Chebfun and numerical quadrature

Chebfun is a Matlab-based software system that overloads Matlab's discrete operations for vectors and matrices to analogous continuous operations for functions and operators.We begin by describing Chebfun's fast capabilities for Clenshaw-Curtis and also Gauss-Legendre,-Jacobi,-Hermite,and-Laguerre quadrature,based on algorithms of Waldvogel and Glaser,Liu and Rokhlin.Then we consider how such methods can be applied to quadrature problems including 2D integrals over rectangles,fractional derivatives and integrals,functions defined on unbounded intervals,and the fast computation of weights for barycentric interpolation.     

Singular value expansion for the Green function of Helmholtz operator

We consider the integral operator defined on a circular disk, and with kernel the Green function of the Helmholtz operator. We present an analytic framework for the explicit computation of the singular system of this kernel. In particular, the main formulas of this framework are given by a characteristic equation for the singular values and explicit expressions for the corresponding singular functions. We provide also a property of the singular values, that gives an important information for the numerical evaluation of the singular system. Finally, we present a simple numerical experiment, where the singular system computed by a simple implementation of these analytic formulas is compared with the singular system obtained by a discretization of the Green function of the Helmholtz operator.     

Singular Value Decomposition of a Finite Hilbert Transform Defined on Several Intervals and the Interior Problem of Tomography: The Riemann‐Hilbert Problem Approach

We study the asymptotics of singular values and singular functions of a finite Hilbert transform (FHT), which is defined on several intervals. Transforms of this kind arise in the study of the interior problem of tomography. We suggest a novel approach based on the technique of the matrix Riemann‐Hilbert problem (RHP) and the steepest‐descent method of Deift‐Zhou. We obtain a family of matrix RHPs depending on the spectral parameter λ and show that the singular values of the FHT coincide with the values of λ for which the RHP is not solvable. Expressing the leading‐order solution as λ → 0 of the RHP in terms of the Riemann Theta functions, we prove that the asymptotics of the singular values can be obtained by studying the intersections of the locus of zeroes of a certain Theta function with a straight line. This line can be calculated explicitly, and it depends on the geometry of the intervals that define the FHT. The leading‐order asymptotics of the singular functions and singular values are explicitly expressed in terms of the Riemann Theta functions and of the period matrix of the corresponding normalized differentials, respectively. We also obtain the error estimates for our asymptotic results. © 2016 Wiley Periodicals, Inc.     

The mixed directional difference–summation algorithm for generating the Bézier net of a trivariate four-direction Box-spline

Trivariate Box-splines lack an efficient and general exact evaluation technique. This paper presents one possible and underexploited approach to solving this problem. The algorithm we propose is based on mixed directional differences and summations for computing the Bézier net coefficients of all trivariate four-direction Box-splines of any degree over tetrahedral tessellations of the domain. A Matlab package, called MDDS, for computing the Bézier net both in the trivariate and bivariate cases, is also provided.     

Games of singular control and stopping driven by spectrally one-sided Lévy processes

We study a zero-sum game where the evolution of a spectrally one-sided Lévy process is modified by a singular controller and is terminated by the stopper. The singular controller minimizes the expected values of running, controlling and terminal costs while the stopper maximizes them. Using fluctuation theory and scale functions, we derive a saddle point and the value function of the game. Numerical examples under phase-type Lévy processes are also given.     

Nonsmooth Analysis of Singular Values. Part II: Applications

In this work we continue the nonsmooth analysis of absolutely symmetric functions of the singular values of a real rectangular matrix. Absolutely symmetric functions are invariant under permutations and sign changes of its arguments. We extend previous work on subgradients to analogous formulae for the proximal subdifferential and Clarke subdifferential when the function is either locally Lipschitz or just lower semicontinuous. We illustrate the results by calculating the various subdifferentials of individual singular values. Another application gives a nonsmooth proof of Lidskii’s theorem for weak majorization. Mathematics Subject Classifications (2000) Primary 90C31, 15A18; secondary 49K40, 26B05.Research supported by NSERC.     

具有整数值伸缩矩阵的三维紧框架小波的存在性

引进了三维紧框架小波的概念,它是由框架多分辨分析中子空间X_1中的若干个三维函数Γ~1(y),Γ~2(y),…,Γ~n(y)构成的.研究了对应于三维尺度函数的三维紧框架小波的存在性.运用时频分析方法、滤波器理论、算子理论,给出这n个三维函数生成小波紧框架的充分条件,得到了由一个尺度函数Ψ(y)构造三维紧框架小波的显式公式.     

A polynomial-time complexity bound for the computation of the singular part of a Puiseux expansion of an algebraic function

In this paper we present a refined version of the Newton polygon process to compute the Puiseux expansions of an algebraic function defined over the rational function field. We determine an upper bound for the bit-complexity of computing the singular part of a Puiseux expansion by this algorithm, and use a recent quantitative version of Eisenstein's theorem on power series expansions of algebraic functions to show that this computational complexity is polynomial in the degrees and the logarithm of the height of the polynomial defining the algebraic function.

