Special Topics II Algorithms and Numerical Recipes of Integration

Hilbert空间中广义变分不等式的近似-似投影算法陈方琴;夏福全【摘要】In this paper, we consider the proximal projected-like method for solving generalized variational inequalities in Hilbert spaces. This method includes proximal method and projected-like method. At first, we obtain temporary iteration points by using proximal method, and then by using the projected-like method, we project the temporary point onto the feasible set of generalized variational inequalities to get the next iterative point. Under the assumptions that the set-valued mapping is maximal monotone, we prove that every weak accumulation point of the sequence is a solution of variational inequalities. Finally, under the condition that the distance-like function is a special function, we prove that the sequence has a unique weak accumulation point.%在Hilbert空间中研究了广义变分不等式解的近似-似投影算法,该算法包含了近似点算法和似投影算法.首先通过近似算法,获得暂时迭代点,然后利用似投影算法将该暂时的迭代点投影到广义变分不等式的可行集上,获得下一步的迭代点.在集值映象为极大单调的条件下,证明了迭代序列的任意弱聚点都是变分不等式的解.最后,在取特殊的似距离泛函的情况下证明了序列具有唯一的弱聚点.【期刊名称】《四川师范大学学报（自然科学版）》【年(卷),期】2012(035)003【总页数】6页(P297-302)【关键词】近似点算法;似投影算法;似距离泛函;极大单调映象【作者】陈方琴;夏福全【作者单位】四川师范大学数学与软件科学学院,四川成都 610066;四川师范大学数学与软件科学学院,四川成都 610066【正文语种】中文【中图分类】O176.3;O178设H为Hilbert空间，X为H中的非空闭凸子集，T：X→2H为集值映射.本文研究下列广义的变分不等式问题：求x*∈X，w*∈T(x*)，使得本文始终假设广义变分不等式问题(1)的解集S非空，并且X∩int(dom(T))≠Ø，或int(X)∩dom(T)≠Ø，其中广义变分不等式问题(1)在经济平衡、运筹学、数学物理等方面都有着广泛的应用［1］.同时，广义变分不等式问题(1)也和许多非线性问题有密切的关系，如相补问题、平衡问题、不动点理论等［2-3］.特别地，当 T是真凸下半连续泛函 f：H→R∪{+∞}的次微分时，广义变分不等式问题(1)退化为下列非光滑约束优化问题因此，对广义变分不等式问题(1)的研究无论是理论还是应用都很有意义.当集值映象T是强单调或者可行集X具有某种特殊结构(比如X是盒子)时，已有很多的有效算法计算广义变分不等式问题(1)的解［4-5］.但是，当集值映象T不是强单调或者可行集X不具有某种特殊结构时，广义变分不等式问题(1)的有效算法却不多.在这种情况下，应用最广泛的算法是投影算法，例如文献［6］.然而，一般情况下投影算子本身难以计算(事实上，必须要求解一个优化问题才能找到投影)，这使得投影型算法难以实现.如何降低投影算子的计算难度或者如何实现投影成为众多数学和应用数学工作者关注的问题.最近，A.Auslender等［7-8］为了克服这一难点，引入了似距离泛函，定义了似投影算子，并在映射T是极大单调以及T在X的有界集上有界的条件下得到迭代序列{xk}的凸组合的极限点是广义变分不等式问题(1)的解.另一方面，广义变分不等式问题(1)也等价于下列的变分包含问题：求x*∈X使得其中，NX是闭凸集X的正规对偶算子，其定义为：显然广义变分不等式问题(1)是下列问题的特例：其中，A是Hilbert空间H到自身的一个集值映射.对于(3)式的求解算法有很多(参见文献［9-11］)，其中最常见的方法之一是近似点算法，它的一般形式为然而，上式的精确解一般难以计算，特别当A为非线性算子时更困难.为了克服上述难点，近年有很多文献提出了非精确的近似点算法.具体方法是在上式中添加容许误差，从而计算上式的近似解(参见文献［12-13］).受上述工作的启发，本文在Hilbert空间中研究了广义变分不等式问题(1)的近似-似投影算法.该算法包含有非精确的近似点算法，即在近似点算法中包含有误差ξ，它满足一个容易验证的条件(5)(见算法2.1).应用非精确的近似点算法，获得暂时的迭代点.然后应用似投影算子，将暂时的迭代点投影到广义变分不等式的可行集上，获得下一步的迭代点，进而构造出变分不等式的迭代序列.本文在集值映象T是极大单调的条件下证明了迭代序列的有界性，也证明了迭代序列的弱聚点都为广义变分不等式问题(1)的解.最后，在取特殊的似距离泛函的情况下证明了序列的弱收敛性.本文只假设T是极大单调映射，去掉了T在X的有界集上有界的条件.因此，本文的结果推广了文献［7］中的相应结果.1 预备知识文中R+代表全体正实数.