History of NumericalAlgorithms      Reva Narasimhan      Kean University, Union, NJ           (c) 2009 Revathi Narasimhan ...
Overview   History   Review of important people and    projects   Examples of importance of    understanding numerical ...
Introduction   What is numerical analysis, also referred to    as scientific computing?An integration of mathematical ana...
Beginnings…   Began with the 1947 paper by John    von Neumann and Herman Goldstine,    "Numerical Inverting of Matrices ...
Beginnings…   ENIAC was the first electronic digital    computer. Funded by the U.S. Army to    help with calculation of ...
Numerical AnalysisSpecialities   numerical linear algebra: used in digital    imaging and compression   numerical method...
Important People and Projects   James Wilkinson : round off error analysis and    solving eigenvalue problems. (Late 50’s...
Important People and Projects   QUADPACK project : numerical    integration package (mid 1970’s)   Bill Gear, Lawrence S...
Math Software Packages   Many of the early software were    incorporated in MATLAB, Maple and    Mathematica   Sophistic...
Numerical algorithmscommonly in use   Simplex method for linear programming   Splines in CAD/CAM design   Matrix comput...
Example: Digital Imaging   A picture can be represented as a m X n    array, with each element of the array    representi...
Images in MATLAB   A=imread(spring_bulbs.jpg);   Name    Size       Bytes    A    480x320x3     460800   Three dimensio...
Read Image from Matrix   The following command displays the image stored    in the matrix A: » imagesc(A)   Further refi...
Image processing in software   This Adobe web page shows how to    use matrices in refining an image             (c) 2009...
Roundoff error and significantdigits   Since machines can support only a    finite number of digits, it is important to  ...
Example: Curve Fitting   Examine data in Example of    population growth              (c) 2009 Revathi Narasimhan All rig...
Teaching with Technology   Nonlinear equation solvers: use of    Newton’s method   Numerical Integration   Numerical so...
Newton’s Method         (c) 2009 Revathi Narasimhan All rights reserved
Role of Mathematical analysis   How does the behavior of the function    affect the root that you find?   What happens i...
Root finding in the TI-84   Left and right bound –    find the interval where    function changes sign;    use bisection ...
Goal Seek in Excel   This is really a nonlinear equation    solver using an iterative method   In the 1970’s and 1980’s,...
Role of mathematical analysis   What is the implication for mathematics    education?   Examine a simple polynomial equa...
Analysis …   Can the root be    found by                                                                       4         ...
Implication for Technology inEducation   Important for students to be familiar with effects of    roundoff error   Sprea...
Summary   Wide use of numerical algorithms    brings about new ideas in teaching    mathematics   Quantitative literacy ...
Contact Information   rnarasim@kean.edu   http://www.mymathspace.net             (c) 2009 Revathi Narasimhan All rights ...
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MELJUN CORTES Numerical Algorithm

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MELJUN CORTES Numerical Algorithm

  1. 1. History of NumericalAlgorithms Reva Narasimhan Kean University, Union, NJ (c) 2009 Revathi Narasimhan All rights reserved
  2. 2. Overview History Review of important people and projects Examples of importance of understanding numerical algorithms Role in teaching with technology (c) 2009 Revathi Narasimhan All rights reserved
  3. 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. (c) 2009 Revathi Narasimhan All rights reserved
  4. 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. (c) 2009 Revathi Narasimhan All rights reserved
  5. 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 (c) 2009 Revathi Narasimhan All rights reserved
  6. 6. Numerical AnalysisSpecialities 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 (c) 2009 Revathi Narasimhan All rights reserved
  7. 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 (c) 2009 Revathi Narasimhan All rights reserved
  8. 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) (c) 2009 Revathi Narasimhan All rights reserved
  9. 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 (c) 2009 Revathi Narasimhan All rights reserved
  10. 10. Numerical algorithmscommonly 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 (c) 2009 Revathi Narasimhan All rights reserved
  11. 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 (c) 2009 Revathi Narasimhan All rights reserved
  12. 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 (c) 2009 Revathi Narasimhan All rights reserved
  13. 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 Narasimhan All rights reserved (c) 2009 Revathi
  14. 14. Image processing in software This Adobe web page shows how to use matrices in refining an image (c) 2009 Revathi Narasimhan All rights reserved
  15. 15. Roundoff error and significantdigits 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 (c) 2009 Revathi Narasimhan All rights reserved
  16. 16. Example: Curve Fitting Examine data in Example of population growth (c) 2009 Revathi Narasimhan All rights reserved
  17. 17. Teaching with Technology Nonlinear equation solvers: use of Newton’s method Numerical Integration Numerical solution of ODE’s (c) 2009 Revathi Narasimhan All rights reserved
  18. 18. Newton’s Method (c) 2009 Revathi Narasimhan All rights reserved
  19. 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 (c) 2009 Revathi Narasimhan All rights reserved
  20. 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 (c) 2009 Revathi Narasimhan All rights reserved
  21. 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 (c) 2009 Revathi Narasimhan All rights reserved
  22. 22. Role of mathematical analysis What is the implication for mathematics education? Examine a simple polynomial equation: x + x +1= 0 3 (c) 2009 Revathi Narasimhan All rights reserved
  23. 23. Analysis … Can the root be found by 4 3 elementary 2 methods? 1 If found −4 −3 −2 −1 1 2 3 4 5 numerically, how −1 do we know there −2 −3 is only one real −4 root? (c) 2009 Revathi Narasimhan All rights reserved
  24. 24. Implication for Technology inEducation 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) (c) 2009 Revathi Narasimhan All rights reserved
  25. 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 (c) 2009 Revathi Narasimhan All rights reserved
  26. 26. Contact Information rnarasim@kean.edu http://www.mymathspace.net (c) 2009 Revathi Narasimhan All rights reserved

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