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Computing the Conditioning of the Components of a Linear Least Squares Solution
Get PDFIn this paper, we address the accuracy of the results for the overdetermined full rank linear least squares
problem. We recall theoretical results obtained in [2] on conditioning of the least squares solution and the
components of the solution when the matrix perturbations are measured in Frobenius or spectral norms. Then we
define computable estimates for these condition numbers and we interpret them in terms of statistical quantities.
In particular, we show that, in the classical linear statistical model, the ratio of the variance of one component of
the solution by the variance of the right-hand side is exactly the condition number of this solution component when
perturbations on the right-hand side are considered. We also provide fragment codes using LAPACK [1] routines
to compute the variance-covariance matrix and the least squares conditioning and we give the corresponding
computational cost. Finally we present a small historical numerical example that was used by Laplace [19] for
computing the mass of Jupiter and experiments from the space industry with real physical data
Netlib and NA-Net: building a scientific computing community
Get PDFThe Netlib software repository was created in 1984 to facilitate quick distribution of
public domain software routines for use in scientific computation. The Numerical
Analysis Net (or "NA Net") had its roots in the same period, beginning as a simple file of
contact information for numerical analysts and evolving into an email forwarding service
for that community. It soon evolved to support a regular electronic mail newsletter, and
eventually an online directory service. Both of these services are still in operation and
enjoy wide use today. While they are and always have been distinct services, Netlib and
NA-Net's histories are intertwined. This document gives the system authors' perspective
on how and why Netlib and NA-Net came to exist and impact on their user communities
Steady and unsteady flame propagation in a premixed counterflow
No full textIn this paper we investigate numerically the influence of heat loss and strain rate on the edges of premixed flames in a two-dimensional counterflow. This work represents a continuation and extension of recent work on the topic. Both steady and non-steady edge behaviour is investigated for certain values of the parameters representing heat loss and strain rate in the governing equations. Combustion waves between strongly and weakly burning flames are also investigated
Involutions in groups of finite Morley rank degenerate type
Get PDFIn a connected group of finite Morley rank, if the Sylow 2-subgroups are finite then they are trivial. The proof involves a combination of model-theoretic ideas with a device originating in black box group theory
Some local definability theory for holomorphic functions
Get PDFLet \mathcal{F} be a collection of holomorphic functions and let \mathbb{R}(PR(\mathcal{F}))
denote the reduct of the structure \mathbb{R}_{an} to the ordered field operations
together with the set of proper restrictions (see below) of the
real and imaginary parts of all functions in \mathcal{F}. We ask the question:
Which holomorphic functions are locally definable (ie have their real
and imaginary parts locally definable) in the structure \mathbb{R}(PR(\mathcal{F}))? It
is easy to see that the collection of all such functions is closed under
composition, partial differentiation, implicit definability (via the
Implicit Function Theorem in one dependent variable) and Schwarz
Reflection. We conjecture that this exhausts the possibilities and we
prove as much in the neighbourhood of generic points. More precisely,
we show that these four operations determine the natural pregeometry
associated with \mathbb{R}(PR(\mathcal{F}))-definable, holomorphic functions
Scaling properties of coating flows in rectangular channels
Get PDFWe present an experimental study of the aspect-ratio dependence of two-phase displacement flows in channels of rectangular cross section. Above a buoyancy-dependent threshold Ca^t, we find that the bulk features of the flow depend only on a modified capillary number Ca^=[1+0.12(alpha-1)+0.018(alpha-1)2]Ca, where Ca is the unmodified capillary number and alpha is the aspect ratio. This novel scaling has tremendous practical significance because it implies that the bulk features of the flows in any rectangular channel can be inferred from those in a square channel for Ca^>Ca^t
Least squares preconditioners for stabilized discretizations of the Navier-Stokes equations
Get PDFThis paper introduces two stabilization schemes for the Least Squares Commutator (LSC) preconditioner developed by Elman, Howle, Shadid, Shuttleworth and Tuminaro [SIAM J. Sci. Comput., 27, 2006, pp. 1651–1668] for the incompressible Navier-Stokes equations. This preconditioning methodology is one of several choices that are effective for Navier-Stokes equations, and it has the advantage of being defined from strictly algebraic considerations. It has previously been limited in its applicability to div-stable discretizations of the Navier-Stokes equations. This paper shows how to extend the same methodology to stabilized low-order mixed finite element approximation methods
Human Dimension of Mathematics
Get PDFA Mathematician's Look at History, Philosophy, Politics of Mathematic
Optimal scaling of generalized and polynomial eigenvalue problems
Get PDFScaling is a commonly used technique for standard
eigenvalue problems to improve the sensitivity of the eigenvalues.
In this paper we investigate scaling for generalized and polynomial
eigenvalue problems (PEPs) of arbitrary degree. It is shown that an optimal diagonal
scaling of a PEP with respect to an eigenvalue can be
described by the ratio of its normwise and componentwise condition
number. Furthermore, the effect of linearization on optimally scaled polynomials is investigated.
We introduce a generalization of the diagonal scaling by Lemonnier and Van Dooren to PEPs that is especially effective if some information about the magnitude of the wanted eigenvalues is available and also discuss variable transformations of the type λ = αμ for PEPs of arbitrary degree
A Grammatical Representation of Visibly Pushdown Languages
Get PDFModel-checking regular properties is well established and a powerful verification technique for regular as well as context-free program behaviours. Recently, through the use of ω-visibly pushdown languages (ωVPLs), defined by ω-visibly pushdown automata, model-checking of properties beyond regular expressiveness was made possible and shown to be still decidable even when the program’s model of behaviour is an ωVPL. In this paper, we give a grammatical representation of ωVPLs and the corresponding finite word languages – VPL. From a specification viewpoint, the grammatical representation provides a more natural representation than the automata approach