A note on subset selection for matrices

In an earlier paper the authors established a result to select subsets of a matrix that are as "non-singular" as possible in a numerical sense. The major result was not constructive. In this note we give a constructive proof and moreover a sharper bound

On the conditioning of multipoint and integral boundary value problems

Linear multipoint boundary value problems are investigated from the point of view of the condition number and properties of the fundamental solution. It is found that when the condition number is not large, the solution space is polychotomic. On the other hand, if the solution space is polychotomic then there exist boundary conditions such that the associated boundary value problem is well conditioned

A note on subset selection for matrices

In an earlier paper the authors established a result to select subsets of a matrix that are as "non-singular" as possible in a numerical sense. The major result was not constructive. In this note we give a constructive proof and moreover a sharper bound

Asymptotic approximations for vibrational modes of helices

The free vibrations in the plane normal to the helical axis are studied under the assumption that the helical pitch is small. Asymptotic approximations for eigenvalues and eigenfunctions are derived for both small and large numbers of helical turns. The analytic approximations reveal interesting features of helix vibrations and the connection between the vibrational modes of a helix and the flexural modes of a curved beam. Comparison with numerical calculations shows that the approximations derived cover with sufficient accuracy a wide range of number of helical turns

A weakly stable algorithm for general Toeplitz systems

We show that a fast algorithm for the QR factorization of a Toeplitz or Hankel matrix A is weakly stable in the sense that R^T.R is close to A^T.A. Thus, when the algorithm is used to solve the semi-normal equations R^T.Rx = A^Tb, we obtain a weakly stable method for the solution of a nonsingular Toeplitz or Hankel linear system Ax = b. The algorithm also applies to the solution of the full-rank Toeplitz or Hankel least squares problem.Comment: 17 pages. An old Technical Report with postscript added. For further details, see http://wwwmaths.anu.edu.au/~brent/pub/pub143.htm

Smoothing and Matching of 3-D Space Curves

International audienceWe present a new approach to the problem of matching 3-D curves. The approach has a low algorithmic complexity in the number of models, and can operate in the presence of noise and partial occlusions. Our method builds upon the seminal work of Kishon et al. (1990), where curves are first smoothed using B-splines, with matching based on hashing using curvature and torsion measures. However, we introduce two enhancements: -- We make use of nonuniform B-spline approximations, which permits us to better retain information at highcurvature locations. The spline approximations are controlled (i.e., regularized) by making use of normal vectors to the surface in 3-D on which the curves lie, and by an explicit minimization of a bending energy. These measures allow a more accurate estimation of position, curvature, torsion, and Frtnet frames along the curve. -- The computational complexity of the recognition process is relatively independent of the number of models and is considerably decreased with explicit use of the Frtnet frame for hypotheses generation. As opposed to previous approaches, the method better copes with partial occlusion. Moreover, following a statistical study of the curvature and torsion covariances, we optimize the hash table discretization and discover improved invariants for recognition, different than the torsion measure. Finally, knowledge of invariant uncertainties is used to compute an optimal global transformation using an extended Kalman filter. We present experimental results using synthetic data and also using characteristic curves extracted from 3-D medical images. An earlier version of this article was presented at the 2nd European Conference on Computer Vision in Italy