Introduction to Numerical Analysis

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Why do we study numerical analysis?
What are errors?

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Introduction to Numerical Analysis

  1. 1. Numerical Analysis - Introduction Mohammad Tawfik #WikiCourses http://WikiCourses.WikiSpaces.com Numerical Analysis An Introduction
  2. 2. Numerical Analysis - Introduction Mohammad Tawfik #WikiCourses http://WikiCourses.WikiSpaces.com References • Steven Chapra and Raymond Canale, "Numerical Methods for Engineers," 5th editions, ISBN 0-07-123140-4
  3. 3. Numerical Analysis - Introduction Mohammad Tawfik #WikiCourses http://WikiCourses.WikiSpaces.com Numerical Methods
  4. 4. Numerical Analysis - Introduction Mohammad Tawfik #WikiCourses http://WikiCourses.WikiSpaces.com Why Numerical? • Ignorance • Readily Available Packages • Need to Develop New Techniques • Good use of your computer! • Re-Understand Mathematics
  5. 5. Numerical Analysis - Introduction Mohammad Tawfik #WikiCourses http://WikiCourses.WikiSpaces.com Mathematical Modeling & Engineering Solutions
  6. 6. Numerical Analysis - Introduction Mohammad Tawfik #WikiCourses http://WikiCourses.WikiSpaces.com Mathematical Models
  7. 7. Numerical Analysis - Introduction Mohammad Tawfik #WikiCourses http://WikiCourses.WikiSpaces.com Example Problem dt dv mmaF  cvmgFFF UD  cvmgvm dt dv m   m cvmg v      mct e c mg tv / 1  
  8. 8. Numerical Analysis - Introduction Mohammad Tawfik #WikiCourses http://WikiCourses.WikiSpaces.com Solution
  9. 9. Numerical Analysis - Introduction Mohammad Tawfik #WikiCourses http://WikiCourses.WikiSpaces.com Approximate Solution 12 12 tt vv t v dt dv       m cvmg tt vv     12 12 m cvmg tt vv 1 12 12    
  10. 10. Numerical Analysis - Introduction Mohammad Tawfik #WikiCourses http://WikiCourses.WikiSpaces.com Approximate Solution   m cvmg ttvv 1 1212  
  11. 11. Numerical Analysis - Introduction Mohammad Tawfik #WikiCourses http://WikiCourses.WikiSpaces.com Approximation and Round-Off Errors
  12. 12. Numerical Analysis - Introduction Mohammad Tawfik #WikiCourses http://WikiCourses.WikiSpaces.com Errors! • Round-off errors are due to the fact that the computers present numbers in as a finite number of bits and bytes! • Truncation Errors are errors that emerge from the approximation of the mathematical model • Model errors are due to the fact that the mathematical model usually is an approximation of the physical reality!
  13. 13. Numerical Analysis - Introduction Mohammad Tawfik #WikiCourses http://WikiCourses.WikiSpaces.com Significant Figures
  14. 14. Numerical Analysis - Introduction Mohammad Tawfik #WikiCourses http://WikiCourses.WikiSpaces.com Accuracy and Precision • Accuracy how close your solution is • Precision is how close your repetition of the solution are!
  15. 15. Numerical Analysis - Introduction Mohammad Tawfik #WikiCourses http://WikiCourses.WikiSpaces.com Error Definition • True Error • True Relative Error • Approximate Error ionApproximatvalueTrueEt  valueTrue Et t  ionApproximatCurrent ApproxPastionApproximatCurrent a . 
  16. 16. Numerical Analysis - Introduction Mohammad Tawfik #WikiCourses http://WikiCourses.WikiSpaces.com Round-off Errors 210 10*210*510*3253  21 10*510*225.0 4 1   ...10*310*310*310*3 ...33333.0 3 1 4321   
  17. 17. Numerical Analysis - Introduction Mohammad Tawfik #WikiCourses http://WikiCourses.WikiSpaces.com Computer Presentation of Numbers
  18. 18. Numerical Analysis - Introduction Mohammad Tawfik #WikiCourses http://WikiCourses.WikiSpaces.com When numbers are too small!
  19. 19. Numerical Analysis - Introduction Mohammad Tawfik #WikiCourses http://WikiCourses.WikiSpaces.com Truncation Errors
  20. 20. Numerical Analysis - Introduction Mohammad Tawfik #WikiCourses http://WikiCourses.WikiSpaces.com Taylor’s Expansion           ... !2 ''' 2 0 0000    xx xfxxxfxfxf          3 2 000 !2 ''' hO h xfhxfxfxf 
  21. 21. Numerical Analysis - Introduction Mohammad Tawfik #WikiCourses http://WikiCourses.WikiSpaces.com Truncation Error!
  22. 22. Numerical Analysis - Introduction Mohammad Tawfik #WikiCourses http://WikiCourses.WikiSpaces.com Conclusion • We need the numerical methods to solve problems that do not have a solution or have a complex mathematical solution • Errors have different sources: Truncation, round off, and model errors • Error is evaluated using approximate solutions

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