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A History of Numerical Analysis Ideas. Prepared for CS 378 History of Computing October 14, 2003. Alan Kaylor Cline Department of Computer Sciences The University of Texas at Austin. What is Different in Numerical Computing?. What is Different in Numerical Computing?. Well, it’s numbers.

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a history of numerical analysis ideas

A History of Numerical Analysis Ideas

Prepared for CS 378

History of Computing

October 14, 2003

Alan Kaylor Cline

Department of Computer Sciences

The University of Texas at Austin

slide6

Your Original CS Department

1966

10 faculty with 2 numerical analysts

slide8

A Small Example

A computation of π

slide10

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 3.141592654807589

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 4.000000000000000

30 0.000000000000000

31 0.000000000000000

Result of 15 digit computation

Red digits are correct

White and green digits are incorrect

slide11

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 3.141592654807589

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 4.000000000000000

30 0.000000000000000

31 0.000000000000000

. . .

Result of 15 digit computation

Red digits are correct

White and green digits are incorrect

π = 0 ?

slide13

Where’s the problem?

is calculated as zero

slide14

Let’s replace

with the algebraically identical expression

slide15

New iteration:

results in …

slide16

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 3.141592652386591

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 3.141592653589793

31 3.141592653589793

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 3.141592654807589

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 4.000000000000000

30 0.000000000000000

31 0.000000000000000

π correct to all digits

slide17

Boring…

Is that all there is

to numerical analysis?

slide19

Not so boring if the result of this computation affects

  • The ability of the next plane you fly to stay in the air
slide20

Not 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
slide21

Not 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
slide22

Not 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 isn’t intended to strike you
slide25

So what are the common problems of numerical analysis?

Application areas:

  • Petroleum modeling
slide26

So what are the common problems of numerical analysis?

Application areas:

  • Petroleum modeling
  • Atomic energy – including weapons
slide27

So what are the common problems of numerical analysis?

Application areas:

  • Petroleum modeling
  • Atomic energy – including weapons
  • Weather modeling
slide28

So what are the common problems of numerical analysis?

Application areas:

  • Petroleum modeling
  • Atomic energy – including weapons
  • Weather modeling
  • Other modeling such as aircraft and automobile
slide31

So what are the common problems of numerical analysis?

Algorithm areas:

  • Linear Equations
  • Nonlinear equations - single and systems
slide32

So what are the common problems of numerical analysis?

Algorithm areas:

  • Linear Equations
  • Nonlinear equations - single and systems
  • Optimization
slide33

So what are the common problems of numerical analysis?

Algorithm areas:

  • Linear Equations
  • Nonlinear equations - single and systems
  • Optimization
  • Data Fitting - interpolation and approximation
slide34

So what are the common problems of numerical analysis?

Algorithm areas:

  • Linear Equations
  • Nonlinear equations - single and systems
  • Optimization
  • Data Fitting - interpolation and approximation
  • Integration
slide35

So what are the common problems of numerical analysis?

Algorithm areas:

  • Linear Equations
  • Nonlinear equations - single and systems
  • Optimization
  • Data Fitting - interpolation and approximation
  • Integration
  • Differential Equations - ordinary and partial
slide37

Didn’t we study that stuff in math classes?

Yes, but as the Pi Example shows, math classes are just the beginning

slide40

Why were computers used primarily for numerical problems initially?

  • Mathematicians and engineers designed them
slide41

Why were computers used primarily for numerical problems initially?

  • Mathematicians and engineers designed them
  • A history of algorithms in that area
slide42

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
slide43

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 computers
slide44

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 computers
  • However, there were non-numerical examples ENIGMA
slide47

What were the major computing ideas that arose in numerical analysis?

  • Backward error analysis

input

output

slide48

What were the major computing ideas that arose in numerical analysis?

  • Backward error analysis

true operation

input

output

slide49

What were the major computing ideas that arose in numerical analysis?

  • Backward error analysis

true operation

approximate operation

input

output

slide50

What were the major computing ideas that arose in numerical analysis?

  • Backward error analysis

true operation

error

approximate operation

input

output

slide51

What were the major computing ideas that arose in numerical analysis?

  • Backward error analysis

true operation

approximate operation

input

output

slide52

What were the major computing ideas that arose in numerical analysis?

  • Backward error analysis

true operation

backward

error

approximate operation

input

output

slide53

What were the major computing ideas that arose in numerical analysis?

  • Backward error analysis
  • FORTRAN
slide54

What were the major computing ideas that arose in numerical analysis?

  • Backward error analysis
  • FORTRAN
  • Mathematical based
  • Computationally Efficient
  • Portable
  • Standardized – 3 times
slide55

What were the major computing ideas that arose in numerical analysis?

  • Backward error analysis
  • FORTRAN
  • Mathematical software packages
slide56

What were the major computing ideas that arose in numerical analysis?

  • Backward error analysis
  • FORTRAN
  • Mathematical software packages
  • IMSL
  • Eispack
  • Linpack
slide57

What were the major computing ideas that arose in numerical analysis?

  • Backward error analysis
  • FORTRAN
  • Mathematical software packages
  • NANET
slide58

What were the major computing ideas that arose in numerical analysis?

  • Backward error analysis
  • FORTRAN
  • Mathematical software packages
  • NANET
  • Weekly information about people, problems, and papers
  • Software repository
slide59

What were the major computing ideas that arose in numerical analysis?

  • Backward error analysis
  • FORTRAN
  • Mathematical software packages
  • NANET
  • Matlab, Mathematica
slide60

What were the major computing ideas that arose in numerical analysis?

  • Backward error analysis
  • FORTRAN
  • Mathematical software packages
  • NANET
  • Matlab, Mathematica

Scientific computing environments

slide61

What were the major computing ideas that arose in numerical analysis?

  • Backward error analysis
  • FORTRAN
  • Mathematical software packages
  • NANET
  • Matlab, Mathematica
  • Super computers - Parallelism