Vlastimil Pták

Sharp error bounds for Newton's process

SummaryThe method of nondiscrete mathematical induction is applied to the Newton process. The method yields a very simple proof of the convergence and sharp apriori estimates; it also gives aposteriori bounds which are, in general, better than those given in [1].

The rate of convergence of Newton's process

SummaryThe author applies the method of nondiscrete mathematical induction (see [2–5]) which involves considering the rate of convergence as a function, not as a number, to Newtons process and proves that the rate of convergence is

Lyapunov, Bézout, and Hankel

\omega (r) = \frac{{r^2 }}{{2(r^2 + d)^{1/2} }}

Explicit expressions for Bezoutians

whered is a positive number depending on the initial data (see Theorem 2.3).

Functions of operators and the spectral radius

In a recent paper ( 6 ) the present author has shown that, for an element a of a Banach algebra A , the condition for all x ∈ A and some constant α is equivalent to [ x , a ]∈Rad a for all x ∈ A ; it turns out that α may be replaced by |α|σ It is the purpose of the present note to investigate a related condition

Bézoutians and intertwining matrices

Abstract Given a polynomial f of degree n , we denote by C its companion matrix, and by S the truncated shift operator of order n . We consider Lyapunov-type equations of the form X − SXC => W and X − CXS = W . We derive some properties of these equations which make it possible to characterize Bezoutian matrices as solutions of the first equation with suitable right-hand sides W (similarly for Hankel and the second equation) and to write down explicit expressions for these solutions. This yields explicit factorization formulae for polynomials in C , for the Schur-Cohn matrix, and for matrices satisfying certain intertwining relations, as well as for Bezoutian matrices.

A maximum problem for matrices

Abstract The author proves a simple general theorem about complete metric spaces which forms an abstract basis of existence theorem in functional analysis and numerical analysis. He shows that this theorem, the so called induction theorem, contains the classical fixed point theorem for contractive mappings as well as the closed graph theorem. He then explains the principles of application of the induction theorem, the method of nondiscrete mathematical induction which consists in reducing the given problem to a system of functional inequalities, to be satisfied by a certain function, called the rate of convergence. The fact that the rate of convergence is defined as a function and not a number makes it possible to obtain sharp estimates valid for the whole iterative process, not only asymptotically. The method of nondiscrete mathematical induction is then illustrated by means of the example of eigenvalues of almost decomposable matrices.

Lifting intertwining relations

Abstract The following commutator identity is proved: [u(S ∗ ), v(S)] = [v 1 (S ∗ ), u 1 (S)] . Here S is the n by n matrix of the truncated shift operator S = (Γi,i+1), i = 0, 1,…, n − 1, and u, v are two polynomials of degree not exceeding n. The reciprocal polynomial f;1 of a polynomial f; of degree ⩽n is defined by f 1 (z) = z n f( 1 z ) . The commutator identity is closely related to some properties of the Bezoutian matrix of a pair of polynomials; it is used to obtain the Bezoutian matrix in the form of a simple expression in terms of S and S∗. To demonstrate the advantage of this expression, we show how it can be used to obtain simple proofs of some interesting corollaries.