Past

In recent times, research into scattering of electromagnetic waves by complex objects

has assumed great importance due to its relevance to radar applications, where the

main objective is to identify targeted objects. In designing stealth weapon systems

such as military aircraft, control of their radar cross section is of paramount

importance. Aircraft in combat situations are threatened by enemy missiles. One

countermeasure which is used to reduce this threat is to minimise the radar cross

section. On the other hand, there is a demand for the enhancement of the radar cross

section of civilian spacecraft. Operators of communication satellites often request

a complicated differential radar cross section in order to assist with the tracking

of the satellite. To control the radar cross section, an essential requirement is a

capability for accurate prediction of electromagnetic scattering from complex objects.

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One difficulty which is encountered in the development of suitable numerical solution

schemes is the existence of constraints which are in excess of those needed for a unique

solution. Rather than attempt to include the constraint in the equation set, the novel

approach which is presented here involves the use of the finite element method and the

construction of a specialised element in which the relevant solution variables are

appropriately constrained by the nature of their interpolation functions. For many

years, such an idea was claimed to be impossible. While the idea is not without its

difficulties, its advantages far outweigh its disadvantages. The presenter has

successfully developed such an element for primitive variable solutions to viscous

incompressible flows and wishes to extend the concept to electromagnetic scattering

problems.

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Dr Mack has first degrees in mathematics and aeronautical engineering, plus a Masters

and a Doctorate, both in computational fluid dynamics. He has some thirty years

experience in this latter field. He pioneered the development of the innovative

solenoidal approach for the finite element solution of viscous incompressible flows.

At the time, such a radical idea was claimed in the literature to be impossible.

Much of this early research was undertaken during a six month sabbatical with the

Numerical Analysis Group at the Oxford University Computing Laboratory. Dr Mack has

since received funding from British Aerospace and the United States Department of

Defense to continue this research.

A new filter method will be presented that attempts to find a feasible

point for sets of nonlinear sets of equalities and inequalities. The

method is intended to work for problems where the number of variables

or the number of (in)equalities is large, or both. No assumption is

made about convexity. The technique used is that of maintaining a list

of multidimensional "filter entries", a recent development of ideas

introduced by Fletcher and Leyffer. The method will be described, as

well as large scale numerical experiments with the corresponding

Fortran 90 module, FILTRANE.

In addition to the announced topic of Pascal Matrices (abstract below) we will speak briefly about more recent work by Per-Olof Persson on generating simplicial meshes on regions defined by a function that gives the distance from the boundary. Our first goal was a short MATLAB code and we just submitted "A Simple Mesh Generator in MATLAB" to SIAM.

This is joint work with Alan Edelman at MIT and a little bit with Pascal. They had all the ideas.

Put the famous Pascal triangle into a matrix. It could go into a lower triangular L or its transpose L' or a symmetric matrix S:

These binomial numbers come from a recursion, or from the formula for i choose j, or functionally from taking powers of (1 + x).

The amazing thing is that L times L' equals S. (OK for 4 by 4) It follows that S has determinant 1. The matrices have other unexpected properties too, that give beautiful examples in teaching linear algebra. The proof of L L' = S comes 3 ways, I don't know which you will prefer:

1. By induction using the recursion formula for the matrix entries.
2. By an identity for the coefficients i+j choose j in S.
3. By applying both sides to the column vector [ 1 x x² x³ ... ]'.

The third way also gives a proof that S³ = -I but we doubt that result.

The rows of the "hypercube matrix" L² count corners and edges and faces and ... in n dimensional cubes.

Seminar moved to Week 8, 19 June 2003.

From the point of view of a numerical analyst, I will describe some algorithms for:

• clustering data points based on pairwise similarity,
• reordering a sparse matrix to reduce envelope, two-sum or bandwidth,
• reordering nodes in a range-dependent random graph to reflect the range-dependency,

and point out some connections between seemingly disparate solution techniques. These datamining problems arise across a range of disciplines. I will mention a particularly new and important application from bioinformatics concerning the analysis of gene or protein interaction data.

