powermetoden

The power method is a fundamental algorithm in numerical linear algebra used to find the dominant eigenvalue and its corresponding eigenvector of a given square matrix. It is an iterative method that converges to the eigenvector associated with the eigenvalue of largest absolute value.

The process begins with an arbitrary non-zero vector, often referred to as the initial guess. In each iteration, this vector is multiplied by the matrix. The resulting vector is then normalized, typically by dividing it by its largest element or its Euclidean norm. This normalized vector becomes the input for the next iteration. As the iterations proceed, the vector progressively aligns itself with the dominant eigenvector.

The dominant eigenvalue can be estimated by observing the scaling factor required to normalize the vector in each step. For instance, if the vector is normalized by dividing by its largest element, that element's value in successive iterations will converge to the dominant eigenvalue. If normalization is done using the Euclidean norm, the Rayleigh quotient can be used to estimate the eigenvalue.

The power method is relatively simple to implement and understand. However, its convergence rate can be slow if the absolute values of the two largest eigenvalues are close to each other. It is guaranteed to converge if the matrix has a unique eigenvalue with the largest absolute value and the initial guess has a non-zero component in the direction of the corresponding eigenvector. Modifications to the basic power method exist to accelerate convergence and handle cases with multiple eigenvalues of the same largest absolute value.