首先介绍A.Auslender等［7］给出的似投影算子的定义及性质：定义1.1 对任给的g∈H，x∈X，定义似投影算子P(g，x)如下：其中d：X×X→R+∪{+∞}为给定的泛函，且对任意的y∈X都有：(d1)d(·，y)是X上的真凸下半连续泛函且有d(y，y)=0，▽1d(y，y)=0，其中▽1d(·，y)是d(·，y)的梯度.(d2)domd(·，y)⊂X，dom∂1d(·，y)=X，其中∂1d(·，y)是d(·，y)的次梯度映象. (d3)d(·，y)在X上ρ强凸，即存在ρ＞0对于任意的y∈X都有设D(X)表示满足条件(d1)～(d3)的所有泛函的集合.易知，若d∈D(X)，则对于任意的g∈H，x∈X都有P(0，x)=x.也需要下面的似距离泛函.定义1.2［8］ X是Hilbert空间H的闭凸子集，d∈D(X)，称泛函F：X×X→R+∪{+∞}为由d诱导的近似距离.若F在X×X上为有限值，且存在σ＞0，γ∈(0，1］，使得对任意的a，b∈X有：引理1.1［8］假设d∈D(X)，F是满足定义1.2的似距离泛函，P是定义2.1中的似投影算子，则对于任意的τ∈X，y∈X都有定义1.3 设X是一个Hilbert空间H的非空子集，T：X→2H为集值映射，称(i)T为单调的，如果对任意的x，y∈X，u∈T(x)，v∈T(y)有(ii)T为极大单调的，如果T为单调映射，并且对于任何的单调映射只要满足都有引理 1.2［14］假设η∈［0，1)且μ=若v=u+ξ，其中‖ξ‖2≤η2(‖u‖2+‖v‖2)，则2 近似-似投影算法及其性质在本节中，首先介绍广义变分不等式问题(1)的近似-似投影算法;然后再研究该算法的一些有用性质.选取正实数序列{λk}和正数η∈［0，1)，构造下列的迭代算法：算法2.11)选取初始点z0∈H.令k=0.2)求xk∈X，使得其中，ξk∈H满足3)若gk+ωk=0，则算法停止，否则令其中4)令k=k+1，然后回到第2步.令A=T+NX，其中NX是由(2)式定义的闭凸集X的正规对偶算子，若T为极大单调映象且dom(NX)∩intdom(T)≠Ø，则A为极大单调映象.从而(I+λkA)-1有意义且是单值的［15］.由(4)式知xk=(I+λkA)-1(zk+ξk)，从而序列{xk}、{zk}有定义. 本文总假设T是极大单调集值映射，η∈［0，1)，现介绍算法2.1产生的迭代序列的一些性质.且序列{λk}满足性质2.1 若则证明令v=λk(gk+ωk)，u=zk-xk，将其带入引理1.2就可以得到性质(i)和(ii).对于(iii)一方面利用Cauchy-Schwarz不等式及(i)有另一方面由(ii)可知注2.1 在算法2.1的第3步中，若gk+ωk= 0，则-ωk∈NX(xk)，从而有因此，xk是广义变分不等式问题(1)的解.另一方面，若gk+ωk≠0，由性质2.1(ii)知由T的伪单调性知对于任意的x*∈S，又因为gk∈NX(xk)则可得性质2.2 设且对任意k都有gk+ωk≠0，则且序列{F(x*，zk)}收敛.证明在引理1.1中，令τ=x*，g=βk(gk+ ωk)，结合(6)式有从而由定义2.2(ii)，上式等于这里由(10)式可得.另一方面所以将(11)～(12)式相结合有由(7)式，上式等于由(7)、(9)式以及可知从而即序列{F(x*，zk)}单调递减.又根据似距离泛函F的定义知对任意的k都有F(x*，zk)≥0，故序列{F(x*，zk)}收敛.性质2.3 假设序列{λk}满足(8)式，则存在一个常数ζ＞0使得证明如果gk+ωk=0，则上式成立.现假设gk +ωk≠0.由性质2.1(ii)有因为λk∈［α1，α2］，所以令性质2.4 假设序列{λk}满足(8)式且1-则证明若gk+ωk≠0，则由(8)、(13)和(14)式以及性质2.1(iii)可知，对于任意的k 有对上式取极限并由序列{F(x*，zk)}的收敛性得性质2.5 假设{xk}、{zk}是由算法2.1产生的两个无限序列，{λk}满足(8)式，则{xk}、{zk}都有界且具有相同的弱聚点.证明由性质2.2和定义1.2(iii)知序列{zk}有界，利用性质2.4和性质2.1(i)，可得所以有因为{zk}有界，从而可得{xk}有界.由(15)式知{xk}和{zk}具有相同的弱聚点.3 收敛性分析定理3.1 如果由算法2.1产生的序列{xk}是有限序列，则序列最后一项为广义变分不等式问题(1)的解.证明若序列{xk}为有限序列，则对于序列的最后一项算法2.1将在第3步停止，故有gk+ωk= 0.由注2.1知xk∈X且是广义变分不等式问题(1)的解.现在假设由算法2.1产生的序列{xk}是无限序列，下面将证明{xk}的弱聚点是广义变分不等式问题(1)的解.定理3.2 设{xk}是由算法2.1产生的序列，则{xk}的任意弱聚点都是广义变分不等式问题(1)的解.证明假设是{}的任意一个弱聚点，由此可以得到一个{xk}的子列弱收敛于.不失一般性，假设xk=(弱收敛).因为{xk}⊂X，所以∈X.由性质2.5知对于所有的v∈H，任意选取u∈T(v)+NX(v)，则存在点ω'∈T(v)和g'∈NX(v)，使得ω'+g'=u.因此，两个不等式相加有因为ω'+g'=u，所以由于‖ωk+gk‖→0，且{xk}有界，故有对(16)式取极限所以故存在使由NX的定义知从而所以是广义变分不等式问题(1)的解.当似距离泛函F(x，y)具有特殊结构时，将证明算法2.1产生的迭代序列{zk}、{xk}弱收敛于广义变分不等式问题(1)的解.下面推论的证明与R.T.Rockafellar［16］中证明序列收敛的方法一样.推论3.1 令F(x，y)=m‖x-y‖2，其中常数则由算法2.1产生的序列{zk}有唯一的弱聚点，从而{xk}和{zk}弱收敛.证明对于任意的x*∈S，由性质2.2知{m‖x*-zk‖2}收敛，下面证明序列{zk}有唯一的弱聚点.假设是{zk}的两个弱聚点，{zkj}和{zki}是{zk}的两个子序列且分别弱收敛于由性质2.5知是序列{xk}的弱聚点.再由定理3.2知根据性质2.2知序列和收敛.令则分别对(17)～(18)式取极限，由于{zkj}、{zki}分别弱收敛于所以〈和都收敛于0.