Immersed interface methods have been developed for a variety of

differential equations on domains containing interfaces or irregular

boundaries. The goal is to use a uniform Cartesian grid (or other fixed

grid on simple domain) and to allow other boundaries or interfaces to

cut through this grid. Special finite difference formulas are developed

at grid points near an interface that incorporate the appropriate jump

conditions across the interface so that uniform second-order accuracy

(or higher) can be obtained. For fluid flow problems with an immersed

deformable elastic membrane, the jump conditions result from a balance

between the singular force imposed by the membrane, inertial forces if

the membrane has mass, and the jump in pressure across the membrane.

A second-order accurate method of this type for Stokes flow was developed

with Zhilin Li and more recently extended to the full incompressible

Navier-Stokes equations in work with Long Lee.

Feedback design for a second order control system leads to an

eigenstructure assignment problem for a quadratic matrix polynomial. It is

desirable that the feedback controller not only assigns specified

eigenvalues to the second order closed loop system, but also that the

system is robust, or insensitive to perturbations. We derive here new

sensitivity measures, or condition numbers, for the eigenvalues of the

quadratic matrix polynomial and define a measure of robustness of the

corresponding system. We then show that the robustness of the quadratic

inverse eigenvalue problem can be achieved by solving a generalized linear

eigenvalue assignment problem subject to structured perturbations.

Numerically reliable methods for solving the structured generalized linear

problem are developed that take advantage of the special properties of the

system in order to minimize the computational work required.

Electricity markets facilitate pricing and delivery of wholesale power.

Generators submit bids to an Independent System Operator (ISO) to indicate

how much power they can produce depending on price. The ISO takes these bids

with demand forecasts and minimizes the total cost of power production

subject to feasibility of distribution in the electrical network.

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Each generator can optimise its bid using a bilevel program or

mathematical program with equilibrium (or complementarity) constraints, by

taking the ISOs problem, which contains all generators bid information, at

the lower level. This leads immediately to a game between generators, where

a Nash equilibrium - at which each generator's bid maximises its profit

provided that none of the other generators changes its bid - is sought.

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In particular, we examine the idealised model of Berry et al (Utility

Policy 8, 1999), which gives a bilevel game that can be modelled as an

"equilibrium problem with complementarity constraints" or EPCC.

Unfortunately, like bilevel games, EPCCs on networks may not have Nash

equilibria in the (common) case when one or more of links of the network is

saturated (at maximum capacity). Nevertheless we explore some theory and

algorithms for this problem, and discuss the economic implications of

numerical examples where equilibria are found for small electricity

networks.

Traditional optimisation theory and -methods on the basis of the

Lagrangian function do not apply to objective or constraint functions

which are defined by means of a combinatorial selection structure. Such

selection structures can be explicit, for example in the case of "min",

"max" or "if" statements in function evaluations, or implicit as in the

case of inverse optimisation problems where the combinatorial structure is

induced by the possible selections of active constraints. The resulting

optimisation problems are typically neither convex nor smooth and do not

fit into the standard framework of nonlinear optimisation. Users typically

treat these problems either through a mixed-integer reformulation, which

drastically reduces the size of tractable problems, or by employing

nonsmooth optimisation methods, such as bundle methods, which are

typically based on convex models and therefore only allow for weak

convergence results. In this talk we argue that the classical Lagrangian

theory and SQP methodology can be extended to a fairly general class of

nonlinear programs with combinatorial constraints. The paper is available

at http://www.eng.cam.ac.uk/~ss248/publications.

Is it possible to construct a computational model of the real numbers in which the sign

of every computed result is corrected determined? The answer is yes, both in theory and in

practice. The resulting viewpoint contrasts strongly with the traditional floating

point model. I will review the theoretical background and software design issues,

discuss previous attempts at implementation and finally demonstrate my own python and

C++ codes.

The pseudospectral method for solving boundary value problems on the interval

consists in replacing the solution by an interpolating polynomial in Lagrangian

form between well-chosen points and collocating at those same points.

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Due to its globality, the method cannot handle steep gradients well (Markov's inequality).

We will present and discuss two means of improving upon this: the attachment of poles to

the ansatz polynomial, on one hand, and conformal point shifts on the other hand, both

optimally adapted to the problem to be solved.

Dynamic optimisation is a tool that enables the process industries to

compute optimal control strategies for important chemical processes.

