BACKWARD ERROR, CONDITION NUMBERS, AND PSEUDOSPECTRUM FOR THE MULTIPARAMETER EIGENVALUE PRO

LECTURE 2 -1MAE3100Prof. D.E. SmithLECTURE 2Round-Off and Truncation ErrorsMotivation:Understanding the concept of error and how it develops in numerical methods is important to the effective use of computer techniques.This study focuses on two major forms of numerical error: Round-offError and Truncation Error.Reading Assignment:Chapra Chapter 4 (required)Chapra and Canale Section 4.1 (optional)LECTURE 2 -2MAE3100Prof. D.E. SmithLECTURE 2 OBJECTIVESThe primary objective of this lecture is to provide a basicunderstanding of computational errors introduced bynumerical methods. More specifically, this lecture will:•Expose potential sources of computational errors.•Compare accuracy and precision.•Review numerical precision associated with digitalcomputing.•Provide specific calculations for evaluating error estimates.•Provide a means for estimating truncation error using theTaylor series.•Explore finite difference approximations for first and secondderivatives.MAE3100Prof. D.E. SmithLECTURE 2 -3You’ve Got A Problem•The numerical methods solution to the bungee jumper problem inChapter 1 employed an approximation to the derivative as•What is known?–Numerical error is introduced byapproximation–Approximation error depends on t–Computers have limited precision•What is needed?– A basic understanding of errors associated with computercalculations– A method for assessing the amount of error introduced withthe approximations used by numerical methodsii i i t t t v t v t v dt dv 11)()(MAE3100Prof. D.E. SmithLECTURE 2 -4Sources of Error•Errors in mathematical modelingResult from simplifying assumptions and approximations madewhen representing physical systems by mathematical equations.•BlundersResult from mathematical modeling errors and/or programmingerrors.•Errors in inputBEWARE: garbage in –garbage out•Machine errorsResult from rounding, chopping, underflow and overflow•Truncation errorsResult from simplifying assumptions and approximations madein the numerical procedureOccurs when truncating an infinite series or numericalevaluation of an improper integral. E.g.,!4!3!21432x x x x e xMAE3100Prof. D.E. SmithLECTURE 2 -5Accuracy and Precisiondata scatterdefinesimprecisionoruncertaintysystematic deviation defines inaccuracy or bias inaccurateandimpreciseaccurateandprecise MAE3100Prof. D.E. SmithLECTURE 2 -6Significant Figures•The significant digits of a number are those that can be used withconfidence.•What is the speed?–48 or 49?–48.8 or 48.9?–48.8642138?–#sig-figs = 3•What is the mileage?–87324.4 or 87324.5?–87324.46?–#sig-figs = 7•All results in MAE3100 must display appropriate units and Sig-Figs–Use no less than 3 sig-figs, 4 is preferred–Use the greatest accuracy possible during calculations–Report result with sig-figs consistent with inputMAE3100Prof. D.E. SmithLECTURE 2 -7Error Definitions•True (Absolute) ErrorNote: Issues arise when working with different orders-of-magnitude•True Percent Relative Error•Approximate Percent Relative Error (true value not always known)• A specified percent tolerance s may be used as a stopping criteria:I.e., stop with n decimal place accuracy when ionapproximat value true t E (100%) value true ionapproximat - value true (100%) value true error true t %)100(ion approximat current ionapproximat previous -ion approximat current (100%)ion approximat errore approximat a )%105.0(2n s a MAE3100Prof. D.E. SmithLECTURE 2 -8Round-Off Error•Recall that numbers are stored in binary format–16 bit integer representationRange: -32,768 to 32,767Shown: -173–Floating point representation•Single precision (32 bit)24 bit mantissa7-8 sig-figs Range: 10-38to 1039•Double precision (64 bit)15-16 sig-figsRange: 10-308to 10308Representation used by MATLAB (223210)272625242330292831….sMAE3100Prof. D.E. SmithLECTURE 2 -9Working with Computer Numbers•Computers work in base 2 making it impossible to accuratelyrepresent numbers of interest, e.g.,•MATLAB commands related to machine precision include realmax ,realmin , eps•Round-off errors can accumulate in large computations.•Loss of significant figures occurs when subtracting numbers ofsimilar magnitude:•Small numbers can be lost when added to large numbers using finitemachine precision 333100001.0107641.0107642.0...,7,,,1.0etc e 4444104000.010*******.010*******.0104000.0chopped MAE3100Prof. D.E. SmithLECTURE 2 -10Truncation Error•Taylor theorem : Any smooth function can be written as a polynomial.•Definition: The Taylor Series expansion of f (x )about x = a iswhere the remainder is•The n -th order Taylor Series with h = x –a becomes:• A Maclaurin Series is a Taylor Series expanded about a = 0 (h = x )nk nk k R a x a f k x f 0)()()(!1)(n -th order Taylor Series approximation f n (x )remainder))(()()()!1(111)1(n n n n a x O a x f n R x a nn n hn a f h a f h a f h a f a f h f !)(!3)(!2)()()()()(3)3(2 nn n xn f x f x f x f f x f !)0(!3)0(!2)0()0()0()()(3)3(2 error is ‘on the order of’(x -a )n +1a x XI (ks-eye)MAE3100Prof. D.E. SmithLECTURE 2 -11Truncation Error (cont.)•Accuracy increases as more terms are added to the Taylor Series expansionconstant)a (i.e., 0a function)linear a (i.e., 10h a a )polynomial order 2nd a (i.e.,2210h a h a a MAE3100Prof. D.E. SmithLECTURE 2 -12Taylor Series ExampleFrom Chapra Figure 4.2, pg. 68:Where) (i.e., 0about Expanded 2.125.05.015.01.0)(234x a x h a x x x x x f 0.1)0(25.0)0(2.1)0(f f f xx f 25.02.1)(1225.025.02.1)(x x x f 2.1)(0x fMAE3100Prof. D.E. SmithLECTURE 2 -13Taylor Series -ExampleDetermine the 4-th order Taylor Series of y (x ) = ln(1+x )about x = a = 0By successive differentiation we obtainThe Taylor Series for |x |≤ 1becomes:Note: The error is proportional to x 5: i.e., when x is reduced by afactor of 2, R 4decreases by a factor of 25= 32)1ln()(x x y 0)0(y x x y 11)(1)0(y 2)1(1)(x x y 1)0(y 3)3()1(2)(x x y 2)0()3(y 4)4()1(6)(x x y 6)0()4(y 4324413121)()(x x x x x y x y )()1(51120)(5555)5(4x O x x y R with 5)5()1(24)(y MAE3100Prof. D.E. SmithLECTURE 2 -14Defining the RemainderRecall that the remainder R n is defined in terms the variable asConsider a zero-order (n = 0)Taylor Seriesthen 1)1()()!1()(n n n a x n f R ha a where )()()(0a f x f x f hR f h f R 00)()(= a = x = a = x)(f xa a+hMAE3100Prof. D.E. SmithLECTURE 2 -15Defining the Remainder (cont)Recall our example y (x ) = ln(1+x )with the 4-th order Taylor Series aboutx = 0The remainder becomesCombining these equations givesFor x = 1, we obtainthen or which satisfies the condition 4432413121)(R x x x x x y 55432)1(151413121)1ln(x x x x x x 555554)1(151)0())1(ln(!51x x x dx d R x 12739.058333.0413121)(69315.0)(4324x x x x x y x y 105544)1()1(15110981.058333.069315.0)1()1(y y R MAE3100Prof. D.E. SmithLECTURE 2 -16Defining the Remainder (cont)NOTE:•Determining the number of terms required for a Taylor series torepresent a function of interest is based on the remainder.•Unfortunately, ξis not known exactly but merely lies somewherebetween a and x = a + h (or x i and x i+1).•Also, R n requires derivatives of f (x ),which are not generally known.•Despite this, R n is still useful for gaining insight into truncationerrors.•Recall that we do have control over h and can therefore assess whathappens as we change h , i.e.,1)1(1)1()!1()()()!1()(n n nn n h n f a x n f R )())((11n n n h O a x O R xa a+h hMAE3100Prof. D.E. SmithLECTURE 2 -17Using the Remainder R n1)Establish order of error (from definition)2)Verify order of error estimate*> at h = 0.5 > at h = 1.03) Provide error estimate when h is increased or reduced*at h = 0.5, we computed R 4= 0.00442344then at h = 1.0 (doubled), we estimate R 4(0.00442344)(25) = 0.141550))(()()!1()(11)1(n n n n a x O a x n f R 0.109814)1()1(44y y R 0.00442344)5.0()5.0(44y y R 2X ~25X note: 25= 32X*For y (x ) = ln(1+x )about x = 0MAE3100Prof. D.E. SmithLECTURE 2 -18Using the Remainder R n (cont)4)Calculate value of ξwhen exact function value is known*Recall for h = 1and 5)Establish upper or lower bound on error*Recall that when h = 1, we obtain and Then the upper bound of the error may becalculated as the maximum R 4over the validrange of ξas 0.109814)1()1(44y y R *For y (x ) = ln(1+x )about x = 0554)1()1(151R Equating andsolving for ξgives12739.01054)1(151R 51)1(151)1(15110max 055max 4R maxξR 4MAE3100Prof. D.E. SmithLECTURE 2 -19Numerical DifferentiationDerivatives may be evaluated numerically withthe Finite (Divided) Difference method.1.Forward Difference (error is O (h ))2.Backward Difference (error is O (h ))3.Central Difference (error is O (h 2))hx f h x f x f )()()(hh x f x f x f )()()(hh x f h x f x f 2)()()(MAE3100Prof. D.E. SmithLECTURE 2 -20Remainder w/ Finite DifferenceConsider the 1st order Taylor Series and Remainder for f (x )expandedabout a = x (note h = x )which can be rearranged to give forward differenceExample problem: consider f (x ) = sin(x ) at x = /4exact derivative:forward finite difference:error bounded by remainder:2!2)()()()(xf x x f x f x x f R 1xf x x f x x f x f 2)()()()(O (x )7071067811.0)cos()(4/x x x f 6706029729.01.0)4/sin()1.04/sin()(x f (with x = 0.1)error = 0.0370781.04/at 0.0387084/at 0.035355)(2)sin(2)(x x f。

