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); }