Aspen DynamicsTM is a well-established commercial engineering software

package containing a dynamic optimisation tool. Its intuitive graphical

user interface and library of robust dynamic models enables engineers to

quickly and easily define a dynamic optimisation problem including

objectives, control vector parameterisations and constraints. However,

this is only one part of the story. The combination of dynamics and

non-linear optimisation can create a problem that can be very difficult

to solve due to a number of reasons, including non-linearities, poor

initial guesses, discontinuities and accuracy and speed of dynamic

integration. In this talk I will begin with an introduction to process

modelling and outline the algorithms and techniques used in dynamic

optimisation. I will move on to discuss the numerical issues that can

give us so much trouble in practice and outline some solutions we have

created to overcome some of them.

Many questions of interest to both mathematicians and physicists relate

to the behavior of eigenvalues and eigenmodes of the Laplace operator

on a polygon. Algorithmic improvements have revived the old "method

of fundamental solutions" associated with Fox, Henrici and Moler; is it

going to end up competitive with the state-of-the-art method of Descloux,

Tolley and Driscoll? This talk will outline the numerical issues but

give equal attention to applications including "can you hear the shape

of a drum?", localization of eigenmodes, eigenvalue avoidance, and the

design of drums that play chords.

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This is very much work in progress -- with graduate student Timo Betcke.

Image processing is an area with many important applications, as well as challenging problems for mathematicians. In particular, Fourier/wavelets analysis and stochastic/statistical methods have had major impact in this area. Recently, there has been increased interest in a new and complementary approach, using partial differential equations (PDEs) and differential-geometric models. It offers a more systematic treatment of geometric features of mages, such as shapes, contours and curvatures, etc., as well as allowing the wealth of techniques developed for PDEs and Computational Fluid Dynamics (CFD) to be brought to bear on image processing tasks.

I'll use two examples from my recent work to illustrate this synergy:

1. A unified image restoration model using Total Variation (TV) which can be used to model denoising, deblurring, as well as image inpainting (e.g. restoring old scratched photos). The TV idea can be traced to shock capturing methods in CFD and was first used in image processing by Rudin, Osher and Fatemi.

2. An "active contour" model which uses a variational level set method for object detection in scalar and vector-valued images. It can detect objects not necessarily defined by sharp edges, as well as objects undetectable in each channel of a vector-valued image or in the combined intensity. The contour can go through topological changes, and the model is robust to noise. The level set method was originally developed by Osher and Sethian for tracking interfaces in CFD.

(The above are joint works with Jackie Shen at the Univ. of Minnesota and Luminita Vese in the Math Dept at UCLA.)

2.00 pm Professor Iain Duff (RAL) Opening remarks

2.15 pm Professor M J D Powell (University of Cambridge)

Some developments of work with Alan on cubic splines

3.00 pm Professor Kevin Burrage (University of Queensland)

Stochastic models and simulations for chemically reacting systems

3.30 pm Tea/Coffee

4.00 pm Professor John Reid (RAL)

Sparse matrix research at Harwell and the Rutherford Appleton Laboratory

4.30 pm Dr Ian Jones (AEA PLC)

Computational fluid dynamics and the role of stiff solvers

5.00 pm Dr Lawrence Daniels (Hyprotech UK Ltd)

Current work with Alan on ODE solvers for HSL

Long-step primal-dual path-following algorithms constitute the

framework of practical interior point methods for

solving linear programming problems. We consider

such an algorithm and a second order variant of it.

We address the problem of the convergence of

the sequences of iterates generated by the two algorithms

to the analytic centre of the optimal primal-dual set.

Several real Lie and Jordan algebras, along with their associated

automorphism groups, can be elegantly expressed in the quaternion tensor

algebra. The resulting insight into structured matrices leads to a class

of simple Jacobi algorithms for the corresponding $n \times n$ structured

eigenproblems. These algorithms have many desirable properties, including

parallelizability, ease of implementation, and strong stability.

A method for computing periodic orbits for the Navier-Stokes

equations will be presented. The method uses a finite-element Galerkin

discretisation for the spatial part of the problem and a spectral

Galerkin method for the temporal part of the problem. The method will

be illustrated by calculations of the periodic flow behind a circular

cylinder in a channel. The problem has a simple reflectional symmetry

and it will be explained how this can be exploited to reduce the cost

of the computations.