     

Rational General Solutions of Trivariate Rational Differential Systems

We generalize the method of Ngô and Winkler (J Symbolic Comput 46:1173–1186, 2011) for finding rational general solutions of a plane rational differential system to the case of a trivariate rational differential system. We give necessary and sufficient conditions for the trivariate rational differential system to have a rational solution based on proper reparametrization of invariant algebraic space curves. In fact, the problem for computing a rational solution of the trivariate rational differential system can be reduced to finding a linear rational solution of an autonomous differential equation. We prove that the linear rational solvability of this autonomous differential equation does not depend on the choice of proper parametrizations of invariant algebraic space curves. In addition, two different rational solutions corresponding to the same invariant algebraic space curve are related by a shifting of the variable. We also present a criterion for a rational solution to be a rational general solution.     

An efficient and accurate parallel algorithm for the singular value problem of bidiagonal matrices

Summary. In this paper we propose an algorithm based on Laguerre's iteration, rank two divide-and-conquer technique and a hybrid strategy for computing singular values of bidiagonal matrices. The algorithm is fully parallel in nature and evaluates singular values to tiny relative error if necessary. It is competitive with QR algorithm in serial mode in speed and advantageous in computing partial singular values. Error analysis and numerical results are presented. Received March 15, 1993 / Revised version received June 7, 1994     

The solution methods for the largest eigenvalue (singular value) of nonnegative tensors and convergence analysis

In this paper we study two solution methods for finding the largest eigenvalue (singular value) of general square (rectangular) nonnegative tensors. For a positive tensor, one can find the largest eigenvalue (singular value) based on the properties of the positive tensor and the power-type method. While for a general nonnegative tensor, we use a series of decreasing positive perturbations of the original tensor and repeatedly recall power-type method for finding the largest eigenvalue (singular value) of a positive tensor with an inexact strategy. We prove the convergence of the method for the general nonnegative tensor. Under a certain assumption, the computing complexity of the method is established. Motivated by the interior-point method for the convex optimization, we put forward a one-step inner iteration power-type method, whose convergence is also established under certain assumption. Additionally, by using embedding technique, we show the relationship between the singular values of the rectangular tensor and the eigenvalues of related square tensor, which suggests another way for finding the largest singular value of nonnegative rectangular tensor besides direct power-type method for this problem. Finally, numerical examples of our algorithms are reported, which demonstrate the convergence behaviors of our methods and show that the algorithms presented are promising.     

Construction of trivariate biorthogonal compactly supported wavelets

In this paper, first we introduce trivariate multiresolution analysis and trivariate biorthogonal wavelets. A sufficient condition on the existence of a pair of trivariate biorthogonal scaling functions is derived. Then, the pair of nonseparable or separable trivariate biorthogonal wavelets can be achieved from the pair of trivariate biorthogonal scaling functions.     

Construction of fundamental solutions of two-dimensional problems of the nonstationary theory of elasticity

We propose a method of constructing the images of the fundamental solutions in the space of the Laplace transform with respect to time, leading to simple formulas. The method is illustrated using three dynamical problems: planar deformation for an anisotropic body; flexural vibrations of an anisotropic plate; and vibrations of a shallow isotropic shell of arbitrary Gaussian curvature. Quadrature formulas are given for computing the values of the fundamental solutions. We give a new interpretation and a new method of computing the values of the special functions used in the construction of singular solutions in problems of the static theory of shells. Translated fromTeoreticheskaya i Prikladnaya Mekhanika, No. 23, 1992, pp. 86–92.     

三元双正交小波滤波器的刻画

研究由三元双正交插值尺度函数构造对应的双正交小波滤波器的矩阵扩充问题.当给定的一对三元双正交尺度函数中有一个为插值函数时,利用提升思想与矩阵多相分解方法,给出一类三元双正交小波滤波器的显示构造公式和一个计算实例.讨论了三元双正交小波包的的性质.     

A singular value decomposition for the Shack-Hartmann based wavefront reconstruction

We study the problem of reconstructing a wavefront from measurements of Shack-Hartmann-type sensors. Mathematically, this leads to the problem of reconstructing a function from a discrete set of averages of the gradient.After choosing appropriate function spaces this is an underdetermined problem for which least squares solutions and generalized inverses can be used. We explore this problem in more detail for the case of periodic functions on a quadratic aperture, where we calculate the singular value decomposition of the associated forward operator. The nonzero singular values can be estimated which shows that asymptotically, with increasing number of measurements, the reconstruction problem becomes an ill-posed problem.     

