Relations Between Forcing Sequences and Inexact Newton Iterates in Banach Space

We use inexact Newton iterates to approximate a solution of a nonlinear equation in a Banach space. Solving a nonlinear equation using Newton iterates at each stage is very expensive in general. That is why we consider inexact Newton methods, where the Newton equations are solved only approximately and in some unspecified manner. In the elegant paper [6] natural assumptions under which the forcing sequence is uniformly less than one were given based on the first-Fréchet derivative of the operator involved. Here, we use assumptions on the second Fréchet-derivative. This way, we essentially reproduce all results found earlier. However, our upper error bounds on the distances involved are smaller. Received: March 3, 1998; revised April 23, 1999     

Minimal Expansions in Redundant Number Systems and Shortest Paths in Graphs

We consider digit expansions in redundant number systems to base q with and consider such an expansion as minimal, if is minimal. We describe an efficient algorithm for determining a minimal representation and give an explicit characterization of optimal representations for odd q. Received: July 20, 1999; revised August 23 1999     

Volume Maximization and Orthoconvex Approximation of Orthogons

Let n axes-parallel hyperparallelepipeds (also called blocks) of the d-dimensional Euclidean space and a positive integer r be given. The volume maximization problem (VMP) selects at most r blocks such that the volume of their union becomes maximum. VMP is shown to be -hard in the 2-dimensional case and polynomially solvable for the line via a constrained critical path problem (CCPP) in an acyclic digraph. This CCPP leads to further well solvable special cases of the maximization problem. In particular, the following approximation problem (OAP) becomes polynomially solvable: given an orthogon P (i.e., a simple polygon in the plane which is a union of blocks) and a positive integer q, find an orthoconvex orthogon with at most q vertices and minimum area, which contains P. Received: September 7, 1998     

On the Multivariate Horner Scheme II: Running Error Analysis

 As an extension of our previous paper which gave forward and backward error estimates, we perform a running error analysis of the multivariate Horner scheme. This leads to a modified algorithm which computes the value of the polynomial together with an error estimate. Received November 9, 1999; revised May 29, 2000     

Analysis of Some Elementary Algorithms for Prime Factorization

For factoring a positive integer n into primes, four variants of the elementary algorithm are analysed. The worst-case time complexities vary from Θ() up to Θ(). The average time complexities vary from Θ() up to Θ(). Received August 21, 1998; revised September 14, 2000     

A Sparse Matrix Arithmetic Based on \Cal H-Matrices. Part I: Introduction to {\Cal H}-Matrices

A class of matrices (-matrices) is introduced which have the following properties. (i) They are sparse in the sense that only few data are needed for their representation. (ii) The matrix-vector multiplication is of almost linear complexity. (iii) In general, sums and products of these matrices are no longer in the same set, but their truncations to the -matrix format are again of almost linear complexity. (iv) The same statement holds for the inverse of an -matrix. This paper is the first of a series and is devoted to the first introduction of the -matrix concept. Two concret formats are described. The first one is the simplest possible. Nevertheless, it allows the exact inversion of tridiagonal matrices. The second one is able to approximate discrete integral operators. Received: July 30, 1998; revised December 28, 1998     

A Note on Transformed Density Rejection

In this paper we describe a version of transformed density rejection that requires less uniform random numbers. Random variates below the squeeze are generated by inversion. For the expensive part between squeeze and density an algorithm that uses a covering with triangles is introduced. Received August 23, 1999; revised March 6, 2000     

Incomplete Cross Approximation in the Mosaic-Skeleton Method

The mosaic-skeleton method was bred in a simple observation that rather large blocks in very large matrices coming from integral formulations can be approximated accurately by a sum of just few rank-one matrices (skeletons). These blocks might correspond to a region where the kernel is smooth enough, and anyway it can be a region where the kernel is approximated by a short sum of separable functions (functional skeletons). Since the effect of approximations is like that of having small-rank matrices, we find it pertinent to say about mosaic ranks of a matrix which turn out to be pretty small for many nonsingular matrices. On the first stage, the method builds up an appropriate mosaic partitioning using the concept of a tree of clusters and some extra information rather than the matrix entries (related to the mesh). On the second stage, it approximates every allowed block by skeletons using the entries of some rather small cross which is chosen by an adaptive procedure. We focus chiefly on some aspects of practical implementation and numerical examples on which the approximation time was found to grow almost linearly in the matrix size. Received February 13, 1999; revised October 26, 1999     

Dynamic Coverage Optimal Control for Multiple Spacecraft Interferometric Imaging

In this paper, we study and generalize a class of optimal control problems known in the literature as τ-elastic variational optimal control problems. In the τ-elastic optimal control, we want to minimize a cost function over trajectories that evolve on a Riemannian manifold and satisfy a second-order differential equation together with some smoothness and motion constraints. The cost function is a weighted sum of the squared norm of the acceleration and the squared norm of the velocity. Here, we generalize the τ-elastic variational problem to the dynamic coverage optimal control problem, which is a class of optimal control problems motivated by multiple spacecraft formation flying for imaging applications. The main novelty of this paper is an interesting connection between multiple spacecraft formation flying and the τ-elastic and coverage optimal control problems. This research is supported by NSF grants DMS-0103895, DMS-0305837, and CMS-0408542.     

热处理温度对掺钕钇铝石榴石陶瓷析晶的影响

采用氧化钕、氧化钇、硝酸铝、氨水以及柠檬酸作为原材料,以低温燃烧法(LCS)制备出纳米粉末材料。该方法解决了固相反应的高温合成及化学沉淀法的粉体团聚问题。通过热重-差热(TG-DTA)、X-射线粉末衍射(XRD)、傅里叶红外透射(FT-IR)和透射电镜(TEM)测试手段研究粉末的特性,采用谢莱方程(Scherrer)根据YAG(420)晶面的衍射曲线半峰宽数据计算出晶粒尺寸,详细研究陶瓷材料在不同热处理条件下的析晶情况。研究结果表明：YAG相的形成温度为850℃,在热处理过程中出现YAP中间相,于1050℃转变成纯YAG相,颗粒在不同的热处理条件下呈现不同的尺寸,在20～50nm范围变化。随着热处理温度的升高,平均晶粒尺寸增加,晶粒尺寸的标准偏差保持在2．0左右,晶格参数逐渐减小。晶粒主要以晶界扩散形式进行生长,晶格参数膨胀是由晶粒表面的排斥偶极距所造成的。