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A More Accurate Briggs Method for the Logarithm
No full textA new approach for computing an expression of the form
is presented that avoids the danger of
subtractive cancellation in floating point arithmetic, where
is a complex number not belonging to the closed negative
real axis and is a nonnegative integer. We also derive a
condition number for the problem. The algorithm therefore
allows highly accurate numerical calculation of using Briggs' method
State constrained reachability for stochastic hybrid systems
No full textMany control problems can be formulated as driving a system to reach some target states while avoiding some unwanted states. We study this problem for systems with regime change operating in uncertain environments. Nowadays, it is a common practice to model such systems in the framework of stochastic hybrid system models. In this casting, the problem is formalized as a mathematical problem named state constrained stochastic reachability analysis. In the state constrained stochastic reachability analysis, this probability is computed by imposing a constraint on the system to avoid the unwanted states. The scope of this paper is twofold. First we define and investigate the state constrained reachability analysis in an abstract mathematical setting. We define the problem for a general model of stochastic hybrid systems, and we show that the reach probabilities can be computed as solutions of an elliptic integro-differential equation. Moreover, we extend the problem by considering randomized targets. We approach this extension using stochastic dynamic programming. The second scope is to define a developmental setting in which the state constrained reachability analysis becomes more tractable. This framework is based on multilayer modelling of a stochastic system using hierarchical viewpoints. Viewpoints represent a method originated from software engineering, where a system is described by multiple models created from different perspectives. Using viewpoints, the reach probabilities can be easily computed, or even symbolically calculated. The reach probabilities computed in one viewpoint can be used in another viewpoint for improving the system control. We illustrate this technique for trajectory design
PVD�s contributions to numerical methods in systems and control
No full textPaul M. Van Dooren received the engineering degree in computer science and the doctoral degree in applied sciences, both from the Katholieke Universiteit Leuven, Belgium, in 1974 and 1979, respectively. He held research and teaching positions at the Katholieke Universiteit Leuven (1974-1979), the University of Southern California (1978-1979), Stanford University (1979-1980), the Australian
National University (1984), Philips Research Laboratory Belgium (1980-1991), the University of Illinois at Urbana-Champaign (1991-1994), Florida State University (1998) and the Universite Catholique de Louvain (1980-1991, 1994-now)
where he is currently a professor of Mathematical Engineering.
Dr. Van Dooren received the IBM-Belgium Informatics Award in 1974, the Householder Award in 1981 and the SIAM Wilkinson Prize of Numerical Analysis and Scientific Computing in 1989. He is a Fellow of IEEE and of SIAM (Society of Industrial and Applied Mathematics). He received the Francqui Chair in Antwerp in 2010. He is an Associate Editor of several journals in numerical analysis and systems and control theory. His main interests lie in the areas of numerical linear algebra, systems and control theory, and in numerical methods for large graphs
and networks.
In this talk we will review some of his main contributions and we will show some rare moments captured on pictures by some of his students and friends
The Two Ball Newton's Cradle
No full textNewton's cradle for two balls with Hertzian interactions is considered as a hybrid system, and this makes it possible to
derive return maps for the motion between collisions in an exact form despite the fact that the three halves interaction law cannot be solved in closed form. The return maps depend on a constant whose value can only be determined numerically, but solutions can be written down explicitly in terms of this parameter, and we compare this with the results of simulations. The results are in fact independent of the details of the interaction potential
A Schur--Pad\'e Algorithm for Fractional Powers of a Matrix
Get PDFA new algorithm is developed for computing arbitrary real powers of a matrix . The algorithm starts with a Schur decomposition, takes square roots of the triangular factor , evaluates an Pad\'e approximant of at , and squares the result times. The parameters and are chosen to minimize the cost subject to achieving double precision accuracy in the evaluation of the Pad\'e approximant, making use of a result that bounds the error in the matrix Pad\'e approximant by the error in the scalar Pad\'e approximant with argument the norm of the matrix. The Pad\'e approximant is evaluated from the continued fraction representation in bottom-up fashion, which is shown to be numerically stable. In the squaring phase the diagonal and first superdiagonal are computed from explicit formulae for , yielding increased accuracy. Since the basic algorithm is designed for , a criterion for reducing an arbitrary real to this range is developed, making use of bounds for the condition number of the problem. How best to compute for a negative integer is also investigated. In numerical experiments the new algorithm is found to be superior in accuracy and stability to several alternatives, including the use of an eigendecomposition and approaches based on the formula
Improved Inverse Scaling and Squaring Algorithms for the Matrix Logarithm
Get PDFA popular method for computing the matrix logarithm is the
inverse scaling and squaring method,
which essentially carries out the steps of the scaling and squaring
method for the matrix exponential in reverse order.
