Skew ray tracing and sensitivity analysis of ellipsoidal optical boundary surfaces

The 4 × 4 homogeneous transformation matrix is one of the most commonly applied mathematical tools in the fields of robotics, mechanisms and computer graphics. Here we extend further this mathematical tool to geometrical optics by addressing the following two topics: (1) skew ray tracing to determine the paths of reflected/refracted skew rays crossing ellipsoidal boundary surfaces; and (2) sensitivity analysis to determine via direct mathematical analysis the differential changes of the incident point and the reflected/refracted vector with respect to changes in the incident light source. The proposed ray tracing and sensitivity analysis are projected as the nucleus of other geometrical optical computations.     

Robustness and applicability of Markov chain Monte Carlo algorithms for eigenvalue problems

In this paper we analyse applicability and robustness of Markov chain Monte Carlo algorithms for eigenvalue problems. We restrict our consideration to real symmetric matrices.

Almost Optimal Monte Carlo (MAO) algorithms for solving eigenvalue problems are formulated. Results for the structure of both – systematic and probability error are presented. It is shown that the values of both errors can be controlled independently by different algorithmic parameters. The results present how the systematic error depends on the matrix spectrum. The analysis of the probability error is presented. It shows that the close (in some sense) the matrix under consideration is to the stochastic matrix the smaller is this error. Sufficient conditions for constructing robust and interpolation Monte Carlo algorithms are obtained. For stochastic matrices an interpolation Monte Carlo algorithm is constructed.

A number of numerical tests for large symmetric dense matrices are performed in order to study experimentally the dependence of the systematic error from the structure of matrix spectrum. We also study how the probability error depends on the balancing of the matrix.     

Skew normal measurement error models

In this paper we define a class of skew normal measurement error models, extending usual symmetric normal models in order to avoid data transformation. The likelihood function of the observed data is obtained, which can be maximized by using existing statistical software. Inference on the parameters of interest can be approached by using the observed information matrix, which can also be computed by using existing statistical software, such as the Ox program. Bayesian inference is also discussed for the family of asymmetric models in terms of invariance with respect to the symmetric normal distribution showing that early results obtained for the normal distribution also holds for the asymmetric family. Results of a simulation study and an analysis of a real data set analysis are provided.     

A combinatorial interpretation of the inverse kostka matrix

The Kostka matrix K relates the. homogeneous and the Schur bases in the ring of symmetric functions where K_λ,μenumerates the number of column strict tableaux of shape λ and type μ. We make use of the Jacobi -Trudi identity to give a combinatorial interpretation for the inverse of the Kostka matrix in terms of certain types of signed rim hook tabloids. Using this interpretation, the matrix identity KK^-1=Iis given a purely combinatorial proof. The generalized Jacobi-Trudi identity itself is also shown to admit a combinatorial proof via these rim hook tabloids. A further application of our combinatorial interpretation is a simple rule for the evaluation of a specialization of skew Schur functions that arises in the computation of plethysms.     

Cost prediction for ray shooting in octrees

The ray shooting problem arises in many different contexts and is a bottleneck of ray tracing in computer graphics. Unfortunately, theoretical solutions to the problem are not very practical, while practical methods offer few provable guarantees on performance.

Attempting to combine practicality with theoretical soundness, we show how to provably measure the average performance of any ray-shooting method based on traversing a bounded-degree spatial decomposition, where the average is taken to mean the expectation over a uniform ray distribution. An approximation yields a simple, easy-to-compute cost predictor that estimates the average performance of ray shooting without running the actual algorithm.

We experimentally show that this predictor provides an accurate estimate of the efficiency of executing ray-shooting queries in octree-induced decompositions, irrespective of whether or not the bounded-degree requirement is enforced, and of the criteria used to construct the octrees. We show similar guarantees for decompositions induced by kd-trees and uniform grids. We also confirm that the performance of an octree while ray tracing or running a radio-propagation simulation is accurately captured by our cost predictor, for ray distributions arising from realistic data.     

Multiplicative Error Analysis of Matrix Transformation Algorithms

The author's recently introduced relative error measure forvectors is applied to the error analysis of algorithms whichproceed by successive transformation of a matrix. Instead ofmodelling the roundoff errors at each stage by A: = T(A)+E onemodels them by A: =e^E T(A) where E is a small linear transformation.This can simplify analyses considerably. Applications to theparallel Jacobi method for eigenvalues, and to Gaussian elimination,are given.     

Three-dimensional stable nonconforming parallelepiped elements for the Stokes problem

Stability result is obtained for the approximation of the stationary Stokes problem with nonconforming elements proposed by Jim Douglas Jr. et al. [Math. Model. Numer. Anal. 33 (4) (1999) 747] for the velocity with conforming bubble functions and discontinuous piecewise linear for the pressure on parallelepiped elements. Optimal order H¹ and L² error estimates are derived.     

