Continued fractions associated with the Newton-Padé table

Summary We discuss first the block structure of the Newton-Padé table (or, rational interpolation table) corresponding to the double sequence of rational interpolants for the data{(z _k, h(z_k)} _k =0. (The (m, n)-entry of this table is the rational function of type (m,n) solving the linearized rational interpolation problem on the firstm+n+1 data.) We then construct continued fractions that are associated with either a diagonal or two adjacent diagonals of this Newton-Padé table in such a way that the convergents of the continued fractions are equal to the distinct entries on this diagonal or this pair of diagonals, respectively. The resulting continued fractions are generalizations of Thiele fractions and of Magnus'sP-fractions. A discussion of an some new results on related algorithms of Werner and Graves-Morris and Hopkins are also given.Dedicated to the memory of Helmut Werner (1931–1985)     

Modified moments for indefinite weight functions

Summary The problem of generating the recurrence coefficients of orthogonal polynomials from the moments or from modified moments of the weight function is well understood for positive weight distributions. Here we extend this theory and the basic algorithms to the case of an indefinite weight function. While the generic indefinite case is formally not much different from the positive definite case, there exist nongeneric degenerate situations, and these require a different more complicated treatment. The understanding of these degenerate situations makes it possible to construct a stable approximate solution of an ill-conditioned problem.The application to adaptive iterative methods for linear systems of equations is anticipated.Dedicated to R.S. Varga on the occasion of his sixtieth birthdayG.H. Golub (USA, Canada), M.H. Gutknecht (other countries)     

The block grade of a block Krylov space

The aim of the paper is to compile and compare basic theoretical facts on Krylov subspaces and block Krylov subspaces. Many Krylov (sub)space methods for solving a linear system Ax=b have the property that in exact computer arithmetic the true solution is found after ν iterations, where ν is the dimension of the largest Krylov subspace generated by A from r₀, the residual of the initial approximation x₀. This dimension is called the grade of r₀ with respect to A. Though the structure of block Krylov subspaces is more complicated than that of ordinary Krylov subspaces, we introduce here a block grade for which an analogous statement holds when block Krylov space methods are applied to linear systems with multiple, say s, right-hand sides. In this case, the s initial residuals are bundled into a matrix R₀ with s columns. The possibility of linear dependence among columns of the block Krylov matrix , which in practical algorithms calls for the deletion (or, deflation) of some columns, requires extra care. Relations between grade and block grade are also established, as well as relations to the corresponding notions of a minimal polynomial and its companion matrix.     

By How Much Can Residual Minimization Accelerate the Convergence of Orthogonal Residual Methods?

We capitalize upon the known relationship between pairs of orthogonal and minimal residual methods (or, biorthogonal and quasi-minimal residual methods) in order to estimate how much smaller the residuals or quasi-residuals of the minimizing methods can be compared to those of the corresponding Galerkin or Petrov–Galerkin method. Examples of such pairs are the conjugate gradient (CG) and the conjugate residual (CR) methods, the full orthogonalization method (FOM) and the generalized minimal residual (GMRES) method, the CGNE and BiCG versions of applying CG to the normal equations, as well as the biconjugate gradient (BiCG) and the quasi-minimal residual (QMR) methods. Also the pairs consisting of the (bi)conjugate gradient squared (CGS) and the transpose-free QMR (TFQMR) methods can be added to this list if the residuals at half-steps are included, and further examples can be created easily.The analysis is more generally applicable to the minimal residual (MR) and quasi-minimal residual (QMR) smoothing processes, which are known to provide the transition from the results of the first method of such a pair to those of the second one. By an interpretation of these smoothing processes in coordinate space we deepen the understanding of some of the underlying relationships and introduce a unifying framework for minimal residual and quasi-minimal residual smoothing. This framework includes the general notion of QMR-type methods.