public Tuple <CCS, double[]> CalculateDisplacementPermutation(CCS a)
{
var totDofCount = a.ColumnCount - 1;
var allEqs = a;
//var empties = allEqs.EmptyRowCount();
//var dns = allEqs.ToDenseMatrix();
#region comment
/*
#region step 2
* //step 2: create adjacency matrix of variables
*
* //step 2-1: find nonzero pattern
* var allEqsNonzeroPattern = allEqs.Clone();
*
* for (var i = 0; i < allEqsNonzeroPattern.Values.Length; i++)
* allEqsNonzeroPattern.Values[i] = 1;
*
* //https://math.stackexchange.com/questions/2340450/extract-independent-sub-systems-from-a-bigger-linear-eq-system
* var tmp = allEqsNonzeroPattern.Transpose();
*
* var variableAdj = tmp.Multiply(allEqsNonzeroPattern);
#endregion
*
#region step 3
* //extract parts
* var parts = EnumerateGraphParts(variableAdj);
*
#endregion
*
#region step 4
* {
*
* allEqs.EnumerateColumns((colNum, vals) =>
* {
* if (vals.Count == 0)
* Console.WriteLine("Col {0} have {1} nonzeros", colNum, vals.Count);
* });
*
* var order = ColumnOrdering.MinimumDegreeAtPlusA;
*
* // Partial pivoting tolerance (0.0 to 1.0)
* double tolerance = 1.0;
*
* var lu = CSparse.Double.Factorization.SparseLU.Create(allEqs, order, tolerance);
*
* }
*
#endregion
*/
#endregion
var rrefFinder = new GaussRrefFinder();
var rref = rrefFinder.CalculateRref(allEqs);
var rrefSys = SparseEqSystem.Generate(rref);
#region generate P_Delta
var pRows = new int[totDofCount]; // pRows[i] = index of equation that its right side is Di (ai*Di = a1*D1+...+an*Dn)
pRows.FillWith(-1);
for (var i = 0; i < rrefSys.RowCount; i++)
{
foreach (var tpl in rrefSys.Equations[i].EnumerateIndexed())
{
if (rrefSys.ColumnNonzeros[tpl.Item1] == 1)
{
if (pRows[tpl.Item1] != -1)
{
throw new Exception();
}
pRows[tpl.Item1] = i;
}
}
}
int cnt = 0;
var lastColIndex = rrefSys.ColumnCount - 1;
var p2Coord = new CoordinateStorage <double>(totDofCount, totDofCount, 1);
var rightSide = new double[totDofCount];
for (var i = 0; i < totDofCount; i++)
{
if (pRows[i] == -1)
{
p2Coord.At(i, i, 1);
continue;
}
var eq = rrefSys.Equations[pRows[i]];
eq.Multiply(-1 / eq.GetMember(i));
var minus1 = eq.GetMember(i);
if (!minus1.FEquals(-1, 1e-9))
{
throw new Exception();
}
//eq.SetMember(i, 0);
foreach (var tpl in eq.EnumerateIndexed())
{
if (tpl.Item1 != lastColIndex)
{
p2Coord.At(i, tpl.Item1, tpl.Item2);
}
else
{
rightSide[i] = tpl.Item2;
}
}
}
cnt = 0;
foreach (var eq in rrefSys.Equations)
{
if (eq.IsZeroRow(1e-9))
{
cnt++;
}
}
#endregion
var p2 = p2Coord.ToCCs();
var colsToRemove = new bool[totDofCount];
var colNumPerm = new int[totDofCount];
colNumPerm.FillWith(-1);
colsToRemove.FillWith(false);
var tmp = 0;
for (var i = 0; i < rrefSys.ColumnNonzeros.Length; i++)
{
if (i != lastColIndex)
{
if (rrefSys.ColumnNonzeros[i] == 1)
{
colsToRemove[i] = true;
}
else
{
colNumPerm[i] = tmp++;
}
}
}
var p3Crd = new CoordinateStorage <double>(totDofCount, totDofCount - colsToRemove.Count(i => i), 1);
foreach (var tpl in p2.EnumerateIndexed2())
{
if (!colsToRemove[tpl.Item2])
{
p3Crd.At(tpl.Item1, colNumPerm[tpl.Item2], tpl.Item3);
}
}
var p3 = p3Crd.ToCCs();
return(Tuple.Create(p3, rightSide));
}
public CCS CalculateRref(CCS a)
{
var sys = SparseEqSystem.Generate((CSparse.Double.CompressedColumnStorage)a);
var dns = a.ToDenseMatrix();
sys.RowsPool = new SortedSet <SparseRow>();
var eqs = sys.Equations;
for (var i = 0; i < sys.Equations.Length; i++)
{
sys.Equations[i].Tag = i;
}
var lastColIndex = a.ColumnCount - 1;
//should have n columns with one nonzero
//var colPivotHistory = new HashSet<int>();
//var rowPivotHistory = new HashSet<int>();
var dependentRows = new bool[sys.RowCount];// HashSet<int>();
var leadingMember = new bool[sys.RowCount];
//var pivotedYet = new bool[a.RowCount];
for (var i = 0; i < sys.Equations.Length; i++)
{
foreach (var tuple in sys.Equations[i].EnumerateIndexed())
{
if (sys.ColumnNonzeros[tuple.Item1] == 1)
{
leadingMember[i] = true;
}
}
}
var maxItter = Math.Max(a.ColumnCount, a.RowCount);
var itter = 0;
while (itter++ < maxItter)
{
//find count
var oneNnzCols = sys.ColumnNonzeros.Count(i => i == 1);
