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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?

Well, it’s numbers

A computation of π

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

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 ?

is calculated as zero

with the algebraically identical expression

results in …

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

Not so boring if the result of this computation affects

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

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

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

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

So what are the common problems of numerical analysis?

Application areas:

So what are the common problems of numerical analysis?

Application areas:

- Petroleum modeling
- Atomic energy – including 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 modeling 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
- Nonlinear equations - single and systems

So 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
- Optimization
- Data Fitting - interpolation and approximation

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

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

Didn’t we study that stuff in math classes?

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

Why were computers used primarily for numerical problems initially?

- Mathematicians and engineers designed them

Why were computers used primarily for numerical problems initially?

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

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

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

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

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

input

output

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

- Backward error analysis

true operation

input

output

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

- Backward error analysis

true operation

approximate operation

input

output

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

- Backward error analysis

true operation

error

approximate operation

input

output

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

- Backward error analysis

true operation

approximate operation

input

output

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

- Backward error analysis

true operation

backward

error

approximate operation

input

output

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

- Backward error analysis
- FORTRAN

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

- Backward error analysis
- FORTRAN

- Mathematical based
- Computationally Efficient
- Portable
- Standardized – 3 times

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

- Backward error analysis
- FORTRAN
- Mathematical software packages

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

- Backward error analysis
- FORTRAN
- Mathematical software packages

- IMSL
- Eispack
- Linpack

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

- Backward error analysis
- FORTRAN
- Mathematical software packages
- NANET

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

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

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

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

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

Scientific computing environments

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

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

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