j = 0; j < sparseMatrix.RowCount - i; j++)
{
if ((count > fillLevel - 1) || (indexSorting[j] == -1))
{
break;
}
_upper[i, indexSorting[j]] = workVector[indexSorting[j]];
count += 1;
upperNonZeroCount += 1;
}
// Simply copy the diagonal element. Next step is to see if we pivot
_upper[i, i] = workVector[i];
// if max(U(i,i + 1: n)) > U(i,i) / pivTol then // pivot if necessary
// {
// pivot by swapping the max and the diagonal entries
// Update L, U
// Update P
// }
// Check if we really need to pivot. If (i+1) >=(mCoefficientMatrix.Rows -1) then
// we are working on the last row. That means that there is only one number
// And pivoting is useless. Also the indexSorting array will only contain
// -1 values.
if ((i + 1) < (sparseMatrix.RowCount - 1))
{
if (Math.Abs(workVector[i]) < _pivotTolerance * Math.Abs(workVector[indexSorting[0]]))
{
// swap columns of u (which holds the values of A in the
// sections that haven't been partitioned yet.
SwapColumns(_upper, i, indexSorting[0]);
// Update P
var temp = _pivots[i];
_pivots[i] = _pivots[indexSorting[0]];
_pivots[indexSorting[0]] = temp;
}
}
// spaceLeft = spaceLeft - nnz(L(i,:)) - nnz(U(i,:))
spaceLeft -= lowerNonZeroCount + upperNonZeroCount;
}
for (var i = 0; i < _lower.RowCount; i++)
{
_lower[i, i] = 1.0f;
}
}
/// <summary>
/// Initializes the preconditioner and loads the internal data structures.
/// </summary>
/// <param name="matrix">
/// The <see cref="Matrix"/> upon which this preconditioner is based. Note that the
/// method takes a general matrix type. However internally the data is stored
/// as a sparse matrix. Therefore it is not recommended to pass a dense matrix.
/// </param>
/// <exception cref="ArgumentNullException"> If <paramref name="matrix"/> is <see langword="null" />.</exception>
/// <exception cref="ArgumentException">If <paramref name="matrix"/> is not a square matrix.</exception>
public void Initialize(Matrix <float> matrix)
{
if (matrix == null)
{
throw new ArgumentNullException("matrix");
}
if (matrix.RowCount != matrix.ColumnCount)
{
throw new ArgumentException(Resources.ArgumentMatrixSquare, "matrix");
}
SparseMatrix sparseMatrix = matrix as SparseMatrix ?? SparseMatrix.OfMatrix(matrix);
// The creation of the preconditioner follows the following algorithm.
// spaceLeft = lfilNnz * nnz(A)
// for i = 1, .. , n
// {
// w = a(i,*)
// for j = 1, .. , i - 1
// {
// if (w(j) != 0)
// {
// w(j) = w(j) / a(j,j)
// if (w(j) < dropTol)
// {
// w(j) = 0;
// }
// if (w(j) != 0)
// {
// w = w - w(j) * U(j,*)
// }
// }
// }
//
// for j = i, .. ,n
// {
// if w(j) <= dropTol * ||A(i,*)||
// {
// w(j) = 0
// }
// }
//
// spaceRow = spaceLeft / (n - i + 1) // Determine the space for this row
// lfil = spaceRow / 2 // space for this row of L
// l(i,j) = w(j) for j = 1, .. , i -1 // only the largest lfil elements
//
// lfil = spaceRow - nnz(L(i,:)) // space for this row of U
// u(i,j) = w(j) for j = i, .. , n // only the largest lfil - 1 elements
// w = 0
//
// if max(U(i,i + 1: n)) > U(i,i) / pivTol then // pivot if necessary
// {
// pivot by swapping the max and the diagonal entries
// Update L, U
// Update P
// }
// spaceLeft = spaceLeft - nnz(L(i,:)) - nnz(U(i,:))
// }
// Create the lower triangular matrix
_lower = new SparseMatrix(sparseMatrix.RowCount);
// Create the upper triangular matrix and copy the values
_upper = new SparseMatrix(sparseMatrix.RowCount);
// Create the pivot array
_pivots = new int[sparseMatrix.RowCount];
for (var i = 0; i < _pivots.Length; i++)
{
_pivots[i] = i;
}
var workVector = new DenseVector(sparseMatrix.RowCount);
