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318
Complementarity Modeling of Hybrid Systems
, 1998
"... A complementarity framework is described for the modeling of certain classes of mixed continuous /discrete dynamical systems. The use of such a framework is wellknown for mechanical systems with inequality constraints, but we give a more general formulation which applies for instance also to sys ..."
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Cited by 88 (15 self)
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A complementarity framework is described for the modeling of certain classes of mixed continuous /discrete dynamical systems. The use of such a framework is wellknown for mechanical systems with inequality constraints, but we give a more general formulation which applies for instance also to systems with relays in a feedback loop. The main theoretical results in the paper are concerned with uniqueness of smooth continuations; the solution of this problem requires the construction of a map from the continuous state to the discrete state. A crucial technical tool is the socalled linear complementarity problem (LCP); we introduce various generalizations of this problem. Specific results are obtained for Hamiltonian systems, passive systems, and linear systems.
An inverse free parallel spectral divide and conquer algorithm for nonsymmetric eigenproblems
, 1997
"... We discuss an inversefree, highly parallel, spectral divide and conquer algorithm. It can compute either an invariant subspace of a nonsymmetric matrix A, or a pair of left and right deflating subspaces of a regular matrix pencil A − λB. This algorithm is based on earlier ones of Bulgakov, Godunov ..."
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Cited by 68 (10 self)
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We discuss an inversefree, highly parallel, spectral divide and conquer algorithm. It can compute either an invariant subspace of a nonsymmetric matrix A, or a pair of left and right deflating subspaces of a regular matrix pencil A − λB. This algorithm is based on earlier ones of Bulgakov, Godunov and Malyshev, but improves on them in several ways. This algorithm only uses easily parallelizable linear algebra building blocks: matrix multiplication and QR decomposition, but not matrix inversion. Similar parallel algorithms for the nonsymmetric eigenproblem use the matrix sign function, which requires matrix inversion and is faster but can be less stable than the new algorithm.
Positivity Problems and Conjectures in Algebraic Combinatorics
 in Mathematics: Frontiers and Perspectives
, 1999
"... Introduction. Algebraic combinatorics is concerned with the interaction between combinatorics and such other branches of mathematics as commutative algebra, algebraic geometry, algebraic topology, and representation theory. Many of the major open problems of algebraic combinatorics are related to p ..."
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Cited by 67 (1 self)
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Introduction. Algebraic combinatorics is concerned with the interaction between combinatorics and such other branches of mathematics as commutative algebra, algebraic geometry, algebraic topology, and representation theory. Many of the major open problems of algebraic combinatorics are related to positivity questions, i.e., showing that certain integers are nonnegative. The significance of positivity to algebraic combinatorics stems from the fact that a nonnegative integer can have both a combinatorial and an algebraic interpretation. The archetypal algebraic interpretation of a nonnegative integer is as the dimension of a vector space. Thus to show that a certain integer m is nonnegative, it suces to nd a vector space Vm of dimension m. Similarly to show that m n, it suces to nd an injective map Vm ! V n or surjective map V n ! Vm . Of course the inequality m n is equivalent to the positivity statement n m 0, while the injectivity of the map ' : Vm ! V n is equivalent to the
Polynomial Roots from Companion Matrix Eigenvalues
 Mathematics of Computation
, 1995
"... In classical linear algebra, the eigenvalues of a matrix are sometimes dened as the roots of the characteristic polynomial. An algorithm to compute the roots of a polynomial by computing the eigenvalues of the corresponding companion matrix turns the tables on the usual denition. We derive a rst or ..."
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Cited by 66 (1 self)
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In classical linear algebra, the eigenvalues of a matrix are sometimes dened as the roots of the characteristic polynomial. An algorithm to compute the roots of a polynomial by computing the eigenvalues of the corresponding companion matrix turns the tables on the usual denition. We derive a rst order error analysis of this algorithm that sheds light on both the underlying geometry of the problem as well as the numerical error of the algorithm. Our error analysis expands on work by Van Dooren and Dewilde in that it states that the algorithm is backwards normwise stable in a sense that must be dened carefully. Regarding the stronger concept of a small componentwise backwards error, our analysis predicts a small such error on a test suite of eight random polynomials suggested by Toh and Trefethen. However, we construct examples for which a small componentwise relative backwards error is neither predicted nor obtained in practice. We extend our results to polynomial matrices, where the result is essentially the same, but the geometry becomes more complicated. 1
Design of a Parallel Nonsymmetric Eigenroutine Toolbox, Part I
, 1993
"... The dense nonsymmetric eigenproblem is one of the hardest linear algebra problems to solve effectively on massively parallel machines. Rather than trying to design a "black box" eigenroutine in the spirit of EISPACK or LAPACK, we propose building a toolbox for this problem. The tools ar ..."