On Skew E–W Matrices

An E–W matrix M is a ( ? 1, 1)‐matrix of order , where t is a positive integer, satisfying that the absolute value of its determinant attains Ehlich–Wojtas' bound. M is said to be of skew type (or simply skew) if is skew‐symmetric where I is the identity matrix. In this paper, we draw a parallel between skew E–W matrices and skew Hadamard matrices concerning a question about the maximal determinant. As a consequence, a problem posted on Cameron's website [7] has been partially solved. Finally, codes constructed from skew E–W matrices are presented. A necessary and sufficient condition for these codes to be self‐dual is given, and examples are provided for lengths up to 52.     

Noncentral matrix quadratic forms of the skew elliptical variables

In this paper, the noncentral matrix quadratic forms of the skew elliptical variables are studied. A family of the matrix variate noncentral generalized Dirichlet distributions is introduced as the extension of the noncentral Wishart distributions, the Dirichlet distributions and the noncentral generalized Dirichlet distributions. Main distributional properties are investigated. These include probability density and closure property under linear transformation and marginalization, the joint distribution of the sub-matrices of the matrix quadratic forms in the skew elliptical variables and the moment generating functions and Bartlett's decomposition of the matrix quadratic forms in the skew normal variables. Two versions of the noncentral Cochran's Theorem for the matrix variate skew normal distributions are obtained, providing sufficient and necessary conditions for the quadratic forms in the skew normal variables to have the matrix variate noncentral generalized Dirichlet distributions. Applications include the properties of the least squares estimation in multivariate linear model and the robustness property of the Wilk's likelihood ratio statistic in the family of the matrix variate skew elliptical distributions.     

New error estimates of nonconforming mixed finite element methods for the Stokes problem

In this paper, we revisit the classical error estimates of nonconforming Crouzeix–Raviart type finite elements for the Stokes equations. By introducing some quasi‐interpolation operators and using the special properties of these nonconforming elements, it is proved that their consistency errors can be bounded by their approximation errors together with a high‐order term, especially which can be of arbitrary order provided that f in the right‐hand side is piecewise smooth enough. Furthermore, we show an interesting result that both in the energy norm and L² norm the consistency errors are dominated by the approximation errors of their finite element spaces. As byproducts, we derive the error estimates in both energy and L² norms under the regularity assumption ( u ,p) ∈ H ^{1 + s}(Ω) × H^s(Ω) with any s ∈ (0,1], which fills the gap in the a priori error estimate of these nonconforming elements with low regularity . Furthermore, a robust convergence is proved with minimal regularity assumption s = 0. These results seem to be missing in the literature. Numerical tests are provided, confirming the analysis, especially the new results on the L² convergence. Copyright © 2013 John Wiley & Sons, Ltd.     

On the Comparisons of Unit Dual Quaternion and Homogeneous Transformation Matrix

This paper reveals the differences and similarities between two popular unified representations, i.e. the UDQ (unit dual quaternion) and the HTM (homogeneous transformation matrix), for transformation in the solution to the kinematic problem, in order to provide a clear, concise and self-contained introduction into dual quaternions and to further present a cohesive view for the UDQ and HTM representations as used in robotics. Specifically, after investigating some fundamental algebraic properties of the UDQ, it is revealed that the kinematical equations represented by the UDQ and the HTM are accordant, and afterwards the direct relationship of UDQ-based error kinematical models in spatialframe and in body-frame are further discussed, with conclusion that either error kinematic model can be chosen for designing kinematical control laws. Finally, the comparative study on the proportional control algorithms based on the logarithmical mapping of the HTM and the UDQ shows that the UDQ-based control law is indeed higher in computational efficiency.     

A posteriori error estimates for hp finite element solutions of convex optimal control problems

In this paper, we present a posteriori error analysis for hp finite element approximation of convex optimal control problems. We derive a new quasi-interpolation operator of Clément type and a new quasi-interpolation operator of Scott-Zhang type that preserves homogeneous boundary condition. The Scott-Zhang type quasi-interpolation is suitable for an application in bounding the errors in L²-norm. Then hp a posteriori error estimators are obtained for the coupled state and control approximations. Such estimators can be used to construct reliable adaptive finite elements for the control problems.     

Relative error propagation in the recursive solution of linear recurrence relations

This paper is concerned with the numerical solution of the general initial value problem for linear recurrence relations. An error analysis of direct recursion is given, based on relative rather than absolute error, and a theory of relative stability developed.Miller's algorithm for second order homogeneous relations is extended to more general cases, and the propagation of errors analysed in a similar manner. The practical significance of the theoretical results is indicated by applying them to particular classes of problem.     

