A rational QZ method

We propose a rational QZ method for the solution of the dense, unsymmetric generalized eigenvalue problem. This generalization of the classical QZ method operates implicitly on a Hessenberg, Hessenberg pencil instead of on a Hessenberg, triangular pencil. Whereas the QZ method performs nested subspace iteration driven by a polynomial, the rational QZ method allows for nested subspace iteration driven by a rational function, this creates the additional freedom of selecting poles. In this article we study Hessenberg, Hessenberg pencils, link them to rational Krylov subspaces, propose a direct reduction method to such a pencil, and introduce the implicit rational QZ step. The link with rational Krylov subspaces allows us to prove essential uniqueness (implicit Q theorem) of the rational QZ iterates as well as convergence of the proposed method. In the proofs, we operate directly on the pencil instead of rephrasing it all in terms of a single matrix. Numerical experiments are included to illustrate competitiveness in terms of speed and accuracy with the classical approach. Two other types of experiments exemplify new possibilities. First we illustrate that good pole selection can be used to deflate the original problem during the reduction phase, and second we use the rational QZ method to implicitly filter a rational Krylov subspace in an iterative method

A multishift, multipole rational QZ method with aggressive early deflation

The rational QZ method generalizes the QZ method by implicitly supporting rational subspace iteration. In this paper we extend the rational QZ method by introducing shifts and poles of higher multiplicity in the Hessenberg pencil, which is a pencil consisting of two Hessenberg matrices. The result is a multishift, multipole iteration on block Hessenberg pencils which allows one to stick to real arithmetic for a real input pencil. In combination with optimally packed shifts and aggressive early deflation as an advanced deflation technique we obtain an efficient method for the dense generalized eigenvalue problem. In the numerical experiments we compare the results with state-of-the-art routines for the generalized eigenvalue problem and show that we are competitive in terms of speed and accuracy

Solving Linear Rational Expectations Models with Lagged Expectations Quickly and Easily

A solution method is derived in this paper for solving a system of linear rationalexpectations equation with lagged expectations (e.g., models incorporating sticky information) using the method of undetermined coefficients for the infinite MA representation. The method applies a combination of a Generalized Schur Decomposition familiar elsewhere in the literature and a simple system of linear equations when lagged expectations are present to the infinite MA representation. Execution is faster, applicability more general, and use more straightforward than with existing algorithms. Current methods of truncating lagged expectations are shown to not generally be innocuous and the use of such methods are rendered obsolete by the tremendous gains in computational efficiency of the method here which allows for a solution to floating-point accuracy in a fraction of the time required by standard methods. The associated computational application of the method provides impulse responses to anticipated and unanticipated innovations, simulations, and frequency-domain and simulated moments.Lagged expectations; linear rational expectations models; block tridiagonal; Generalized Schur Form; QZ decomposition; LAPACK

The inverse moment problem for convex polytopes: implementation aspects

We give a detailed technical report on the implementation of the algorithm presented in Gravin et al. (Discrete & Computational Geometry'12) for reconstructing an $N$-vertex convex polytope $P$ in $\mathbb{R}^d$ from the knowledge of $O(Nd)$ of its moments

On a q-difference Painlev\'e III equation: II. Rational solutions

Rational solutions for a $q$-difference analogue of the Painlev\'e III equation are considered. A Determinant formula of Jacobi-Trudi type for the solutions is constructed.Comment: Archive version is already official. Published by JNMP at http://www.sm.luth.se/math/JNMP

Applicability of the qq-Analogue of Zeilberger's Algorithm

The applicability or terminating condition for the ordinary case of Zeilberger's algorithm was recently obtained by Abramov. For the $q$-analogue, the question of whether a bivariate $q$-hypergeometric term has a $qZ$-pair remains open. Le has found a solution to this problem when the given bivariate $q$-hypergeometric term is a rational function in certain powers of $q$. We solve the problem for the general case by giving a characterization of bivariate $q$-hypergeometric terms for which the $q$-analogue of Zeilberger's algorithm terminates. Moreover, we give an algorithm to determine whether a bivariate $q$-hypergeometric term has a $qZ$-pair.Comment: 15 page

On pole-swapping algorithms for the eigenvalue problem

Pole-swapping algorithms, which are generalizations of the QZ algorithm for the generalized eigenvalue problem, are studied. A new modular (and therefore more flexible) convergence theory that applies to all pole-swapping algorithms is developed. A key component of all such algorithms is a procedure that swaps two adjacent eigenvalues in a triangular pencil. An improved swapping routine is developed, and its superiority over existing methods is demonstrated by a backward error analysis and numerical tests. The modularity of the new convergence theory and the generality of the pole-swapping approach shed new light on bi-directional chasing algorithms, optimally packed shifts, and bulge pencils, and allow the design of novel algorithms