由α1、α2、θ的定义可得由(19)和(20)式可得从而θ=0，故所以{zk}的所有子列具有相同的弱聚点，从而{zk}弱收敛，由性质2.5知{xk}弱收敛.参考文献［1］Fang Y P，Huang N J.Variational-like inequalities with generalized monotone mappings in Banach spaces［J］.Optim Theo Appl，2003，118(2)：327-338.［2］吴定平.随机变分不等式和随机相补问题［J］.四川师范大学学报：自然科学版，2005，28(5)：535-537.［3］张石生.变分不等式和相补问题理论及应用［M］.上海：科学技术文献出版社，1991.［4］Auslender A，Teboulle M.Interior gradient and proximal methods for convex and coinc optimization［J］.SIAM J Optim，2006，16：697-725. ［5］Xia F Q，Huang N J，Liu Z B.A projected subgradient method for solving generalized mixed variational inequalities［J］.Oper Research Lett，2008，36：637-642.［6］Solodov M V，Svaiter B F.A new projection method for variational inequality problems［J］.SIAM J Control Optim，1999，37(3)：765-776. ［7］Auslender A，Teboulle M.Projected subgradient menthods whih non-euclidean distances for non-differentiable convex minimization and variational inequalities［J］.Math Program，2009，120：27-48.［8］Auslender A，Teboulle M.Interior projection-like methods for monotone variational inequalities［J］.Math Program，2005，A104：39-68.［9］Eckstein J，Bertsekas D P.On the Douglas-Rachford splitting method and the proximal point algorithm for maximal monotone operators［J］.Math Program，1992，55：293-318.［10］Eckstein J，Svaiter B F.A family of projective splitting methods forthe sum of two maximal monotone operators［J］.Math Pro-gram，2008，111：173-199.［11］Eckstein J，Ferris M.Smooth methods of mulitipliers forcomplementarity problems［J］.Math Program，1999，86：65-90. ［12］Solodov M V，Svaiter B F.Error bounds for proximal point subproblems and associated inexact proximal point algorithms［J］.Math Program，2002，88：371-389.［13］Solodov M V，Svaiter B F.A hybrid projection-proximal point algorithm［J］.J Convex Anal，1999，6：59-70.［14］Solodov M V，Svaiter B F.A unified framework for some inexact proximal point algorithms［J］.Numer Funct Anal Optim，2001，22：1013-1035.［15］Minty G.A theorem on monotone sets in Hilbert spaces［J］.J Math Anal Appl，1967，97：434-439.［16］Rockafellar R T.Monotone operators and the proximal point algorithm［J］.SIAM J Control Optim，1976，14：877-898.［17］Ding X P，Xia F Q.A new class of completely generalized quasi-variational inclusions in Banach spaces［J］.J Comput Appl Math，2002，147：369-383.［18］Xia F Q，Huang N J.Variational inclusions with general H-monotone operators in Banach spaces［J］.Comput Math Appl，2007，54：24-30. ［19］Rockafellar R T，Bets R J.Variational Analysis［M］.New York：Springer-Verlag，1988.［20］Teboulle M.Convergence of proximal-like algorithms［J］.SIAM J Optim，1997，7：1069-1083.。

