The document discusses the history and importance of numerical algorithms. It began with early papers in the 1940s and projects like ENIAC for calculating trajectories. Over time, important people and projects advanced areas like linear algebra, differential equations, and approximation methods. Today, numerical algorithms are widely used in applications such as digital imaging, engineering, finance, and more. Understanding numerical methods is crucial for teaching mathematics with technology, as it ensures students have proper conceptual knowledge to analyze problems.

1. History of Numerical Algorithms Reva Narasimhan Kean University, Union, NJ

2. Overview History Review of important people and projects Examples of importance of understanding numerical algorithms Role in teaching with technology

3. 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.

4. 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.

5. 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

6. Numerical Analysis Specialities numerical linear algebra: used in digital imaging and compression numerical methods for ordinary and partial differential equations: Aircraft and automobile design Computational finance Computational biology Weather forecasting methods of approximation of functions: used in approximating curves in CAD/CAM design

7. 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’s EISPACK 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

8. 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)

9. 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

10. Numerical algorithms commonly in use Simplex method for linear programming Splines in CAD/CAM design Matrix computations for digital imaging Digital animation ( Toy Story , Shrek …) Numerical fluid dynamics for simulation of blood flow Options pricing models in finance

11. 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 point Using the language of linear algebra, this array can be manipulated in many ways Algorithm in numerical linear algebra play an important role in the manipulation of digital images In addition to numerical linear algebra, statistics and signal processing tools are also used

12. 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

13. 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

14. Image processing in software This Adobe web page shows how to use matrices in refining an image

15. 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

16. Example: Curve Fitting Examine data in Example of population growth

17. Teaching with Technology Nonlinear equation solvers: use of Newton’s method Numerical Integration Numerical solution of ODE’s

18. Newton’s Method

19. 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

20. Root finding in the TI-84 Left and right bound – find the interval where function changes sign; use bisection method Use of guess – employing variation of Newton’s Method

21. Goal Seek in Excel This is really a nonlinear equation solver using an iterative method In the 1970’s and 1980’s, numerical computations were done on mainframes Now, a lot of quantitative analysis takes place on the desktop PC, using Excel

22. Role of mathematical analysis What is the implication for mathematics education? Examine a simple polynomial equation:

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

24. 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 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)

25. Summary Wide use of numerical algorithms brings about new ideas in teaching mathematics Quantitative literacy involves knowing how to use technology such as a spreadsheet to analyze a problem Conceptual understanding is a necessity for proper use of technology

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