MELJUN Numerical Algorithm
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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
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MELJUN CORTES,MBA,MPA,BSCS,ACSMELJUN CORTES,MBA,MPA,BSCS,ACS
Over view
History
Rev
iew of important people andprojects
Examples of importance of
understanding numerical algorithms
Role in teaching with technology
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MELJUN CORTES,MBA,MPA,BSCS,ACSMELJUN CORTES,MBA,MPA,BSCS,ACS
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.
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MELJUN CORTES,MBA,MPA,BSCS,ACSMELJUN CORTES,MBA,MPA,BSCS,ACS
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.
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MELJUN CORTES,MBA,MPA,BSCS,ACSMELJUN CORTES,MBA,MPA,BSCS,ACS
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
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MELJUN CORTES,MBA,MPA,BSCS,ACSMELJUN CORTES,MBA,MPA,BSCS,ACS
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
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MELJUN CORTES,MBA,MPA,BSCS,ACSMELJUN CORTES,MBA,MPA,BSCS,ACS
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
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MELJUN CORTES,MBA,MPA,BSCS,ACSMELJUN CORTES,MBA,MPA,BSCS,ACS
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)
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MELJUN CORTES,MBA,MPA,BSCS,ACSMELJUN CORTES,MBA,MPA,BSCS,ACS
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
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MELJUN CORTES,MBA,MPA,BSCS,ACSMELJUN CORTES,MBA,MPA,BSCS,ACS
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
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MELJUN CORTES,MBA,MPA,BSCS,ACSMELJUN CORTES,MBA,MPA,BSCS,ACS
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
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MELJUN CORTES,MBA,MPA,BSCS,ACSMELJUN CORTES,MBA,MPA,BSCS,ACS
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
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ReadImage from Matrix
The following command displays the image stored
in the matrix A: » imagesc(A)
Further refinements require image processing
toolbox in MATLAB
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MELJUN CORTES,MBA,MPA,BSCS,ACSMELJUN CORTES,MBA,MPA,BSCS,ACS
Image processing in software
This Adobe web page shows how to
use matrices in refining an image
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MELJUN CORTES,MBA,MPA,BSCS,ACSMELJUN CORTES,MBA,MPA,BSCS,ACS
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
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MELJUN CORTES,MBA,MPA,BSCS,ACSMELJUN CORTES,MBA,MPA,BSCS,ACS
Example: Cur ve Fitting
Examine data in Example of
population growth
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MELJUN CORTES,MBA,MPA,BSCS,ACSMELJUN CORTES,MBA,MPA,BSCS,ACS
Teaching with Technology
Nonlinear equation solvers: use of
Newton¶s method
Numerical Integration
Numerical solution of ODE¶s
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Newton¶s Method
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MELJUN CORTES,MBA,MPA,BSCS,ACSMELJUN CORTES,MBA,MPA,BSCS,ACS
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
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MELJUN CORTES,MBA,MPA,BSCS,ACSMELJUN CORTES,MBA,MPA,BSCS,ACS
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
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MELJUN CORTES,MBA,MPA,BSCS,ACSMELJUN CORTES,MBA,MPA,BSCS,ACS
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
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MELJUN CORTES,MBA,MPA,BSCS,ACSMELJUN CORTES,MBA,MPA,BSCS,ACS
Role of mathematical analysis What is the implication for mathematics
education?
Examine a simple polynomial equation:
013
! x x
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MELJUN CORTES,MBA,MPA,BSCS,ACSMELJUN CORTES,MBA,MPA,BSCS,ACS
Analysis « Can the root be
found by
elementarymethods?
If found
numerically, how
do we know thereis only one real
root?
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MELJUN CORTES,MBA,MPA,BSCS,ACSMELJUN CORTES,MBA,MPA,BSCS,ACS
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)
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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
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MELJUN CORTES,MBA,MPA,BSCS,ACSMELJUN CORTES,MBA,MPA,BSCS,ACS
Contact Information
J_cortes2005J_cortes2005@[email protected]
http://www.meljun.ning.comhttp://www.meljun.ning.com