一类改进的谱共轭梯度法景书杰;李亚敏;牛海峰【摘要】谱共轭梯度法有两个方向控制参数,是解决大规模无约束优化问题的有效方法.本文提出了一个改进的谱参数θk,它不同于现有的θk.新算法在任何线搜索下都满足著名的共轭条件:dTk yk-1=0.新方法的搜索方向在任何线搜索下都是充分下降的.在一般假设下,我们证明该方法在改进的Wolfe线搜索是全局收敛的.【期刊名称】《洛阳师范学院学报》【年(卷),期】2019(038)002【总页数】5页(P1-5)【关键词】无约束优化;谱共轭梯度法;下降条件;谱参数;Wolfe线搜索【作者】景书杰;李亚敏;牛海峰【作者单位】河南理工大学数学与信息科学学院,河南焦作454000;河南理工大学数学与信息科学学院,河南焦作454000;河南理工大学数学与信息科学学院,河南焦作454000【正文语种】中文【中图分类】O221.20 引言考虑无约束优化问题(0.1)其中f(x)在Rn→R上是连续可微的函数，Rn表示n维欧式空间. 我们定义g(x)=▽f(x)是f(x)在xk处的梯度向量，且令gk=g(xk).由于非线性共轭梯度法(简称CG法)迭代简单有效，全局收敛性和低内存需求，故它是解决问题(0.1)的最有效的迭代方法之一，特别是在科学和工程计算中的大规模优化问题中. 在解决问题(0.1)的迭代算法中得到序列{xk}，它的一般迭代格式如下xk+1=xk+αkdk(0.2)其中xk是当前迭代点，αk为步长.这里βk∈Rn为共轭参数，不同的CG法是由不同形式的共轭参数βk决定. 本文被以下共轭参数所吸引：它们的βk公式[1-4]如下这里代表Euclidean范数，yk:=gk+1-gk.PRP和HS是公认的最有效的两个CG法，但它们的收敛性都不是很好. 已有很多关于收敛性的研究[5-11]. 这些CG法都有良好的收敛性和数值表现，然而它们构造复杂且难以理解，不像经典的CG法[1-4,12-15], 形式简单，容易应用，所以工程师们也很少把它们应用到科学和生产等研究中. 因此，Rivaie等[16]给出了一个形式简单的共轭参数为方便起见，我们称它为RMIL法.2012年，Rivaie等[16]提出RMIL法的共轭参数，定义为(0.3)这里yk-1=gk-gk-1.显然(0.4)Rivaie等[16]验证了该方法产生的搜索方向dk是充分下降的，并在精确线搜索下建立了该算法的全局收敛性. 数值试验表明，RMIL法具有线性收敛速率，比其它CG法更有效.2001年，Birgin 和Martinez[17]提出了谱共轭梯度法(SCG法)，即将谱梯度方法和CG法的思想结合起来，搜索方向dk的迭代格式如下(0.5)其中这里θk是谱参数；sk-1=xk-xk-1；yk-1=gk-gk-1. 令人惊奇的是, SCG法在很多情况下优于经典的CG法. 但SCG法产生的搜索方向dk不满足下降条件并且没有证明算法是否是全局收敛性的. 故已有学者对此进行研究，使其修正的SCG法产生搜索方向dk是下降方向，并在一般假设下建立算法的全局收敛性.Zhang等在文献 [18] 给出一个修正的FR共轭梯度法(MFR)，搜索方向dk如下dk=-θkgk+βkdk-1其中显然，对k≥1，有成立. 即搜索方向dk是不依赖于任何线搜索的充分下降方向. Zhang等[18]证明了MFR法对于一般的目标函数在Wolfe线搜索或Armijo线搜索下也具有全局收敛性.2008年，Yu等[19]修正谱Perry共轭梯度法得到一个新的SCG法，称为DSP-CG法.的公式如下这里数值试验表明，对于任何的线搜索DSP-CG法都是下降方法. Yu等[19]证明了DSP-CG法对一般目标函数在Wolfe线搜索下是全局收敛性的.最近，Deng等[20]改进了SCG算法，给出混合的θk和βk公式，定义为:这里η是一个给定的小常数. 参数θk和βk的选择使得搜索方向dk既是充分下降的也是拟牛顿方向. 在Armijo线搜索下验证了改进的SCG算法的全局收敛性. 数值试验证实了改进的SCG算法比现存的算法更有效和稳定.本文将展示一个改进的谱参数θk，进而结合文献[16] 中的构造一个新的SCG法，我们称它为SRMIL法. 该方法的搜索方向dk不需要任何线搜索都是充分下降的.我们建立了在修正的Wolfe线搜索下SRMIL法的全局收敛性.1 谱参数θk及算法下降性下面我们给出谱参数θk的选取方法. 我们给出的谱参数θk不依赖于任何线搜索而满足著名的共轭条件：给式(0.3)的两边同乘yk-1，可得因此所以(1.1)本文用SRMIL法解决问题(0.1)，该方法中xk和dk的迭代格式分别选用(0.2)和(0.5). 用式(0.3)计算βk，用式(1.1)计算θk. 故有SRMIL法满足著名的共轭条件. 算法:Step 0：给定初始值x0∈Rn，ε>0，令0<ρ<σ<1，令k:=0,d0=-g0.Step 1：计算gk；若则停止，否则转Step 2.Step 2：计算步长αk>0，使其满足修正的Wolfe线搜索[21]：(1.2)Step 3：利用式(0.3)，式(0.5)，式(1.1)，分别计算Step 4：令xk+1=xk+αkdk，求gk+1，并用(0.3)试求令令k:=k+1，转Step 1.基本假设H[22](H1)目标函数f(x)在水平集l0={x∈Rn|f(x)≤f(x0)} 上有下界，其中x0为初始点.(H2)目标函数f(x)在水平集l0的一个邻域N内连续可微，且梯度函数g(x)满足Lipschitz连续，即存在常数L>0，使(1.3)引理1.1 若假设 H 成立，则修正的 Wolfe 线搜索(1.2)是可行的，故必存在αk>0满足条件(1.2).证明类似于文献 [19] 中引理1的证明，这个结果的证明是显然的.下面给出算法的充分下降条件.引理1.2 设序列{gk}和{dk}由算法生成，则对任意k≥0，(1.4)和(1.5)成立.证明用数学归纳法证明.(i)当k=0时，有d0=-g0，则有成立.(ii)假设有成立. 当k=k+1时，由式(0.4)，式(0.5)和式(1.1)有(1.6)综上，式(1.4)得证.由式(1.6)，显然有式(1.5)成立.2 全局收敛性引理2.1[23] 若假设H成立，则由算法生成的序列{gk}和{dk}满足Zoutendijk条件(2.1)证明由式(1.2)和式(1.3)，可得因此将上式的两边取平方得由式(1.2)和假设H，可得<+∞定理2.1 若假设H成立，序列{gk}由迭代算法(0.2)和(0.5)产生，则有(2.2)证明我们用反证法证明，反设结论不成立，则必存在常数γ>0，使得对式(0.5)变形得(2.3)把(2.3)的两边取平方模，并移项得上式两边除以得再利用式(1.5)，得(2.4)注意到当k=0时，d0=-g0，所以，由式(2.4)得所以显然，这与引理2.1中的(2.1)矛盾，故参考文献【相关文献】[1] Polak E,Ribiere G. 