/// <summary>
/// Solves the system of equations analytically.
/// </summary>
/// <param name="b">The b.</param>
/// <param name="IsASymmetric">if set to <c>true</c> [a is symmetric].</param>
/// <param name="potentialDiagonals">The potential diagonals.</param>
/// <returns>System.Double[].</returns>
public IList <double> SolveAnalytically(IList <double> b, bool IsASymmetric = false)
{
if (IsASymmetric)
{
if (ValuesChanged || TopologyChanged)
{
MatrixInCCS = convertToCCS(this);
}
if (TopologyChanged)
{
symbolicFactorizationMat = CSparse.SymbolicAnalysisLDL(MatrixInCCS);
}
if (ValuesChanged || TopologyChanged)
{
CSparse.FactorizeLDL(MatrixInCCS, symbolicFactorizationMat, out D, out FactorizationMatrix);
}
TopologyChanged = ValuesChanged = false;
return(CSparse.SolveLDL(b, FactorizationMatrix, D, symbolicFactorizationMat.InversePermute));
}
else
{
var ccs = convertToCCS(this);
var columnPermutation = ApproximateMinimumDegree.Generate(
new SymbolicColumnStorage(ccs), NumCols);
CompressedColumnStorage L, U;
int[] pinv;
// Numeric LU factorization
CSparse.FactorizeLU(ccs, columnPermutation, out L, out U, out pinv);
var x = CSparse.ApplyInverse(pinv, b, NumCols); // x = b(p)
CSparse.SolveLower(L, x); // x = L\x.
CSparse.SolveUpper(U, x); // x = U\x.
return(CSparse.ApplyInverse(columnPermutation, x, NumCols)); // b(q) = x
}
}
/// <summary>
/// Solve Gx=b(:,k), where G is either upper (lo=false) or lower (lo=true)
/// triangular.
/// </summary>
/// <param name="G">lower or upper triangular matrix in column-compressed form</param>
/// <param name="B">right hand side, b=B(:,k)</param>
/// <param name="k">use kth column of B as right hand side</param>
/// <param name="xi">size 2*n, nonzero pattern of x in xi[top..n-1]</param>
/// <param name="x">size n, x in x[xi[top..n-1]]</param>
/// <param name="pinv">mapping of rows to columns of G, ignored if null</param>
/// <param name="lo">true if lower triangular, false if upper</param>
/// <returns>top, -1 in error</returns>
private static int SolveSp(CompressedColumnStorage G, CompressedColumnStorage B,
int k, int[] xi, double[] x, int[] pinv, bool lo)
{
if (xi == null || x == null)
{
return(-1);
}
var gColPointers = G.ColumnPointers;
var gRowIndices = G.RowIndices;
var gValues = G.Values;
var bColPointers = B.ColumnPointers;
var bRowIndices = B.RowIndices;
var bValues = B.Values;
var n = G.ncols;
// xi[top..n-1]=Reach(B(:,k))
var top = ApproximateMinimumDegree.Reach(gColPointers, gRowIndices, bColPointers, bRowIndices, n, k, xi,
pinv);
for (var i = top; i < n; i++)
{
x[xi[i]] = 0; // clear x
}
for (var i = bColPointers[k]; i < bColPointers[k + 1]; i++)
{
x[bRowIndices[i]] = bValues[i]; // scatter B
}
for (var px = top; px < n; px++)
{
var j = xi[px];
var J = pinv != null ? pinv[j] : j;
if (J < 0)
{
continue; // column J is empty
}
x[j] /= gValues[lo ? gColPointers[J] : gColPointers[J + 1] - 1]; // x(j) /= G(j,j)
var p = lo ? gColPointers[J] + 1 : gColPointers[J]; // lo: L(j,j) 1st entry
var q = lo ? gColPointers[J + 1] : gColPointers[J + 1] - 1;
for (; p < q; p++)
{
x[gRowIndices[p]] -= gValues[p] * x[j]; // x(i) -= G(i,j) * x(j)
}
}
// Return top of stack.
return(top);
}
/// <summary>
/// Symbolics the analysis LDL.
/// </summary>
/// <param name="A">a.</param>
/// <returns>SymbolicFactorization.</returns>
internal static SymbolicFactorization SymbolicAnalysisLDL(CompressedColumnStorage A)
{
var n = A.ncols;
var aColPointers = A.ColumnPointers;
var aRowIndices = A.RowIndices;
// P = amd(A+A') or natural
var permute = ApproximateMinimumDegree.Generate(A);
var invPermute = Invert(permute);
// Output: column pointers and elimination tree.
var lp = new int[n + 1];
var parent = new int[n];
// Workspace
var lnz = new int[n];
var flag = new int[n];
for (var k = 0; k < n; k++)
{
// L(k,:) pattern: all nodes reachable in etree from nz in A(0:k-1,k)
parent[k] = -1; // parent of k is not yet known
flag[k] = k; // mark node k as visited
lnz[k] = 0; // count of nonzeros in column k of L
var kk = permute != null ? permute[k] : k;
var p2 = aColPointers[kk + 1];
int p;
for (p = aColPointers[kk]; p < p2; p++)
{
// A(i,k) is nonzero (original or permuted A)
var i = invPermute != null ? invPermute[aRowIndices[p]] : aRowIndices[p];
if (i < k)
{
// follow path from i to root of etree, stop at flagged node
for (; flag[i] != k; i = parent[i])
{
// find parent of i if not yet determined
if (parent[i] == -1)
{
parent[i] = k;
}
lnz[i]++; // L(k,i) is nonzero
flag[i] = k; // mark i as visited
}
}
}
}
// construct Lp index array from Lnz column counts
lp[0] = 0;
for (var k = 0; k < n; k++)
{
lp[k + 1] = lp[k] + lnz[k];
}
return(new SymbolicFactorization
{
ParentIndices = parent,
ColumnPointers = lp,
PermutationVector = permute,
InversePermute = invPermute
});
}