I have noticed, it is possible to evaluate one eigenvalue and the coresponding eigenvector by newton's method as it has been explained on :

https://www.vulcanhammer.org/2012/12/30/use-of-netwons-method-to-determine-matrix-eigenvalues/

Iteration in a loop, knowing the jacobian matrix, converge an initial guess to the closest "vector of answer".

The equation to solve is as: M[x]=\lambda [x] but the [vector of answer] arranged as X(i)=[x1(i) x2(i) x3(i) ... xn(i) \lambda (i)] and [x]=[x1(i) x2(i) x3(i) ... xn(i)]. M is a n by n matrix. It has been mentioned " The method is better at finding eigenvalues than finding eigenvectors".

I need to know if this method has been developed to acheive (correct by iteration) all eigenvalue-eigenvectors simultaneously. I believe the simulatenous correction in Newton's method is the most strongest point about this wonderful numerical method.

I would appreciate, if experts would kindly express their opinions on this problem.

Regards

Vahid