C/C++编译器错误代码大全-编译器错误C2001 错误消息常数中有换行符字符串常数不能继续到第二行，除非进行下列操作：•用反斜杠结束第一行。

•用一个双引号结束第一行上的字符串，并在下一行用另一个双引号开始该字符串。

用\n 结束第一行是不够的。

编译器错误C2002 错误消息无效的宽字符常数多字节字符常数是非法的。

通过检查下面的可能原因进行修复1.宽字符常数包含的字节比需要的多。

2.未包括标准头文件STDDEF.h。

3.宽字符不能与一般字符串连接。

4.宽字符常数之前必须是字符“L”：编译器错误C2003 错误消息应输入“defined id”标识符必须跟在预处理器关键字之后。

编译器错误C2004 错误消息应为“defined(id)”标识符必须出现在预处理器关键字之后的括号中。

也可能由于为V isual Studio .NET 2003 进行的编译器一致性工作生成此错误：在预处理器指令中缺少括号。

如果预处理器指令缺少右括号，则编译器将生成一个错误。

编译器错误C2005 错误消息#line 应跟一个行号，却找到“token”#line 指令后面必须跟行号。

编译器错误C2006 错误消息“directive”应输入文件名，却找到“token”诸如#include 或#import 等指令需要文件名。

若要解决该错误，请确保token 是一个有效文件名。

并且将该文件名放在双引号或尖括号中。

编译器错误C2007 错误消息#define 语法#define 后未出现标识符。

若要解决该错误，请使用标识符。

编译器错误C2008 错误消息“character”: 宏定义中的意外该字符紧跟在宏名之后。

若要解决该错误，宏名之后必须有一个空格。

编译器错误C2009 错误消息宏形式“identifier”重复使用宏定义的形参表多次使用该标识符。

宏的参数列表中的标识符必须是唯一的。

编译器错误C2010 错误消息“character”: 宏形参表中的意外该字符在宏定义的形参表中使用不正确。

C++出错提示英汉对照表Ambiguous operators need parentheses -----------不明确的运算需要用括号括起Ambiguous symbol ''xxx'' ----------------不明确的符号Argument list syntax error ----------------参数表语法错误Array bounds missing ------------------丢失数组界限符Array size toolarge -----------------数组尺寸太大Bad character in paramenters ------------------参数中有不适当的字符Bad file name format in include directive --------------------包含命令中文件名格式不正确Bad ifdef directive synatax ------------------------------编译预处理ifdef有语法错Bad undef directive syntax ---------------------------编译预处理undef有语法错Bit field too large ----------------位字段太长Call of non-function -----------------调用未定义的函数Call to function with no prototype ---------------调用函数时没有函数的说明Cannot modify a const object ---------------不允许修改常量对象Case outside of switch ----------------漏掉了case 语句Case syntax error ------------------ Case 语法错误Code has no effect -----------------代码不可述不可能执行到Compound statement missing{ --------------------分程序漏掉"{"Conflicting type modifiers ------------------不明确的类型说明符Constant expression required ----------------要求常量表达式Constant out of range in comparison -----------------在比较中常量超出范围Conversion may lose significant digits -----------------转换时会丢失意义的数字Conversion of near pointer not allowed -----------------不允许转换近指针Could not find file ''xxx'' -----------------------找不到XXX文件Declaration missing ; ----------------说明缺少"；"Declaration syntax error -----------------说明中出现语法错误Default outside of switch ------------------ Default 出现在switch语句之外Define directive needs an identifier ------------------定义编译预处理需要标识符Division by zero ------------------用零作除数Do statement must have while ------------------ Do-while语句中缺少while部分Enum syntax error ---------------------枚举类型语法错误Enumeration constant syntax error -----------------枚举常数语法错误Error directive :xxx ------------------------错误的编译预处理命令Error writing output file ---------------------写输出文件错误Expression syntax error -----------------------表达式语法错误Extra parameter in call ------------------------调用时出现多余错误File name too long ----------------文件名太长Function call missing -----------------函数调用缺少右括号Fuction definition out of place ------------------函数定义位置错误Fuction should return a value ------------------函数必需返回一个值Goto statement missing label ------------------ Goto语句没有标号Hexadecimal or octal constant too large ------------------16进制或8进制常数太大Illegal character ''x'' ------------------非法字符xIllegal initialization ------------------非法的初始化Illegal octal digit ------------------非法的8进制数字Illegal pointer subtraction ------------------非法的指针相减Illegal structure operation ------------------非法的结构体操作Illegal use of floating point -----------------非法的浮点运算Illegal use of pointer --------------------指针使用非法Improper use of a typedefsymbol ----------------类型定义符号使用不恰当In-line assembly not allowed -----------------不允许使用行间汇编Incompatible storage class -----------------存储类别不相容Incompatible type conversion --------------------不相容的类型转换Incorrect number format -----------------------错误的数据格式Incorrect use of default --------------------- Default使用不当Invalid indirection ---------------------无效的间接运算Invalid pointer addition ------------------指针相加无效Irreducible expression tree -----------------------无法执行的表达式运算Lvalue required ---------------------------需要逻辑值0或非0值Macro argument syntax error -------------------宏参数语法错误Macro expansion too long ----------------------宏的扩展以后太长Mismatched number of parameters in definition ---------------------定义中参数个数不匹配Misplaced break ---------------------此处不应出现break语句Misplaced continue ------------------------此处不应出现continue语句Misplaced decimal point --------------------此处不应出现小数点Misplaced elif directive --------------------不应编译预处理elifMisplaced else ----------------------此处不应出现elseMisplaced else directive ------------------此处不应出现编译预处理elseMisplaced endif directive -------------------此处不应出现编译预处理endifMust be addressable ----------------------必须是可以编址的Must take address of memory location ------------------必须存储定位的地址No declaration for function ''xxx'' -------------------没有函数xxx的说明No stack ---------------缺少堆栈No type information ------------------没有类型信息Non-portable pointer assignment --------------------不可移动的指针（地址常数）赋值Non-portable pointer comparison --------------------不可移动的指针（地址常数）比较Non-portable pointer conversion ----------------------不可移动的指针（地址常数）转换Not a valid expression format type ---------------------不合法的表达式格式Not an allowed type ---------------------不允许使用的类型Numeric constant too large -------------------数值常太大Out of memory -------------------内存不够用Parameter ''xxx'' is never used ------------------能数xxx没有用到Pointer required on left side of -> -----------------------符号->的左边必须是指针Possible use of ''xxx'' before definition -------------------在定义之前就使用了xxx（警告）Possibly incorrect assignment ----------------赋值可能不正确Redeclaration of ''xxx'' -------------------重复定义了xxxRedefinition of ''xxx'' is not identical ------------------- xxx的两次定义不一致Register allocation failure ------------------寄存器定址失败Repeat count needs an lvalue ------------------重复计数需要逻辑值Size of structure or array not known ------------------结构体或数给大小不确定Statement missing ; ------------------语句后缺少"；"Structure or union syntax error --------------结构体或联合体语法错误Structure size too large ----------------结构体尺寸太大Sub scripting missing ] ----------------下标缺少右方括号Superfluous & with function or array ------------------函数或数组中有多余的"&" Suspicious pointer conversion ---------------------可疑的指针转换Symbol limit exceeded ---------------符号超限Too few parameters in call -----------------函数调用时的实参少于函数的参数不Too many default cases ------------------- Default太多(switch语句中一个)Too many error or warning messages --------------------错误或警告信息太多Too many type in declaration -----------------说明中类型太多Too much auto memory in function -----------------函数用到的局部存储太多Too much global data defined in file ------------------文件中全局数据太多Two consecutive dots -----------------两个连续的句点Type mismatch in parameter xxx ----------------参数xxx类型不匹配Type mismatch in redeclaration of ''xxx'' ---------------- xxx重定义的类型不匹配Unable to create output file ''xxx'' ----------------无法建立输出文件xxxUnable to open include file ''xxx'' ---------------无法打开被包含的文件xxxUnable to open input file ''xxx'' ----------------无法打开输入文件xxxUndefined label ''xxx'' -------------------没有定义的标号xxxUndefined structure ''xxx'' -----------------没有定义的结构xxxUndefined symbol ''xxx'' -----------------没有定义的符号xxxUnexpected end of file in comment started on line xxx ----------从xxx行开始的注解尚未结束文件不能结束Unexpected end of file in conditional started on line xxx ----从xxx 开始的条件语句尚未结束文件不能结束Unknown assemble instruction ----------------未知的汇编结构Unknown option ---------------未知的操作Unknown preprocessor directive: ''xxx'' -----------------不认识的预处理命令xxx Unreachable code ------------------无路可达的代码Unterminated string or character constant -----------------字符串缺少引号User break ----------------用户强行中断了程序Void functions may not return a value ----------------- Void类型的函数不应有返回值Wrong number of arguments -----------------调用函数的参数数目错''xxx'' not an argument ----------------- xxx不是参数''xxx'' not part of structure -------------------- xxx不是结构体的一部分xxx statement missing ( -------------------- xxx语句缺少左括号xxx statement missing ) ------------------ xxx语句缺少右括号xxx statement missing ; -------------------- xxx缺少分号xxx'' declared but never used -------------------说明了xxx但没有使用xxx'' is assigned a value which is never used ----------------------给xxx赋了值但未用过Zero length structure ------------------结构体的长度为零。

中山大学本科教学大纲Undergraduate Course Syllabus学院（系）：数据科学与计算机学院School (Department)：School of Data and Computer Science课程名称：离散数学基础Course Title：Discrete Mathematics二〇二〇年离散数学教学大纲Course Syllabus: Discreate Mathematics（编写日期：2020 年12 月）(Date: 19/12/2020)一、课程基本说明I. Basic Information二、课程基本内容 II. Course Content（一）课程内容i. Course Content1、逻辑与证明（22学时） Logic and Proofs (22 hours)1.1 命题逻辑的语法和语义(4学时) Propositional Logic (4 hours)命题的概念、命题逻辑联结词和复合命题，命题的真值表和命题运算的优先级，自然语言命题的符号化Propositional Logic, logic operators (negation, conjunction, disjunction, implication, bicondition), compound propositions, truth table, translating sentences into logic expressions1.2 命题公式等值演算(2学时) Logical Equivalences (2 hours)命题之间的关系、逻辑等值和逻辑蕴含，基本等值式，等值演算Logical equivalence, basic laws of logical equivalences, constructing new logical equivalences1.3 命题逻辑的推理理论（2学时）论断模式，论断的有效性及其证明，推理规则，命题逻辑中的基本推理规则（假言推理、假言易位、假言三段论、析取三段论、附加律、化简律、合取律）,构造推理有效性的形式证明方法Argument forms, validity of arguments, inference rules, formal proofs1.4 谓词逻辑的语法和语义 (4学时) Predicates and Quantifiers (4 hours)命题逻辑的局限，个体与谓词、量词、全程量词与存在量词，自由变量与约束变量，谓词公式的真值，带量词的自然语言命题的符号化Limitations of propositional logic, individuals and predicates, quantifiers, the universal quantification and conjunction, the existential quantification and disjunction, free variables and bound variables, logic equivalences involving quantifiers, translating sentences into quantified expressions.1.4 谓词公式等值演算(2学时) Nested Quantifiers (2 hours)谓词公式之间的逻辑蕴含与逻辑等值，带嵌套量词的自然语言命题的符号化，嵌套量词与逻辑等值Understanding statements involving nested quantifiers, the order of quantifiers, translating sentences into logical expressions involving nested quantifiers, logical equivalences involving nested quantifiers.1.5谓词逻辑的推理规则和有效推理(4学时) Rules of Inference (4 hours)证明的基本含和证明的形式结构，带量词公式的推理规则（全程量词实例化、全程量词一般化、存在量词实例化、存在量词一般化），证明的构造Arguments, argument forms, validity of arguments, rules of inference for propositional logic (modus ponens, modus tollens, hypothetical syllogism, disjunctive syllogism, addition, simplication, conjunction), using rules of inference to build arguments, rules of inference for quantified statements (universal instantiation, universal generalization, existential instantiation, existential generalization)1.6 数学证明简介(2学时) Introduction to Proofs (2 hours)数学证明的相关术语、直接证明、通过逆反命题证明、反证法、证明中常见的错误Terminology of proofs, direct proofs, proof by contraposition, proof by contradiction, mistakes in proofs1.7 数学证明方法与策略初步(2学时) Proof Methods and Strategy (2 hours)穷举法、分情况证明、存在命题的证明、证明策略（前向与后向推理）Exhaustive proof, proof by cases, existence proofs, proof strategies (forward and backward reasoning)2、集合、函数和关系（18学时）Sets, Functions and Relations(18 hours)2.1 集合及其运算(3学时) Sets (3 hours)集合与元素、集合的表示、集合相等、文氏图、子集、幂集、笛卡尔积Set and its elements, set representations, set identities, Venn diagrams, subsets, power sets, Cartesian products.集合基本运算（并、交、补）、广义并与广义交、集合基本恒等式Unions, intersections, differences, complements, generalized unions and intersections, basic laws for set identities.2.2函数(3学时) Functions (3 hours)函数的定义、域和共域、像和原像、函数相等、单函数与满函数、函数逆与函数复合、函数图像Functions, domains and codomains, images and pre-images, function identity, one-to-one and onto functions, inverse functions and compositions of functions.2.3. 集合的基数（1学时）集合等势、有穷集、无穷集、可数集和不可数集Set equinumerous, finite set, infinite set, countable set, uncountable set.2.4 集合的归纳定义、归纳法和递归（3学时）Inductive sets, inductions and recursions (3 hours)自然数的归纳定义，自然数上的归纳法和递归函数；数学归纳法（第一数学归纳法）及应用举例、强归纳法（第二数学归纳法）及应用举例；集合一般归纳定义模式、结构归纳法和递归函数。

