Consider the linear system of equations Bx = f, where B is an N x N singular matrix, but the system is consistent. In this work we show that iterative techniques coupled with vector extrapolation methods can be used to obtain (approximations to) a solution of Bx = f. We do this by extending the results of some previous work on vector extrapolation methods as they apply to nonsingular matrices B...

We prove that the binary complexity of solving ordinary polynomial differential equations in terms of Puiseux series is single exponential in the number of terms in the series. Such a bound was given by Grigoriev [10] for Riccatti differential polynomials associated to ordinary linear differential operators. In this paper, we get the same bound for arbitrary differential polynomials. The algori...

Given a system of polynomial equations and inequations with coefficients in the field of rational numbers, we show how to compute a geometric resolution of the set of common roots of the system over the field of complex numbers. A geometric resolution consists of a primitive element of the algebraic extension defined by the set of roots, its minimal polynomial, and the parametrizations of the c...

Two linear numeration systems, with characteristic polynomial equal to the minimal polynomial of two Pisot numbers β and γ respectively, such that β and γ are multiplicatively dependent, are considered. It is shown that the conversion between one system and the other one is computable by a finite automaton. We also define a sequence of integers which is equal to the number of periodic points of...

1.1. Field extensions. Suppose K is a field containing a subfield F . An element α ∈ K is called algebraic over F if there exists a non-zero polynomial f ∈ F [x] such that f(α) = 0. If α is algebraic over F , the set Iα(F ) := {f ∈ F [x] | f(α) = 0} is a non-zero ideal of F [x] (this is easy, but do check it). Since F [x] is a PID (it is in fact a Euclidean ring with respect to the degree funct...

Pseudorandom sequences with certain properties are widely used in simulation, software testing, ranging systems, global positioning systems, code-division multiple-access systems, radar systems, spread-spectrum communication systems, and especially in stream ciphers. The cyclotomic sequences [1] have a number of attractive randomness properties. In this letter we first describe generalized cycl...

Let k(~x) := k(x1, . . . , xn) be a finitely generated extension field of some field k, and denote by k(~g) := k(g1, . . . , gr) an intermediate field of k(~x)/k generated over k by some elements g1, . . . , gr ∈ k(~x). So geometrically, we may take ~g for rational functions on the variety determined by the generic point (~x). To determine whether the extension k(~x)/k(~g) is transcendental or ...

Let β > 1 be an algebraic number. The relations between the coefficient vector of its minimal polynomial and the digits of the Rényi β-expansion of unity are investigated in terms of the germ of curve associated with β, which is constructed from the Salem parametrization, and the Parry Upper function fβ(z). If β is a Parry number, the Parry Upper function fβ(z) is simply related to the dynamica...

In this paper we analyze the bi-conjugate gradient algorithm in finite precision arithmetic, and suggest reasons for its often observed robustness. By using a tridiagonal structure, which is preserved by the finite precision bi-conjugate gradient iteration, we are able to bound its residual norm by a minimum polynomial of a perturbed matrix (i.e. the residual norm of the exact GMRES applied to ...