/// <summary> /// Given we want to find x such as A * x = b /// 1. Decompose A = L * U, where L is the lower triangular and U is the upper triangular /// 2. We have L * U * x = b, which can be rewritten as L * y = b, where U * x = y /// 3. Solve y for L * y = b using forward substitution /// 4. Solve x for U * x = y using backward substitution /// </summary> /// <param name="A"></param> /// <param name="b"></param> /// <returns></returns> public static IVector Solve(IMatrix A, IVector b) { IMatrix L; Cholesky.Factorize(A, out L); IMatrix U = L.Transpose(); // upper triangular matrix IVector y = ForwardSubstitution.Solve(L, b); IVector x = BackwardSubstitution.Solve(U, y); return(x); }
/// <summary> /// This is used for data fitting / regression /// A is a m x n matrix, where m >= n /// b is a m x 1 column vector /// The method solves for x, which is a n x 1 column vectors such that A * x is closest to b /// /// The method works as follows: /// 1. Let C = A.transpose * A, we have A.transpose * A * x = C * x = A.transpose * b /// 2. Decompose C : C = L * L.transpose = L * U, we have L * U * x = A.transpose * b /// 3. Let z = U * x, we have L * z = A.transpose * b /// 4. Solve z using forward substitution /// 5. Solve x from U * x = z using backward substitution /// </summary> /// <param name="A"></param> /// <param name="b"></param> /// <returns></returns> public static IVector SolveLeastSquare(IMatrix A, IVector b) { IMatrix At = A.Transpose(); IMatrix C = At.Multiply(A); //C is a n x n matrix IVector d = At.Multiply(b); IMatrix L; Cholesky.Factorize(C, out L); IMatrix U = L.Transpose(); IVector z = ForwardSubstitution.Solve(L, d); IVector x = BackwardSubstitution.Solve(U, z); return(x); }