2002年专业英语八级考试真题试卷●试卷一(95 min)Part Ⅰ Listening Comprehension (40 min)In Sections A, B and C you will hear everything ONCE ONLY. Listen carefully and then answer the questions that follow. Mark the correct answer to each question on your Coloured Answer Sheet.SECTION A TALKQuestions 1 to 5 refer to the talk in this section .At the end of the talk you will be given 15 seconds to answer each of the following five questions. Now listen to the talk.1. According to the passage, during the 18th and 19th centuries cities were small in size mainly because ___.A. the urban population was stableB. few people lived in citiesC. transport was backwardD. it was originally planned2. Cities survived in those days largely as a result of ___.A. the trade activities they undertookB. the agricultural activities in the nearby areasC. their relatively small sizeD. the non-economic roles they played3. City dwellers were engaged in all the following economic activitiesEXCEPT ___.A. commerceB. distributionC. processingD. transportation4. Urban people left cities for the following reasons EXCEPT ___.A. more economic opportunitiesB. a freer social and political environmentC. more educational opportunitiesD. a more relaxed religious environment5. Why did the early cities fail to grow as quickly as expected throughout the 18th century?A. Because the countryside attracted more people.B. Because cities did not increase in number.C. Because the functions of the cities changed.D. Because the number of city people was stable.SECTION B INTERVIEWQuestions 6 to 10 are based on an interview .At the end of the interview you will be given 15 seconds to answer each of the following five questions. Now listen to the interview.6. According to Janet, the factor that would most affect negotiations is ___.A. English language proficiencyB. different cultural practicesC. different negotiation tasksD. the international Americanized style7. J anet‟s attitude towards the Americanized style as a model for business negotiations is ___.A. supportiveB. negativeC. ambiguousD. cautious8. Which of the following can NOT be seen as a difference between Brazilian and American negotiators?A. Americans prepare more points before negotiations.B. Americans are more straightforward during negotiations.C. Brazilians prefer more eye contact during negotiations.D. Brazilians seek more background information.9. Which group of people seems to be the most straightforward?A. The British.B. Germans.C. Americans.D. Not mentioned.10. Which of the following is NOT characteristic of Japanese negotiators?A. Reserved.B. Prejudiced.C. Polite.D. Prudent.SECTION C NEWS BROADCASTQuestion 11 is based on the following news. At the end of the news item, you will be given 15 seconds to answer the question. Now listen to the news.11. The news item is mainly about ___.A. a call for research papers to be read at the conferenceB. an international conference on traditional Tibetan medicineC. the number of participants at the conference and their nationalitiesD. the preparations made by the sponsors for the international conference Questions 12 and 13 are based on the following news. At the end of the news item, you will be given 30 seconds to answer the questions. Now listen to the news.12. The news item mainly concerns ___ in Hong Kong.A. Internet centresB. an IBM seminarC. e-governmentD. broadcasting13. The aims of the three policy objectives include all the followingEXCEPT ___.A. improvement of government efficiencyB. promotion of e-commerceC. integration of service deliveryD. formulation of Digital 21 StrategyQuestions 14 and 15 are based on the following news .At the end of the news item , you will be given 30 seconds to answer the questions. Now listen to the news.14. Which of the following records was the second best time of the year by Donovan Bailey?A.9.98.B.9.80.C.9.91.D.9.95.15. The record shows that Bailey was ___.A. still suffering from an injuryB. getting back in shapeC. unable to compete with GreeneD. less confident than beforeSECTION D NOTE-TAKING AND GAP-FILLINGFill in each of the gaps with ONE word. You may refer to your notes. Make sure the word you fill in is both grammatically and semantically acceptable. Study Activities in University In order to help college and university students in the process of learning, four key study activities have been designed and used to encourage them to make knowledge their own.1. Essay writing: central focus of university work esp. in thehumanities, e.g.(l) ___.benefits: 1) helping to select interesting content in books and to express understanding.2 ) enabling teachers to know progress and to offer(2) ___.3) familiarizing students with exam forms.2. Seminars and classroom discussion: another form to internalize knowledge in specialized contexts.benefits: 1) (3)___ enables you to know the effectiveness of and others‟ response to your speech immediately.2) Within the same period of time, more topics can be dealt with thanin(4) ___.3) The use of a broader range of knowledge is encouraged.3. Individual tutorials: a substitute for group discussion.format: from teacher (5)___ to flexible conversation.benefit: encouraging ideas and interaction.4. Lectures: a most (6) ___.used study activity.disadvantages: 1 ) less (7)___ than discussions or tutorials.2) more demanding in note-taking.advantages: 1) providing a general (8)___ of a subject under discussion.2) offering more easily understood versions of a theory.3) updating students on (9)___ developments.4) allowing students to follow different (10) ___.1.___2.___3.___4.___5.___6.___7.___8.___9.___10.___Part Ⅱ Proofreading and Error Correction (15 min)The following passage contains TEN errors. Each line contains a maximum of ONE error. In each case, only ONE word is involved. You should proofread the passage and correct it in the following way. For a wrong word,underline the wrong word and write the correct one in the blank provided at the end of the line.For a missing word, mark the position of the missing word with a “∧” sign and write the word you believe to be missing in the blank provided at the end of the line. For an unnecessary word cross out the unnecessary word with a slash “/‟ and put the word in the blank provided at the end of the line.ExampleWhen∧art museum wants a new exhibit, (1) anit never／buys things in finished form and hangs (2) neverthem on the wall. When a natural history museumwants an exhibition, it must often build it. (3) exhibitThere are great impediments to the general use of a standardin pronunciation comparable to that existing in spelling(orthography). One is the fact that pronunciation is learnt…naturally‟ and unconsciously, and orthography is learnt 1.___deliberately and consciously. Large numbers of us, in fact,remain throughout our lives quite unconscious with what our 2.___speech sounds like when we speak out, and it often comes as a 3.___shock when we firstly hear a recording of ourselves. It is not a 4.___voice we recognize at once, whereas our own handwriting issomething which we almost always know. We begin the 5.___…natural‟ learning of pronunciation long before we start learningto read or write, and in our early years we went on unconsciously 6.___ imitating and practicing the pronunciation of those around us formany more hours per every day than we ever have to spend 7.___learning even our difficult English spelling. This is …natural‟,8.___therefore, that our speech-sounds should be those of ourimmediate circle; after all, as we have seen, speech operates as ameans of holding a community and giving a sense of 9.___‟belonging‟. We learn quite early to recognize a ‟stranger‟,someone who speaks with an accent of a different community-perhaps only a few miles far. 10.___Part Ⅲ Reading Comprehension (40 min)SECTION A READING COMPREHENSION (30 min)In this section there are four reading passages followed by a total of fifteenmultiple-choice questions. Read the passages and then mark your answers on your Coloured Answer Sheet.TEXT ADo you ever feel as though you spend all your time in meetings?Henry Mintzberg, in his book The Nature of Managerial Work , found that in large organizations managers spent 22 per cent of their time at their desk, 6 percent on the telephone, 3 per cent on other activities, but a whopping 69 per cent in meetings.There is a widely-held but mistaken belief that meetings are for “solving problems” and “making decisions. ” For a start, the number of people attending a meeting tends to be inversely proportional to their collective ability to reach conclusions and make decisions. And these are the least important elements.Instead hours are devoted to side issues, playing elaborate games with one another. It seems, therefore, that meetings serve some purpose other than just making decisions.All meetings have one thing in common: role-playing. The most formal role is that of chairman. He sets the agenda, and a good chairman will keep the meeting running on time and to the point. Sadly, the other, informal role-players are often able to gain the upper hand. Chief is the “constant talker”, who just loves to hear his or her own voice.Then there are the “can‟t do” types who want to maintain the status quo. Since they have often been in the organization for a long time, they frequently quote historical experience as an excuse to block change: “It won‟t work, we tried that last year and it was a disaster.”A more subtle version of the “can‟t do” type, the “yes, but… ” has emerged recently. They have learnt about the need to sound positive, but they still can‟t bear to have things change.Another whole sub-set of characters are people who love meetings and want them to continue until 5:30 pm or beyond. Irrelevant issues are their specialty. They need to call or attend meetings, either to avoid work, or to justify their lack of performance, or simply because they do not have enough to do.Then ther e are the “counter-dependents”, those who usually disagree with everything that is said, particularly if it comes from the chairman or through consensus from the group. These people need to fight authority in whatever form.Meetings can also provide attenders with a sense of identification of their status and power. In this ease, managers arrange meetings as a means of communicating to others the boundaries of their exclusive club: who is “in”, and who is not.Because so many meetings end in confusion and without a decision, another game is played at the end of meetings, called reaching a false consensus. Since it is important for the chairman to appear successful in problem-solving and making a decision, the group reaches a false consensus. Everyone is happy, having spent their time productively. The reality is that the decision is so ambiguous that it is never acted upon, or, if it is, there is continuing conflict, for which another meeting is necessary.In the end, meetings provide the opportunity for social intercourse, to engage in battle in front of our bosses, to avoid unpleasant or unsatisfying work, to highlight our social status and identity. They are, in fact, a necessary though not necessarily productive psychological sideshow. Perhaps it is our civilized way to moderating, if not preventing, change.16. On role-playing, the passage seems to indicate that chairman ___.A. talks as much as participantsB. is usually a “constant talker”C. prefers to take the role of an observerD. is frequently outshone by participants17. Which of the following is NOT a distinct characteristic of the three types of participants?A. Submissiveness.B. Stubbornness.C. Disobedience.D. Lack of focus.18. The passage suggests that a false consensus was reached at the end of a meeting in order to ___.A. make room for another meetingB. bring an illusory sense of achievementC. highlight the importance of a meetingD. go ahead with the agreed programmeTEXT BCooperative competition. Competitive cooperation. Confused? Airline alliances have travellers scratching their heads over what‟ s going on in the skies. Some folks view alliances as a blessing to travellers, offering seamless travel, reduced fares and enhanced frequent-flyer benefits. Others see a conspiracy of big businesses, causing decreased competition, increased fares and fewer choices. Whatever your opinion, there‟s no escaping airline alliances: the marketing hype is unrelenting, with each of the two mega-groupings, Oneworld and Star Alliance, promoting itself as the best choice for all travellers. And, even if you turn away from their ads, chances are they will figure in any of your travel plans. By the end of the year, Oneworld and Star Alliance will between them control more than 40% of the traffic in the sky. Some pundits predict that figure will be more like 75% in 10 years.But why, after years of often ferocious competition, have airlines decided to band together? Let‟s just say the timing is mutually convenient. North American airlines, having exhausted all means of earning customer loyalty at home, have been looking for ways to reach out to foreign flyers. Asian carders are stillhurting from the region-wide economic downturn that began two years ago——just when some of the airlines were taking delivery of new aircraft. Alliances also allow carriers to cut costs and increase profits by pooling manpower resources on the ground (rather than each airline maintaining its own ground crew) and code-sharing——the practice of two partners selling tickets and operating only one aircraft.So alliances are terrific for airlines-but are they good for the passenger? Absolutely, say the airlines: think of the lounges, the joint FFP (frequent flyer programme) benefits, the round-the-world fares, and the global service networks. Then there‟s the promise of “seamless” travel : the ability to, say, travel from Singapore to Rome to New York to Rio de Janiero, all on one ticket, without having to wait hours for connections or worry about your bags. Sounds utopian? Peter Buecking, Cathay Pacific‟ s director of sales and marketing, thinks that seamless travel is still evolving. “It‟s fair to say that t hese links are only in