In this paper, a new algorithm for the eigenvalue problem of matrices is given. Namerical examples show that it could be a remarkable approach for practical purposes. Some open problems are listed.

The quantum phase estimation algorithm efficiently finds the eigenvalue of a given eigenvector but requires fully coherent evolution.

Abstract Computing the roots of a scalar polynomial, or the eigenvalues of a matrix polynomial, expressed in the Chebyshev basis {𝑇𝑘 (𝑥)} is a fundamental problem that arises in many applications.