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《Python编程实用技巧》IntroductionAre you a Python enthusiast? Do you want to enhance your Python programming skills and become a more proficient developer? Look no further because "《Python编程实用技巧》" is here to help you! In this article, we will explore the contents of this book, discuss its significance, and present you with a comprehensive review. Whether you are a beginner or an experienced programmer, "《Python编程实用技巧》" is a must-read resource that will greatly benefit you in your Python journey. Chapter 1: Introduction to the Book1.1 What is "《Python编程实用技巧》"?"《Python编程实用技巧》" is a groundbreaking book that aims to provide practical tips and techniques for Python programming. It covers a wide range of topics, from fundamental concepts to advanced tricks, enabling readers to enhance their Python skills effectively. This book is authored by experienced Python developers who have a deep understanding of the language and its unique features.1.2 Importance of "《Python编程实用技巧》"Python is a popular programming language known for its simplicity and versatility. However, mastering Python requires more than just a basic understanding of its syntax. "《Python编程实用技巧》" fills the gap by providing readers with actionable advice and expert insights into Python programming. Whether you are writing scripts, developing web applications, or working on data analysis, this book will equip you with the necessary skills to solve complex problems efficiently. Chapter 2: Key Features of "《Python编程实用技巧》"2.1 Comprehensive Coverage of Python Language Features"《Python编程实用技巧》" covers a wide range of Python language features, including variables, data types, control flow, functions, classes, and modules. Each topic is explained in detail, ensuring readers have a solid foundation in Python programming.2.2 Advanced Techniques and Best PracticesApart from the basics, this book also delves into advanced techniques and best practices that separate an ordinary programmer from an exceptional one. It covers topics such as error handling, debugging,code optimization, and writing efficient algorithms. By mastering these techniques, readers can write cleaner, more optimized code that performs better and is easier to maintain.Chapter 3: Learning Python with Practical Examples 3.1 Practical Examples and Hands-on Approach "《Python编程实用技巧》" follows a practical approach to learning Python. Each concept is illustrated with real-world examples, making it easier for readers to understand and apply the knowledge in their own projects. Learning by doing is an effective method, and this book capitalizes on it to ensure readers can immediately put their newly acquired skills to practice.3.2 Strengthening Python Skills Through ProjectsTo further reinforce the concepts learned, "《Python编程实用技巧》" provides project-based exercises. These exercises challenge readers to apply their knowledge and think critically to solve problems. By completing these projects, readers can build confidence in their abilities and gain valuable hands-on experience in Python programming.Chapter 4: Strategies for Effective Python Development4.1 Code Organization