their infancy. The key to seamlessness rests in infrastructure and information sharing. We‟re working on this. ” Henry Ma, spokesperson for Star Alliance in Hong Kong, lists some of the other benefits for consumers: “Global traveller s have an easier time making connections and planning their itineraries.” Ma claims alliances also assure passengers consistent service standards.Critics of alliances say the much-touted benefits to the consumer are mostly pie in the sky, that alliances are all about reducing costs for the airlines, rationalizing services and running joint marketing programmes. Jeff Blyskal, associate editor of Consumer Reports magazine, says the promotional ballyhoo overalliances is much ado about nothing. “I don‟t see much of a gain forconsumers:alliances are just a marketing gimmick. And as far as seamless travel goes, I‟ll believe it when I see it. Most airlines can‟t even get their own connections under control, let alone coordinate with another airline.”Blyskal believes alliances will ultimately result in decreased flight choices and increased costs for consumers. Instead of two airlines competing and each operating a flight on the same route at 70% capacity, the allied pair will share the route and run one full flight. Since fewer seats will be available, passengers will be obliged to pay more for tickets.The truth about alliances and their merits probably lies somewhere between the travel utopia presented by the players and the evil empires portrayed by their critics. And how much they affect you depends on what kind of traveller you are.Those who‟ve already made the elite grade in the FFP of a major airline stand to benefit the most when it joins an alliance: then they enjoy the FFP perks and advantages on any and all of the member carriers. For example, if you‟re a Marco Polo Club “gold” member of Cathay Pacific‟s Asia Miles FFP, you will auto matically be treated as a valuable customer by all members of Oneworld, of which Cathay Pacific is a member—even if you‟ve never flown with them before.For those who haven‟t made the top grade in any FFP, alliances might be a way of simplifying the earning of frequent flyer miles. For example, I belong to United Airline‟s Mileage Plus and generally fly less than 25, 000 miles a year. But I earn miles with every flight I take on Star Alliance member — All Nippon Airways and Thai Airways.If you fly less than I do, you might be smarter to stay out of the FFP game altogether. Hunt for bargains when booking flights and you might be able to save enough to take that extra trip anyway. The only real benefit infrequent flyers can draw from an alliance is an inexpensive round-the-world fare.The bottom line: for all the marketing hype, alliances aren‟t all things to all people-but everybody can get some benefit out of them.19. Which is the best word to describe air travellers‟ reaction to airline alliances?A. Delight.B. Indifference.C. Objection.D. Puzzlement20. According to the passage, setting up airline alliances will chiefly benefit ___.A. North American airlines and their domestic travellersB. North American airlines and their foreign counterpartsC. Asian airlines and their foreign travellersD. Asian airlines and their domestic travellers21. Which of the following is NOT a perceived advantage of alliances?A. Baggage allowance.B. Passenger comfort.C. Convenience.D. Quality.22. One disadvantage of alliances foreseen by the critics is that air travel may be mere expensive as a result of ___.A. less convenienceB. higher operation costsC. less competitionD. more joint marketing23. According to the passage, which of the following categories of travellers will gain most from airline alliances?A. Travellers who fly frequently economy class.B. Travellers who fly frequently business class.C. Travellers who fly occasionally during holidays.D. Travellers who fly economy class once in a while.TEXT CIt is nothing new that English use is on the rise around the world, especially in business circles. This also happens in France, the headquarters of the global battle against American cultural hegemony. If French guys are giving in toEnglish, something really big must be going on. And something big is going on.Partly, it‟s that American hegemony. Didier Benchimol, CEO of a Frenche-commerce software company, feels compelled to speak English perfectly because the Internet software business is dominated by Americans. He and other French businessmen also have to speak English because they want to get their message out to American investors, possessors of the world‟s deepest pockets.The triumph of English in France and elsewhere in Europe, however, may rest on something more enduring. As they become entwined with each other politically and economically, Europeans need a way to talk to one another and to the rest of the world. And for a number of reasons, they‟ve decided upon English as their common tongue.So when German chemical and pharmaceutical company Hoechst merged with French competitor Rhone Poulenc last year, the companies chose the vaguely Latinate Aventis as the new company name—and settled on E nglish as the company‟scommon language. When monetary policymakers from around Europe began meeting at the European Central Bank in Frankfurt last year to set interest rates for the new Euroland, they held their deliberations in English. Even the European Commission, with 11 official languages and a traditionally French-speaking bureaucracy, effectively switched over to English as its working language last year.How did this happen? One school attributes English‟s great success to the sheer weight of its merit. It‟s a Germanic language, brought to Britain around the fifth century A. D. During the four centuries of French-speaking rule that followed Norman Conquest of 1066, the language morphed into something else entirely. French words were added wholesale, and most of the complications of Germanic grammar were shed while few of the complications of French were added. Theresult is a language with a huge vocabulary and a simple grammar that can express most things more efficiently than either of its pa rents. What‟s more, English has remained ungoverned and open to change—foreign words, coinages, and grammatical shifts—in a way that French, ruled by the purist Academic Francaise, has not.So it‟s a swell language, especially for business. But the rise of English over the past few centuries clearly owes at least as much to history and economics as to the language‟s ability to economically express the concept win-win. What happened is that the competition— first Latin, then French, then, briefly, German—faded with the waning of the political, economic, and military fortunes of, respectively, the Catholic Church, France, and Germany. All along, English was increasing in importance: Britain was the birthplace of the Industrial Revolution, and London th e world‟s most important financial centre, which made English a key language for business. England‟s colonies around the world also made it the language with the most global reach. And as that former colony the U. S. rose to the status of the world‟s preem inent political, economic, military, and cultural power, English became the obvious second language to learn.In the 1990s more and more Europeans found themselves forced to use English. The last generation of business and government leaders who had n‟t studied English in school was leaving the stage. The European Community was adding new members and evolving from a paper-shuffling club into a serious regional government that would need a single common language if it were ever to get anything done. Meanwhile, economic barriers between European nations have been disappearing, meaning that more and more companies are beginning tolook at the whole continent as their domestic market. And then the Internetcame along.The Net had two big impacts. One was that it was an exciting, potentially lucrative new industry that had its roots in the U.S., so if you wanted to get in on it, you had to speak some English. The other was that by surfing the Web, Europeans who had previously encountered English only in school and in pop songs were now coming into contact with it daily.None of this means English has taken over European life. According to the European Union, 47% of Western Europeans (including the British and Irish)speakEnglish well enough to carry on a conversation. That‟s a lot more than those who can speak German (32% ) or French (28%), but it still means more Europeans don‟ t speak the language. If you want to sell shampoo or cell phones, you have to do it in French or German or Spanish or Greek. Even the U. S. and British media companies that stand to benefit most from the spread of English have been hedging their bets—CNN broadcasts in Spanish; the Financial Times has recently launched a daily German-language edition.But just look at who speaks English: 77% of Western European college students, 69% of managers, and 65% of those aged 15 to 24. In the secondary schools of the European Union‟s non-English-speaking countries, 91 % of students study English, all of which means that the transition to English as the language of European business hasn‟t been all that traumatic, and it‟s only going to get easier in the future.24. In the author‟s opinion, what really underlies the rising status of English in France and Europe is ___.A. American dominance in the Internet software businessB. a practical need for effective communication among EuropeansC. Europeans‟ eagerness to do business with American businessmenD. the recent trend for foreign companies to merge with each other25. Europeans began to favour English for all the following reasonsEXCEPT its ___.A. inherent linguistic propertiesB. association with the business worldC. links with the United StatesD. disassociation from political changes26. Which of the following statements forecasts the continuous rise of English in the future?A. About half of Western Europeans are now proficient in English.B. U. S. and British media companies are operating in Western Europe.C. Most secondary school students in Europe study English.D. Most Europeans continue to use their own language.27. The passage mainly examines the factors related to ___.A. the rising status of English in EuropeB. English learning in non-English-speaking E. U. nationsC. the preference for English by European businessmenD. the switch from French to English in the European CommissionTEXT DAs humankind moves into the third millennium, it can rightfully claim to have broken new ground in its age-old quest to master the environment. The fantastic achievements of modern technology and the speed at which scientific discoveries are translated into technological applications attest to the triumph of human endeavour.At the same time, however, some of these applications threaten to unleash forces over which we have no control. In other words, the new technology man now believes allows him to dominate this wider cosmos could well be a Frankenstein monster waiting to turn on its master.This is an entirely new situation that promises to change many of the perceptions governing life on the planet. The most acute challenges facing the future are likely to be not only those pitting man against his fellow man, but those involving humankind‟s struggle to preserve the enviro nment and ensure the sustainability of life on earth.A conflict waged to ensure the survival of the human species is bound to bring humans closer together. Technological progress has thus proved to be a double-edged sword, giving rise to a new form of conflict: a clash between Man and Nature.The new conflict is more dangerous than the traditional one between man and his fellow man, where the protagonists at least shared a common language. But when it comes to the reactions of the ecosystems to the onslaught of modern technology, there is no common language.Nature reacts with weather disturbances, with storms and earthquakes,with mutant viruses and bacteria—that is, with phenomena having no apparentcause and effect relationship with the modern technology that supposedly triggers them.As technology becomes ever more potent and nature reacts ever more violently, there is an urgent need to rethink how best to deal with the growing contradictions between Man and Nature.For a start, the planet, and hence all its inhabitants, must be perceived as an integral whole, not as a dichotomous mass divided geographically into the rich and developed and the poor and underdeveloped.Today, globalization encompasses the whole world and deals with it as an integral unit. It is no longer possible to say that conflict has shifted from its traditional east-west axis to a north-south axis. The real divide today is between summit and base, between the higher echelons of the international politicalstructure and its grassroots level, between governments and NGOs, between state and civil society, between public and private enterprise.The mesh structure is particularly obvious on the Internet. While it is true that to date the Internet seems to be favouring the most developed sectors of the international community over the less developed, this need not always be the case. Indeed, it could eventually overcome the disparities between the privileged and the underdeveloped.On the other hand, the macro-world in which we live is exposed to distortions because of the unpredictable side-effects of a micro-world we do not and cannot totally control.This raises the need for a global system of checks and balances, for mandatory rules and constraints in our dealings with Nature, in short, for a new type of veto designed to manage what is increasingly becoming a main contradiction of our time: the one between technology and ecology.A new type of international machinery must be set in place to cope with the new challenges. We need a new look at the harnessing of scientific discoveries, to maximize their positive effects for the promotion of humanity as a whole and to minimize their negative effects. We need an authority with veto powers to forbid。

