Tridiagonal matrix algorithm

Abstract: We introduce particular Poisson algebras designed to describe (after quantization) generalized eigenvalue problems for two tridiagonal matrices. These algebras are naturally implemented upon various elliptic curves.

Abstract: A block parallel partitioning method for computing the eigenvalues of symmetric tridiagonal matrix is presented. The algorithm is based on partitioning, in a way that ensures load balance during computation. This method is applicable to both shared memory- and distributed memory-MIMD systems. Compared with other parallel tridiagonal eigenvalue algorithms existing in the literature, the proposed algorithm achieves a higher speedup of O( p ) on a parallel computer with p -fold parallelism, which is linear, and the data communication between processors is less than that required for other methods. The results were tested and evaluated on an MIMD machine, and were within 62% to 98% of the predicted performance.

Abstract: In many numerical problems the solution of tridiagonal systems of equations consumes an important part of the computation time. For their efficient solution on vector or parallel computers the recursive Gauss algorithm has often to be replaced by a method with a higher degree of parallelism. Among other methods cyclic reduction has been widely discussed. In the present paper we discuss some aspects of the numerical treatment of tridiagonal systems with interval coefficients which arise, for example, as part of interval arithmetic Newton-like methods combined with a “fast Poisson solver” [8, 9]. We have discussed interval arithmetic cyclic reduction in [10]. Here we introduce a truncated version dedicated to reduce the computation time. In contrast to the non-interval case we have to preserve inclusion properties. Instead of really truncating steps, we replace them by easily computable intervals. In contrast to the non-interval case we can “truncate” both the reduction and the solution phase.

Abstract: In this short note, we present a novel method for computing exact lower and upper bounds of eigenvalues of a symmetric tridiagonal interval matrix. Compared to the known methods, our approach is fast, simple to present and to implement, and avoids any assumptions. Our construction explicitly yields those matrices for which particular lower and upper bounds are attained.