and ModularityOne key aspect of effective Python development is code organization and modularity. "《Python编程实用技巧》" teaches readers how to structure their code for better maintainability and reusability. It covers topics such as choosing appropriate variable names, writing modular code with functions and classes, and applying design patterns for better code structure.4.2 Debugging and Error HandlingDebugging is an inevitable part of software development, and "《Python编程实用技巧》" equips readers with effective debugging techniques. It explains how to use Python's built-in debugging tools and third-party libraries to identify and fix errors. Furthermore, the book provides insights into error handling, teaching readers how to handle exceptions gracefully and build robust applications.Chapter 5: Optimizing Python Code for Performance5.1 Profiling and Performance Analysis"《Python编程实用技巧》" recognizes the importance of writing code that performs well. To optimize Python code for performance, the book introduces readers to profiling and performance analysis techniques. It explains how to identify bottlenecks in code execution and provides strategies for optimizing performance, such as using appropriate data structures and algorithms.5.2 Utilizing Python's Special FeaturesPython offers numerous powerful features that can enhance code performance. "《Python编程实用技巧》" explores these features, including list comprehensions, generator expressions, and built-in functions like map, filter, and reduce. By leveraging these features effectively, readers can write Python code that executes faster and utilizes system resources more efficiently.Chapter 6: Exploring Advanced Python Topics6.1 Metaprogramming and ReflectionFor those seeking to dive deeper into Python, "《Python编程实用技巧》" covers advanced topics such as metaprogramming and reflection.These topics allow developers to write code that modifies itself and introspects its own structure. By understanding metaprogramming and reflection, readers can unlock new possibilities in Python development.6.2 Concurrency and ParallelismAs applications become more complex, the need for concurrent and parallel execution arises. "《Python编程实用技巧》" introduces readers to Python's concurrency and parallelism capabilities, such as threading, multiprocessing, and asynchronous programming. By utilizing these techniques, developers can write applications that take advantage of modern computer architectures and provide better performance. Chapter 7: ConclusionIn conclusion, "《Python编程实用技巧》" is an invaluable resource for anyone looking to enhance their Python programming skills. With its comprehensive coverage of Python language features, practical examples, and expert insights, this book caters to both beginners and experienced programmers. By applying the knowledge gained from this book, readers can write cleaner, more optimized Python code and develop robust applications efficiently. So, what are you waiting for? Dive into "《Python编程实用技巧》" and take your Python programming to the next level!。

