/// <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>
/// Solve x for Ax = b, where A is a invertible matrix
///
/// The method works as follows:
/// 1. QR factorization A = Q * R
/// 2. Let: c = Q.transpose * b, we have R * x = c, where R is a triangular matrix if A is n x n
/// 3. Solve x using backward substitution
/// </summary>
/// <param name="A">An m x n matrix with linearly independent columns, where m >= n</param>
/// <param name="b">An m x 1 column vector </param>
/// <returns>An n x 1 column vector, x, such that Ax = b when m = n and Ax ~ b when m > n </returns>
public static IVector Solve(IMatrix A, IVector b)
{
// Q is a m x m matrix, R is a m x n matrix
// Q = [Q1 Q2] Q1 is a m x n matrix, Q2 is a m x (m-n) matrix
// R = [R1; 0] R1 is a n x n upper triangular matrix matrix
// A = Q * R = Q1 * R1
IMatrix Q, R;
QR.Factorize(A, out Q, out R);
IVector c = Q.Transpose().Multiply(b);
IVector x = BackwardSubstitution.Solve(R, c);
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);
}
/// <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 = Q * R, we have Q * R * x = A.transpose * b
/// 3. Multiply both side by Q.transpose = Q.inverse, we have Q.transpose * Q * R * x = Q.transpose * A.transpose * b
/// 4. Since Q.tranpose * Q = I, we have R * x = Q.transpose * A.transpose * b
/// 5. Solve x from R * x = Q.transpose * A.transpose * b 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
IMatrix Q, R;
QR.Factorize(C, out Q, out R);
IMatrix Qt = Q.Transpose();
IVector d = Qt.Multiply(At).Multiply(b);
IVector x = BackwardSubstitution.Solve(R, d);
return(x);
}
