Related papers: Minimal Polynomial Algorithms for Finite Sequences

We propose a slight modification of the Berlekamp-Massey Algorithm for obtaining the minimal polynomial of a given linearly recurrent sequence. Such a modification enables to explain it in a simpler way and to adapt it to lazy evaluation.

Let $s$ be a finite sequence over a field of length $n$. It is well-known that if $s$ satisfies a linear recurrence of order $d$ with non-zero constant term, then the reverse of $s$ also satisfies a recurrence of order $d$ (with…

In this paper, we propose a new and simple approach to the approximation algorithms that are modified and improved from our published results. The computational and graphical examples are presented with the aid of Maple procedures.

We present a complete algorithm for finding an exact minimal polynomial from its approximate value by using an improved parameterized integer relation construction method. Our result is superior to the existence of error controlling on…

We present a new algorithm for reconstructing an exact algebraic number from its approximate value using an improved parameterized integer relation construction method. Our result is consistent with the existence of error controlling on…

We consider the problem of computing the minimum of a polynomial function g on a basic closed semialgebraic set E in R^n. We present a probabilistic symbolic algorithm to find a finite set of sample points of the subset E^{min} of E where…

Let $S=(s_1,s_2,...,s_m,...)$ be a linear recurring sequence with terms in $GF(q^n)$ and $T$ be a linear transformation of $GF(q^n)$ over $GF(q)$. Denote $T(S)=(T(s_1),T(s_2),...,T(s_m),...)$. In this paper, we first present counter…

We present a novel iterative algorithm for approximating the linear least squares solution with low complexity. After a motivation of the algorithm we discuss the algorithm's properties including its complexity, and we present theoretical…

The analysis of solutions to algebraic equations is further simplified. A couple of functions and their analytic continuation or root findings are required.

This paper considers fast algorithms for operations on linearized polynomials. We propose a new multiplication algorithm for skew polynomials (a generalization of linearized polynomials) which has sub-quadratic complexity in the polynomial…

We utilize the same technique as in [arXiv:2205.04254 (2022)] to provide some representations of polynomials non-negative on a basic semi-algebraic set, defined by polynomial inequalities, under more general conditions. Based on each…

We propose a very simple preprocessing algorithm for semidefinite programming. Our algorithm inspects the constraints of the problem, deletes redundant rows and columns in the constraints, and reduces the size of the variable matrix. It…

Minimal annihilating polynomials are very useful in a wide variety of algorithms in exact linear algebra. A new efficient method is proposed for calculating the minimal annihilating polynomials for all the unit vectors, for a square matrix…

We present a recursive minimal polynomial theorem for finite sequences over a commutative integral domain $D$. This theorem is relative to any element of $D$. The ingredients are: the arithmetic of Laurent polynomials over $D$, a recursive…

The celebrated minimum residual method (MINRES), proposed in the seminal paper of Paige and Saunders, has seen great success and widespread use in solving Hermitian (and complex-symmetric) linear systems. Unless the system is consistent,…

We show how polynomial path orders can be employed efficiently in conjunction with weak innermost dependency pairs to automatically certify polynomial runtime complexity of term rewrite systems and the polytime computability of the…

We present and analyse a Monte-Carlo algorithm to compute the minimal polynomial of an $n\times n$ matrix over a finite field that requires $O(n^3)$ field operations and O(n) random vectors, and is well suited for successful practical…

Modifying an idea of E. Brietzke we give simple proofs for the recurrence relations of some sequences of binomial sums which have previously been obtained by other more complicated methods.

We describe a provably quasi-polynomial algorithm to compute discrete logarithms in the multiplicative groups of finite fields of small characteristic, that is finite fields whose characteristic is logarithmic in the order. We partially…

Short integer linear programs are programs with a relatively small number of constraints. We show how recent improvements on the running-times of solvers for such programs can be used to obtain fast pseudo-polynomial time algorithms for…