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Cited by 65 (12 self)
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The dense nonsymmetric eigenproblem is one of the hardest linear algebra problems to solve effectively on massively parallel machines. Rather than trying to design a "black box" eigenroutine in the spirit of EISPACK or LAPACK, we propose building a toolbox for this problem. The tools are meant to be used in different combinations on different problems and architectures. In this paper, we will describe these tools which include basic block matrix computations, the matrix sign function, 2dimensional bisection, and spectral divide and conquer using the matrix sign function to find selected eigenvalues. We also outline how we deal with illconditioning and potential instability. Numerical examples are included. A future paper will discuss error analysis in detail and extensions to the generalized eigenproblem.
Motif Statistics
, 1999
"... We present a complete analysis of the statistics of number of occurrences of a regular expression pattern in a random text. This covers "motifs" widely used in computational biology. Our approach is based on: (i) a constructive approach to classical results in theoretical computer science ..."
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Cited by 64 (5 self)
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We present a complete analysis of the statistics of number of occurrences of a regular expression pattern in a random text. This covers "motifs" widely used in computational biology. Our approach is based on: (i) a constructive approach to classical results in theoretical computer science (automata and formal language theory), in particular, the rationality of generating functions of regular languages; (ii) analytic combinatorics that is used for deriving asymptotic properties from generating functions; (iii) computer algebra for determining generating functions explicitly, analysing generating functions and extracting coefficients efficiently. We provide constructions for overlapping or nonoverlapping matches of a regular expression. A companion implementation produces multivariate generating functions for the statistics under study. A fast computation of Taylor coefficients of the generating functions then yields exact values of the moments with typical application to random t...
A geometric approach to perturbation theory of matrices and matrix pencils. Part II: A stratificationenhanced staircase algorithm
 SIAM J. Matrix Anal. Appl
, 1997
"... Computing the Jordan form of a matrix or the Kronecker structure of a pencil is a wellknown illposed problem. We propose that knowledge of the closure relations, i.e., the stratification, of the orbits and bundles of the various forms may be applied in the staircase algorithm. Here we discuss and ..."
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Cited by 62 (11 self)
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Computing the Jordan form of a matrix or the Kronecker structure of a pencil is a wellknown illposed problem. We propose that knowledge of the closure relations, i.e., the stratification, of the orbits and bundles of the various forms may be applied in the staircase algorithm. Here we discuss and complete the mathematical theory of these relationships and show how they may be applied to the staircase algorithm. This paper is a continuation of our Part I paper on versal deformations, but may also be read independently.
Nearly Optimal Algorithms For Canonical Matrix Forms
, 1993
"... A Las Vegas type probabilistic algorithm is presented for finding the Frobenius canonical form of an n x n matrix T over any field K. The algorithm requires O~(MM(n)) = MM(n) (log n) ^ O(1) operations in K, where O(MM(n)) operations in K are sufficient to multiply two n x n matrices over K. This nea ..."
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Cited by 62 (13 self)
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A Las Vegas type probabilistic algorithm is presented for finding the Frobenius canonical form of an n x n matrix T over any field K. The algorithm requires O~(MM(n)) = MM(n) (log n) ^ O(1) operations in K, where O(MM(n)) operations in K are sufficient to multiply two n x n matrices over K. This nearly matches the lower bound of \Omega(MM(n)) operations in K for this problem, and improves on the O(n^4) operations in K required by the previously best known algorithms. We also demonstrate a fast parallel implementation of our algorithm for the Frobenius form, which is processorefficient on a PRAM. As an application we give an algorithm to evaluate a polynomial g(x) in K[x] at T which requires only O~(MM(n)) operations in K when deg g < n^2. Other applications include sequential and parallel algorithms for computing the minimal and characteristic polynomials of a matrix, the rational Jordan form of a matrix, for testing whether two matrices are similar, and for matrix powering, which are substantially faster than those previously known.
A restarted Krylov subspace method for the evaluation of matrix functions
 SIAM J. Numer. Anal
"... Abstract. We show how the Arnoldi algorithm for approximating a function of a matrix times a vector can be restarted in a manner analogous to restarted Krylov subspace methods for solving linear systems of equations. The resulting restarted algorithm reduces to other known algorithms for the recipro ..."
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Cited by 58 (8 self)
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Abstract. We show how the Arnoldi algorithm for approximating a function of a matrix times a vector can be restarted in a manner analogous to restarted Krylov subspace methods for solving linear systems of equations. The resulting restarted algorithm reduces to other known algorithms for the reciprocal and the exponential functions. We further show that the restarted algorithm inherits the superlinear convergence property of its unrestarted counterpart for entire functions and present the results of numerical experiments.
Canonical Forms for Hamiltonian and Symplectic Matrices and Pencils
, 1998
"... We study canonical forms for Hamiltonian and symplectic matrices or pencils under equivalence transformations which keep the class invariant. In contrast to other canonical forms our forms are as close as possible to a triangular structure in the same class. We give necessary and sufficient conditio ..."
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Cited by 58 (27 self)
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We study canonical forms for Hamiltonian and symplectic matrices or pencils under equivalence transformations which keep the class invariant. In contrast to other canonical forms our forms are as close as possible to a triangular structure in the same class. We give necessary and sufficient conditions for the existence of Hamiltonian and symplectic triangular Jordan, Kronecker and Schur forms. The presented results generalize results of Lin and Ho [17] and simplify the proofs presented there.