Rational Krylov approximation of matrix functions: Numerical methods and optimal pole selection

Matrix functions are a central topic of linear algebra, and problems of their numerical approximation appear increasingly often in scientific computing. We review various rational Krylov methods for the computation of large-scale matrix functions. Emphasis is put on the rational Arnoldi method and variants thereof, namely, the extended Krylov subspace method and the shift-and-invert Arnoldi method, but we also discuss the nonorthogonal generalized Leja point (or PAIN) method. The issue of optimal pole selection for rational Krylov methods applied for approximating the resolvent and exponential function, and functions of Markov type, is treated in some detail

Convergence of linear barycentric rational interpolation for analytic functions

Polynomial interpolation to analytic functions can be very accurate, depending on the distribution of the interpolation nodes. However, in equispaced nodes and the like, besides being badly conditioned, these interpolants fail to converge even in exact arithmetic in some cases. Linear barycentric rational interpolation with the weights presented by Floater and Hormann can be viewed as blended polynomial interpolation and often yields better approximation in such cases. This has been proven for differentiable functions and indicated in several experiments for analytic functions. So far, these rational interpolants have been used mainly with a constant parameter usually denoted by d, the degree of the blended polynomials, which leads to small condition numbers but to merely algebraic convergence. With the help of logarithmic potential theory we derive asymptotic convergence results for analytic functions when this parameter varies with the number of nodes. Moreover, we present suggestions on how to choose d in order to observe fast and stable convergence, even in equispaced nodes where stable geometric convergence is provably impossible. We demonstrate our results with several numerical examples

A black-box rational Arnoldi variant for Cauchy-Stieltjes matrix functions

Rational Arnoldi is a powerful method for approximating functions of large sparse matrices times a vector. The selection of asymptotically optimal parameters for this method is crucial for its fast convergence. We present and investigate a novel strategy for the automated parameter selection when the function to be approximated is of Cauchy-Stieltjes (or Markov) type, such as the matrix square root or the logarithm. The performance of this approach is demonstrated by numerical examples involving symmetric and nonsymmetric matrices. These examples suggest that our black-box method performs at least as well, and typically better, as the standard rational Arnoldi method with parameters being manually optimized for a given matrix

Some observations on weighted GMRES

We investigate the convergence of the weighted GMRES method for solving linear systems. Two different weighting variants are compared with unweighted GMRES for three model problems, giving a phenomenological explanation of cases where weighting improves convergence, and a case where weighting has no effect on the convergence. We also present new alternative implementations of the weighted Arnoldi algorithm which may be favorable in terms of computational complexity, and examine stability issues connected with these implementations. Two implementations of weighted GMRES are compared for a large number of examples. We find that weighted GMRES may outperform unweighted GMRES for some problems, but more often this method is not competitive with other Krylov subspace methods like GMRES with deflated restarting or BICGSTAB, in particular when a preconditioner is used

Robust Padé approximation via SVD

Padé approximation is considered from the point of view of robust methods of numerical linear algebra, in particular the singular value decomposition. This leads to an algorithm for practical computation that bypasses most problems of solution of nearly-singular systems and spurious pole-zero pairs caused by rounding errors; a Matlab code is provided. The success of this algorithm suggests that there might be variants of Padé approximation that would be pointwise convergent as the degrees of the numerator and denominator increase to infinity, unlike traditional Padé approximants, which converge only in measure or capacity

Zolotarev Quadrature Rules and Load Balancing for the FEAST Eigensolver

The FEAST method for solving large sparse eigenproblems is equivalent to subspace iteration with an approximate spectral projector and implicit orthogonalization. This relation allows to characterize the convergence of this method in terms of the error of a certain rational approximant to an indicator function. We propose improved rational approximants leading to FEAST variants with faster convergence, in particular, when using rational approximants based on the work of Zolotarev. Numerical experiments demonstrate the possible computational savings especially for pencils whose eigenvalues are not well separated and when the dimension of the search space is only slightly larger than the number of wanted eigenvalues. The new approach improves both convergence robustness and load balancing when FEAST runs on multiple search intervals in parallel.Comment: 22 pages, 8 figure

Nationality and Colonial Strategies: Germany and America – How the American Expansion Resonated in Germany

We all tend to see what we want to see — in ourselves, in our friends, in our culture, and in other cultures. In his dissertation, Jens-Uwe Guettel takes a penetrating look at how Germany viewed America over the course of the 19th century, the period of America’s great expansion westward. In the following interview and excerpt, you will find highlights of Prof. Guettel’s wide-ranging consideration of the many authors, themes and images which were part of this cultural “moment.” In the dissertation itself, you will find a deeper look at the novels and writings which reflect the complex attitudes and ideas of the times. Germans certainly noticed what Americans were doing as they expanded the nation westward, but not always the same we saw ourselves. What makes this dissertation so explosive (to me, anyway) is what comes next – what is off-screen, so to speak. When Prof. Guettel brings up the concept of lebensraum, we realize that his thesis is by no means an obscure topic of study: the colonial attitudes of the 19th century can be seen to lead directly to the German nationalism of the modern era and to the rise of the Third Reich. Most certainly, German views of American colonialism formed the roots of the two world wars which dominated the 20th century. Understanding the deeper cultural roots of war is important to all of us. As I write this, our entire nation is at war – two wars, actually – and each and every citizen is part of that decision. We need to understand why these conflicts have happened in the past, are happening today, and may break out again soon

Scaled and squared subdiagonal Padé approximation for the matrix exponential

The scaling and squaring method is the most widely used algorithm for computing the exponential of a square matrix A. We introduce an efficient variant that uses a much smaller squaring factor when ||A|| » 1 and a subdiagonal Padé approximant of low degree, thereby significantly reducing the overall cost and avoiding the potential instability caused by overscaling, while giving forward error of the same magnitude as that of the standard algorithm. The new algorithm performs well if a rough estimate of the rightmost eigenvalue of A is available and the rightmost eigenvalues do not have widely varying imaginary parts, and it achieves significant speedup over the conventional algorithm especially when A is of large norm. Our algorithm uses the partial fraction form to evaluate the Padé approximant, which makes it suitable for parallelization and directly applicable to computing the action of the matrix exponential exp(A)b, where b is a vector or a tall skinny matrix. For this problem the significantly smaller squaring factor has an even more pronounced benefit for efficiency when evaluating the action of the Padé approximant