Recursive interpolation,extrapolation and projection

The recursive projection algorithm derived in a previous paper is related to several well-known methods of numerical analysis such as the conjugate gradient method, Rosen's method and Henrici's. It is connected with the general interpolation problem, with extrapolation methods, with orthogonal projection on a subspace and with Fourier expansions. Several other connections and applications are presented.     

Prediction properties of some extrapolation methods

Convergence acceleration methods consist in the construction of a sequence converging faster than the initial sequence. Each member of the new sequence is a guess for the limit and it is computed from a restricted number of terms of the initial sequence. It is shown herein how convergence acceleration methods can be used to predict the next (unknown) term of the initial sequence instead of its limit. Particular emphasis on Aitken's △² process and the E-algorithm is placed.     

Transpose-free Lanczos-type algorithms for nonsymmetric linear systems

The method of Lanczos for solving systems of linear equations is implemented by using recurrence relationships between formal orthogonal polynomials. A drawback is that the computation of the coefficients of these recurrence relationships usually requires the use of the transpose of the matrix of the system. Due to the indirect addressing, this is a costly operation. In this paper, a new procedure for computing these coefficients is proposed. It is based on the recursive computation of the products of polynomials appearing in their expressions and it does not involve the transpose of the matrix. Moreover, our approach allows to implement simultaneously and at a low price a Lanczos-type product method such as the CGS or the BiCGSTAB. This revised version was published online in June 2006 with corrections to the Cover Date.     

Limiting relationships and comparison theorems for sequences

The aim of this paper is to define some mathematical concepts which are useful to measure the speed of convergence of a sequence and to compare two converging sequences. In that way we define the order, the relative order and the α-equivalence of sequences. The asymptotic expansion of a series is studied and an application to Aitken acceleration process is given. A theorem similar to l'Hospital's rule is also proved for sequences.     

Nonlinear functional equations satisfied by orthogonal polynomials

Let c be a linear functional defined by its moments c(xⁱ)=c_i for i=0,1,…. We proved that the nonlinear functional equations P(t)=c(P(x)P(αx+t)) and P(t)=c(P(x)P(xt)) admit polynomial solutions which are the polynomials belonging to the family of formal orthogonal polynomials with respect to a linear functional related to c. This equation relates the polynomials of the family with those of the scaled and shifted family. Other types of nonlinear functional equations whose solutions are formal orthogonal polynomials are also presented. Applications to Legendre and Chebyshev polynomials are given. Then, orthogonality with respect to a definite inner product is studied. When c is an integral functional with respect to a weight function, the preceding functional equations are nonlinear integral equations, and these results lead to new characterizations of orthogonal polynomials on the real line, on the unit circle, and, more generally, on an algebraic curve.     

Copper and iron interactions with angiotensin-converting enzyme inhibitors. A study by fast-atom bombardment tandem mass spectrometry

Fast atom bombardment, combined with high-energy collision-induced tandem mass spectrometry, has been used to investigate gas-phase metal-ion interactions with captopril, enalaprilat and lisinopril, all angiotensin-converting enzyme inhibitors.Suggestions for the location of metal-binding sites are presented. For captopril, metal binding occurs most likely at both the sulphur and the nitrogen atom. For enalaprilat and lisinopril, binding preferably occurs at the amine nitrogen. Copyright 1999 John Wiley & Sons, Ltd.     

Krylov subspace methods, biorthogonal polynomials and Padé-type approximants

It is shown that Krylov subspace methods for solving systems of linear equations can be based on formal biorthogonal polynomials and on Padé-type and Padé approximants. New algorithms for their implementation are derived. This revised version was published online in June 2006 with corrections to the Cover Date.