Here we make several improvements to the method,
putting its development on a par with our recent version
[\emph{SIAM J. Matrix Anal.\ Appl.}, 31 (2009), pp.\ 970--989]
of the scaling and squaring method for the exponential.
In particular,
we introduce backward error analysis to replace the previous forward
error analysis;
obtain backward error bounds in terms of the quantities
, for several small integer , instead of ;
and use special techniques to compute the argument of the
Pad\'e approximant more accurately.
We derive one algorithm
that employs a Schur decomposition,
and thereby works with triangular matrices,
and another that requires only matrix multiplications and the solution
of multiple right-hand side linear systems.
Numerical experiments show the new algorithms to be generally faster and more
accurate than their existing counterparts
and suggest that the Schur-based method is the method of choice for computing
the matrix logarithm
Characterisation of multiple substrate-specific (d)ITP/(d)XTPase and modelling of deaminated purine nucleotide metabolism
No full textAccumulation of modified nucleotides is defective to various cellular processes, especially those involving DNA and RNA. To be viable, organisms possess a number of (deoxy)nucleotide phosphohydrolases, which hydrolyze these nucleotides removing them from the active NTP and dNTP pools. Deamination of purine bases can result in accumulation of such nucleotides as ITP, dITP, XTP and dXTP. E. coli RdgB has been characterised as a deoxyribonucleoside triphosphate pyrophosphohydrolase that can act on these nucleotides. S. cerevisiae homologue encoded by YJR069C was purified and its (d)NTPase activity was assayed using fifteen nucleotide substrates. ITP, dITP, and XTP were identified as major substrates and kinetic parameters measured. Inhibition by ATP, dATP and GTP were established. On the basis of experimental and published data, modelling and simulation of ITP, dITP, XTP and dXTP metabolism was performed. (d)ITP/ (d)XTPase is a new example of enzyme with multiple substrate-specificity demonstrating that multispecificity is not a rare phenomenon
Fiedler companion linearizations for rectangular matrix polynomials
Get PDFThe development of new classes of linearizations of
square matrix polynomials that generalize
the classical first and second Frobenius companion forms has attracted much attention in the last decade.
Research in this area has two main goals:
finding linearizations that retain whatever structure
the original polynomial might possess,
and improving properties that are essential
for accurate numerical computation,
such as eigenvalue condition numbers and backward errors.
However, all recent progress on linearizations
has been restricted to square matrix polynomials.
Since rectangular polynomials arise in many applications, it is natural to investigate
if the new classes of linearizations
can be extended to rectangular polynomials.
In this paper, the family of Fiedler linearizations
is extended
from square to rectangular matrix polynomials,
and it is shown that minimal indices and bases
of polynomials can be recovered
from those of any linearization in this class
via the same simple procedures
developed previously for square polynomials.
Fiedler linearizations are
one of the most important classes of linearizations introduced in recent years,
but their generalization to rectangular polynomials
is nontrivial,
and requires a completely different approach
to the one used in the square case.
To the best of our knowledge,
this is the first class of new linearizations
that has been generalized to rectangular polynomials