var eqNeeded = Math.Min(a.ColumnCount - 1, a.RowCount - dependentRows.Count(i => i == true));
if (oneNnzCols == eqNeeded)
{
break;
}
//select pivot
//minimum row nnz x col nnz
//select row strategy: row with minimum nnz, where nnz is nonzero count except last column (right side) and nnz > 1
var minRowNnz = int.MaxValue; // sys.RowNonzeros.Where(i => i > 1).Min();
var minRowNnzIndex = -1; // sys.RowNonzeros.FirstIndexOf(minRowNnz);
{
for (var j = 0; j < sys.RowCount; j++)
{
if (leadingMember[j])
{
continue;
}
if (dependentRows[j])
{
continue;
}
//if (sys.RowNonzeros[j] <= 1)
// continue;
if (sys.RowNonzeros[j] < minRowNnz)
{
minRowNnzIndex = j;
minRowNnz = sys.RowNonzeros[j];
}
}
}
if (minRowNnzIndex == -1)
{
throw new Exception();
}
//var minColNnz = sys.ColumnNonzeros.Where(i => i > 1).Min();
var minColNnz = int.MaxValue;// sys.ColumnNonzeros.Where(i => i > 1).Min();
var minColNnzIndex = -1;
{
for (var jj = 0; jj < sys.Equations[minRowNnzIndex].Size; jj++)
{
var j = sys.Equations[minRowNnzIndex].Indexes[jj];
//if (colPivotHistory.Contains(j))
// continue;
if (sys.ColumnNonzeros[j] <= 1)
{
continue;
}
if (sys.ColumnNonzeros[j] < minColNnz)
{
minColNnzIndex = j;
minColNnz = sys.ColumnNonzeros[j];
}
}
}
//var minColNnzIndex = sys.ColumnNonzeros.FirstIndexOf(minColNnz);
if (minColNnzIndex == -1)
{
//var t = sys.ToCcs().ToDenseMatrix();
throw new Exception();
}
var col = minColNnzIndex;
var c1 = sys.Equations.Where(i => i.ContainsIndex(col) && !leadingMember[i.Tag]);
var rw =
c1.MinBy(j => j.Size, null);
//sys.Equations[minRowNnzIndex];
if (dependentRows[sys.Equations.IndexOfReference(rw)])
{
throw new Exception();
}
Console.WriteLine("Pivot: {0},{1}", sys.Equations.IndexOfReference(rw), col);
//Console.ReadKey();
// eqs.Equations.Where(i => i.Size > 1).MinBy(new SparseRowLengthComparer());
var eliminator = rw;// sys.Equations[rw];
//eliminate all rows with pivot
for (var i = 0; i < sys.Equations.Length; i++)
{
var canditate = eqs[i];
if (canditate.ContainsIndex(col) && !ReferenceEquals(eliminator, canditate))
{
var oldOne = eqs[i];//to be removed
foreach (var enm in oldOne.EnumerateIndexed())
{
sys.ColumnNonzeros[enm.Item1]--;
}
var newOne = SparseRow.Eliminate(eliminator, oldOne, col);
if (newOne.CalcNnz(lastColIndex) == 0 && newOne.GetRightSideValue(lastColIndex) != 0)
{
throw new Exception("Inconsistent system");
}
if (newOne.CalcNnz(lastColIndex) == 0 && newOne.GetRightSideValue(lastColIndex) == 0)
{
//fully zero equation
dependentRows[i] = true;// newOne.Tag);
}
newOne.Tag = oldOne.Tag;
foreach (var enm in newOne.EnumerateIndexed())
{
sys.ColumnNonzeros[enm.Item1]++;
if (sys.ColumnNonzeros[enm.Item1] == 1)
{
leadingMember[i] = true;
}
}
if (sys.ColumnNonzeros.Contains(0))
{
Guid.NewGuid();
}
eqs[i] = newOne;
sys.RowNonzeros[i] = newOne.CalcNnz(lastColIndex);
}
}
leadingMember[sys.Equations.IndexOfReference(rw)] = true;
Console.WriteLine("elimination done for all eqs, Col[{0}] : {1} nnzs", col, sys.ColumnNonzeros[col]);
if (sys.ColumnNonzeros[col] != 1)
{
Guid.NewGuid();
}
}
SparseEqSystem newSys;// = new SparseEqSystem();
{
////////start final check
var tol = 1e-9;
var dependentEquationCount = sys.Equations.Count(i => i.CalcNnz(lastColIndex) == 0 && i.GetRightSideValue(lastColIndex).FEquals(0, tol));
var incosistentEquationCount = sys.Equations.Count(i => i.CalcNnz(lastColIndex) == 0 && !i.GetRightSideValue(lastColIndex).FEquals(0, tol));
var NnzMemberInCol = new int[sys.ColumnCount - 1];
foreach (var eq in sys.Equations)
{
foreach (var idx in eq.EnumerateIndexed())
{
if (idx.Item1 != lastColIndex)
{
if (!idx.Item2.FEquals(0, tol))
{
NnzMemberInCol[idx.Item1]++;
}
}
}
}
var leadingMembers = NnzMemberInCol.Count(i => i == 1);
if (leadingMembers != Math.Min(sys.ColumnCount - 1, sys.RowCount - dependentEquationCount))
{
throw new Exception();
}
if (incosistentEquationCount != 0)
{
throw new Exception();
}
var allExceptDependentEqs = sys.Equations.Where(i => !(i.CalcNnz(lastColIndex) == 0 && i.GetRightSideValue(lastColIndex).FEquals(0, tol))).ToArray();
newSys = SparseEqSystem.Generate(allExceptDependentEqs, a.ColumnCount);
}
return(newSys.ToCcs());
}