Algorithm Design Techniques and Analysis: English VersionExercise with AnswersIntroductionAlgorithms are an essential aspect of computer science. As such, students who are part of this field must master the art of algorithm design and analysis. Algorithm design refers to the process of creating algorithms that solve computational problems. Algorithm analysis, on the other hand, focuses on evaluating the resources required to execute those algorithms. This includes computational time and memory consumption.This document provides students with helpful algorithm design and analysis exercises. The exercises are in the formof questions with step-by-step solutions. The document is suitable for students who have completed the English versionof the Algorithm Design Techniques and Analysis textbook. The exercises cover various algorithm design techniques, such as divide-and-conquer, dynamic programming, and greedy approaches.InstructionEach exercise comes with a question and its solution. Read the question carefully and try to find a solution withoutlooking at the answer first. If you get stuck, look at the solution. Lastly, try the exercise agn without referring to the answer.Exercise 1: Divide and ConquerQuestion:Given an array of integers, find the maximum possible sum of a contiguous subarray.Example:Input: [-2, -3, 4, -1, -2, 1, 5, -3]Output: 7 (the contiguous subarray [4, -1, -2, 1, 5]) Solution:def max_subarray_sum(arr):if len(arr) ==1:return arr[0]mid =len(arr) //2left_arr = arr[:mid]right_arr = arr[mid:]max_left_sum = max_subarray_sum(left_arr)max_right_sum = max_subarray_sum(right_arr)max_left_border_sum =0left_border_sum =0for i in range(mid-1, -1, -1):left_border_sum += arr[i]max_left_border_sum =max(max_left_border_sum, left_b order_sum)max_right_border_sum =0right_border_sum =0for i in range(mid, len(arr)):right_border_sum += arr[i]max_right_border_sum =max(max_right_border_sum, righ t_border_sum)return max(max_left_sum, max_right_sum, max_left_border_s um+max_right_border_sum)Exercise 2: Dynamic ProgrammingQuestion:Given a list of lengths of steel rods and a corresponding list of prices, determine the maximum revenue you can get by cutting these rods into smaller pieces and selling them. Assume the cost of each cut is 0.Lengths: [1, 2, 3, 4, 5, 6, 7, 8]Prices: [1, 5, 8, 9, 10, 17, 17, 20]If the rod length is 4, the maximum revenue is 10.Solution:def max_revenue(lengths, prices, n):if n ==0:return0max_val =float('-inf')for i in range(n):max_val =max(max_val, prices[i] + max_revenue(length s, prices, n-i-1))return max_valExercise 3: Greedy AlgorithmQuestion:Given a set of jobs with start times and end times, find the maximum number of non-overlapping jobs that can be scheduled.Start times: [1, 3, 0, 5, 8, 5]End times: [2, 4, 6, 7, 9, 9]Output: 4Solution:def maximum_jobs(start_times, end_times):job_list =sorted(zip(end_times, start_times))count =0end_time =float('-inf')for e, s in job_list:if s >= end_time:count +=1end_time = ereturn countConclusionThe exercises presented in this document provide a practical way to master essential algorithm design and analysis techniques. Solving the problems without looking at the answers will expose students to the type of problems they might encounter in real life. The document’s solutionsprovide step-by-step instructions to ensure that students can approach the problems with confidence.。

