Cuthill–McKee algorithm

Abstract: An approximate minimum degree (AMD) ordering algorithm for preordering a symmetric sparse matrix prior to numerical factorization is presented. We use techniques based on the quotient graph for matrix factorization that allow us to obtain computationally cheap bounds for the minimum degree. We show that these bounds are often equal to the actual degree. The resulting algorithm is typically much faster than previous minimum degree ordering algorithms and produces results that are comparable in quality with the best orderings from other minimum degree algorithms.

Abstract: We consider the problem of approximating a given m × n matrix A by another matrix of specified rank k, which is smaller than m and n. The Singular Value Decomposition (SVD) can be used to find the "best" such approximation. However, it takes time polynomial in m, n which is prohibitive for some modern applications. In this article, we develop an algorithm that is qualitatively faster, provided we may sample the entries of the matrix in accordance with a natural probability distribution. In many applications, such sampling can be done efficiently. Our main result is a randomized algorithm to find the description of a matrix D* of rank at most k so that holds with probability at least 1 − δ (where v·vF is the Frobenius norm). The algorithm takes time polynomial in k,1/e, log(1/δ) only and is independent of m and n. In particular, this implies that in constant time, it can be determined if a given matrix of arbitrary size has a good low-rank approximation.

Abstract: A new algorithm for reducing the bandwidth and profile of a sparse matrix is described. Extensive testing on finite element matrices indicates that the algorithm typically produces bandwidth and profile which are comparable to those of the commonly-used reverse Cuthill–McKee algorithm, yet requires significantly less computation time.