科学计数法与近似数

一、方法导学1 .科学记数法　把一个数写成ａ× 1 0 ｎ 的形式 (其中 1≤ａ<1 0 ,ｎ是整数 ) ,这种记数法叫做科学记数法 .记数的方法 :①确定ａ .ａ是只有一位整数数位的数 .②确定ｎ .当原数≥ 1时 ,ｎ等于原数的整数位数减 1 ;当原数 <1时 ,ｎ是负整数 ,它的绝对值等于原数中左起第一个非零数字前零的个数 (含整数位上的零 ) .2 .近似数　一般地 ,一个近似数四舍五入到哪一位就说这个近似数精确到哪一位 .一个近似数 ,从左边起第一个非 0数字起 ,直到精确到的数位 ,所有的数字都叫做这个近似数的有效数字 .注意 :2 .0 5与 2 .0 50 0的区…     

环面上一般有根地图的计数

这篇文章给出了环面上以内面个数，根面次和非根节点个数为参数的一般有根地图的计数方程，导出了以内面个数和非根节点个数为参数的这类地图的计数方程的精确解。作为推论，推出了以边数为参数的这类地图的个数，其近似解在文献[2]中已讨论。     

Jaulent-Miodek方程组和长水波近似方程组的新精确解

首先,利用直接代数法给出了一类非线性方程的四组显式精确解的公式.进而,很方便地得到了Jaulent-Miodek方程组和长水波近似方程组的若干新精确解.     

整除在初等数论中的应用

利用格点图内因子的分布规律,推导出素数判断函数,孪生素数判断函数,歌德巴赫素数判断函数;推导出可计算不大于某正整数的素数个数,不大于某正整数的孪生素数个数和大偶数包含的歌德巴赫素数个数精确和近似的计算公式.     

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对试验次数较小的情形, 我们可以运用穷举法来寻找其可能存在的正交平衡区组设计. 这样做一方面可以尽可能多地找到正交平衡区组设计, 另一方面也可以为今后我们构造试验次数较大的正交平衡区组设计奠定基础. 本文最后给出了试验次数n=9以内的部分正交平衡区组设计.     

W_2~1［a，b」空间中线性变系数常微分方程组的精确解

该文利用再生核空间的技巧，在Ｗ［ａ，ｂ］空间中给出了微分方程组：的精确解，利用精确解给出了便于用计算机计算的近似解．     

具有复杂约束混料试验的渐近D-最优设计

复杂约束条件下试验设计区域极不规则,通常难以得到精确的最优设计.本文构造一种针对混料试验设计的随机搜索算法(MDRS),在具有复杂约束的区域内由Monte-Carlo方法产生一组初始点集,并通过MDRS算法迭代至逼近最优点集.通过实例验证,这种方法是有效的.它可以作为衡量其他设计的一个标准,即只有当给出的其他设计优于近似的最优解时才是有效.     

利用近似数确定准确的范围

由于生活实践的需要,产生了准确数和近似数.与实际完全符合的数称为准确数;与实际接近的数称为近似数.如果给出一个准确数,那么我们会由四舍五入法得到近似数;如果给出由四舍五入法得到的近似数,那么如何求出这个近似数所表示的精确数的范围呢?     

再生核空间W_2~2(*)中积分-微分方程精确解的表示

该文定义了一个再生核空间W_2~2(*),在其中讨论了积分-微分方程解的存在唯一性,给出了积分-微分方程一个定解问题的精确解的表达式及由精确解得出近似解的性质.     

一阶数值微分的局部正则化方法

本文研究了一阶数值微分问题,将其等价转化为第一类积分方程的求解问题,给出了求解该问题的局部正则化方法.在精确导数的一定假设条件下,讨论了正则化参数的先验选取策略及相应近似导数的误差估计.相对于经典的正则化方法,数值实验表明局部正则化方法能在有效抑制噪声的同时,保证近似导数逼近精确导数的效果,尤其是在精确导数有间断或急剧变化时.     

Block Krylov Subspace Methods for Large Algebraic Riccati Equations

In the present paper, we present numerical methods for the computation of approximate solutions to large continuous-time and discrete-time algebraic Riccati equations. The proposed methods are projection methods onto block Krylov subspaces. We use the block Arnoldi process to construct an orthonormal basis of the corresponding block Krylov subspace and then extract low rank approximate solutions. We consider the sequential version of the block Arnoldi algorithm by incorporating a deflation technique which allows us to delete linearly and almost linearly dependent vectors in the block Krylov subspace sequences. We give some theoretical results and present numerical experiments for large problems.     

