a history of numerical analysis ideas[数值分析思想史](61)

1、A History of Numerical Analysis IdeasAlan Kaylor ClineDepartment of Computer SciencesThe University of Texas at AustinPrepared for CS 378History of ComputingOctober 14, 2003What is Different in Numerical Computing?What is Different in Numerical Computing?Well, its numbersScientific Computing vs.Nume

2、rical AnalysisYour Original CS Department1966Your Original CS Department196610 faculty with 2 numerical analystsIs there anything special with numerical computing?A Small ExampleA computation of Simple iteration: 2 2.828427124746190 3 3.061467458920719 4 3.121445152258053 5 3.136548490545941 6 3.140

3、331156954739 7 3.141277250932757 8 3.141513801144145 9 3.141572940367883 10 3.141587725279961 11 3.141591421504635 12 3.141592345611077 13 3.141592576545004 14 3.141592633463248 15 3.141592654807589 16 3.141592645321215 17 3.141592607375720 18 3.141592910939673 19 3.141594125195191 20 3.141596553704

4、820 21 3.141596553704820 22 3.141674265021758 23 3.141829681889202 24 3.142451272494134 25 3.142451272494134 26 3.162277660168380 27 3.162277660168380 28 3.464101615137754 29 4.000000000000000 30 0.000000000000000 31 0.000000000000000 Result of 15 digit computationRed digits are correctWhite and gre

5、en digits are incorrect 2 2.828427124746190 3 3.061467458920719 4 3.121445152258053 5 3.136548490545941 6 3.140331156954739 7 3.141277250932757 8 3.141513801144145 9 3.141572940367883 10 3.141587725279961 11 3.141591421504635 12 3.141592345611077 13 3.141592576545004 14 3.141592633463248 15 3.141592

6、654807589 16 3.141592645321215 17 3.141592607375720 18 3.141592910939673 19 3.141594125195191 20 3.141596553704820 21 3.141596553704820 22 3.141674265021758 23 3.141829681889202 24 3.142451272494134 25 3.142451272494134 26 3.162277660168380 27 3.162277660168380 28 3.464101615137754 29 4.000000000000

7、000 30 0.000000000000000 31 0.000000000000000 . . .Result of 15 digit computationRed digits are correctWhite and green digits are incorrect = 0 ?Wheres the problem?Wheres the problem?is calculated as zeroLets replacewith the algebraically identical expressionNew iteration:results in 2 2.828427124746

8、190 3 3.061467458920719 4 3.121445152258053 5 3.136548490545941 6 3.140331156954739 7 3.141277250932757 8 3.141513801144145 9 3.141572940367883 10 3.141587725279961 11 3.141591421504635 12 3.141592345611077 13 3.141592576545004 14 3.141592633463248 15 3.141592654807589 16 3.141592645321215 17 3.1415

9、92607375720 18 3.141592910939673 19 3.141594125195191 20 3.141596553704820 21 3.141596553704820 22 3.141674265021758 23 3.141829681889202 24 3.142451272494134 25 3.142451272494134 26 3.162277660168380 27 3.162277660168380 28 3.464101615137754 29 4.000000000000000 30 0.000000000000000 31 0.0000000000

10、00000 2 2.828427124746190 3 3.061467458920719 4 3.121445152258053 5 3.136548490545940 6 3.140331156954753 7 3.141277250932773 8 3.141513801144301 9 3.141572940367091 10 3.141587725277160 11 3.141591421511200 12 3.141592345570118 13 3.141592576584872 14 3.141592634338563 15 3.141592648776985 16 3.141

11、592652386591 17 3.141592653288992 18 3.141592653514593 19 3.141592653570993 20 3.141592653585093 21 3.141592653588618 22 3.141592653589499 23 3.141592653589719 24 3.141592653589774 25 3.141592653589788 26 3.141592653589792 27 3.141592653589793 28 3.141592653589793 29 3.141592653589793 30 3.141592653

12、589793 31 3.141592653589793 correct to all digitsBoringIs that all there isto numerical analysis?Not so boring if the result of this computation affectsNot so boring if the result of this computation affects The ability of the next plane you fly to stay in the airNot so boring if the result of this

