/// <summary>
/// Solves Ax=b where A is symmetric positive definite; b is overwritten with
/// solution.
/// </summary>
/// <param name="b">right hand side, b is overwritten with solution</param>
/// <returns>true if successful, false on error</returns>
public bool Solve(Complex[] b)
{
if (b == null || numFactor == null)
{
return(false); // check inputs
}
Complex[] x = new Complex[n];
Common.InversePermute(symFactor.pinv, b, x, n); // x = P*b
Triangular.SolveL(numFactor.L, x); // x = L\x
Triangular.SolveLt(numFactor.L, x); // x = L'\x
Common.Permute(symFactor.pinv, x, b, n); // b = P'*x
return(true);
}
/// <summary>
/// Solves Ax=b, where A is square and nonsingular. b overwritten with
/// solution. Partial pivoting if tol = 1.
/// </summary>
/// <param name="b">size n, b on input, x on output</param>
/// <returns>true if successful, false on error</returns>
public bool Solve(Complex[] b)
{
if (b == null || numFactor == null)
{
return(false); // check inputs
}
Complex[] x = new Complex[n]; // get workspace
Common.InversePermute(numFactor.pinv, b, x, n); // x = b(p)
Triangular.SolveL(numFactor.L, x); // x = L\x
Triangular.SolveU(numFactor.U, x); // x = U\x
Common.InversePermute(symFactor.q, x, b, n); // b(q) = x
return(true);
}
/// <summary>
/// Solve a least-squares problem (min ||Ax-b||_2, where A is m-by-n with m
/// >= n) or underdetermined system (Ax=b, where m < n)
/// </summary>
/// <param name="b">size max(m,n), b (size m) on input, x(size n) on output</param>
/// <returns>true if successful, false on error</returns>
public bool Solve(Complex[] b)
{
Complex[] x;
if (b == null)
{
return(false); // check inputs
}
if (m >= n)
{
x = new Complex[symFactor != null ? symFactor.m2 : 1]; // get workspace
Common.InversePermute(symFactor.pinv, b, x, m); // x(0:m-1) = b(p(0:m-1)
for (int k = 0; k < n; k++) // apply Householder refl. to x
{
ApplyHouseholder(numFactor.L, k, numFactor.B[k], x);
}
Triangular.SolveU(numFactor.U, x); // x = R\x
Common.InversePermute(symFactor.q, x, b, n); // b(q(0:n-1)) = x(0:n-1)
}
else
{
x = new Complex[symFactor != null ? symFactor.m2 : 1]; // get workspace
Common.Permute(symFactor.q, b, x, m); // x(q(0:m-1)) = b(0:m-1)
Triangular.SolveUt(numFactor.U, x); // x = R'\x
for (int k = m - 1; k >= 0; k--) // apply Householder refl. to x
{
ApplyHouseholder(numFactor.L, k, numFactor.B[k], x);
}
Common.Permute(symFactor.pinv, x, b, n); // b(0:n-1) = x(p(0:n-1))
}
return(true);
}
// [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
Complex[] x = new Complex[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;
Complex pivot;
double a, t;
int[] Li, Ui, Lp = L.p, Up = U.p;
Complex[] 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 = Complex.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 && Complex.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
}