Tridiagonal matrix algorithm

Abstract: We present a method for the determination of eigenvalues of a symmetric tridiagonal matrix which combines Givens' Sturm bisection [4, 5] with interpolation, to accelerate convergence in high precision cases. By using an appropriate root of the absolute value of the determinant to derive the interpolation weight, results are obtained which compare favorably with the Barth, Martin, Wilkinson algorithm [1].

Abstract: In this paper we describe a new tridiagonal equation solver, based on a rank-one updating strategy and the repeated partitioning of the system matrix into 2 × 2 submatrices. On this basis, a recursive decoupling method is developed [2,3], which operates on the tridiagonal linear system, enabling the solution to be expressed in explicit form and solved independently on a multiprocessor system. We will show, in fact, that the Recursive Decoupling method is intrinsically parallel and can be implemented as an efficient parallel algorithm.

Abstract: Gaussian elimination with partial pivoting is the standard numerical algorithm for solving unstructured linear systems. Here it is shown that Gaussian elimination with partial pivoting or complete pivoting is log-space complete for P. This provides theoretical evidence that these algorithms cannot be efficiently implemented on a highly parallel computer with a large number of processors. Since other algorithms for linear systems that are efficient on parallel computers are already known, this suggests that elimination-based approaches should not be pursued in a parallel environment with many processors.

Abstract: For a given real or complex polynomial p of degree n we modify the Euclidean algorithm to find a general tridiagonal matrix representation T of the monic version of p and then use the tridiagonal DQR eigenvalue algorithm on T in order to find all roots ofp with their multiplicities in O(n 2) operations and 0(n) storage. We include details of the implementation and comparisons with several, standard and recent, essentially 0(n 3) polynomial root finders.