The convergence of Krylov subspace methods like conjugate gradients

depends on the eigenvalues of the underlying matrix. In many cases

the exact location of the eigenvalues is unknown, but one has some

information about the distribution of eigenvalues in an asymptotic

sense. This could be the case for linear systems arising from a

discretization of a PDE. The asymptotic behavior then takes place

when the meshsize tends to zero.

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We discuss two possible approaches to study the convergence of

conjugate gradients based on such information.

The first approach is based on a straightforward idea to estimate

the condition number. This method is illustrated by means of a

comparison of preconditioning techniques.

The second approach takes into account the full asymptotic

spectrum. It gives a bound on the asymptotic convergence factor

which explains the superlinear convergence observed in many situations.

This method is mathematically more involved since it deals with

potential theory. I will explain the basic ideas.

We develop an algorithm for estimating the local Sobolev regularity index

of a given function by monitoring the decay rate of its Legendre expansion

coefficients. On the basis of these local regularities, we design and

implement an hp--adaptive finite element method based on employing

discontinuous piecewise polynomials, for the approximation of nonlinear

systems of hyperbolic conservation laws. The performance of the proposed

adaptive strategy is demonstrated numerically.

The study of the finite precision behaviour of numerical algorithms dates back at least as far as Turing and Wilkinson in the 1940s. At the start of the 21st century, this area of research is still very active.

We focus on some topics of current interest, describing recent developments and trends and pointing out future research directions. The talk will be accessible to those who are not specialists in numerical analysis.

Specific topics intended to be addressed include

• Floating point arithmetic: correctly rounded elementary functions, and the fused multiply-add operation.
• The use of extra precision for key parts of a computation: iterative refinement in fixed and mixed precision.
• Gaussian elimination with rook pivoting and new error bounds for Gaussian elimination.
• Automatic error analysis.
• Application and analysis of hyperbolic transformations.

We describe "avoidance of crossing" and its explanation by von

Neumann and Wigner. We show Lax's criterion for degeneracy and then

discover matrices whose determinants give the discriminant of the

given matrix. This yields a simple proof of the bound given by

Ilyushechkin on the number of terms in the expansion of the discriminant

as a sum of squares. We discuss the 3 x 3 case in detail.

The talk will discuss unsymmetric sparse LU factorization based on

the Markowitz pivot selection criterium. The key question for the

author is the following: Is it possible to implement a sparse

factorization where the overhead is limited to a constant times

the actual numerical work? In other words, can the work be bounded

by o(sum(k, M(k)), where M(k) is the Markowitz count in pivot k.

The answer is probably NO, but how close can we get? We will give

several bad examples for traditional methods and suggest alternative

methods / data structure both for pivot selection and for the sparse

update operations.

We discuss two filters that are frequently used to smooth data.

One is the (nonlinear) median filter, that chooses the median

of the sample values in the sliding window. This deals effectively

with "outliers" that are beyond the correct sample range, and will

never be chosen as the median. A straightforward implementation of

the filter is expensive for large windows, particularly in two dimensions

(for images).

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The second filter is linear, and known as "Savitzky-Golay". It is

frequently used in spectroscopy, to locate positions and peaks and

widths of spectral lines. This filter is based on a least-squares fit

of the samples in the sliding window to a polynomial of relatively

low degree. The filter coefficients are unlike the equiripple filter

that is optimal in the maximum norm, and the "maxflat" filters that

are central in wavelet constructions. Should they be better known....?

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We will discuss the analysis and the implementation of both filters.

The asymptotic rate of convergence of the trapezium rule is

defined, for smooth functions, by the Euler-Maclaurin expansion.

The extension to other methods, such as Gauss rules, is straightforward;

this talk begins with some special cases, such as Periodic functions, and

functions with various singularities.

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Convergence rates for ODEs (Initial and Boundary value problems)

and for PDEs are available, but not so well known. Extension to singular

problems seems to require methods specific to each situation. Some of

the results are unexpected - to me, anyway.

In the past few years we have developed some expertise in solving optimization

problems that involve large scale simulations in various areas of Computational

Geophysics and Engineering. We will discuss some of those applications here,

namely: inversion of seismic data, characterization of piezoelectrical crystals

material properties, optimal design of piezoelectrical transducers and

opto-electronic devices, and the optimal design of steel structures.