国外计算机编程经典书籍1.《代码大全》（Code Complete）作者Steve McConnell。

这本书是软件开发领域的经典之作，涵盖了软件构建过程中的各个方面，包括设计、编码、调试等。

2.《计算机程序的构造和解释》（Structure and Interpretation of Computer Programs）作者Harold Abelson和Gerald Jay Sussman。

这本书被誉为计算机科学教育的经典教材，深入讲解了程序设计的基本原理和方法。

3.《算法导论》（Introduction to Algorithms）作者ThomasH. Cormen、Charles E. Leiserson、Ronald L. Rivest和Clifford Stein。

这本书是关于算法和数据结构的权威指南，被广泛应用于计算机科学教育和专业领域。

4.《设计模式，可复用面向对象软件的基础》（Design Patterns: Elements of Reusable Object-Oriented Software）作者Erich Gamma、Richard Helm、Ralph Johnson和John Vlissides。

这本书介绍了面向对象设计中的23种设计模式，对软件开发具有重要的指导作用。

5.《Clean Code: A Handbook of Agile Software Craftsmanship》作者Robert C. Martin。

这本书强调编写整洁、可读、可维护代码的重要性，是软件工程师必读的经典之作。

6.《编程珠玑》（Programming Pearls）作者Jon Bentley。

这本书以一系列有趣的问题和解决方案展示了高效编程的技巧和方法，对提高编程技能有很大帮助。

以上列举的书籍只是众多优秀计算机编程书籍中的一部分，它们涵盖了计算机科学领域的各个方面，对于想要深入了解编程和软件开发的人来说，都是非常值得阅读和学习的经典之作。

六年级计算机科学英语阅读理解30题1<背景文章>A computer is an amazing device that has become an essential part of our daily lives. It consists of several important components, each with its own unique function.One of the most crucial parts is the Central Processing Unit (CPU). The CPU is often considered the "brain" of the computer. It executes instructions and performs calculations. It fetches data from the memory, processes it, and then stores the results back in the memory.Memory, also known as Random - Access Memory (RAM), is another vital component. RAM is a temporary storage space. When you open a program or a file, it is loaded into the RAM. This allows the CPU to access the data quickly. However, the data stored in RAM is lost when the computer is turned off.The hard disk drive (HDD) or solid - state drive (SSD) is a long - term storage device. It stores all your programs, files, and operating system. The hard disk has a large capacity, which means it can hold a huge amount of data. For example, you can store your photos, videos, and documents on it.The motherboard is like a big circuit board that connects all the components together. It provides the electrical connections and pathwaysfor data to travel between different parts of the computer.1. <问题1>What is the function of the CPU in a computer?A. It only stores data permanently.B. It executes instructions and does calculations.C. It is only used to connect other components.D. It only displays the information on the screen.答案：B。