Potensmetoden

The Potensmetoden, often translated as the "power method," is a numerical algorithm used to find the dominant eigenvalue and corresponding eigenvector of a square matrix. This method is iterative and relies on repeated multiplication of a matrix by an arbitrary non-zero vector.

The core idea of the Potensmetoden is that as a matrix is repeatedly applied to a vector, the resulting vector will increasingly align itself with the direction of the eigenvector associated with the largest (in magnitude) eigenvalue. The ratio of corresponding elements in successive iterations of the vector converges to this dominant eigenvalue.

To implement the method, one starts with an initial guess for an eigenvector, usually a vector with all ones or a random vector. This vector is then multiplied by the matrix. The resulting vector is then normalized, typically by dividing it by its largest component or its Euclidean norm, to prevent its components from becoming too large or too small. This normalized vector then becomes the input for the next iteration. This process is repeated until the vector converges, meaning that the difference between successive vectors is below a predefined tolerance. The eigenvalue is then approximated by the ratio of corresponding components of the vector from two consecutive iterations, or by the normalization factor.

The Potensmetoden is computationally efficient for large, sparse matrices, especially when only the dominant eigenvalue is of interest. However, it converges slowly if the dominant eigenvalue is close in magnitude to the next largest eigenvalue. It also requires the dominant eigenvalue to be unique.