We introduce a new iterative root-finding method for complex polynomials, dubbed Newton-Ellipsoid method. It is inspired by the Ellipsoid method, a classical method in optimization, and a property of Newton’s Method derived in [7], according to which at each complex number a half-space can be found containing a root. Newton-Ellipsoid method combines this property, bounds on zeros, together with...

The Ellipsoid NDF is a new normal distribution function (NDF) that we introduce and use in the accompanying paper. It can be used with micro-facet BRDF to model the reflection and refraction of light from surfaces. This new distribution is a generalization of the widely used isotropic GGX/Trowbridge-Reitz distribution. The Ellipsoid model is based on the surface statistics of an arbitrary 3D el...

Similarity retrieval mechanisms should utilize generalized quadratic form distance functions as well as the Euclidean distance function since ellipsoid queries parameters may vary with the user and situation. In this paper, we present the spatial transformation technique that yields a new search method for adaptive ellipsoid queries with quadratic form distance functions. The basic idea is to t...

Two FORTRAN IV algorithms are given for determining the two axial ratios of a macromolecule (as modelled by a tri-axial ellipsoid) from its hydrodynamic parameters. The first involves a simple graphical inversion procedure of the volume independent V and R functions but can only be applied to a restricted range of macromolecules. The second algorithm is more general and involves an R-function c...

In this paper, we provide the first linear programming formulations for the stable set problem in claw-free graphs, together with polynomial time separation routines for those formulations (they are not compact). We then exploit one of those extended formulations and propose a new polytime algorithm for solving the separation problem for the stable set polytope of claw-free graphs. This routine...

Equal-time comparison. Stochastic light culling has a trade-off between the performance and quality, which is controlled by the user-specified parameter δ. Therefore, our ellipsoid-based approach can render higher-quality images than the sphere-based approach when comparing at the same computation time. Fig. 2 shows the equal-time comparison between the bounding sphere and our bounding ellipsoi...

We give a new proof of the complete integrability of the geodesic flow on the ellipsoid (in Euclidean, spherical or hyperbolic space). The proof is based on the construction of a metric on the ellipsoid whose non-parameterized geodesics coincide with those of the standard metric. This new metric is induced by the hyperbolic metric inside the ellipsoid (Klein’s model). Mathematics Subject Classi...

An iterative algorithm is given for solving a system Ax= k of n linear equations in n unknowns. The solution is given in n steps . It is shown that this method is a special case of a very general met hod which also includes Gaussian elimination . These general algorithms are essentially algorithms for findin g an n dimensional ellipsoid . Connections a re m ade wit h the theory of orthogonal po...

This paper presents a technique for automatically approximating a given mesh model with an ellipsoidal blobby model. Firstly, an ellipsoid decomposition algorithm is introduced to approximate given models by ellipsoids. After that, a blobby implicit surface employing ellipsoidal blobs is modelled to fit the sampling points on the given mesh. Finally, the reconstructed ellipsoidal blobby model i...

This paper deals with the performance of the recursive least squares algorithm when it is applied to problems where the measured signal is corrupted by bounded noise. Using ideas from bounding ellipsoid algorithms we derive an asymptotic expression for the bound on the uncertainty of the parameter estimate for a simple choice of design variables. This bound is also transformed to a bound on the...