hatalomiteráció

hatalomiteráció, also known as the power iteration, is a simple algorithm used in numerical linear algebra to approximate the dominant eigenvalue and its corresponding eigenvector of a square matrix. The method starts with an arbitrary non-zero vector and repeatedly multiplies it by the matrix, normalizing the result at each step. Over successive iterations, the vector converges to the eigenvector associated with the eigenvalue of greatest magnitude, while the Rayleigh quotient of the iterates approaches the dominant eigenvalue.

The convergence rate of the power iteration depends on the ratio between the absolute values of the largest and the second-largest eigenvalues. If this ratio is close to one, many iterations are required for accurate results. The method can be inefficient for large sparse matrices, where more advanced techniques such as the Arnoldi or Lanczos algorithms are preferred. Nevertheless, power iteration remains a useful pedagogical tool and a building block for more sophisticated eigensolvers, particularly when only the leading eigenpair is needed.

In practice, the algorithm is implemented in numerical libraries across scientific computing environments, often with safeguards like orthogonal re‑normalization to maintain numerical stability. The term has Hungarian origins where “hatalom” means “power” and “iteráció” means “iteration,” reflecting the method’s iterative amplification of a matrix’s dominant eigenvector.