13 research outputs found
Asymptotic correction of Numerov's eigenvalue estimates with general boundary conditions
The error in the estimate of the kth eigenvalue of ?y??+qy=?y, y(0)=y(?)=0, obtained by Numerov's method with uniform step length h, is O(k 6 h 4 ). The author and J. Paine showed that a correction technique of Paine, de Hoog and Anderssen reduced this to O(k 4 h 5 /sin(kh)), with negligible extra effort. Later the author extended the method to deal with boundary conditions of the form y?(a)=0. This paper shows how a similar increase in accuracy can be obtained, with a little more effort, for problems with one or more boundary conditions of the form y?(a)=?y(a) where ? ? 0
A Householder-based algorithm for Hessenberg-triangular reduction
The QZ algorithm for computing eigenvalues and eigenvectors of a matrix
pencil requires that the matrices first be reduced to
Hessenberg-triangular (HT) form. The current method of choice for HT reduction
relies entirely on Givens rotations regrouped and accumulated into small dense
matrices which are subsequently applied using matrix multiplication routines. A
non-vanishing fraction of the total flop count must nevertheless still be
performed as sequences of overlapping Givens rotations alternately applied from
the left and from the right. The many data dependencies associated with this
computational pattern leads to inefficient use of the processor and poor
scalability.
In this paper, we therefore introduce a fundamentally different approach that
relies entirely on (large) Householder reflectors partially accumulated into
block reflectors, by using (compact) WY representations. Even though the new
algorithm requires more floating point operations than the state of the art
algorithm, extensive experiments on both real and synthetic data indicate that
it is still competitive, even in a sequential setting. The new algorithm is
conjectured to have better parallel scalability, an idea which is partially
supported by early small-scale experiments using multi-threaded BLAS. The
design and evaluation of a parallel formulation is future work
DESCRIPTOR APPROACH FOR ELIMINATING SPURIOUS EIGENVALUES IN HYDRODYNAMIC EQUATIONS
Abstract. We describe a general framework for avoiding spurious eigenvalues -unphysical unstable eigenvalues that often occur in hydrodynamic stability problems. In two example problems, we show that when system stability is analyzed numerically using descriptor notation, spurious eigenvalues are eliminated. Descriptor notation is a generalized eigenvalue formulation for differential-algebraic equations that explicitly retains algebraic constraints. We propose that spurious eigenvalues are likely to occur when algebraic constraints are used to analytically reduce the number of independent variables in a differential-algebraic system of equations before the system is approximated numerically. In contrast, the simple and easily generalizable descriptor framework simultaneously solves the differential equations and algebraic constraints and is well-suited to stability analysis in these systems. Key words. spurious eigenvalue, descriptor, differential algebraic, spectral method, incompressible flow, hydrodynamic stability, generalized eigenvalue, collocation 1. Introduction. Spurious eigenvalues are unphysical, numerically-computed eigenvalues with large positive real parts that often occur in hydrodynamic stability problems. We propose that these unphysical eigenvalues occur when the incompressible Navier Stokes equations are analytically reduced -i.e., the algebraic constraints are used to reduce the number of independent variables before the system is approximated using spectral methods. An alternative approach to analyzing differential-algebraic equations is the descriptor framework, posed as a generalized eigenvalue problem, which explicitly retains the algebraic constraints during the numerical computation of eigenvalues. We reformulate two common hydrodynamic stability problems using descriptor notation and show that this method of computation avoids the spurious eigenvalues generated by other methods. The descriptor formulation is a simple, robust framework for eliminating spurious eigenvalues that occur in hydrodynamic stability analysis. Additionally, this formulation reduces the order of the numerically approximated differential operators and accommodates complex boundary conditions(BCs), such as a fluid interacting with a flexible wall. Resolving the spectrum of hydrodynamic operators is critical for time integration, linear stability Researchers have developed special methods to avoid or filter these modes and uncover the true spectrum of the model problem. Perhaps the first description of these unphysical values is given by Gottlieb and Orsza
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
Generalized Pseudospectral Shattering and Inverse-Free Matrix Pencil Diagonalization
We present a randomized, inverse-free algorithm for producing an approximate
diagonalization of any matrix pencil . The bulk of the
algorithm rests on a randomized divide-and-conquer eigensolver for the
generalized eigenvalue problem originally proposed by Ballard, Demmel, and
Dumitriu [Technical Report 2010]. We demonstrate that this divide-and-conquer
approach can be formulated to succeed with high probability as long as the
input pencil is sufficiently well-behaved, which is accomplished by
generalizing the recent pseudospectral shattering work of Banks, Garza-Vargas,
Kulkarni, and Srivastava [Foundations of Computational Mathematics 2022]. In
particular, we show that perturbing and scaling regularizes its
pseudospectra, allowing divide-and-conquer to run over a simple random grid and
in turn producing an accurate diagonalization of in the backward error
sense. The main result of the paper states the existence of a randomized
algorithm that with high probability (and in exact arithmetic) produces
invertible and diagonal such that and in at most
operations, where is the asymptotic complexity of matrix
multiplication. This not only provides a new set of guarantees for highly
parallel generalized eigenvalue solvers but also establishes nearly matrix
multiplication time as an upper bound on the complexity of exact arithmetic
matrix pencil diagonalization.Comment: 58 pages, 8 figures, 2 table
Multishift variants of the QZ algorithm with aggressive early deflation
New variants of the QZ algorithm for solving the generalized eigenvalue problem are proposed. An extension of the small-bulge multishift QR algorithm is developed, which chases chains of many small bulges instead of only one bulge in each QZ iteration. This allows the effective use of level 3 BLAS operations, which in turn can provide efficient utilization of high performance computing systems with deep memory hierarchies. Moreover, an extension of the aggressive early deflation strategy is proposed, which can identify and de. ate converged eigenvalues long before classic deflation strategies would. Consequently, the number of overall QZ iterations needed until convergence is considerably reduced. As a third ingredient, we reconsider the deflation of infinite eigenvalues and present a new deflation algorithm, which is particularly effective in the presence of a large number of infinite eigenvalues. Combining all these developments, our implementation significantly improves existing implementations of the QZ algorithm. This is demonstrated by numerical experiments with random matrix pairs as well as with matrix pairs arising from various applications