解平面弹性力学问题的边界积分方程的机械求积方法及其外推

１．引 言 考虑平面弹性力学内或外位移边值问题和内或外应力边值问题这里Ω是平面有界开集,Ωｃ是闭包Ω的补集,Γ是Ω或Ωｃ的边界,ｕ＝（ｕ１,ｕ２）是位移,ｎ＝（ｎ１,ｎ２）是Γ的外法向单位向量,δｉｊ＝（ｕｉ,ｊ＋ｕｊ,ｉ）／２是应变张量,λ和μ是Ｌａｍｅ常数,并且按张量计算规则：重复下标蕴含对该下标从１到２的求和． 使用直接边界元方法（１．１）与（１．２）皆可被转换为边界积分方程组这里αｉｊ（ｙ）是取决于ｙ∈Γ的常数,当ｙ是Γ的光滑点时,;式中是ｋｅｌｖｉｎ基本解,有以下表达式［５,７］这里ｒ＝…     

Geometric methods for nonlinear many-body quantum systems

Geometric techniques have played an important role in the seventies, for the study of the spectrum of many-body Schrödinger operators. In this paper we provide a formalism which also allows to study nonlinear systems. We start by defining a weak topology on many-body states, which appropriately describes the physical behavior of the system in the case of lack of compactness, that is when some particles are lost at infinity. We provide several important properties of this topology and use them to write a simple proof of the famous HVZ theorem in the repulsive case. In the second step we recall the method of geometric localization in Fock space as proposed by Dereziński and Gérard, and we relate this tool to our weak topology. We then provide several applications. We start by studying the so-called finite-rank approximation which consists in imposing that the many-body wavefunction can be expanded using finitely many one-body functions. We thereby emphasize geometric properties of Hartree-Fock states and prove nonlinear versions of the HVZ theorem, in the spirit of works of Friesecke. In the last section we study translation-invariant many-body systems comprising a nonlinear term, which effectively describes the interactions with a second system. As an example, we prove the existence of the multi-polaron in the Pekar-Tomasevich approximation, for certain values of the coupling constant.     

计算部分奇异值分解的隐式重新启动的双对角化Lanczos方法和精化的双对角化Lanczos方法

The singular value decomposition problem is mathematically equivalent to the eigenproblem of an argumented matrix. Golub et al. give a bidiagonalization Lanczos method for computing a number of largest or smallest singular values and corresponding singular vertors, but the method may encounter some convergence problems. In this paper we analyse the convergence of the method and show why it may fail to converge. To correct this possible nonconvergence, we propose a refined bidiagonalization Lanczos method and apply the implicitly restarting technique to it, and we then present an implicitly restarted bidiagonalization Lanczos algorithm(IRBL) and an implicitly restarted refined bidiagonalization Lanczos algorithm (IRRBL). A new implicitly restarting scheme and a reliable and efficient algorithm for computing refined shifts are developed for this special structure eigenproblem.Theoretical analysis and numerical experiments show that IRRBL performs much better than IRBL.     

A <Emphasis Type="Italic">QR</Emphasis>–method for computing the singular values via semiseparable matrices

Summary. The standard procedure to compute the singular value decomposition of a dense matrix, first reduces it into a bidiagonal one by means of orthogonal transformations. Once the bidiagonal matrix has been computed, the QR–method is applied to reduce the latter matrix into a diagonal one. In this paper we propose a new method for computing the singular value decomposition of a real matrix. In a first phase, an algorithm for reducing the matrix A into an upper triangular semiseparable matrix by means of orthogonal transformations is described. A remarkable feature of this phase is that, depending on the distribution of the singular values, after few steps of the reduction, the largest singular values are already computed with a precision depending on the gaps between the singular values. An implicit QR–method for upper triangular semiseparable matrices is derived and applied to the latter matrix for computing its singular values. The numerical tests show that the proposed method can compete with the standard method (using an intermediate bidiagonal matrix) for computing the singular values of a matrix.Mathematics Subject Classification (2000): 65F15, 15A18The research of the first two authors was partially supported by the Research Council K.U.Leuven, project OT/00/16 (SLAP: Structured Linear Algebra Package), by the Fund for Scientific Research–Flanders (Belgium), projects G.0078.01 (SMA: Structured Matrices and their Applications), G.0176.02 (ANCILA: Asymptotic aNalysis of the Convergence behavior of Iterative methods in numerical Linear Algebra), G.0184.02 (CORFU: Constructive study of Orthogonal Functions) and G.0455.0 (RHPH: Riemann-Hilbert problems, random matrices and Padé-Hermite approximation), and by the Belgian Programme on Interuniversity Poles of Attraction, initiated by the Belgian State, Prime Ministers Office for Science, Technology and Culture, project IUAP V-22 (Dynamical Systems and Control: Computation, Identification & Modelling). The work of the third author was partially supported by MIUR, grant number 2002014121. The scientific responsibility rests with the authors.Acknowledgments.We thank the referees for their suggestions which increased the readability of the paper.