13、computation affects The ability of the next plane you fly to stay in the air The integrity of the next bridge you crossNot so boring if the result of this computation affects The ability of the next plane you fly to stay in the air The integrity of the next bridge you cross The state of the economy

14、on which you liveNot so boring if the result of this computation affects The ability of the next plane you fly to stay in the air The integrity of the next bridge you cross The state of the economy on which you live The path of a missile that isnt intended to strike youSo what are the common problem

15、s of numerical analysis?So what are the common problems of numerical analysis?Application areas:So what are the common problems of numerical analysis?Application areas: Petroleum modelingSo what are the common problems of numerical analysis?Application areas: Petroleum modeling Atomic energy includi

16、ng weapons So what are the common problems of numerical analysis?Application areas: Petroleum modeling Atomic energy including weapons Weather modeling So what are the common problems of numerical analysis?Application areas: Petroleum modeling Atomic energy including weapons Weather modeling Other m

17、odeling such as aircraft and automobile So what are the common problems of numerical analysis?Algorithm areas:So what are the common problems of numerical analysis?Algorithm areas: Linear Equations So what are the common problems of numerical analysis?Algorithm areas: Linear Equations Nonlinear equa

18、tions - single and systemsSo what are the common problems of numerical analysis?Algorithm areas: Linear Equations Nonlinear equations - single and systems Optimization So what are the common problems of numerical analysis?Algorithm areas: Linear Equations Nonlinear equations - single and systems Opt

19、imization Data Fitting - interpolation and approximationSo what are the common problems of numerical analysis?Algorithm areas: Linear Equations Nonlinear equations - single and systems Optimization Data Fitting - interpolation and approximation IntegrationSo what are the common problems of numerical

20、 analysis?Algorithm areas: Linear Equations Nonlinear equations - single and systems Optimization Data Fitting - interpolation and approximation Integration Differential Equations - ordinary and partialDidnt we study that stuff in math classes?Didnt we study that stuff in math classes?Yes, but as th

21、e Pi Example shows, math classes are just the beginningLets get back to historyWhy were computers used primarily for numerical problems initially?Why were computers used primarily for numerical problems initially? Mathematicians and engineers designed themWhy were computers used primarily for numeri

22、cal problems initially? Mathematicians and engineers designed them A history of algorithms in that areaWhy were computers used primarily for numerical problems initially? Mathematicians and engineers designed them A history of algorithms in that area Immediate war-time and post-war-time applications

23、Why were computers used primarily for numerical problems initially? Mathematicians and engineers designed them A history of algorithms in that area Immediate war-time and post-war-time applications Applications did not depend upon having a large number of computersWhy were computers used primarily f

24、or numerical problems initially? Mathematicians and engineers designed them A history of algorithms in that area Immediate war-time and post-war-time applications Applications did not depend upon having a large number of computers However, there were non-numerical examples ENIGMAWhat were the major

25、computing ideas that arose in numerical analysis?What were the major computing ideas that arose in numerical analysis? Backward error analysis What were the major computing ideas that arose in numerical analysis? Backward error analysis inputoutputWhat were the major computing ideas that arose in nu

26、merical analysis? Backward error analysis inputoutputtrue operationWhat were the major computing ideas that arose in numerical analysis? Backward error analysis inputoutputtrue operationapproximate operationWhat were the major computing ideas that arose in numerical analysis? Backward error analysis

27、 inputoutputtrue operationapproximate operationerrorWhat were the major computing ideas that arose in numerical analysis? Backward error analysis inputoutputtrue operationapproximate operationWhat were the major computing ideas that arose in numerical analysis? Backward error analysis inputoutputtru

28、e operationapproximate operationbackward errorWhat were the major computing ideas that arose in numerical analysis? Backward error analysis FORTRANWhat were the major computing ideas that arose in numerical analysis? Backward error analysis FORTRANMathematical basedComputationally EfficientPortableStandardized 3 timesWhat were the major computing ideas that arose in numerical analysis? Backward error analysis FORTRAN Mathematical software packagesWhat were the major computing ideas that arose in numerical analysis? Backwar