In this paper, we describe progress on the development of algorithms for triangular decomposition of approximate systems. We begin with the treatment of linear, homogeneous systems with positive-dimensional solution spaces, and approximate coefficients. We use the Singular Value Decomposition to decompose such systems into a stable form, and discuss condition numbers for approximate triangular ...

In this paper, we investigate GPU based parallel triangular solvers systematically. The parallel triangular solvers are fundamental to incomplete LU factorization family preconditioners and algebraic multigrid solvers. We develop a new matrix format suitable for GPU devices. Parallel lower triangular solvers and upper triangular solvers are developed for this new data structure. With these solv...

in this paper, an effective and simple numerical method is proposed for solving systems of integral equations using radial basis functions (rbfs). we present an algorithm based on interpolation by radial basis functions including multiquadratics (mqs), using legendre-gauss-lobatto nodes and weights. also a theorem is proved for convergence of the algorithm. some numerical examples are presented...

Computer memory is linear and contiguous, while the values that enter it are always structured. Typically, is array of ndimensions, upper and lower triangular matrices, or symmetric and skewed symmetric matrices. Functions that map the upper and lower triangular matrices exist in most literatures, however its applications are limited since they cannot handle all order of traversals. This paper ...

We present a zero decomposition theorem and an algorithm based on Wu’s method, which computes a zero decomposition with multiplicity for a given zerodimensional polynomial system. If the system satisfies some condition, the zero decomposition is of triangular form.

We consider wreath product decompositions for semigroups of triangular matrices. We exhibit an explicit wreath product decomposition for the semigroup of all n× n upper triangular matrices over a given field k, in terms of aperiodic semigroups and affine groups over k. In the case that k is finite this decomposition is optimal, in the sense that the number of group terms is equal to the group c...

We describe and analyze a novel symmetric triangular factorization algorithm. The algorithm is essentially a block version of Aasen’s triangular tridiagonalization. It factors a dense symmetric matrix A as the product A = PLTLP where P is a permutation matrix, L is lower triangular, and T is block tridiagonal and banded. The algorithm is the first symmetric-indefinite communication-avoiding fac...

LetX = LσU be the Gelfand-Naimark decomposition of X ∈ GLn(C), where L is unit lower triangular, σ is a permutation matrix, and U is upper triangular. Call u(X) := diagU the u-component of X. We show that in a Zariski dense open subset of the ω-orbit of certain Bruhat decomposition, lim m→∞ |u(X)| = diag (|λω(1)|, · · · , |λω(n)|). The other situations where lim m→∞ |u(X)| converge to different...