Let \(A(\lambda, \rho)\) be an \(n \times n\) matrix which is nonlinear in \(\lambda\) and \(\rho\). Consider the nonlinear eigenvalue problem \[ A(\lambda (\rho), \rho) x (\rho) = 0, \quad y (\rho)^T A(\lambda (\rho), \rho) = 0^T \] together with some desirable scaling schemes for the right and left eigenvectors \(x\) and \(y\). Following the general approach that constructs a scalar value function \(g\) whose zeros are the eigenvalues of the matrix \(A(\lambda (\rho), \rho)\) and computes these zeros by Newton's method, some new algorithms are presented. A unified analysis for such kind of methods is given in terms of the corresponding matrix equations, as well as their condition and the calculation of the partial derivatives of the eigensolutions. Numerical examples are also presented.