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A common theme among these different applications is that the goal functional

is very expensive to evaluate, often, no derivatives are readily available, and

some times the dimensionality can be large.

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Thus parallelism is a need, and when no derivatives are present, search type

methods have to be used for the optimization part. Additional difficulties can

be ill-conditioning and non-convexity, that leads to issues of global

optimization. Another area that has not been extensively explored in numerical

optimization and that is important in real applications is that of

multiobjective optimization.

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As a result of these varied experiences we are currently designing a toolbox

to facilitate the rapid deployment of these techniques to other areas of

application with a minimum of retooling.

A-Posteriori Error estimates for high order Godunov finite

volume methods are presented which exploit the two solution

representations inherent in the method, viz. as piecewise

constants $u_0$ and cell-wise $q$-th order reconstructed

functions $R^0_q u_0$. The analysis provided here applies

directly to schemes such as MUSCL, TVD, UNO, ENO, WENO or any

other scheme that is a faithful extension of Godunov's method

to high order accuracy in a sense that will be made precise.

Using standard duality arguments, we construct exact error

representation formulas for derived functionals that are

tailored to the class of high order Godunov finite volume

methods with data reconstruction, $R^0_q u_0$. We then consider

computable error estimates that exploit the structure of higher

order Godunov finite volume methods. The analysis technique used

in this work exploits a certain relationship between higher

order Godunov methods and the discontinuous Galerkin method.

Issues such as the treatment of nonlinearity and the optional

post-processing of numerical dual data are also discussed.

Numerical results for linear and nonlinear scalar conservation

laws are presented to verify the analysis. Complete details can

be found in a paper appearing in the proceedings of FVCA3,

Porquerolles, France, June 24-28, 2002.

SMP (Symmetric Multi-Processors) hardware technologies are very popular

with vendors and end-users alike for a number of reasons. However, true

shared memory parallelism has experienced somewhat slower to take up

amongst the scientific-programming community. NAG has been at the

forefront of SMP technology for a number of years, and the NAG SMP

Library has shown the potential of SMP systems.

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At the very high end, SMP hardware technologies are used as building

blocks of modern supercomputers, which truly consist of clusters of SMP

systems, for which no dedicated model of parallelism yet exists.

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The aim of this talk is to introduce SMP systems and their potential.

Results from our work at NAG will also be introduced to show how SMP

parallelism, based on a shared memory paradigm, can be used to very

good effect and can produce high performance, scalable software. The

talk also aims to discuss some aspects of the apparent slow take up of

shared memory parallelism and the potential competition from PC (i.e.

Intel)-based cluster technology. The talk then aims to explore the

potential of SMP technology within "hybrid parallelism", i.e. mixed

distributed and shared memory modes, illustrating the point with some

preliminary work carried out by the author and others. Finally, a

number of potential future challenges to numerical analysts will be

discussed.

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The talk is aimed at all who are interested in SMP technologies for

numerical computing, irrespective of any previous experience in the

field. The talk aims to stimulate discussion, by presenting some ideas,

backing these with data, not to stifle it in an ocean of detail!

It is well known that discrete solutions to the convection-diffusion

equation contain nonphysical oscillations when boundary layers are present

but not resolved by the discretisation. For the Galerkin finite element

method with linear elements on a uniform 1D grid, a precise statement as

to exactly when such oscillations occur can be made, namely, that for a

problem with mesh size h, constant advective velocity and different values

at the left and right boundaries, oscillations will occur if the mesh

P\'{e}clet number $P_e$ is greater than one. In 2D, however, the situation

is not so well understood. In this talk, we present an analysis of a 2D

model problem on a square domain with grid-aligned flow which enables us

to clarify precisely when oscillations occur, and what can be done to

prevent them. We prove the somewhat surprising result that there are

oscillations in the 2D problem even when $P_e$ is less than 1. Also, we show that there

are distinct effects arising from differences in the top and bottom

boundary conditions (equivalent to those seen in 1D), and the non-zero

boundaries parallel to the flow direction.

