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An algorithm for quadratic eigenproblems with low rank damping
Get PDFWe consider quadratic eigenproblems
, where all coefficient
matrices are real and positive semidefinite, is regular and is
of low rank. Matrix polynomials of this form appear in the analysis of
vibrating structures with discrete dampers. We develop an algorithm for
such problems, which first solves the undamped problem
and then accommodates for the low
rank term . For the first part, we modify an algorithm
proposed by Wang and Zhao [SIAM J. Matrix Anal. Appl. 12-4 (1991), pp.
654--660]. The modified algorithm is then used to solve the undamped
problem such that all eigenvalues are computed in a backward stable
manner. We then use the solution to the undamped problem to compute all eigenvalues of the original problem, and the associated eigenvectors if requested. To this end, we use an Ehrlich-Aberth iteration that works exclusively with vectors and tall skinny matrices and contributes only with lower order terms to the flop count. Numerical experiments show that the proposed algorithm is both fast and accurate. Finally we discuss the application to the large scale case and the possibility of generalizations
A Rational Krylov Toolbox for MATLAB
Get PDFThe Rational Krylov Toolbox contains MATLAB implementations of Ruhe's rational Krylov sequence method (and its block variants), algorithms for the implicit and explicit relocation of the poles of a rational Krylov space, an implementation of RKFIT, a robust algorithm for rational least squares fitting, and the RKFUN/RKFUNM/RKFUNB/BARYFUN classes for numerical computations with rational functions
Functions of Matrices
Get PDFMatrix functions are used in many areas of linear algebra and arise
in numerous applications in science and engineering.
The most common matrix function is the matrix inverse;
it is not treated specifically in this chapter,
but is covered in Section~1.5 and Section~51.3.
This chapter is concerned with general matrix functions as
well as specific cases such as matrix square roots,
trigonometric functions,
and the exponential and logarithmic functions
Möbius Transformations of Matrix Polynomials
No full textWe discuss Möbius transformations for general matrix polynomials over
arbitrary fields, analyzing their influence on regularity, rank,
determinant, constructs such as compound matrices, and on structural
features including sparsity and symmetry. Results on the preservation
of spectral information contained in elementary divisors, partial
multiplicity sequences, invariant pairs, and minimal indices are
presented. The effect on canonical forms such as Smith forms and local
Smith forms, on relationships of strict equivalence and spectral
equivalence, and on the property of being a linearization or
quadratification are investigated. We show that many important
transformations are special instances of Möbius transformations, and
analyze a Möbius connection between alternating and palindromic matrix
polynomials. Finally, the use of Möbius transformations in solving
polynomial inverse eigenproblems is illustrated
An algorithm for quadratic eigenproblems with low rank damping
No full textWe consider quadratic eigenproblems
, where all coefficient
matrices are real and positive semidefinite, is regular and is
of low rank. Matrix polynomials of this form appear in the analysis of
vibrating structures with discrete dampers. We develop an algorithm for
such problems, which first solves the undamped problem
and then accommodates for the low
rank term . For the first part, we modify an algorithm
proposed by Wang and Zhao [SIAM J. Matrix Anal. Appl. 12-4 (1991), pp.
654--660]. The modified algorithm is then used to solve the undamped
problem such that all eigenvalues are computed in a backward stable
manner. We then use the solution to the undamped problem to compute all eigenvalues of the original problem, and the associated eigenvectors if requested. To this end, we use an Ehrlich-Aberth iteration that works exclusively with vectors and tall skinny matrices and contributes only with lower order terms to the flop count. Numerical experiments show that the proposed algorithm is both fast and accurate. Finally we discuss the application to the large scale case and the possibility of generalizations
Testing matrix function algorithms using identities
Get PDFAlgorithms for computing matrix functions are typically tested by comparing the forward error with the product of the condition number and the unit roundoff. The forward error is computed with the aid of a reference solution, typically computed at high precision. An alternative approach is to use functional identities such as the ``round trip tests'' and , as are currently employed in a SciPy test module.
We show how a linearized perturbation analysis for a functional identity allows the determination of a maximum residual consistent with backward stability of the constituent matrix function evaluations. Comparison of this maximum residual with a computed residual provides a necessary test for backward stability. We also show how the actual linearized backward error for these relations can be computed. Our approach makes use of Fr\'echet derivatives and estimates of their norms. Numerical experiments show that the proposed approaches are able both to detect instability and to confirm stability
A Catalogue of Software for Matrix Functions. Version 1.0
No full textA catalogue of software for computing matrix functions
and their Fr\'echet derivatives is presented.
For a wide variety of languages and for software ranging from
commercial products to open source packages
we describe what matrix function codes are available
and which algorithms they implement
Collective Reasoning under Uncertainty and Inconsistency
No full textIn this thesis we investigate some global desiderata for probabilistic knowledge merging given several possibly jointly inconsistent, but individually consistent knowledge bases. We show that the most naive methods of merging, which combine applications of a single expert inference process with the application of a pooling operator, fail to satisfy certain basic consistency principles.
We therefore adopt a different approach. Following recent developments in machine learning where Bregman divergences appear to be powerful, we define several probabilistic merging operators which minimise the joint divergence between merged knowledge and given knowledge bases. In particular we prove that in many cases the result of applying such operators coincides with the sets of fixed points of averaging projective procedures - procedures which combine knowledge updating with pooling operators of decision theory.
We develop relevant results concerning the geometry of Bregman divergences and prove new theorems in this field. We show that this geometry connects nicely with some desirable principles which have arisen in the epistemology of merging. In particular, we prove that the merging operators which we define by means of convex Bregman divergences satisfy analogues of the principles of merging due to Konieczny and Pino-Perez. Additionally, we investigate how such merging operators behave with respect to principles concerning irrelevant information, independence and relativisation which have previously been intensively studied in case of single-expert probabilistic inference.
Finally, we argue that two particular probabilistic merging operators which are based on Kullback-Leibler divergence, a special type of Bregman divergence, have overall the most appealing properties amongst merging operators hitherto considered. By investigating some iterative procedures we propose algorithms to practically compute them
A review of some recent work on hypercyclicity
No full textEven linear operators on infinite-dimensional spaces can display interesting dynamical properties and yield important links among functional analysis, differential and global geometry and dynamical systems, with a wide range of applications. In particular, hypercyclicity is an essentially infinite-dimensional property, when iterations of the operator generate a dense subspace. A Frechet space admits a hypercyclic operator if and only if it
is separable and infinite-dimensional. However, by considering the semigroups generated by multiples of operators, it is possible to obtain hypercyclic behaviour on finite dimensional spaces. The main part of this article gives a brief review of some recent work on hypercyclicity of operators on Banach, Hilbert and Frechet spaces