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Iterative Solution of a Nonsymmetric Algebraic Riccati Equation
No full textWe study the nonsymmetric algebraic Riccati equation whose four
coefficient matrices are the blocks of
a nonsingular -matrix or an irreducible singular
-matrix . The solution of practical interest is the minimal nonnegative
solution. We show that Newton's method with zero initial guess can be used to
find this solution without any further assumptions. We also present a
qualitative perturbation analysis for the minimal solution, which is
instructive in designing algorithms for finding more accurate approximations.
For the most practically important case, in
which is an irreducible singular -matrix with zero row sums,
the minimal solution is either stochastic or substochastic and
the Riccati equation can be transformed into a unilateral matrix equation by
a procedure of Ramaswami. The minimal solution of the Riccati equation can
then be found by computing the minimal nonnegative solution of
the unilateral equation using
the Latouche--Ramaswami algorithm.
When the minimal solution of the Riccati equation is stochastic,
we show that the Latouche--Ramaswami algorithm, combined with a shift
technique suggested by He, Meini, and Rhee,
is breakdown-free and is able to find the minimal solution
more efficiently and more accurately than the algorithm without a shift.
When the minimal solution of the Riccati equation is substochastic,
we show how the substochastic minimal solution can be found by computing
the stochastic minimal solution of a related Riccati equation of the same
type
Stochastic Modeling of Gene Regulatory Networks
Get PDFGene Regulatory Networks (GRNs) describe how chemical species within a cell interact with one another, thereby governing the rates at which key genes are expressed. This thesis is concerned with modeling a particular GRN, Arabidopsis thaliana Circadian Clock, by considering three different approaches; discrete stochastic, continuous stochastic and parameter variation. By considering these different methods we will see if the desired behavior required from our network is robust to biological noise. Through employing stochastic approaches we found the GRN under question is robust to biological noise to a point; the results of our study led to a couple of interesting questions to people within the field. When the number of molecules involved in the reactions were reduced sufficiently the biological noise in the system destroyed the desired circadian rhythm. To the biologists we would ask how low are the molecule numbers involved in such reactions and to the modelers how appropriate is it to use Michaelis-Menten type kinetics for low molecule numbers
The Solution of S exp(S) = A is Not Always the Lambert W Function of A
No full textWe study the solutions of the matrix equation .
Our motivation comes from the study of systems of delay differential equations
, which occur in some models of practical
interest, especially in mathematical biology. This paper
concentrates on the distinction between \emph{evaluating a matrix
function} and \emph{solving a matrix equation}.
In particular,
it shows that the matrix Lambert function evaluated at the
matrix does not represent all possible solutions of . These results can easily be extended to more general matrix
equations
Limitations of the PlayStation 3 for High Performance Cluster Computing
Get PDFPower consumption, heat dissipation and other physical limitations are pushing
the microprocessor industry towards multicore design patterns. Most of the
processor manufacturers, such as Intel and AMD, are following more conventional
approaches, which consist of homogeneous, symmetric multicores where
execution units are replicated on the same dime; multiple execution units share
some cache level (generally L2 and L3) and the bus to memory. Other manufacturers
proposed still homogeneous approaches but with a stronger emphasis on
parallelism and hyperthreading. This is, for example, the case of Sun with the
UltraSPARC T1 (known as “Niagara”). The UltraSPARC T1 [25,24] can have
up to eight homogeneous cores each of which is four-way hyperthreaded which
delivers a maximum parallelism degree of thirty-two. The Niagara processor is
mostly developed for web servers and database applications since it provides
high computational power for integer operations, which are used considerably in
pointer arithmetics and string processing. Yet other chip manufacturers started
exploring heterogeneous designs where cores have different architectural features.
One such example is the Cell Broadband Engine [22,17,19,18] developed by STI,
a consortium formed by Sony, Toshiba and IBM. The Cell BE has outstanding
floating-point computational power, which makes it a considerable candidate for
high performance computing systems. IBM shipped the first Cell-based system,
the BladeCenter QS20, on September 12th 2006. This blade is equipped with
two Cell processors with a 512 MB memory each and connected in a NUMA
configuration; the external connectivity is achieved through a Gigabit and an
Infiniband network interface. The BladeCenter QS20 has impressive computational
power that, coupled with its high speed network interfaces, makes it a good
candidate for high performance cluster computing. At almost the same period
(November 11th), Sony released the PlayStation 3 (PS3) gaming console. Even
if this console is not meant for high performance computing, it is still equipped
with a (stripped down) Cell processor and its price ( $600) definitely makes
it an attractive solution for building a Cell-based cluster. This document aims
at evaluating the performance and the limitations of the PS3 platform for high
performance cluster computing
Permutation groups of finite Morley rank: an Oberwolfach talk
Get PDFThe principal result of the talk bounds the Morley rank of a definably primitive permutation group of finite Morley rank in terms of the rank of the set on which it acts
Mixed Precision Iterative Refinement Techniques for the Solution of Dense Linear Systems
Get PDFBy using a combination of 32-bit and 64-bit floating point arithmetic, the performance
of many dense and sparse linear algebra algorithms can be significantly
enhanced while maintaining the 64-bit accuracy of the resulting solution. The approach
presented here can apply not only to conventional processors but also to
exotic technologies such as Field Programmable Gate Arrays (FPGA), Graphical
Processing Units (GPU), and the Cell BE processor. Results on modern processor
architectures and the Cell BE are presented
On the Krohn-Rhodes complexity of semigroups of upper triangular matrices
Get PDFWe consider the Krohn–Rhodes complexity of certain semigroups of upper triangular matrices over finite fields. We show that for any n > 1 and finite field k, the semigroups of all n × n upper triangular matrices over k and of all n × n unitriangular matrices over k have complexity n - 1. A consequence is that the complexity c > 1 of a finite semigroup places a lower bound of c + 1 on the dimension of any faithful triangular representation of that semigroup over a finite field
Conjugacy problem in HNN-extensions: regular elements and black holes
Get PDFWe discuss the complexity of conjugacy problem in HNN-extensions of groups. We stratify the groups in question and show that for ``almost all'', in some explicit sense, elements, the conjugacy search problem is decidable
Definite Matrix Polynomials and their Linearization by Definite Pencils
Get PDFHyperbolic matrix polynomials
are an important class of Hermitian matrix polynomials
that contain overdamped quadratics as a special case.
They share with definite pencils the spectral property that their eigenvalues
are real and semisimple.
We extend the definition of hyperbolic matrix polynomial
in a way that relaxes the requirement of definiteness of
the leading coefficient matrix,
yielding what we call definite polynomials.
We show that this class of polynomials has an elegant characterization in terms
of definiteness intervals on the extended real line,
and that it includes definite pencils as a special case.
A fundamental question is whether a definite
matrix polynomial can be linearized
in a structure-preserving way.
We show that the answer to this question is affirmative:
is definite if and only if it has a
definite linearization in ,
a certain vector space of Hermitian pencils;
and for definite we give a complete characterization of all the
linearizations in that are definite. For the important
special case of quadratics, we show how a definite quadratic
polynomial can be transformed into a definite linearization with a
positive definite leading coefficient matrix---a form that is
particularly attractive numerically