Optimal second order rectangular elasticity elements with weakly symmetric stress

We present new second order rectangular mixed finite elements for linear elasticity where the symmetry condition on the stress is imposed weakly with a Lagrange multiplier. The key idea in constructing the new finite elements is enhancing the stress space of the Awanou’s rectangular elements (rectangular Arnold–Falk–Winther elements) using bubble functions. The proposed elements have only 18 and 63 degrees of freedom for the stress in two and three dimensions, respectively, and they achieve the optimal second order convergence of errors for all the unknowns. We also present a new simple a priori error analysis and provide numerical results illustrating our analysis.     

Central limit theorem behavior in the skew tent map

In this paper we study and establish central limit theorem behavior in the skew (generalized) tent map transformation T: Y →Y originally considered by Billings and Bollt [Billings L, Bollt EM. Probability density functions of some skew tent maps. Chaos, Solitons & Fractals 2001; 12: 365–376] and Ito et al. [Ito S, Tanaka S, Nakada H. On unimodal linear transformations and chaos. II. Tokyo J Math 1979; 2: 241–59]. When the measure ν is invariant under T, the transfer operator governing the evolution of densities f under the action of the skew tent map, as well as the unique stationary density, are given explicitly for specific transformation parameters. Then, using this development, we solve the Poisson equation for two specific integrable observables and explicitly calculate the variance σ()²=∫_Y²(y)ν(dy).     

On Lyapunov equivalence of linear differential systems with unbounded coefficients

We show that any linear homogeneous differential system can be reduced by some linear piecewise differentiable transformation whose matrix, together with its inverse, is bounded on the half-line to a system with piecewise constant coefficients of the same growth order, and any system with a uniformly small perturbation can be reduced by this linear transformation to the same system with a piecewise constant perturbation of the same smallness.     

-curved finite elements and applications to plate and shell problems

Many thin-plate and thin-shell problems are set on plane reference domains with a curved boundary. Their approximation by conforming finite-elements methods requires ¹-curved finite elements entirely compatible with the associated ¹-rectilinear finite elements. In this contribution we introduce a ¹-curved finite element compatible with the P₅-Argyris element, we study its approximation properties, and then, we use such an element to approximate the solution of thin-plate or thin-shell problems set on a plane-curved boundary domain. We prove the convergence and we get a priori asymptotic error estimates which show the very high degree of accuracy of the method. Moreover we obtain criteria to observe when choosing the numerical integration schemes in order to preserve the order of the error estimates obtained for exact integration.     

Forward error analysis of Gaussian elimination

Summary Part I of this work deals with the forward error analysis of Gaussian elimination for general linear algebraic systems. The error analysis is based on a linearization method which determines first order approximations of the absolute errors exactly. Superposition and cancellation of error effects, structure and sparsity of the coefficient matrices are completely taken into account by this method. The most important results of the paper are new condition numbers and associated optimal component-wise error and residual estimates for the solutions of linear algebraic systems under data perturbations and perturbations by rounding erros in the arithmetic floating-point operations. The estimates do not use vector or matrix norms. The relative data and rounding condition numbers as well as the associated backward and residual stability constants are scaling-invariant. The condition numbers can be computed approximately from the input data, the intermediate results, and the solution of the linear system. Numerical examples show that by these means realistic bounds of the errors and the residuals of approximate solutions can be obtained. Using the forward error analysis, also typical results of backward error analysis are deduced. Stability theorems and a priori error estimates for special classes of linear systems are proved in Part II of this work.     

广义耦合Sylvester矩阵方程的对称-反对称最小二乘解

本文主要研究极小残差问题‖(A1XB1+C1YD1A2XB2+C2YD2)-(M1M2)‖=min关于X对称-Y反对称解的迭代算法.本文首先给出等价于极小残差问题的规范方程,然后,提出求解此规范方程的对称-反对称解的迭代算法.在不考虑舍入误差的情况下,任取一个初始的对称-反对称矩阵对(X0,Y0),该算法都可以在有限步内求得该极小残差问题的对称-反对称解.最后讨论该问题的极小范数对称-反对称解.     

Improving and estimating the accuracy of Strassen's algorithm

Fast matrix multiplication algorithms of Strassen and Winograd are known to have weaker numerical accuracy than usual (inner product) multiplication. In this paper, we show that scaling usually improves accuracy when operands have elements of widely varying magnitude. We also propose estimators for numerical errors, based on samples of the result. All these estimators can be computed in operations. Experiments prove the effectiveness of the scaling idea and of the absolute error estimator. Received February 20, 1996/ Revised version received July 1, 1997