附，常见错误号对照表MySQL error code SQLSTATE code Error message1011 HY000 Error on delete of '%s' (errn %d)1021 HY000 Disk full (%s); waiting for someone to free some space . . . 1022 23000 Can't write; duplicate key in table '%s'1027 HY000 '%s' is locked against change1036 HY000 Table '%s' is read only1048 23000 Column '%s' cannot be null1062 23000 Duplicate entry '%s' for key %d1099 HY000 Table '%s' was locked with a READ lock and can't be updated 1100 HY000 Table '%s' was not locked with LOCK TABLES1104 42000 The SELECT would examine more than MAX_JOIN_SIZE rows; check your WHERE and use SET SQL_BIG_SELECTS=1 or SET SQL_MAX_JOIN_SIZE=# if the SELECT is okay1106 42000 Incorrect parameters to procedure '%s'1114 HY000 The table '%s' is full1150 HY000 Delayed insert thread couldn't get requested lock for table %s 1165 HY000 INSERT DELAYED can't be used with table '%s' because it is locked with LOCK TABLES1242 21000 Subquery returns more than 1 row1263 22004 Column set to default value; NULL supplied to NOT NULL column '%s' at row %ld1264 22003 Out of range value adjusted for column '%s' at row %ld 1265 1000 Data truncated for column '%s' at row %ld1312 0A000 SELECT in a stored program must have INTO1317 70100 Query execution was interrupted1319 42000 Undefined CONDITION: %s1325 24000 Cursor is already open1326 24000 Cursor is not open1328 HY000 Incorrect number of FETCH variables1329 2000 No data to FETCH1336 42000 USE is not allowed in a stored program1337 42000 Variable or condition declaration after cursor or handler declaration1338 42000 Cursor declaration after handler declaration1339 20000 Case not found for CASE statement1348 HY000 Column '%s' is not updatable1357 HY000 Can't drop a %s from within another stored routine1358 HY000 GOTO is not allowed in a stored program handler1362 HY000 Updating of %s row is not allowed in %s trigger1363 HY000 There is no %s row in %s trigger优先级:当同时使用MySQl错误码,标准SQLSTATE错误码,命名条件(SQLEXCEPTION)来定义错误处理时，其捕获顺序是（只可捕获一条错误）:MYSQL码->SQLSTATE->命名条件。

