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Square matrix
About: Square matrix is a research topic. Over the lifetime, 5000 publications have been published within this topic receiving 92428 citations.
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TL;DR: This work generalizes the basis to allow the deficient case, characterized as one that has columns fewer than rows of the coefficient matrix, and develops a two-phase method based on such a basis, the number of whose columns varies dynamically in the solution process.
Abstract: h one of the most important and fundamental concepts in the simplex methodology, basis is restricted to being a square matrix of the order exactly equal to the number of rows of the coefficient matrix. Such inflexibility might have been the source of too many zero steps taken by the simplex method in solving real-world linear programming problems, which are usually highly degen- erate. To dodge this difficulty, we first generalize the basis to allow the deficient case, characterized as one that has columns fewer than rows of the coefficient matrix. Variations of the primal and dual simplex procedures are then made, and used to form a two-phase method based on such a basis, the number of whose columns varies dynamically in the solution process. Generally speaking, the more degenerate a problem to be handled is, the fewer columns the basis will have; so, thii renders the possibility of efficiently solving highly degenerate problems. We also provide a valuable insight into the distinctive and favorable behavior of the proposed method by reporting our computational experiments. @ 1998 Elsevier Science Ltd. All rights reserved.
67 citations
TL;DR: An extension of the Diffie-Hellman public key distribution system to matrix rings is described and it is shown that the number of possible secret keys is much greater for a given prime p compared to the original system.
Abstract: An extension of the Diffie-Hellman public key distribution system to matrix rings is described. Using rings of non-singular matrices over Z/pZ and upper triangular matrices with invertible elements along the diagonal over Z/pZ, it is shown that the number of possible secret keys is much greater for a given prime p compared to the original system. An outline of a method to construct the base matrix used in the system is given.
67 citations
TL;DR: The proposed method proves to be much faster and has significantly better accuracy than the already proposed methods, while works for full and sparse matrices.
67 citations
Journal Article•
TL;DR: In this paper, two approaches to circumvent this difficulty are presented: zero-padding the rectangular matrices to make them square and extending the rectangular matrix to a square circulant.
Abstract: Tikhonov regularization of linear discrete ill-posed prob lems often is applied with a finite differ- ence regularization operator that approximates a low-order derivative. These operators generally are represented by a banded rectangular matrix with fewer rows than columns. They therefore cannot be applied in iterative meth- ods that are based on the Arnoldi process, which requires the regularization operator to be represented by a square matrix. This paper discusses two approaches to circumvent this difficulty: zero-padding the rectangular matrices to make them square and extending the rectangular matrix to a square circulant. We also describe how to com- bine these operators by weighted averaging and with orthogonal projection. Applications to Arnoldi and Lanczos bidiagonalization-based Tikhonov regularization, as well as to truncated iteration with a range-restricted minimal residual method, are presented.
67 citations
TL;DR: A substantial acceleration of randomized computation of scalar, univariate, and multivariate matrix determinants, in terms of the output-sensitive bit operation complexity bounds, including computation modulo a product of random primes from a fixed range is accelerated.
67 citations