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1.
The class of symmetric positive definite matrices is an important class both in theory and application. Although this class is well studied, little is known about how to efficiently interpolate such data within the class. We extend the 4-point interpolatory subdivision scheme, as a method of interpolation, to data consisting of symmetric positive definite matrices. This extension is based on an explicit formula for calculating a binary “geodetic average”. Our method generates a smooth curve of matrices, which retain many important properties of the interpolated matrices. Furthermore, the scheme is robust and easy to implement. 相似文献
2.
J.M. Peña 《Linear and Multilinear Algebra》2013,61(1):91-97
A matrix is called sign regular of order k if every minor of order i has the same sign for each i = 1,2,<, k . If an m × n matrix is sign regular of order k for k = min { m,n } then it is called sign regular. This paper studies some properties of sign regular matrices of order two. Remarkable properties are proved when the row sums of these matrices form a monotone vector. 相似文献
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We reprove and extend results for uniform convergence of non-negative subdivision using results from infinite products of stochastic matrices as they appear in the study of non-homogeneous finite Markov processes. Geometric convergence of such products to rank one matrices, based on the notions of SIA matrices and of the so-called ergodic coefficient, is discussed. We also point to the properties of the directed graphs of such matrices. In this way the existing convergence results for non-negative subdivision are put into the context of such processes. 相似文献
5.
Consider the ensemble of real symmetric Toeplitz matrices, each independent entry an i.i.d. random variable chosen from a
fixed probability distribution p of mean 0, variance 1, and finite higher moments. Previous investigations showed that the limiting spectral measure (the
density of normalized eigenvalues) converges weakly and almost surely, independent of p, to a distribution which is almost the standard Gaussian. The deviations from Gaussian behavior can be interpreted as arising
from obstructions to solutions of Diophantine equations. We show that these obstructions vanish if instead one considers real
symmetric palindromic Toeplitz matrices, matrices where the first row is a palindrome. A similar result was previously proved
for a related circulant ensemble through an analysis of the explicit formulas for eigenvalues. By Cauchy’s interlacing property
and the rank inequality, this ensemble has the same limiting spectral distribution as the palindromic Toeplitz matrices; a
consequence of combining the two approaches is a version of the almost sure Central Limit Theorem. Thus our analysis of these
Diophantine equations provides an alternate technique for proving limiting spectral measures for certain ensembles of circulant
matrices.
A. Massey’s current address: Department of Mathematics, UCLA, Los Angeles, CA 90095, USA. e-mail: amassey3102@math.ucla.edu. 相似文献
6.
The present paper deals with subdivision schemes associated with irregular grids. We first give a sufficient condition concerning the difference scheme to obtain convergence. This condition generalizes a necessary and sufficient condition for convergence known in the case of uniform and stationary schemes associated with a regular grid. Through this sufficient condition, convergence of a given subdivision scheme can be proved by comparison with another scheme. Indeed, when two schemes are equivalent in some sense, and when one satisfies the sufficient condition for convergence, the other also satisfies it and it therefore converges too. We also study the smoothness of the limit functions produced by a scheme which satisfies the sufficient condition. Finally, the results are applied to the study of Lagrange interpolating subdivision schemes of any degree, with respect to particular irregular grids. 相似文献
7.
This paper is concerned with a family of nonstationary, interpolatory subdivision schemes that have the capability of reproducing
functions in a finite-dimensional subspace of exponential polynomials. We give conditions for the existence and uniqueness
of such schemes, and analyze their convergence and smoothness. It is shown that the refinement rules of an even-order exponentials
reproducing scheme converge to the Dubuc—Deslauriers interpolatory scheme of the same order, and that both schemes have the
same smoothness. Unlike the stationary case, the application of a nonstationary scheme requires the computation of a different
rule for each refinement level. We show that the rules of an exponentials reproducing scheme can be efficiently derived by
means of an auxiliary orthogonal scheme , using only linear operations. The orthogonal schemes are also very useful tools in fitting an appropriate space of exponential
polynomials to a given data sequence. 相似文献
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Consider the ensemble of real symmetric Toeplitz matrices whose entries are i.i.d. random variables chosen from a fixed probability distribution p of mean 0, variance 1, and finite higher moments. Previous work (Bryc et al., Ann. Probab. 34(1):1–38, 2006; Hammond and Miller, J. Theor. Probab. 18(3):537–566, 2005) showed that the spectral measures (the density of normalized eigenvalues) converge almost surely to a universal distribution almost that of the Gaussian, independent of p. The deficit from the Gaussian distribution is due to obstructions to solutions of Diophantine equations and can be removed (see Massey et al., J. Theor. Probab. 20(3):637–662, 2007) by making the first row palindromic. In this paper we study the case where there is more than one palindrome in the first row of real symmetric Toeplitz matrices. Using the method of moments and an analysis of the resulting Diophantine equations, we show that the spectral measures converge almost surely to a universal distribution. Assuming a conjecture on the resulting Diophantine sums (which is supported by numerics and some theoretical arguments), we prove that the limiting distribution has a fatter tail than any previously seen limiting spectral measure. 相似文献
10.
