Conference on the Occasion of Richard Varga’s 80th Birthday

Bernstein’s inequality for Legendre polynomials Pn, as generalized by Baratella, Chow, Gatteschi, and Wong to Jacobi polynomials P n , (α, β) ∈ calR1/2 = {|α| ≤ 1/2, |β| ≤ 1/2}, is analyzed analytically and computationally with regard to validity and sharpness in larger domains Rs = {−1/2 ≤ α ≤ s,−1/2 ≤ β ≤ s}, s > 1/2. Title: Bivariate B-splines used as basis functions for data fitting Speaker: Daniel E. Gonsor, The Boeing Company Abstract: We present results summarizing the utility of bivariate B-splines for solving data fitting and related problems. We present results summarizing the utility of bivariate B-splines for solving data fitting and related problems. Title: IDR explained Speaker: Martin H. Gutknecht, ETH Zurich Abstract: Under the section heading “A preconditioned Lanczos type method” the Induced Dimension Reduction (IDR) method was introduced on three and a half pages of a 1980 proceedings paper [5] by Wesseling and Sonneveld. Few people may have noticed it. Soon after IDR, Sonneveld must have found his widely applied Conjugate Gradient Squared (CGS) algorithm, published in SISSC in 1989 [2], but submitted to that journal on April 24, 1984, already. Then, at the Householder Symposium 1990 in Tylosand, van der Vorst suggested with CGSTAB an improvement of both these methods. It was published as the Bi-CGSTAB method in SISSC in 1992 [4]. Under the section heading “A preconditioned Lanczos type method” the Induced Dimension Reduction (IDR) method was introduced on three and a half pages of a 1980 proceedings paper [5] by Wesseling and Sonneveld. Few people may have noticed it. Soon after IDR, Sonneveld must have found his widely applied Conjugate Gradient Squared (CGS) algorithm, published in SISSC in 1989 [2], but submitted to that journal on April 24, 1984, already. Then, at the Householder Symposium 1990 in Tylosand, van der Vorst suggested with CGSTAB an improvement of both these methods. It was published as the Bi-CGSTAB method in SISSC in 1992 [4]. Bi-CGSTAB has become one of the methods of choice for nonsymmetric linear systems, and it has been generalized in various ways in the hope of further improving its reliability and speed. Among these generalizations there is the ML(k)BiCGSTAB method of Yeung and Chan, published in SISC in 1999 [6], which in the framework of block Lanczos methods can be understood as a variation of Bi-CGSTAB with right-hand side block size 1 and left-hand side block size k. Probably only few readers studied this paper completely, since the algorithm turned out to be rather complicated. Last year Sonneveld and van Gijzen [3] reconsidered IDR and generalized it to IDR(s), claiming that IDR is equally fast but preferable to Bi-CGSTAB, and that IDR(s) may be much faster than IDR = IDR(1). It turned out that IDR(s) is closely related to ML(s)BiCGSTAB, but at no step mathematically equivalent. In fact, the recurrences of IDR and IDR(s) are simpler than those of BiCGSTAB and ML(s)BiCGSTAB, respectively, and they offer some flexibility that can be capitalized upon for increasing the stability. A similar flexibility is also available in the block Lanczos process but has not been made use of in ML(k)BiCGSTAB. In this talk we first try to explain the basic, seemingly quite general IDR approach, which differs completely from traditional approaches to Krylov space methods. Then we compare the basic properties of the above mentioned methods and discuss some of their connections. Most of what we say is taken from [3] and [1], but we also provide some further explanations.