/// <summary>
/// Find a pivot in a matrix.
/// </summary>
/// <param name="markowitz">The Markowitz pivot strategy.</param>
/// <param name="matrix">The matrix</param>
/// <param name="eliminationStep">The current elimination step.</param>
/// <returns>
/// The pivot element, or null if no pivot was found.
/// </returns>
/// <exception cref="ArgumentNullException">
/// markowitz
/// or
/// matrix
/// </exception>
/// <exception cref="ArgumentException">Invalid elimination step</exception>
public override MatrixElement <T> FindPivot(Markowitz <T> markowitz, SparseMatrix <T> matrix, int eliminationStep)
{
if (markowitz == null)
{
throw new ArgumentNullException(nameof(markowitz));
}
if (matrix == null)
{
throw new ArgumentNullException(nameof(matrix));
}
if (eliminationStep < 1)
{
throw new ArgumentException("Invalid elimination step");
}
// No singletons left, so don't bother
if (markowitz.Singletons == 0)
{
return(null);
}
// Find the first valid singleton we can use
int singletons = 0, index;
for (var i = matrix.Size + 1; i >= eliminationStep; i--)
{
// First check the current pivot, else
// search from last to first as it tends to push the higher markowitz
// products downwards.
index = i > matrix.Size ? eliminationStep : i;
// Not a singleton, let's skip this one...
if (markowitz.Product(index) != 0)
{
continue;
}
// Keep track of how many singletons we have found
singletons++;
/*
* NOTE: In the sparse library of Spice 3f5 first the diagonal is checked,
* then the column is checked (if no success go to row) and finally the
* row is checked for a valid singleton pivot.
* The diagonal should actually not be checked, as the checking algorithm is the same
* as for checking the column (FindBiggestInColExclude will not find anything in the column
* for singletons). The original author did not find this.
* Also, the original algorithm has a bug in there that renders the whole code invalid...
* if (ChosenPivot != NULL) { break; } will throw away the pivot even if it was found!
* (ref. Spice 3f5 Libraries/Sparse/spfactor.c, line 1286)
*/
// Find the singleton element
MatrixElement <T> chosen;
if (markowitz.ColumnCount(index) == 0)
{
// The last element in the column is the singleton element!
chosen = matrix.GetLastInColumn(index);
// Check if it is a valid pivot
var magnitude = markowitz.Magnitude(chosen.Value);
if (magnitude > markowitz.AbsolutePivotThreshold)
{
return(chosen);
}
}
// Check if we can still use a row here
if (markowitz.RowCount(index) == 0)
{
// The last element in the row is the singleton element
chosen = matrix.GetLastInRow(index);
// When the matrix has an empty row, and an RHS element, it is possible
// that the singleton is not a singleton
if (chosen == null || chosen.Column < eliminationStep)
{
// The last element is not valid, singleton failed!
singletons--;
continue;
}
// First find the biggest magnitude in the column, not counting the pivot candidate
var element = chosen.Above;
var largest = 0.0;
while (element != null && element.Row >= eliminationStep)
{
largest = Math.Max(largest, markowitz.Magnitude(element.Value));
element = element.Above;
}
element = chosen.Below;
while (element != null)
{
largest = Math.Max(largest, markowitz.Magnitude(element.Value));
element = element.Below;
}
// Check if the pivot is valid
var magnitude = markowitz.Magnitude(chosen.Value);
if (magnitude > markowitz.AbsolutePivotThreshold &&
magnitude > markowitz.RelativePivotThreshold * largest)
{
return(chosen);
}
}
// Don't continue if no more singletons are available
if (singletons >= markowitz.Singletons)
{
break;
}
}
// All singletons were unacceptable...
return(null);
}
/// <summary>
/// Search the entire matrix for a suitable pivot
/// In order to preserve sparsity, Markowitz counts are used to find the largest valid
/// pivot with the smallest number of elements.
