-gradient algorithms, indicating that I~QR is the most reliable algorithm when A is ill-conditioned. Categories and Subject Descriptors: G.1.2 [Numerical Analysis]: ApprorJmation--least squares approximation; G.1.3 [Numerical Analysis]: Numerical Linear Algebra--linear systems (direct and

This report compares the performance of different computer systems in solving dense systems of linear equations. The comparison involves approximately a hundred computers, ranging from the Earth Simulator to personal computers.

We consider linear equations y = Φα where y is a given vector in R n, Φ is a given n by m matrix with n < m ≤ An, and we wish to solve for α ∈ R m. We suppose that the columns of Φ are normalized to unit ℓ 2 norm 1 and we place uniform measure on such Φ. We prove the existence of ρ = ρ(A) so

Abstract. We consider a class of iterative algorithms for solving systems of linear equations where the coefficient matrix is nonsymmetric with positive-definite symmetric part. The algorithms are modelled after the conjugate gradient method, and are well suited for large sparse systems. They do

now appears in Section V, Sub-section A, of the Task Group’s report on Conceptual

Finding the sparsest solution to underdetermined systems of linear equations y = Φx is NP-hard in general. We show here that for systems with ‘typical’/‘random ’ Φ, a good approximation to the sparsest solution is obtained by applying a fixed number of standard operations from linear algebra. Our

fraction of the output is corrupted. This work is related to the problem of finding sparse solutions to vastly underdetermined systems of linear equations. There are also significant connections with the problem of recovering signals from highly incomplete measurements. In fact, the results introduced

A simple algorithm is described which is well adapted to the effective solution of large systems of linear algebraic equations by a succession of well-convergent approximations. 1.

illustrate these principles using current architectures and software systems, and by showing how one would implement matrix multiplication. Then, we present direct and iterative algorithms for solving linear systems of equations, linear least squares problems, the symmetric eigenvalue problem

simultaneous linear system of equations, in which all unknowns are required to be prime. In this paper we (conditionally) verify this asymptotic under the assumption that no two of the affine-linear forms ψ1,..., ψt are affinely related; this excludes the important “binary ” cases such as the twin prime