/// <summary>
/// Compute the Numeric Cholesky factorization, L = chol (A, [pinv parent cp]).
/// </summary>
/// <returns>Numeric Cholesky factorization</returns>
public static NumericFactorization Factorize(SparseMatrix A, SymbolicFactorization S)
{
double d, lki;
double[] Lx, x, Cx;
int top, i, p, k;
int[] Li, Lp, cp, pinv, s, c, parent, Cp, Ci;
SparseMatrix L, C;
NumericFactorization N = new NumericFactorization(); // allocate result
int n = A.n;
c = new int[n]; // get int workspace
s = new int[n];
x = new double[n]; // get double workspace
cp = S.cp;
pinv = S.pinv;
parent = S.parent;
C = pinv != null?A.PermuteSym(pinv, true) : A;
Cp = C.p;
Ci = C.i;
Cx = C.x;
N.L = L = new SparseMatrix(n, n, cp[n], true); // allocate result
Lp = L.p;
Li = L.i;
Lx = L.x;
for (k = 0; k < n; k++)
{
Lp[k] = c[k] = cp[k];
}
for (k = 0; k < n; k++) // compute L(k,:) for L*L' = C
{
// Nonzero pattern of L(k,:)
top = Common.EReach(C, k, parent, s, c); // find pattern of L(k,:)
x[k] = 0; // x (0:k) is now zero
for (p = Cp[k]; p < Cp[k + 1]; p++) // x = full(triu(C(:,k)))
{
if (Ci[p] <= k)
{
x[Ci[p]] = Cx[p];
}
}
d = x[k]; // d = C(k,k)
x[k] = 0; // clear x for k+1st iteration
// Triangular solve
for (; top < n; top++) // solve L(0:k-1,0:k-1) * x = C(:,k)
{
i = s[top]; // s [top..n-1] is pattern of L(k,:)
lki = x[i] / Lx[Lp[i]]; // L(k,i) = x (i) / L(i,i)
x[i] = 0; // clear x for k+1st iteration
for (p = Lp[i] + 1; p < c[i]; p++)
{
x[Li[p]] -= Lx[p] * lki;
}
d -= lki * lki; // d = d - L(k,i)*L(k,i)
p = c[i]++;
Li[p] = k; // store L(k,i) in column i
Lx[p] = lki;
}
// Compute L(k,k)
if (d <= 0)
{
return(null); // not pos def
}
p = c[k]++;
Li[p] = k; // store L(k,k) = sqrt (d) in column k
Lx[p] = Math.Sqrt(d);
}
Lp[n] = cp[n]; // finalize L
return(N); // success
}
/// <summary>
/// Sparse QR factorization [V,beta,pinv,R] = qr (A)
/// </summary>
static NumericFactorization Factorize(SparseMatrix A, SymbolicFactorization S)
{
double[] Beta;
int i, p, p1, top, len, col;
int m = A.m;
int n = A.n;
int[] Ap = A.p;
int[] Ai = A.i;
double[] Ax = A.x;
int[] q = S.q;
int[] parent = S.parent;
int[] pinv = S.pinv;
int m2 = S.m2;
int vnz = S.lnz;
int rnz = S.unz;
int[] leftmost = S.leftmost;
int[] w = new int[m2 + n]; // get int workspace
double[] x = new double[m2]; // get double workspace
int s = m2; // offset into w
SparseMatrix R, V;
NumericFactorization N = new NumericFactorization(); // allocate result
N.L = V = new SparseMatrix(m2, n, vnz, true); // allocate result V
N.U = R = new SparseMatrix(m2, n, rnz, true); // allocate result R
N.B = Beta = new double[n]; // allocate result Beta
int[] Rp = R.p;
int[] Ri = R.i;
double[] Rx = R.x;
int[] Vp = V.p;
int[] Vi = V.i;
double[] Vx = V.x;
for (i = 0; i < m2; i++)
{
w[i] = -1; // clear w, to mark nodes
}
rnz = 0; vnz = 0;
for (int k = 0; k < n; k++) // compute V and R
{
Rp[k] = rnz; // R(:,k) starts here
Vp[k] = p1 = vnz; // V(:,k) starts here
w[k] = k; // add V(k,k) to pattern of V
Vi[vnz++] = k;
top = n;
col = q != null ? q[k] : k;
for (p = Ap[col]; p < Ap[col + 1]; p++) // find R(:,k) pattern
{
i = leftmost[Ai[p]]; // i = min(find(A(i,q)))
for (len = 0; w[i] != k; i = parent[i]) // traverse up to k
{
//len++;
w[s + len++] = i;