冯·诺依曼（John von Neumann）数学奇才、计算机之父回顾20世纪科学技术的辉煌发展时，不能不提及20世纪最杰出的数学家之一的冯·诺依曼．众所周知，1946年发明的电子计算机，大大促进了科学技术的进步，大大促进了社会生活的进步．鉴于冯·诺依曼在发明电子计算机中所起到关键性作用，他被西方人誉为"计算机之父"．约翰·冯·诺依曼（J o h n v o n N e u m a n n，1903－1957），美藉匈牙利人，1903年12月28日生于匈牙利的布达佩斯，父亲是一个银行家，家境富裕，十分注意对孩子的教育．冯·诺依曼从小聪颖过人，兴趣广泛，读书过目不忘．据说他6岁时就能用古希腊语同父亲闲谈，一生掌握了七种语言．最擅德语，可在他用德语思考种种设想时，又能以阅读的速度译成英语．他对读过的书籍和论文．能很快一句不差地将内容复述出来，而且若干年之后，仍可如此．1911年一1921年，冯·诺依曼在布达佩斯的卢瑟伦中学读书期间，就崭露头角而深受老师的器重．在费克特老师的个别指导下并合作发表了第一篇数学论文，此时冯·诺依曼还不到18岁．1921年一1923年在苏黎世大学学习．很快又在1926年以优异的成绩获得了布达佩斯大学数学博士学位，此时冯·诺依曼年仅22岁．1927年一1929年冯·诺依曼相继在柏林大学和汉堡大学担任数学讲师。

