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History of Numerical Algorithms

Reva NarasimhanKean University, Union, NJ

Overview

HistoryReview of important people and projectsExamples of importance of understanding numerical algorithmsRole 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 in applications

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 include discussion of what today is called scientific computing.

Beginnings…

ENIAC 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 compressionnumerical methods for ordinary and partial differential equations:

Aircraft and automobile designComputational financeComputational biologyWeather forecasting

methods of approximation of functions: used in approximating curves in CAD/CAM design

Important People and Projects

James Wilkinson : round off error analysis and solving eigenvalue problems. (Late 50’s, Early 60’s)Cooley-Tukey: FFT algorithm (1960’s)Peter Lax (Courant Institute) – numerical PDE’sEISPACK and LINPACK projects run by the Argonne National Laboratory to produce high quality, tested and portable mathematical software during the early- to mid-1970s.

These were linear algebra packages written in 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 – widely used in engineering and business

Numerical algorithms commonly in use

Simplex method for linear programmingSplines in CAD/CAM designMatrix computations for digital imagingDigital animation (Toy Story, Shrek…) Numerical fluid dynamics for simulation of blood flowOptions pricing models in finance

Example: Digital Imaging

A picture can be represented as a m X n array, with each element of the array representing the color value at that pointUsing the language of linear algebra, this array can be manipulated in many waysAlgorithm in numerical linear algebra play an important role in the manipulation of digital imagesIn addition to numerical linear algebra, statistics and signal processing tools are also used

Images in MATLAB

A=imread('spring_bulbs.jpg');

Name Size Bytes A 480x320x3 460800

Three dimensional array to store RGB value

Grand total is 460800 elements using 460800 bytes

Read Image 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 errorTo understand rounding error, examine a simple problem

Example: Curve Fitting

Examine data in Example of population growth

Teaching with Technology

Nonlinear equation solvers: use of Newton’s methodNumerical IntegrationNumerical 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 is near a local maximum or minimum?How many roots are there anyway?Thus, proper use of technology requires a higher level of conceptual understanding

Root finding in the TI-84

Left and right bound –find the interval where function changes sign; use bisection methodUse of guess –employing variation of Newton’s Method

Goal Seek in Excel

This is really a nonlinear equation solver using an iterative methodIn the 1970’s and 1980’s, numerical computations were done on mainframesNow, a lot of quantitative analysis takes place on the desktop PC, using Excel

Role of mathematical analysis

What is the implication for mathematics education?Examine a simple polynomial equation:

013 =++ xx

Analysis …

Can the root be found by elementary methods?If found numerically, how do we know there is only one real root?

−4 −3 −2 −1 1 2 3 4 5

−4

−3

−2

−1

1

2

3

4

Implication for Technology in Education

Important for students to be familiar with effects of roundoff errorSpreadsheets are used in analysis of large amounts of data and is a tool for all commercial and governmental decision-makers.A robust quantitative curriculum: numerical methods, introduction to computer simulation, and statistics. “Quantitative arguments are underpinning successful business and political decisions. Students of commerce and government must become equally skilled consumers of quantitative information.” (Deborah Hughes-Hallett, 2000)

Summary

Wide use of numerical algorithms brings about new ideas in teaching mathematicsQuantitative literacy involves knowing how to use technology such as a spreadsheet to analyze a problemConceptual understanding is a necessity for proper use of technology

Contact Information

rnarasim@kean.eduMaterial downloadable at

www.collegemath.info