Hermite subdivision schemes have been studied by Merrien, Dyn, and Levin
and they appear to be very different from subdivision schemes analyzed before since the rules depend on the subdivision level.
As suggested by Dyn and Levin, it is possible to transform the initial scheme into a uniform stationary vector subdivision
scheme which can be handled more easily.With this transformation, the study of convergence of Hermite subdivision schemes
is reduced to that of vector stationary subdivision schemes. We propose a first criterion for C0-convergence for a large class of vector subdivision schemes. This gives a criterion for C1-convergence of Hermite subdivision schemes. It can be noticed that these schemes do not have to be interpolatory. We conclude
by investigating spectral properties of Hermite schemes and other necessary/sufficient conditions of convergence. 相似文献
11.
We propose a general study of the convergence of a Hermite subdivision scheme ℋ of degree d>0 in dimension 1. This is done by linking Hermite subdivision schemes and Taylor polynomials and by associating a so-called
Taylor subdivision (vector) scheme
. The main point of investigation is a spectral condition. If the subdivision scheme of the finite differences of
is contractive, then
is C
0 and ℋ is C
d
. We apply this result to two families of Hermite subdivision schemes. The first one is interpolatory; the second one is a
kind of corner cutting. Both of them use the Tchakalov-Obreshkov interpolation polynomial.
相似文献
12.
We construct local subdivision schemes that interpolate functional univariate data and that preserve convexity. The resulting
limit function of these schemes is continuous and convex for arbitrary convex data. Moreover this class of schemes is restricted
to a subdivision scheme that generates a limit function that is convex and continuously differentiable for strictly convex
data. The approximation order of this scheme is four. Some generalizations, such as tension control and piecewise convexity
preservation, are briefly discussed.
November 29, 1996. Date revised: May 28, 1997. 相似文献
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Foundations of Computational Mathematics - Hermite interpolation property is desired in applied and computational mathematics. Hermite and vector subdivision schemes are of interest in CAGD for... 相似文献
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Andreas Weinmann 《Constructive Approximation》2010,31(3):395-415
The present article deals with convergence and smoothness analysis of geometric, nonlinear subdivision schemes in the presence
of extraordinary points. We discuss when the existence of a proximity condition between a linear scheme and its nonlinear
analogue implies convergence of the nonlinear scheme (for dense enough input data). Furthermore, we obtain C
1 smoothness of the nonlinear limit function in the vicinity of an extraordinary point over Reif’s characteristic parametrization.
The results apply to the geometric analogues of well-known subdivision schemes such as Doo–Sabin or Catmull–Clark schemes. 相似文献
18.
本文讨论了与一般伸缩矩阵M相关的Subdivsion算法的L^p-收敛性,多个细分方程的解可由一个给定的细分函数通过迭代算法得到。 相似文献
19.
We present generalized and unified families of $(2n)$-point and $(2n-1)$-point $p$-ary interpolating subdivision schemes originated from Lagrange polynomial for any integers $n ≥ 2$ and $p ≥ 3$. Almost all existing even-point and odd-point interpolating schemes of lower and higher arity belong to this family of schemes. We also present tensor product version of generalized and unified families of schemes. Moreover, error bounds between limit curves and control polygons of schemes are also calculated. It has been observed that error bounds decrease when complexity of the scheme decrease and vice versa. Furthermore, error bounds decrease with the increase of arity of the schemes. We also observe that in general the continuity of interpolating scheme do not increase by increasing complexity and arity of the scheme. 相似文献
20.
Johannes Wallner 《Constructive Approximation》2006,24(3):289-318
Linear curve subdivision schemes may be perturbed in various ways, for example, by modifying them such as to work in a manifold,
surface, or group. The analysis of such perturbed and often nonlinear schemes "T" is based on their proximity to the linear
schemes "S" which they are derived from. This paper considers two aspects of this problem: One is to find proximity inequalities
which together with Ck smoothness of S imply Ck smoothness of T. The other is to verify these proximity inequalities for several ways to construct the nonlinear scheme T
analogous to the linear scheme S. The first question is treated for general k, whereas the second one is treated only in
the case k = 2. The main result of the paper is that convergent geodesic/projection/Lie group analogues of a certain class
of factorizable linear schemes have C2 limit curves. 相似文献