块三对角阵分解因子的估值与应用

1.引 言 许多物理应用问题归结为求微分方程数值解,而这可以通过离散化为求解稀疏线性方程组,所以稀疏线性方程组求解的有效性在很大程度上决定了原问题求解算法的有效性.直接     

Approximate convex hull of affine iterated function system attractors

In this paper, we present an algorithm to construct an approximate convex hull of the attractors of an affine iterated function system (IFS). We construct a sequence of convex hull approximations for any required precision using the self-similarity property of the attractor in order to optimize calculations. Due to the affine properties of IFS transformations, the number of points considered in the construction is reduced. The time complexity of our algorithm is a linear function of the number of iterations and the number of points in the output approximate convex hull. The number of iterations and the execution time increases logarithmically with increasing accuracy. In addition, we introduce a method to simplify the approximate convex hull without loss of accuracy.     

An Approach to Finding a Global Minimization with Equality and Inequality Constraints

We give an approach for finding a global minimization with equality and inequality Constraints.Our approach is to construct an exact penalty function, and prove that the global minimal points of this exact penalty function are the primal constrained global minimal points. Thus we convert the problem of global constrained optimization into a problem of global unconstrained optimization. Furthermore, the integral approach for finding a global minimization for a class of discontinuous functions is used and an implementable algorithm is given.     

Four pairwise balanced designs

We construct pairwise balanced designs on 49, 57, 93, and 129 points of index unity, with block sizes 5, 9, 13, and 29. This completes the determination of the unique minimal finite basis for the PBD-closed set which consists of the integers congruent to 1 modulo 4. The design on 129 points has been used several times by a number of different authors but no correct version has previously appeared in print.     

An application of a multipoint differential-difference scheme to a boundary-value problem

For the boundary-value problem (2) we construct a scheme of the method of lines with a central-difference approximation of the derivative for any odd pattern. In particular cases we investigate the behavior at the net refinement of the direct solution of the boundary-value problem for the determination of the difference between the approximate solution obtained by the method of lines and the exact solution of the problem (1), (2). We also consider some modifications of the method of lines: the number of the lines of the net is taken to be equal to that of the pattern. We give an estimate for the norm of the difference between the approximate solution obtained by this method and the exact solution of the problem (1), (2).Translated from Zapiski Nauchnykh Seminarov Leningradskogo Instituta im. V. A. Steklova AN SSSR, Vol. 70, pp. 76–88, 1977.     

Polyharmonic functions in structures of exact solutions in the elasticity theory

We introduce an asymptotic algorithm that allows us to construct both approximate and exact solutions to a set of equations in the linear elasticity theory. The exact solutions are expressed by polynomials in one of coordinates, while their coefficients include polyharmonic functions that depend on two other coordinates. For the sake of ordering of solutions, one can associate every exact solution with the number of the asymptotic approximation.     

可加混料模型参数估计的D-最优正交区组设计

对于含有过程变量的二阶可加混料模型,利用正交拉丁方,研究了其参数估计的D-最优正交区组设计,一般性地给出了q分量时的D-最优正交区组设计的谱点结构,并以此推广得到含有相同或不同下界约束时的最优正交设计的谱点。     

On theoretical and numerical aspects of the bang-bang-principle

Summary In the first part of this paper we are dealing with theoretical statements and conditions which finally lead to bang-bang-principles. A careful analysis of these theorems is used for the development of a numerical method. This method consists of two stages: During the first iterations the number and approximate location of the switching points of the optimal control are determined. In the second phase a rapidly convergent algorithm determines the exact location. We apply this method successfully to a parabolic boundary control problem and give an extensive discussion of numerical results.The work of the second author on this paper was partially done during his stay at North Carolina State University, Graduate Program in Operations Research and Department of Mathematics, Raleigh, USA     

Obtaining exact value by approximate computations

Numerical approximate computations can solve large and complex problems fast.They have the advantage of high efficiency.However they only give approximate results,whereas we need exact results in some fields.There is a gap between approximate computations and exact results. In this paper,we build a bridge by which exact results can be obtained by numerical approximate computations.