《数值分析》课程PPT教学课件（Numerical Analysis）Chapter 1 Introduction

Mumerical AnalysisWuhan University ofTransportation

Numerical Analysis Wuhan University of Transportation

MTeaching material《Numerical Analysis 》 (9th)Richard L. Burden, J. Douglas Faires, 2011 by Brooks-Cole,Cengage Learningm Bibliography《 Theoretical Numerical Analysis》Kendall Atkinson, Weimin Han, 2009 by SpringerDordrecht Heidelberg London New York《A Theoretical Introduction to Numerical Analysis》Victor S. Rvaben'kii, Semvon V. Tsvnkov, 2007 by Taylor &Francis Group, LLC

Teaching material 《Numerical Analysis 》(9th) Richard L. Burden, J. Douglas Faires, 2011 by Brooks-Cole, Cengage Learning  Bibliography 《 Theoretical Numerical Analysis》 Kendall Atkinson, Weimin Han，2009 by Springer Dordrecht Heidelberg London New York 《A Theoretical Introduction to Numerical Analysis》 Victor S. Rvaben'kii, Semvon V. Tsvnkov, 2007 by Taylor & Francis Group, LLC

Chapter 1 Introduction1 The Object and Character of Numerical Analysis2 Computer Machine Number System and FloatingPointArithmetic3TheErrorofNumericalAnalysis4 QualitativeError Analysis and Avoid ErrorHarm

Chapter 1 Introduction 1 The Object and Character of Numerical Analysis 2 Computer Machine Number System and Floating Point Arithmetic 4 Qualitative Error Analysis and Avoid Error Harm 3 The Error of Numerical Analysis

s1 The Objects and Characters of Numerical Analysis1)The study objectsNumerical analvsis is also called calculation method.Itcan propose numericalcalculation method and theory tosolve problems according to mathematicalmodel ofpracticalproblems.2)ContentNumericalapproximation offunction;Numerical differentialand numericalintegral;Numerical solution of nonlinear equations;Numerical linearalgebra;Ordinary differential and partial differentialnumerical solution

§1 The Objects and Characters of Numerical Analysis 1) The study objects Numerical analysis is also called calculation method. It can propose numerical calculation method and theory to solve problems according to mathematical model of practical problems. 2) Content Numerical approximation of function； Numerical differential and numerical integral； Numerical solution of nonlinear equations； Numerical linear algebra； Ordinary differential and partial differential numerical solution

3)CharacteristicNumerical analysis not only includes characters of highlyabstract of pure mathematics and rigorous scientificitybut also has the wide range of applications and the hightechnology of the actual test. It is a mathematicscurriculum combined with the use of computer withhighlypracticability4)Thechartersof numericalanalysisOFaced to the computer, supply the feasible and effectivearithmetic;OHaving reliable theory, doing error analysis toarithmetic and can get the precision demand;

3) Characteristic Numerical analysis not only includes characters of highly abstract of pure mathematics and rigorous scientificity but also has the wide range of applications and the high technology of the actual test. It is a mathematics curriculum combined with the use of computer with highly practicability. 4)The charters of numerical analysis ⚫Faced to the computer, supply the feasible and effective arithmetic； ⚫Having reliable theory，doing error analysis to arithmetic and can get the precision demand;

Having great computational complexity and arithmeticcould be realizedinthe computerBythe numerical test toprovearithmeticis effictiveFor example, Getting the solution of a n order linearequations by Cramer ‘s Rule, we need do n!(n-1)(n+1)multiplicativecalculations.If n-20,we need do 9.7X1020 multiplicative calculations.With a ten million times floating-pointcomputer persecond to count,it need to use3oo thousand years.5)CommonlyusedmethodsDiscretization ; Recursion ; Replace Approximatively

⚫Having great computational complexity and arithmetic could be realized in the computer ⚫By the numerical test to prove arithmetic is effictive For example, Getting the solution of a n order linear equations by Cramer‘s Rule, we need do n!(n-1)(n+1) multiplicative calculations. If n=20，we need do 9.7×1020 multiplicative calculations. With a ten million times floating-point computer per second to count, it need to use 300 thousand years. 5) Commonly used methods Discretization ; Recursion ; Replace Approximatively

6)PracticalapplicationRealizing numerical calculation in computer based onMATLAB or other software , at the same time ,solvingthe practical problems

6) Practical application Realizing numerical calculation in computer based on MATLAB or other software , at the same time ,solving the practical problems

Question : What is calculationmthod is used to do ?MathematicalPracticalConstructionalgorithmmodelingproblemsProgramTo calculate the result in the computerdesign

• Question：What is calculation method is used to do ? Mathematical modeling Construction algorithm Program design To calculate the result in the computer Practical problems

s 2 Computer Machine Number System andFloating Point Arithmetic1. Binary digit system and computer machine digitsystem>In a computer-With thebinary representation ofreal numbers>In a computerConvert inputed decimal to binarynumber-Calculateinbinarynumber systemConvert result to decimal number

§2 Computer Machine Number System and Floating Point Arithmetic 1. Binary digit system and computer machine digit system ➢In a computer ➢In a computer ——With the binary representation of real numbers ——Convert inputed decimal to binary number ——Calculate in binary number system ——Convert result to decimal number

Example 1. Please x = 237 expresses as a binary numberSolutionFirst,xexpressed asx=237=1×27+1×26+1×25+0×24+1×23+1×22+0×2l+1×20Sothebinarynumber style ofxis:x = (11101101)

Solution First, x expressed as So the binary number style of x is: 7 6 5 4 3 2 1 0 x = =  +  +  +  +  +  +  +  237 1 2 1 2 1 2 0 2 1 2 1 2 0 2 1 2 2 x = (11101101) Example 1. Please expresses as a binary number x = 237