1930年接受了普林斯顿大学客座教授的职位，西渡美国．1931年成为该校终身教授．1933年转到该校的高级研究所，成为最初六位教授之一，并在那里工作了一生．冯·诺依曼是普林斯顿大学、宾夕法尼亚大学、哈佛大学、伊斯坦堡大学、马里兰大学、哥伦比亚大学和慕尼黑高等技术学院等校的荣誉博士．他是美国国家科学院、秘鲁国立自然科学院和意大利国立林且学院等院的院士.1954年他任美国原子能委员会委员；1951年至1953年任美国数学会主席．1954年夏，冯·诺依曼被使现患有癌症，1957年2月8日，在华盛顿去世，终年54岁．冯·诺依曼在数学的诸多领域都进行了开创性工作，并作出了重大贡献．在第二次世界大战前，他主要从事算子理论、鼻子理论、集合论等方面的研究．1923年关于集合论中超限序数的论文，显示了冯·诺依曼处理集合论问题所特有的方式和风格．他把集会论加以公理化，他的公理化体系奠定了公理集合论的基础．他从公理出发，用代数方法导出了集合论中许多重要概念、基本运算、重要定理等．特别在1925年的一篇论文中，冯·诺依曼就指出了任何一种公理化系统中都存在着无法判定的命题．1933年，冯·诺依曼解决了希尔伯特第5问题，即证明了局部欧几里得紧群是李群．1934年他又把紧群理论与波尔的殆周期函数理论统一起来．他还对一般拓扑群的结构有深刻的认识，弄清了它的代数结构和拓扑结构与实数是一致的．他对其子代数进行了开创性工作，并莫定了它的理论基础，从而建立了算子代数这门新的数学分支．这个分支在当代的有关数学文献中均称为冯·诺依曼代数．这是有限维空间中矩阵代数的自然推广．冯·诺依曼还创立了博奕论这一现代数学的又一重要分支．1944年发表了奠基性的重要论文《博奕论与经济行为》．论文中包含博奕论的纯粹数学形式的阐述以及对于实际博奕应用的详细说明．文中还包含了诸如统计理论等教学思想．冯·诺依曼在格论、连续几何、理论物理、动力学、连续介质力学、气象计算、原子能和经济学等领域都作过重要的工作．冯·诺依曼对人类的最大贡献是对计算机科学、计算机技术和数值分析的开拓性工作．现在一般认为E N I A C机是世界第一台电子计算机，它是由美国科学家研制的，于1946年2月14日在费城开始运行．其实由汤米、费劳尔斯等英国科学家研制的"科洛萨斯"计算机比E N I A C机问世早两年多，于1944年1月10日在布莱奇利园区开始运行．E N I A C机证明电子真空技术可以大大地提高计算技术，不过，E N I A C机本身存在两大缺点：（1）没有存储器；（2）它用布线接板进行控制，甚至要搭接见天，计算速度也就被这一工作抵消了．E N I A C机研制组的莫克利和埃克特显然是感到了这一点，他们也想尽快着手研制另一台计算机，以便改进．冯·诺依曼由E N I A C机研制组的戈尔德斯廷中尉介绍参加E N I A C机研制小组后，便带领这批富有创新精神的年轻科技人员，向着更高的目标进军．1945年，他们在共同讨论的基础上，发表了一个全新的"存储程序通用电子计算机方案"--E D V A C（E l e c t r o n i cD i s c r e t e V a r i a b l e A u t o m a t i c C o m p U t e r的缩写）．在这过程中，冯·诺依曼显示出他雄厚的数理基础知识，充分发挥了他的顾问作用及探索问题和综合分析的能力．E D V A C方案明确奠定了新机器由五个部分组成，包括：运算器、逻辑控制装置、存储器、输入和输出设备，并描述了这五部分的职能和相互关系．E D V A C机还有两个非常重大的改进，即：（1）采用了二进制，不但数据采用二进制，指令也采用二进制；（2建立了存储程序，指令和数据便可一起放在存储器里，并作同样处理．简化了计算机的结构，大大提高了计算机的速度．1946年7，8月间，冯·诺依曼和戈尔德斯廷、勃克斯在E D V A C方案的基础上，为普林斯顿大学高级研究所研制I A S计算机时，又提出了一个更加完善的设计报告《电子计算机逻辑设计初探》．以上两份既有理论又有具体设计的文件，首次在全世界掀起了一股"计算机热"，它们的综合设计思想，便是著名的"冯·诺依曼机"，其中心就是有存储程序原则--指令和数据一起存储．这个概念被誉为‘计算机发展史上的一个里程碑"．它标志着电子计算机时代的真正开始，指导着以后的计算机设计．自然一切事物总是在发展着的，随着科学技术的进步，今天人们又认识到"冯·诺依曼机"的不足，它妨碍着计算机速度的进一步提高，而提出了"非冯·诺依曼机"的设想．冯·诺依曼还积极参与了推广应用计算机的工作，对如何编制程序及搞数值计算都作出了杰出的贡献．冯·诺依曼于1937年获美国数学会的波策奖；1947年获美国总统的功勋奖章、美国海军优秀公民服务奖；1956年获美国总统的自由奖章和爱因斯坦纪念奖以及费米奖．冯·诺依曼逝世后，未完成的手稿于1958年以《计算机与人脑》为名出版．他的主要著作收集在六卷《冯·诺依曼全集》中，1961年出版.(摘自名人网w w w.m i n g r e n.b i z)James Hardy Wilkinson( 09/27/1919 --10/05/1986) ：数值分析和数值计算的开拓者和奠基人，第五位图灵奖获得者J a m e s H a r d y W i l k i n s o n作为数值分析和数值计算的开拓者和奠基人是当之无愧的.J.H.w i l k i n s o n1919年9月27日生于英国S t r o o d,16岁进入英国的剑桥大学三一学院学习，受到H a r d y和L i t t l e w o o d等数学大师的深刻影响，成绩优异. 1940年由于战争的需要，他开始研究弹道的数学模型与数值计算。

