public void TestQR()
{
IMatrix Q, R;
QR.Factorize(A, out Q, out R);
IMatrix Api = Q.Multiply(R);
for (int r = 0; r < A.RowCount; ++r)
{
for (int c = 0; c < A.ColCount; ++c)
{
Assert.True(System.Math.Abs(A[r, c] - Api[r, c]) < 1e-10);
}
}
Assert.Equal(A, Q.Multiply(R));
}
public override double[] Solve()
{
int m = A.RowCount;
int n = A.ColCount;
Debug.Assert(m >= n);
IVector s = new SparseVector(n);
IVector sy = new SparseVector(n);
for (int i = 0; i < n; ++i)
{
s[i] = 0;
}
IVector t = new SparseVector(m);
for (int i = 0; i < m; ++i)
{
t[i] = 0;
}
double[] g = new double[m];
double[] gprime = new double[m];
IMatrix Q;
IMatrix R;
QR.Factorize(A, out Q, out R); // A is m x n, Q is m x n orthogonal matrix, R is n x n (R will be upper triangular matrix if m == n)
IMatrix Qt = Q.Transpose();
IVector W = null;
for (int j = 0; j < mMaxIters; ++j)
{
IVector z = new SparseVector(m);
for (int k = 0; k < m; ++k)
{
g[k] = mLinkFunc.GetInvLink(t[k]);
gprime[k] = mLinkFunc.GetInvLinkDerivative(t[k]);
z[k] = t[k] + (b[k] - g[k]) / gprime[k];
}
W = new SparseVector(m);
double w_kk_min = double.MaxValue;
for (int k = 0; k < m; ++k)
{
double g_variance = GetVariance(g[k]);
double w_kk = gprime[k] * gprime[k] / (g_variance);
W[k] = w_kk;
w_kk_min = System.Math.Min(w_kk, w_kk_min);
}
if (w_kk_min < System.Math.Sqrt(double.Epsilon))
{
Console.WriteLine("Warning: Tiny weights encountered, min(diag(W)) is too small");
}
IVector s_old = s;
IMatrix WQ = new SparseMatrix(m, n); // W * Q
IVector Wz = new SparseVector(m); // W * z
for (int k = 0; k < m; k++)
{
Wz[k] = z[k] * W[k];
for (int k2 = 0; k2 < m; ++k2)
{
WQ[k, k2] = Q[k, k2] * W[k];
}
}
IMatrix QtWQ = Qt.Multiply(WQ); // a n x n positive definite matrix, therefore can apply Cholesky
IVector QtWz = Qt.Multiply(Wz);
IMatrix L;
Cholesky.Factorize(QtWQ, out L);
IMatrix Lt = L.Transpose();
// (Qt * W * Q) * s = Qt * W * z;
// L * Lt * s = Qt * W * z (Cholesky factorization on Qt * W * Q)
// L * sy = Qt * W * z, Lt * s = sy
// Now forward solve sy for L * sy = Qt * W * z
// Now backward solve s for Lt * s = sy
s = new SparseVector(n);
for (int i = 0; i < n; ++i)
{
s[i] = 0;
sy[i] = 0;
}
//forward solve sy for L * sy = Qt * W * z
//Console.WriteLine(L);
for (int i = 0; i < n; ++i)
{
double cross_prod = 0;
for (int k = 0; k < i; ++k)
{
cross_prod += L[i, k] * sy[k];
}
sy[i] = (QtWz[i] - cross_prod) / L[i, i];
}
//backward solve s for U * s = sy
for (int i = n - 1; i >= 0; --i)
{
double cross_prod = 0;
for (int k = i + 1; k < n; ++k)
{
cross_prod += Lt[i, k] * s[k];
}
s[i] = (sy[i] - cross_prod) / Lt[i, i];
}
t = Q.Multiply(s);
double cost = (s_old.Minus(s)).Norm(2);
if (j % 100 == 0)
{
Console.WriteLine("Iteration: {0}, Cost: {1}", j, cost);
}
if (cost < mTol)
{
break;
}
}
mX = new double[n];
//backsolve x for R * x = Qt * t
IVector c = Qt.Multiply(t);
for (int i = n - 1; i >= 0; --i) // since m >= n
{
double cross_prod = 0;
for (int j = i + 1; j < n; ++j)
{
cross_prod += R[i, j] * mX[j];
}
mX[i] = (c[i] - cross_prod) / R[i, i];
}
UpdateStatistics(W);
return(X);
}
