MELJUN Numerical Algorithm

8/9/2019 MELJUN Numerical Algorithm

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MELJUN CORTES,MBA,MPA,BSCS,ACSMELJUN CORTES,MBA,MPA,BSCS,ACS

History of Numerical

Algorithms

MELJUN CORTES,MBA,MPAMELJUN CORTES,MBA,MPA

EE--mail: J_cortes2005mail: J_cortes2005@[email protected]://www.meljun.ning.comhttp://www.meljun.ning.com




Over view

History

Rev

iew of important people andprojects

Examples of importance of

understanding numerical algorithms

Role in teaching with technology




Introduction What is numerical analysis, also referred to

as scientific computing ?

An integration of mathematical analysis, software,and large, complex problems inapplications

Why is it important?Most equations cannot be solved by analytical

methods.




Beginnings«

Began with the 1947 paper by John

von Neumann and Herman Goldstine,

"Numerical Inverting of Matrices of

High Order" (Bulletin of the AMS, Nov.

1947). It is one of the first papers to

study rounding error and includediscussion of what today is called

scientific computing.




Beginnings«

ENI AC was the first electronic digital

computer. Funded by the U.S. Army to

help with calculation of trajectories of

ballistics (early 1940¶s)

At that time, computer time was

extremely expensive




Numerical Analysis

Specialities numerical linear algebra: used in digital

imaging and compression

numerical methods for ordinary and partialdifferential equations: Aircraft and automobile design

Computational finance

Computational biology

Weather forecasting methods of approximation of functions: used

in approximating cur ves in CAD/CAM design




Important People and Projects James Wilkinson : round off error analysis and

solving eigenvalue problems. (Late 50¶s, Early60¶s)

Cooley-Tukey: FFT algorithm (1960¶s) Peter Lax (Courant Institute) ± numerical PDE¶s

EISPACK and LINPACK projects run by the Argonne National Laboratory to produce high

quality, tested and portable mathematicalsoftware during the early- to mid-1970s.

These were linear algebra packages writtenin FORTRAN




Important People and Projects QUADPACK project : numerical

integration package (mid 1970¶s)

Bill Gear , Lawrence Shampine:Numerical ODE¶s (1980¶s)

Cleve Moler : founder of MATLAB;

(late 1980¶s) Stephen Wolfram : founder of

Mathematica (early 1990¶s)




Math Software Packages

Many of the early software were

incorporated in MATLAB, Maple and

Mathematica

Sophisticated mathematical analysis

can also be done by Excel ± widelyused in engineering and business




Numerical algorithms

commonly in use

Simplex method for linear programming

Splines in CAD/CAM design Matrix computations for digital imaging

Digital animation (T oy Story , Shrek «)

Numerical fluid dynamics for simulation of blood flow

Options pricing models in finance




Example: DigitalImaging

A picture can be represented as a m X narray, with each element of the arrayrepresenting the color value at that point

Using the language of linear algebra, thisarray can be manipulated in many ways

Algorithm in numerical linear algebra playan important role in the manipulation of

digital images In addition to numerical linear algebra,

statistics and signal processing tools arealso used




Images in MATLA

B

A=imread('spring_bulbs.jpg');

Name Size Bytes A 480x320x3 460800

Three dimensional array tostore RGB value

Grand total is 460800elements using 460800 bytes




ReadImage from Matrix

The following command displays the image stored

in the matrix A: » imagesc(A)

Further refinements require image processing

toolbox in MATLAB




Image processing in software

This Adobe web page shows how to

use matrices in refining an image




Roundoff error and significant

digits

Since machines can support only a

finite number of digits, it is important to

know the effect of rounding error

To understand rounding error ,

examine a simple problem




Example: Cur ve Fitting

Examine data in Example of

population growth




Teaching with Technology

Nonlinear equation solvers: use of

Newton¶s method

Numerical Integration

Numerical solution of ODE¶s




Newton¶s Method




Role of Mathematical analysis How does the behavior of the function

affect the root that you find?

What happens if your initial guess isnear a local maximum or minimum?

How many roots are there anyway?

Thus, proper use of technologyrequires a higher level of conceptualunderstanding




Root finding in the TI-84

Left and right bound ±

find the inter val wherefunction changes sign;

use bisection method

Use of guess ±

employing variation of

Newton¶s Method




Goal Seek in Excel This is really a nonlinear equation

solver using an iterative method

In the 1970¶s and 1980¶s, numericalcomputations were done onmainframes

Now, a lot of quantitative analysistakes place on the desktop PC, using

Excel




Role of mathematical analysis What is the implication for mathematics

education?

Examine a simple polynomial equation:

013

! x x




Analysis « Can the root be

found by

elementarymethods?

If found

numerically, how

do we know thereis only one real

root?




Implication for Technology in

Education

Important for students to be familiar with effects of roundoff error

Spreadsheets are used in analysis of large amounts

of data and is a tool for all commercial andgovernmental decision-makers.

A robust quantitative curriculum: numericalmethods, introduction to computer simulation, andstatistics.

³Quantitative arguments are underpinningsuccessful business and political decisions.

Students of commerce and government mustbecome equally skilled consumers of quantitativeinformation.´ (Deborah Hughes-Hallett, 2000)




Summary

Wide use of numerical algorithmsbrings about new ideas in teaching

mathematics Quantitative literacy involves knowing

how to use technology such as aspreadsheet to analyze a problem

Conceptual understanding is anecessity for proper use of technology




Contact Information

J_cortes2005J_cortes2005@[email protected]

http://www.meljun.ning.comhttp://www.meljun.ning.com

MELJUN Numerical Algorithm

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