w[i] = k;
}
while (len > 0)
{
--top;
--len;
w[s + top] = w[s + len]; // push path on stack
}
i = pinv[Ai[p]]; // i = permuted row of A(:,col)
x[i] = Ax[p]; // x (i) = A(:,col)
if (i > k && w[i] < k) // pattern of V(:,k) = x (k+1:m)
{
Vi[vnz++] = i; // add i to pattern of V(:,k)
w[i] = k;
}
}
for (p = top; p < n; p++) // for each i in pattern of R(:,k)
{
i = w[s + p]; // R(i,k) is nonzero
ApplyHouseholder(V, i, Beta[i], x); // apply (V(i),Beta(i)) to x
Ri[rnz] = i; // R(i,k) = x(i)
Rx[rnz++] = x[i];
x[i] = 0;
if (parent[i] == k)
{
vnz = V.Scatter(i, 0, w, null, k, V, vnz);
}
}
for (p = p1; p < vnz; p++) // gather V(:,k) = x
{
Vx[p] = x[Vi[p]];
x[Vi[p]] = 0;
}
Ri[rnz] = k; // R(k,k) = norm (x)
Rx[rnz++] = CreateHouseholder(Vx, p1, ref Beta[k], vnz - p1); // [v,beta]=house(x)
}
Rp[n] = rnz; // finalize R
Vp[n] = vnz; // finalize V
return(N); // success
}
// [L,U,pinv]=lu(A, [q lnz unz]). lnz and unz can be guess
static NumericFactorization Factorize(SparseMatrix A, SymbolicFactorization S, double tol)
{
int i, n = A.n;
int[] q = S.q;
int lnz = S.lnz;
int unz = S.unz;
int[] pinv;
NumericFactorization N = new NumericFactorization(); // allocate result
SparseMatrix L, U;
N.L = L = new SparseMatrix(n, n, lnz, true); // allocate result L
N.U = U = new SparseMatrix(n, n, unz, true); // allocate result U
N.pinv = pinv = new int[n]; // allocate result pinv
double[] x = new double[n]; // get double workspace
int[] xi = new int[2 * n]; // get int workspace
for (i = 0; i < n; i++)
{
pinv[i] = -1; // no rows pivotal yet
}
lnz = unz = 0;
int ipiv, top, p, col;
double pivot, a, t;
int[] Li, Ui, Lp = L.p, Up = U.p;
double[] Lx, Ux;
for (int k = 0; k < n; k++) // compute L(:,k) and U(:,k)
{
// Triangular solve
Lp[k] = lnz; // L(:,k) starts here
Up[k] = unz; // U(:,k) starts here
if ((lnz + n > L.nzmax && !L.Resize(2 * L.nzmax + n)) ||
(unz + n > U.nzmax && !U.Resize(2 * U.nzmax + n)))
{
return(null);
}
Li = L.i;
Lx = L.x;
Ui = U.i;
Ux = U.x;
col = q != null ? (q[k]) : k;
top = Triangular.SolveSp(L, A, col, xi, x, pinv, true); // x = L\A(:,col)
// Find pivot
ipiv = -1;
a = -1;
for (p = top; p < n; p++)
{
i = xi[p]; // x(i) is nonzero
if (pinv[i] < 0) // row i is not yet pivotal
{
if ((t = Math.Abs(x[i])) > a)
{
a = t; // largest pivot candidate so far
ipiv = i;
}
}
else // x(i) is the entry U(pinv[i],k)
{
Ui[unz] = pinv[i];
Ux[unz++] = x[i];
}
}
if (ipiv == -1 || a <= 0)
{
return(null);
}
if (pinv[col] < 0 && Math.Abs(x[col]) >= a * tol)
{
ipiv = col;
}
// Divide by pivot
pivot = x[ipiv]; // the chosen pivot
Ui[unz] = k; // last entry in U(:,k) is U(k,k)
Ux[unz++] = pivot;
pinv[ipiv] = k; // ipiv is the kth pivot row
Li[lnz] = ipiv; // first entry in L(:,k) is L(k,k) = 1
Lx[lnz++] = 1;
for (p = top; p < n; p++) // L(k+1:n,k) = x / pivot
{
i = xi[p];
if (pinv[i] < 0) // x(i) is an entry in L(:,k)
{
Li[lnz] = i; // save unpermuted row in L
Lx[lnz++] = x[i] / pivot; // scale pivot column
}
x[i] = 0; // x [0..n-1] = 0 for next k
}
}
// Finalize L and U
Lp[n] = lnz;
Up[n] = unz;
Li = L.i; // fix row indices of L for final pinv
for (p = 0; p < lnz; p++)
{
Li[p] = pinv[Li[p]];
}
L.Resize(0); // remove extra space from L and U
U.Resize(0);
return(N); // success
}