var rowVector = new DenseVector(sparseMatrix.ColumnCount);
var indexSorting = new int[sparseMatrix.RowCount];
// spaceLeft = lfilNnz * nnz(A)
var spaceLeft = (int)_fillLevel * sparseMatrix.NonZerosCount;
// for i = 1, .. , n
for (var i = 0; i < sparseMatrix.RowCount; i++)
{
// w = a(i,*)
sparseMatrix.Row(i, workVector);
// pivot the row
PivotRow(workVector);
var vectorNorm = workVector.InfinityNorm();
// for j = 1, .. , i - 1)
for (var j = 0; j < i; j++)
{
// if (w(j) != 0)
// {
// w(j) = w(j) / a(j,j)
// if (w(j) < dropTol)
// {
// w(j) = 0;
// }
// if (w(j) != 0)
// {
// w = w - w(j) * U(j,*)
// }
if (workVector[j] != 0.0)
{
// Calculate the multiplication factors that go into the L matrix
workVector[j] = workVector[j] / _upper[j, j];
if (Math.Abs(workVector[j]) < _dropTolerance)
{
workVector[j] = 0.0f;
}
// Calculate the addition factor
if (workVector[j] != 0.0)
{
// vector update all in one go
_upper.Row(j, rowVector);
// zero out columnVector[k] because we don't need that
// one anymore for k = 0 to k = j
for (var k = 0; k <= j; k++)
{
rowVector[k] = 0.0f;
}
rowVector.Multiply(workVector[j], rowVector);
workVector.Subtract(rowVector, workVector);
}
}
}
// for j = i, .. ,n
for (var j = i; j < sparseMatrix.RowCount; j++)
{
// if w(j) <= dropTol * ||A(i,*)||
// {
// w(j) = 0
// }
if (Math.Abs(workVector[j]) <= _dropTolerance * vectorNorm)
{
workVector[j] = 0.0f;
}
}
// spaceRow = spaceLeft / (n - i + 1) // Determine the space for this row
var spaceRow = spaceLeft / (sparseMatrix.RowCount - i + 1);
// lfil = spaceRow / 2 // space for this row of L
var fillLevel = spaceRow / 2;
FindLargestItems(0, i - 1, indexSorting, workVector);
// l(i,j) = w(j) for j = 1, .. , i -1 // only the largest lfil elements
var lowerNonZeroCount = 0;
var count = 0;
for (var j = 0; j < i; j++)
{
if ((count > fillLevel) || (indexSorting[j] == -1))
{
break;
}
_lower[i, indexSorting[j]] = workVector[indexSorting[j]];
count += 1;
lowerNonZeroCount += 1;
}
FindLargestItems(i + 1, sparseMatrix.RowCount - 1, indexSorting, workVector);
// lfil = spaceRow - nnz(L(i,:)) // space for this row of U
fillLevel = spaceRow - lowerNonZeroCount;
// u(i,j) = w(j) for j = i + 1, .. , n // only the largest lfil - 1 elements
var upperNonZeroCount = 0;
count = 0;
for (var j = 0; j < sparseMatrix.RowCount - i; j++)
{
if ((count > fillLevel - 1) || (indexSorting[j] == -1))
{
break;
}
_upper[i, indexSorting[j]] = workVector[indexSorting[j]];
count += 1;
upperNonZeroCount += 1;
}
// Simply copy the diagonal element. Next step is to see if we pivot
_upper[i, i] = workVector[i];
// if max(U(i,i + 1: n)) > U(i,i) / pivTol then // pivot if necessary
// {
// pivot by swapping the max and the diagonal entries
// Update L, U
// Update P
// }
// Check if we really need to pivot. If (i+1) >=(mCoefficientMatrix.Rows -1) then
// we are working on the last row. That means that there is only one number
// And pivoting is useless. Also the indexSorting array will only contain
// -1 values.
if ((i + 1) < (sparseMatrix.RowCount - 1))
{
if (Math.Abs(workVector[i]) < _pivotTolerance * Math.Abs(workVector[indexSorting[0]]))
{
// swap columns of u (which holds the values of A in the
// sections that haven't been partitioned yet.
SwapColumns(_upper, i, indexSorting[0]);
// Update P
var temp = _pivots[i];
_pivots[i] = _pivots[indexSorting[0]];
_pivots[indexSorting[0]] = temp;
}
}
// spaceLeft = spaceLeft - nnz(L(i,:)) - nnz(U(i,:))
spaceLeft -= lowerNonZeroCount + upperNonZeroCount;
}
for (var i = 0; i < _lower.RowCount; i++)
{
_lower[i, i] = 1.0f;
}
}