Lecture 7Floating point arithmetic and stability2.5Machine representation of numbers•Scientific notation:− 3.14159265 ×10 −23signmantissa base exponent (significand)s m βe •A floating point number system is a (finite)subset F of R :(−1)s ·m ·βe −tA more common way:x =(−1)s ·(0.a 1a 2···a t )·βe =(−1)s(a 1β+a 2β2+···+a t βt )•Exponent range [e min ,e max ]•Normalized if a 1=e e =e min −1to represent 0.•Floating points are not evenly spaced.For example β=2,t =3,e min =−1and e max =3.The 21nonnegative floating point numbers are0,0.25,0.3125,0.3750,0.4375,0.5,0.625,0.750,0.875,1.0,1.25,1.50,1.75,2.0,2.5,3.0,3.5,4.0,5.0,6.0,7.0Spacing of the floating point numbers jumps by a factor 2at each power of 2.1Machine precision•Machine epsilon: MM=distance between1and the next largerfloating point numberM=β1−tIt is the spacing of thefloating point numbers between1.0 andβ.The spacing between1.0and1/βisβ−t=O( M)/β.•roundoffunit u=12 M,the most useful quantity in error analy-sis.RoundingIn converting x∈R tofl(x)∈F.•Rounding:fl(x)rounds up if(t+1)st digit is≥β/2and down otherwise.•error due to thefinite representation infloating point form of real(and complex)numbers in digital computers.•The main causes of error are round-offerror and data error. Property2.1If x∈R lies in the range of F,thenfl(x)=x(1+ ),| |≤u2IEEE Standards(rounding):•single precision,β=2,t=24, M≈10−8,e min=−125,e max=128•double precision,β=2,t=53, M≈10−16,e min=−1021,e max=1024•MATLAB’s eps gives M≈2.220446049250313×10−16with format long.single(x)to convert x to single precision.•Single precision:00000000000000000000000000000000s e m•Overflow and underflow•IEEEfloating point data typesSingle precision Double precisiont24bits53bitse8bits11bitss1bit1bite max1271023e min−126−1022Smallest normalized2−126≈10−382−1022≈10−308Largest normalized e127≈103821023≈308M2−24≈6×2−82−53≈10−16•Read p.47-553Special Quantities•∞is returned when an operation overflows•x/±∞=0for any number x,x/0=±∞for any nonzero number x•Operations with infinity are defined as limits,e.g.4−∞=limx→∞4−x=−∞•NaN(Not a Number)is returned when the an operation has no well-definedfinite or infinite result.•Examples:∞−∞,∞/∞,0/0,√−1,NaN 4.4Two fundamental issues:stability and conditioning1.Stability pertains to the perturbation behavior of an algorithm.2.Conditioning pertains to the perturbation behavior of a math-ematical problem.Algorithms•A mathematical problem may be defined as a function f: X→Y from a vector space X of data to a vector space Y of solutions.•An algorithm˜f for a problem f can be viewed as another map ˜f:X→Y.•The problem of solving Ax=b:input data A and b and the solution is x.The computed solution is denoted by˜xAccuracy•An algorithm˜f for a problem f is accurate if(for each x)˜f(x)−f(˜x)=O( M)f(x)•If the problem f is ill-conditioned,the goal of accuracy is unreasonably ambitious.Rounding of the input data is un-avoidable on a computer.The perturbation alone might lead to a significant change in the result.5Stability•An algorithm˜f for a problem f is stable if(for all x)˜f(x)−f(˜x)=O( M)f(˜x)for some˜x with˜x−x=O( M)x•“Nearly the right answer to nearly the right question”•The definitions of stability given here are useful in many parts of numerical linear algebra,the condition O( M)is probably too strict to be appropriate for all numerical problems in other areas such as differential equations.Backward stable•An algorithm˜f for a problem f is backward stable if(for all x)˜f(x)=f(˜x)for some˜x with ˜x−x=O( M)x•“exactly the right answer to nearly the right question”•Stronger and simpler than stability:a tightening of the defi-nition of stability.6Conditioning and condition number•Well condition problem means small perturbations of x lead to only small changes in f (x ).An ill-conditioned problem means some small perturbation of x leads to a large change in f (x ).•The relative condition number is defined by K =supδxδf f (x ) δx x•Example 2.3(condition of a system of equations)Let b ∈R n be fixed.The condition number of computing the solution Ax =b where A is nonsingular isK (A )= A A −1 .Conditioning vs.Stability•Condition =sensitivity of the solution to perturbations in the data•Backward stable algorithm produces accurate results on well conditioned problems.•Backward stable algorithm could produce inaccurate answers for sensitive (ill-conditioned)problems,but the inaccuracy is proportional to the sensitivity.•We concentrate on designing backward stable algorithms!7Accuracy of a backward stable algorithm•If a backward stable algorithm is used to solve a problem f with condition number K,then˜f(x)−f(x)=O(K(x) M)f(x)•It is accurate if the problem is well conditioned.•Rule of thumb:forward error condition number x backward error Stability offloating point arithmetic•x→fl(x)is backward stable.For each x∈R,there is ≤ M such thatfl(x)=x(1+ )•For allfloating points x,y,(x,y)→x∗y is backward stable.There exists ≤ M such thatfl(x∗y)=(x∗y)(1+ )•For allfloating points x,y,(x,y)→fl(x+y)=fl(x)+fl(y)is backward stable•Inner product(x,y)→fl(x T y)is backward stable,x,y∈R n.•Outer product(x,y)→fl(xy T)is not backward stable,x,y∈R n.•x→fl(x+1)is not backward stable(consider x≈0)8。

题目：Unit 6 Useful numbers题目：Unit 6 Useful numbers2. 老师说出数字及物品，学生快速展示。

Teacher: I will say the number and the thing, please show me. If I say two books, please show me two books. Ready? One pencil box, three pencils, two rulers...3. 播放韵句，学生认真倾听。

然后讨论韵句内容，学生跟读，自由练习之后学习表演韵句。

Teacher: It’s time for a chant. Listen carefully first.Teacher: What are they doing? They are jumping. Look at me, jump, jump, jump. Say with me, please, jump. （通过运动示范让学生明白jump的意思）Can you jump high?（使用手势让学生明白high的意思）Is it fun? Do you like jumping?4. Letters and sounds: V v , W w , X x , Y y, Z z.Step 5 Summary1. 齐唱韵句，结束课程。

Teacher: Boys and girls, let’s chant together.板书设计Unit 6 Useful numbersPart A When do we use numbers?one, two, three, four, five课堂作业新设计一、将下列数字与相应的单词连线。

one two three four five三、数一数，填一填I have _______________. I have _______________________.I have _______________. I have _______________________.题目：Unit 6 Useful numbersTeacher: Let’s act the dialogue out. Who wants to try?4. 顺序数单词。