There are several iterative methods for solving problems of the  form in the Eigen library. The LeastSquaresConjugateGradient class is one of them, which allows us to solve linear regression problems with the conjugate gradient algorithm. The ConjugateGradient algorithm can converge more quickly to the function's minimum than regular GD but requires that matrix A is positively defined to guarantee numerical stability. The LeastSquaresConjugateGradient class has two main settings: the maximum number of iterations and a tolerance threshold value that is used as a stopping criteria as an upper bound to the relative residual error, as illustrated in the following code block:

typedef float DType;
using Matrix = Eigen::Matrix<DType, Eigen::Dynamic, Eigen::Dynamic>;
int n = 10000;
Matrix x(n,1);
Matrix y(n,1);
Eigen::LeastSquaresConjugateGradient<Matrix> gd;
gd.setMaxIterations(1000);
gd.setTolerance(0.001) ;
gd.compute(x);
auto b = dg.solve(y);

For new x inputs, we can predict new y values with matrices operations, as follows:

Eigen::Matrixxf new_x(5, 2);
new_x << 1, 1, 1, 2, 1, 3, 1, 4, 1, 5;
auto new_y = new_x.array().rowwise() * b.transpose().array();

Also, we can calculate parameter's b vector (the linear regression task solution) by solving the normal equation directly, as follows:

auto b = (x.transpose() * x).ldlt().solve(x.transpose() * y);