Spline filters and their corresponding robust filters are commonly used filters recommended in ISO (the International Organization for Standardization) standards for surface evaluation. Generally, these linear and non-linear spline filters, composed of symmetric, positive-definite matrices, are solved in an iterative fashion based on a Cholesky decomposition. They have been demonstrated to be r...

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 ...

We study the eigenvalues and eigenspaces of the quadratic matrix polynomial Mλ + sDλ + K as s → ∞, where M and K are symmetric positive definite and D is symmetric positive semi-definite. The work is motivated by its application to modal analysis of finite element models with strong linear damping. Our results yield a mathematical explanation of why too strong damping may lead to practically un...

Exploiting Lower Precision Arithmetic in Solving Symmetric Positive Definite Linear Systems and Least Squares Problems

where A is symmetric positive definite (SPD), C is symmetric semi-positive definite, and B is of full rank. System of the form (1) arises in a variety of scientific and engineering applications, such as constrained optimization, least squares. We refer the reader to [1] for a more detailed list of applications and numerical solution techniques of (1). Recently, drawing on previous works: [2, 3]...

We show in this note how support preconditioners can be applied to a class of linear systems arising from use of the finite element method to solve linear elliptic problems. Our technique reduces the problem, which is symmetric and positive definite, to a symmetric positive definite diagonally dominant problem. Significant theory has already been developed for preconditioners in the diagonally ...

We investigate here the representability of integers as sums of triangular numbers, where the n-th triangular number is given by Tn = n(n + 1)/2. In particular, we show that f(x1, x2, . . . , xk) = b1Tx1 + · · · + bkTxk , for fixed positive integers b1, b2, . . . , bk, represents every nonnegative integer if and only if it represents 1, 2, 4, 5, and 8. Moreover, if ‘cross-terms’ are allowed in ...