Numerical Algorithms


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Summary of history of numerical analysis with implication for teaching math with technology.

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Numerical Algorithms

  1. 1. History of Numerical Algorithms Reva Narasimhan Kean University, Union, NJ
  2. 2. Overview <ul><li>History </li></ul><ul><li>Review of important people and projects </li></ul><ul><li>Examples of importance of understanding numerical algorithms </li></ul><ul><li>Role in teaching with technology </li></ul>
  3. 3. Introduction <ul><li>What is numerical analysis , also referred to as scientific computing ? </li></ul><ul><li>An integration of mathematical analysis, software,and large, complex problems in applications </li></ul><ul><li>Why is it important? </li></ul><ul><li>Most equations cannot be solved by analytical methods. </li></ul>
  4. 4. Beginnings… <ul><li>Began with the 1947 paper by John von Neumann and Herman Goldstine, &quot;Numerical Inverting of Matrices of High Order&quot; (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.   </li></ul>
  5. 5. Beginnings… <ul><li>ENIAC was the first electronic digital computer. Funded by the U.S. Army to help with calculation of trajectories of ballistics (early 1940’s) </li></ul><ul><li>At that time, computer time was extremely expensive </li></ul>
  6. 6. Numerical Analysis Specialities <ul><li>numerical linear algebra: used in digital imaging and compression </li></ul><ul><li>numerical methods for ordinary and partial differential equations: </li></ul><ul><ul><li>Aircraft and automobile design </li></ul></ul><ul><ul><li>Computational finance </li></ul></ul><ul><ul><li>Computational biology </li></ul></ul><ul><ul><li>Weather forecasting </li></ul></ul><ul><li>methods of approximation of functions: used in approximating curves in CAD/CAM design </li></ul>
  7. 7. Important People and Projects <ul><li>James Wilkinson : round off error analysis and solving eigenvalue problems. (Late 50’s, Early 60’s) </li></ul><ul><li>Cooley-Tukey: FFT algorithm (1960’s) </li></ul><ul><li>Peter Lax (Courant Institute) – numerical PDE’s </li></ul><ul><li>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. </li></ul><ul><ul><li>These were linear algebra packages written in FORTRAN </li></ul></ul>
  8. 8. Important People and Projects <ul><li>QUADPACK project : numerical integration package (mid 1970’s) </li></ul><ul><li>Bill Gear, Lawrence Shampine: Numerical ODE’s (1980’s) </li></ul><ul><li>Cleve Moler : founder of MATLAB; (late 1980’s) </li></ul><ul><li>Stephen Wolfram : founder of Mathematica (early 1990’s) </li></ul>
  9. 9. Math Software Packages <ul><li>Many of the early software were incorporated in MATLAB, Maple and Mathematica </li></ul><ul><li>Sophisticated mathematical analysis can also be done by Excel – widely used in engineering and business </li></ul>
  10. 10. Numerical algorithms commonly in use <ul><li>Simplex method for linear programming </li></ul><ul><li>Splines in CAD/CAM design </li></ul><ul><li>Matrix computations for digital imaging </li></ul><ul><li>Digital animation ( Toy Story , Shrek …) </li></ul><ul><li>Numerical fluid dynamics for simulation of blood flow </li></ul><ul><li>Options pricing models in finance </li></ul>
  11. 11. Example: Digital Imaging <ul><li>A picture can be represented as a m X n array, with each element of the array representing the color value at that point </li></ul><ul><li>Using the language of linear algebra, this array can be manipulated in many ways </li></ul><ul><li>Algorithm in numerical linear algebra play an important role in the manipulation of digital images </li></ul><ul><li>In addition to numerical linear algebra, statistics and signal processing tools are also used </li></ul>
  12. 12. Images in MATLAB <ul><li>A=imread('spring_bulbs.jpg'); </li></ul><ul><li>Name Size Bytes </li></ul><ul><li>A 480x320x3 460800 </li></ul><ul><li>Three dimensional array to store RGB value </li></ul><ul><li>Grand total is 460800 elements using 460800 bytes </li></ul>
  13. 13. Read Image from Matrix <ul><li>The following command displays the image stored in the matrix A: » imagesc(A) </li></ul><ul><li>Further refinements require image processing toolbox in MATLAB </li></ul>
  14. 14. Image processing in software <ul><li>This Adobe web page shows how to use matrices in refining an image </li></ul>
  15. 15. Roundoff error and significant digits <ul><li>Since machines can support only a finite number of digits, it is important to know the effect of rounding error </li></ul><ul><li>To understand rounding error, examine a simple problem </li></ul>
  16. 16. Example: Curve Fitting <ul><li>Examine data in Example of population growth </li></ul>
  17. 17. Teaching with Technology <ul><li>Nonlinear equation solvers: use of Newton’s method </li></ul><ul><li>Numerical Integration </li></ul><ul><li>Numerical solution of ODE’s </li></ul>
  18. 18. Newton’s Method
  19. 19. Role of Mathematical analysis <ul><li>How does the behavior of the function affect the root that you find? </li></ul><ul><li>What happens if your initial guess is near a local maximum or minimum? </li></ul><ul><li>How many roots are there anyway? </li></ul><ul><li>Thus, proper use of technology requires a higher level of conceptual understanding </li></ul>
  20. 20. Root finding in the TI-84 <ul><li>Left and right bound – find the interval where function changes sign; use bisection method </li></ul><ul><li>Use of guess – employing variation of Newton’s Method </li></ul>
  21. 21. Goal Seek in Excel <ul><li>This is really a nonlinear equation solver using an iterative method </li></ul><ul><li>In the 1970’s and 1980’s, numerical computations were done on mainframes </li></ul><ul><li>Now, a lot of quantitative analysis takes place on the desktop PC, using Excel </li></ul>
  22. 22. Role of mathematical analysis <ul><li>What is the implication for mathematics education? </li></ul><ul><li>Examine a simple polynomial equation: </li></ul>
  23. 23. Analysis … <ul><li>Can the root be found by elementary methods? </li></ul><ul><li>If found numerically, how do we know there is only one real root? </li></ul>
  24. 24. Implication for Technology in Education <ul><li>Important for students to be familiar with effects of roundoff error </li></ul><ul><li>Spreadsheets are used in analysis of large amounts of data and is a tool for all commercial and governmental decision-makers. </li></ul><ul><li>A robust quantitative curriculum: numerical methods, introduction to computer simulation, and statistics. </li></ul><ul><li>“ 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) </li></ul>
  25. 25. Summary <ul><li>Wide use of numerical algorithms brings about new ideas in teaching mathematics </li></ul><ul><li>Quantitative literacy involves knowing how to use technology such as a spreadsheet to analyze a problem </li></ul><ul><li>Conceptual understanding is a necessity for proper use of technology </li></ul>
  26. 26. Contact Information <ul><li>[email_address] </li></ul><ul><li> </li></ul>
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