Algebra based modeling systems are becoming essential elements in the

application of large and complex mathematical programs. These systems

enable the abstraction, expression and translation of practical

problems into reliable and effective operational systems. They provide

the bridged between algorithms and real world problems by automating

the problem analysis and translation into specific data structures and

provide computational services required by different solvers. The

modeling system GAMS will be used to illustrate the design goals and

main features of such systems. Applications in use and under

development will be used to provide the context for discussing the

changes in user focus and future requirements. This presents new sets

of opportunities and challenges to the supplier and implementer of

mathematical programming solvers and modeling systems.

A systematic approach to error control and mesh adaptation for

optimal control of systems governed by PDEs is presented.

Starting from a coarse mesh, the finite element spaces are successively

enriched in order to construct suitable discrete models.

This process is guided by an a posteriori error estimator which employs

sensitivity factors from the adjoint equation.

We consider different examples with the stationary Navier-Stokes

equations as state equation.

In this presentation we examine the convergence characteristics of a

Krylov subspace solver preconditioned by a new indefinite

constraint-type preconditioner, when applied to discrete systems

arising from low-order mixed finite element approximation of the

classical biharmonic problem. The preconditioning operator leads to

preconditioned systems having an eigenvalue distribution consisting of

a tightly clustered set together with a small number of outliers. We

compare the convergence characteristics of a new approach with the

convergence characteristics of a standard block-diagonal Schur

complement preconditioner that has proved to be extremely effective in

the context of mixed approximation methods.

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In the second part of the presentation we are concerned with the

efficient parallel implementation of proposed algorithm on modern

shared memory architectures. We consider use of the efficient parallel

"black-box'' solvers for the Dirichlet Laplacian problems based on

sparse Cholesky factorisation and multigrid, and for this purpose we

use publicly available codes from the HSL library and MGNet collection.

We compare the performance of our algorithm with sparse direct solvers

from the HSL library and discuss some implementation related issues.

(Joint work with Felipe Cucker and Dennis Cheung, City University of Hong Kong.)

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Condition numbers are important complexity-theoretic tools to capture

a "distillation" of the input aspects of a computational problem that

determine the running time of algorithms for its solution and the

sensitivity of the computed output. The motivation for our work is the

desire to understand the average case behaviour of linear programming

algorithms for a large class of randomly generated input data in the

computational model of a machine that computes with real numbers. In

this model it is not known whether linear programming is polynomial

time solvable, or so-called "strongly polynomial". Closely related to

linear programming is the problem of either proving non-existence of

or finding an explicit example of a point in a polyhedral cone defined

in terms of certain input data. A natural condition number for this

computational problem was developed by Cheung and Cucker, and we analyse

its distributions under a rather general family of input distributions.

We distinguish random sampling of primal and dual constraints

respectively, two cases that necessitate completely different techniques

of analysis. We derive the exact exponents of the decay rates of the

distribution tails and prove various limit theorems of complexity

theoretic importance. An interesting result is that the existence of

the k-th moment of Cheung-Cucker's condition number depends only very

mildly on the distribution of the input data. Our results also form

the basis for a second paper in which we analyse the distributions of

Renegar's condition number for the randomly generated linear programming

problem.

Toeplitz matrices enjoy the dual virtues of ubiquity and beauty. We begin this talk by surveying some of the interesting spectral properties of such matrices, emphasizing the distinctions between infinite-dimensional Toeplitz matrices and the large-dimensional limit of the corresponding finite matrices. These basic results utilize the algebraic Toeplitz structure, but in many applications, one is forced to spoil this structure with some perturbations (e.g., by imposing boundary conditions upon a finite difference discretization of an initial-boundary value problem). How do such

perturbations affect the eigenvalues? This talk will address this question for "localized" perturbations, by which we mean perturbations that are restricted to a single entry, or a block of entries whose size remains fixed as the matrix dimension grows. One finds, for a broad class of matrices, that sufficiently small perturbations fail to alter the spectrum, though the spectrum is exponentially sensitive to other perturbations. For larger real single-entry

perturbations, one observes the perturbed eigenvalues trace out curves in the complex plane. We'll show a number of illustrations of this phenomenon for tridiagonal Toeplitz matrices.

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This talk describes collaborative work with Albrecht Boettcher, Marko Lindner, and Viatcheslav Sokolov of TU Chemnitz.