1. What is Numerical Computation?
Numerical Computation in the real world.
A Brief History.
Questions and Answers.
Numerical Computation
Duo Wang
ESL 598 Section 05
Applied Math and Statistics Department
December 11, 2007
Duo Wang Numerical Computation

2. What is Numerical Computation?
Numerical Computation in the real world.
A Brief History.
Questions and Answers.
Why do we need a Computer?
Duo Wang Numerical Computation

3. What is Numerical Computation?
Numerical Computation in the real world.
A Brief History.
Questions and Answers.
Why do we need a Computer?
for me
Dealing with E-documents
Surfing on the internet
Playing video games
Communicating with friends
Duo Wang Numerical Computation

4. What is Numerical Computation?
Numerical Computation in the real world.
A Brief History.
Questions and Answers.
Why do we need a Computer?
for me
Dealing with E-documents
Surfing on the internet
Playing video games
Communicating with friends
Giving ESL 598 Final Presentation
Duo Wang Numerical Computation

5. What is Numerical Computation?
Numerical Computation in the real world.
A Brief History.
Questions and Answers.
Why do we need a Computer?
for me
Dealing with E-documents
Surfing on the internet
Playing video games
Communicating with friends
Giving ESL 598 Final Presentation
But the origin of computer is for
Numerical Computation!
Duo Wang Numerical Computation

6. What is Numerical Computation?
Numerical Computation in the real world.
A Brief History.
Questions and Answers.
Outline
What is Numerical Computation?
Numerical Computation in the real world.
A Brief History.
Questions and Answers.
Duo Wang Numerical Computation

7. What is Numerical Computation?
Numerical Computation in the real world.
A Brief History.
Questions and Answers.
What is Numerical Computation?
Definition
(Numerical Analysis or Scientific Computing):The study of
approximation techniques for numerically solving mathematical
problems.
Duo Wang Numerical Computation

8. What is Numerical Computation?
Numerical Computation in the real world.
A Brief History.
Questions and Answers.
What is Numerical Computation?
Definition
(Numerical Analysis or Scientific Computing):The study of
approximation techniques for numerically solving mathematical
problems.
Numerical Computation is necessary for problem solving in that
not all the(in fact, very few) mathematical problems have
closed-form solutions. We can only obtain numerical solutions.
For example:
The root of a polynomial with degree 5 or more
The eigenvalues of a matrix
Most ODEs and PDEs
Duo Wang Numerical Computation

9. What is Numerical Computation?
Numerical Computation in the real world.
A Brief History.
Questions and Answers.
What is Numerical Computation?
Definition
(Numerical Analysis or Scientific Computing):The study of
approximation techniques for numerically solving mathematical
problems.
Numerical Computation is necessary for problem solving in that
not all the(in fact, very few) mathematical problems have
closed-form solutions. We can only obtain numerical solutions.
For example:
The root of a polynomial with degree 5 or more
The eigenvalues of a matrix
Most ODEs and PDEs
Duo Wang Numerical Computation

10. What is Numerical Computation?
Numerical Computation in the real world.
A Brief History.
Questions and Answers.
Main branches:
Main branches of Numerical Computation:
Linear equations: Solve a system of n
linear equations in n unknowns.
Linear programming: Minimize a linear
function subject to m linear constraints.
Optimization: Minimize a function of
several variables.
Numerical PDE: Numerically solve a
Partial Differential Equation.
Duo Wang Numerical Computation

11. What is Numerical Computation?
Numerical Computation in the real world.
A Brief History.
Questions and Answers.
What is the difference from Pure Mathematics?
Focus on: Focus on:
Concept Problem solving
Theory Algorithm
Reasoning Stability, Convergence
Duo Wang Numerical Computation

12. What is Numerical Computation?
Numerical Computation in the real world.
A Brief History.
Questions and Answers.
Numerical Computation in the real world
Some real word applications:
The World’s Largest Matrix Computation:
Google’s PageRank is an eigenvector of a
matrix of order about 3 billion
Airlines use optimization algorithms to decide
ticket prices, airplane and crew assignments
and fuel needs.
Duo Wang Numerical Computation

13. What is Numerical Computation?
Numerical Computation in the real world.
A Brief History.
Questions and Answers.
Numerical Computation in the real world(cont.)
Car companies can improve the crash safety by
using computer simulations of car crashes.
These simulations are essentially solving partial
differential equations numerically.
Hedge funds (private investment funds) use
tools from all fields of numerical analysis to
calculate the value of stocks and derivatives.
Duo Wang Numerical Computation

14. What is Numerical Computation?
Numerical Computation in the real world.
A Brief History.
Questions and Answers.
Brief History
Before 1950s,
Computation methods already known:
Linear interpolation
Newton’s method
Gaussian elimination
Gauss(1777-1855)
Duo Wang Numerical Computation

15. What is Numerical Computation?
Numerical Computation in the real world.
A Brief History.
Questions and Answers.
Brief History
Before 1950s,
Computation methods already known:
Linear interpolation
Newton’s method
Gaussian elimination
Computation tools:
Hand computation
Abacus
Gauss(1777-1855) Formula handbook and Data tables
Duo Wang Numerical Computation

16. What is Numerical Computation?
Numerical Computation in the real world.
A Brief History.
Questions and Answers.
Brief History
Before 1950s,
Computation methods already known:
Linear interpolation
Newton’s method
Gaussian elimination
Computation tools:
Hand computation
Abacus
Gauss(1777-1855) Formula handbook and Data tables
The need for large scale computation predates
the invention of modern computers
Duo Wang Numerical Computation

17. What is Numerical Computation?
Numerical Computation in the real world.
A Brief History.
Questions and Answers.
After Computers’ Invention
The revolution was sparked by invention of
computer.
By Moore’s Law, the computational
ability is increasing exponentially,
doubling approximately every two years.
Mathematics and algorithms have also
developed after 1950s.
These two effects generated a combined
increase in speed of almost incomprehensible
scale.
Duo Wang Numerical Computation

18. What is Numerical Computation?
Numerical Computation in the real world.
A Brief History.
Questions and Answers.
Summary
Half a century on, numerical computation has grown into
one of the largest branches of mathematics.
Duo Wang Numerical Computation

19. What is Numerical Computation?
Numerical Computation in the real world.
A Brief History.
Questions and Answers.
Summary
Half a century on, numerical computation has grown into
one of the largest branches of mathematics.
Today, we have reached a point where most of the classical
mathematical problems can be solved numerically to high
accuracy.
Duo Wang Numerical Computation

20. What is Numerical Computation?
Numerical Computation in the real world.
A Brief History.
Questions and Answers.
Questions and Answers
Thanks for listening!
Duo Wang Numerical Computation

21. What is Numerical Computation?
Numerical Computation in the real world.
A Brief History.
Questions and Answers.
Questions and Answers
Thanks for listening! Any Questions?
Duo Wang Numerical Computation