VANDERMONDE-LIKE MATRICES LUCA GEMIGNANI Abstract. This paper is concerned with the solution of linear systems with coe cient matrices which are Vandermonde-like matrices modi ed by adding low-rank corrections. Hereafter we refer to these matrices as to modi ed Vandermonde-like matrices. The solution of modi ed Vandermondelike linear systems arises in the approximation theory both when we use R...

||||||||||||||||||||||||||||||||||||||| ABSTRACT In this paper we introduce a new class of what we shall call polynomial Vandermonde-like matrices. This class generalizes the polynomial Vandermonde matrices studied earlier by various authors, who derived explicit inversion formulas and fast algorithms for inversion and for solving the associated linear systems. A displacement structure approach...

The accurate solution of some of the main problems in numerical linear algebra (linear system solving, eigenvalue computation, singular value computation and the least squares problem) for a totally positive Bernstein–Vandermonde matrix is considered. Bernstein–Vandermonde matrices are a generalization of Vandermonde matrices arising when considering for the space of the algebraic polynomials o...

In this paper we introduce a new class of what we shall call polynomial Vandermonde-like matrices. This class generalizes the polynomial Vandermonde matrices studied earlier by various authors, who derived explicit inversion formulas and fast algorithms for inversion and for solving the associated linear systems. A displacement structure approach allows us to carry over all these results to the...

The accurate solution of some of the main problems in numerical linear algebra (linear system solving, eigenvalue computation, singular value computation and the least squares problem) for a totally positive Bernstein–Vandermonde matrix is considered. Bernstein–Vandermonde matrices are a generalization of Vandermonde matrices arising when considering for the space of the algebraic polynomials o...

Since every even power of the Vandermonde determinant is a symmetric polynomial, we want to understand its decomposition in terms of the basis of Schur functions. We investigate several combinatorial properties of the coefficients in the decomposition. In particular, I will give a recursive approach for computing the coefficient of the Schur function sμ in the decomposition of an even power of ...

In this paper we propose a new fast O(n) algorithm for solving Cauchy linear systems of equations. An a priori rounding error analysis, which provides bounds for the forward, backward and residual errors associated with the computed solution, indicates that for the class of totally positive Cauchy matrices the proposed algorithm is forward and backward stable, producing a remarkably high relati...

Various problems in pure and applied mathematics and engineering can be reformulated as linear algebra problems involving dense structured matrices. The structure of these dense matrices is understood in the sense that their n2 entries can be completeley described by a smaller number O(n) of parameters. Manipulating directly on these parameters allows us to design efficient fast algorithms. One...