/// </summary>
/// <param name="markowitz">Markowitz object</param>
/// <param name="matrix">Matrix</param>
/// <param name="eliminationStep">Step</param>
/// <returns></returns>
public override MatrixElement <T> FindPivot(Markowitz <T> markowitz, SparseMatrix <T> matrix, int eliminationStep)
{
if (markowitz == null)
{
throw new ArgumentNullException(nameof(markowitz));
}
if (matrix == null)
{
throw new ArgumentNullException(nameof(matrix));
}
if (eliminationStep < 1)
{
throw new ArgumentException("Invalid elimination step");
}
MatrixElement <T> chosen = null;
long minMarkowitzProduct = long.MaxValue;
double largestMagnitude = 0.0, acceptedRatio = 0.0;
MatrixElement <T> largestElement = null;
int ties = 0;
// Start search of matrix on column by column basis
for (int i = eliminationStep; i <= matrix.Size; i++)
{
// Find the biggest magnitude in the column for checking valid pivots later
double largest = 0.0;
var element = matrix.GetLastInColumn(i);
while (element != null && element.Row >= eliminationStep)
{
largest = Math.Max(largest, markowitz.Magnitude(element.Value));
element = element.Above;
}
if (largest.Equals(0.0))
{
continue;
}
// Restart search for a pivot
element = matrix.GetLastInColumn(i);
while (element != null && element.Row >= eliminationStep)
{
// Find the magnitude and Markowitz product
double magnitude = markowitz.Magnitude(element.Value);
int product = markowitz.RowCount(element.Row) * markowitz.ColumnCount(element.Column);
// In the case no valid pivot is available, at least return the largest element
if (magnitude > largestMagnitude)
{
largestElement = element;
largestMagnitude = magnitude;
}
// test to see if the element is acceptable as a pivot candidate
if (product <= minMarkowitzProduct &&
magnitude > markowitz.RelativePivotThreshold * largest &&
magnitude > markowitz.AbsolutePivotThreshold)
{
// Test to see if the element has the lowest Markowitz product yet found,
// or whether it is tied with an element found earlier
if (product < minMarkowitzProduct)
{
// Notice strict inequality
// This is a new smallest Markowitz product
chosen = element;
minMarkowitzProduct = product;
acceptedRatio = largest / magnitude;
ties = 0;
}
else
{
// This case handles Markowitz ties
ties++;
double ratio = largest / magnitude;
if (ratio < acceptedRatio)
{
chosen = element;
acceptedRatio = ratio;
}
if (ties >= minMarkowitzProduct * TiesMultiplier)
{
return(chosen);
}
}
}
element = element.Above;
}
}
// If a valid pivot was found, return it
if (chosen != null)
{
return(chosen);
}
// Singular matrix
// If we can't find it while searching the entire matrix, then we definitely have a singular matrix...
if (largestElement == null || largestElement.Value.Equals(0.0))
{
throw new SingularException(eliminationStep);
}
return(largestElement);
}
/// <inheritdoc/>
public override Pivot <ISparseMatrixElement <T> > FindPivot(Markowitz <T> markowitz, ISparseMatrix <T> matrix, int eliminationStep, int max)
{
markowitz.ThrowIfNull(nameof(markowitz));
matrix.ThrowIfNull(nameof(matrix));
if (eliminationStep < 1 || eliminationStep > max)
{
return(Pivot <ISparseMatrixElement <T> > .Empty);
}
ISparseMatrixElement <T> chosen = null;
var minMarkowitzProduct = long.MaxValue;
double largestMagnitude = 0.0, acceptedRatio = 0.0;
ISparseMatrixElement <T> largestElement = null;
var ties = 0;
// Start search of matrix on column by column basis
for (var i = eliminationStep; i <= max; i++)
{
// Find an entry point to the interesting part of the column
var lowest = matrix.GetLastInColumn(i);
while (lowest != null && lowest.Row > max)
{
lowest = lowest.Above;
}
if (lowest == null || lowest.Row < eliminationStep)
{
continue;
}
// Find the biggest magnitude in the column for checking valid pivots later
var largest = 0.0;
var element = lowest;
while (element != null && element.Row >= eliminationStep)
{
largest = Math.Max(largest, markowitz.Magnitude(element.Value));
element = element.Above;
}
if (largest.Equals(0.0))
{
continue;
}
// Restart search for a pivot
element = lowest;
while (element != null && element.Row >= eliminationStep)
{
// Find the magnitude and Markowitz product
var magnitude = markowitz.Magnitude(element.Value);
var product = markowitz.RowCount(element.Row) * markowitz.ColumnCount(element.Column);
// In the case no valid pivot is available, at least return the largest element
if (magnitude > largestMagnitude)
{
largestElement = element;
largestMagnitude = magnitude;
}
// test to see if the element is acceptable as a pivot candidate
if (product <= minMarkowitzProduct &&
magnitude > markowitz.RelativePivotThreshold * largest &&
magnitude > markowitz.AbsolutePivotThreshold)
{
// Test to see if the element has the lowest Markowitz product yet found,
// or whether it is tied with an element found earlier
if (product < minMarkowitzProduct)
{
// Notice strict inequality
// This is a new smallest Markowitz product
chosen = element;
minMarkowitzProduct = product;
acceptedRatio = largest / magnitude;
ties = 0;
}
else
{
// This case handles Markowitz ties
ties++;
var ratio = largest / magnitude;
if (ratio < acceptedRatio)
{
chosen = element;
acceptedRatio = ratio;
}
if (ties >= minMarkowitzProduct * _tiesMultiplier)
{
return(new Pivot <ISparseMatrixElement <T> >(chosen, PivotInfo.Suboptimal));
}
}
}
element = element.Above;
}
}
// If a valid pivot was found, return it
if (chosen != null)
{
return(new Pivot <ISparseMatrixElement <T> >(chosen, PivotInfo.Suboptimal));
}
// Else just return the largest element
if (largestElement == null || largestElement.Value.Equals(default))
