Trigonometric Approximation of SO(N) Loops

Journal of Dynamical and Control Systems, 1999

We examine the De Casteljau algorithm in the context of Riemannian symmetric spaces. This algorithm, whose classical form is used to generate interpolating polynomials in $$\mathbb{R}^n $$ , was also generalized to arbitrary Riemannian manifolds by others. However, the implementation of the generalized algorithm is difficult since detailed structure, such as boundary value expressions, has not been available. Lie groups

Linear Algebra and its Applications, 1994

The operation of multiplication of a vector by a matrix can be represented by a computational scheme (or model) that acts sequentially on the entries of the vector. The number of intermediate quantities ("states") that are needed in the computations is a measure of the complexity of the model. If that complexity is low, then not only multiplication, but also other operations such as inversion, can be carried out efficiently using the model rather than the original matrix. In the introductory sections, we describe an algorithm to derive a computational model of minimal complexity that gives an exact representation of an arbitrary upper triangular matrix. The main result of the paper is an algorithm for computing an approximating matrix with a model of (much) lower complexity than the original-as low as possible for a given tolerance on the approximation error. As measure for the tolerance we use a strong norm which we will call the Hankel norm. It is a generalization of the Hankel norm which is used in the classical model approximation theory for complex analytical functions. 1.

2005

multiplication operator on a Hilbert space may be approximated with finite sections by choosing an orthonormal basis of the Hilbert space. Nonzero multiplication operators on $L^2$ spaces of functions are never compact and then such approximations cannot converge in the norm topology. Instead, we consider how well the spectra of the finite sections approximate the spectrum of the multiplication operator whose expression is simply given by the essential range of the symbol (i.e. the multiplier). We discuss the case of real orthogonal polynomial bases and the relations with the classical Fourier basis whose choice leads to well studied Toeplitz case. The use of circulant approximations leads to constructive algorithms working for the separable multivariate and matrix-valued cases as well.

Journal of Scientific Computing, 1989

Journal of Computational and Applied Mathematics, 2010

In this paper we explore the computation of the matrix exponential in a manner that is consistent with Lie group structure. Our point of departure is the decomposition of Lie algebra as the semidirect product of two Lie subspaces and an application of the Baker-Campbell-Hausdorff formula. Our results extend the results in Iserles and Zanna (2005) [2], Zanna and Munthe-Kaas(2001/02) [4] to a range of Lie groups: the Lie group of all solid motions in Euclidean space, the Lorentz Lie group of all solid motions in Minkowski space and the group of all invertible (upper) triangular matrices. In our method, the matrix exponential group can be computed by a less computational cost and is more accurate than the current methods. In addition, by this method the approximated matrix exponential belongs to the corresponding Lie group.

Lecture Notes in Computer Science, 2002

In the last years several numerical methods have been d e veloped to integrate matrix differential equations which preserve certain features of the theoretical solution such as orthogonality, eigenvalues, first integrals, etc. In this paper we approach the numerical solution of a second order matrix differential system whose solution evolves on the Lie-group of the orthogonal matrices 0,. We study the orthogonality properties of classical Runge Kutta Nystrom methods and non standard numerical procedures for second order ordinary differential equations.

Computational Geometry, 1997

Several algorithms are presented for approximating an orthogonal rotation matrix M in three dimensions by an orthogonal matrix with rational entries. The first algorithm generates an approximation M2(M, e) with accuracy e and (2b + 2)-bit numerators and a common (2b + 2)-bit denominator (bit-size 2b + 2), where b = [-lg e] (e ~ 2-b). The second algorithm uses basis reduction to generate an approximation M,(M, e) with accuracy e ~/1'5 and bit-size vb for some 1.5 ~< v ~< 6 (but v cannot be controlled except by trial and error). A third algorithm, based on integer programming, generates optimal Mopt(M, 6) with accuracy e and bit-size proven to be no more than 1.5b. In practice, the second algorithm generates an approximation with v ~ 1.5 and is much faster than the third algorithm. The best bit-sizes which one could obtain using previously known results in two dimensions (Canny et al., 1992) are more than 3b bits for numerator and denominator. Applications are described for the approximation functions in the area of solid modeling.

Procedia Engineering, 2012

In this paper we obtain the degree of approximation of signals (functions) belonging to (, ,) Lip class introduced in [8], through (;), (;) n n t f x N f x and (;) n R f x means of their trigonometric Fourier series. Our results generalize some results of Chandra [2] for 1 p and Guven [3].

Usando propiedades estructurales de los polinomios ortogonales clásicos (Hermite, Laguerre, Jacobi y Bessel), se implementa el algoritmo de Leverrier-Fadeev para obtener el polinomio característico de una matriz cuadrada de elementos complejos.

2000

We study the matrix factorization problem associated with an SO(2) spinning top by using the algebro-geometric approach. We derive the explicit expressions in terms of Riemann theta functions and discus some related problems including a non-compact extension and the case when the Lax matrix contains higher-order powers of the spectral parameter.