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Numerical Analysis 2. Condition and Stability

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Slides based on "Inleiding tot de numerieke wiskunde" by Prof. Dr. Adhemar Bultheel and course notes by Prof. Dr. Marc Van Barel.

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Numerical Analysis 2. Condition and Stability

  1. 1. Numerical Analysis Fault analysis: condition and stability
  2. 2. Overview • • • • • • • • Description of a numerical problem Condition Condition : example Numerical stability Numerical stability : forward stability Numerical stability : weak stability Numerical stability : backward stability Numerical stability : example
  3. 3. Overview • • • • • • • • Description of a numerical problem Condition Condition : example Numerical stability Numerical stability : forward stability Numerical stability : weak stability Numerical stability : backward stability Numerical stability : example
  4. 4. Description of a numerical problem • A relation F between data g and results r r = F(g) ▫ F is an exact mathematical description of the relation ▫ Different methods may apply to the same description
  5. 5. Overview • • • • • • • • Description of a numerical problem Condition Condition : example Numerical stability Numerical stability : forward stability Numerical stability : weak stability Numerical stability : backward stability Numerical stability : example
  6. 6. Condition • Definition: “the condition of a numerical problem indicates how much the result r is being influenced if the data g are altered” • Exact relationship • Characteristic to a certain problem • Independent of the method
  7. 7. Condition • Definitions:
  8. 8. Condition
  9. 9. Condition • Condition number: ▫ Ratio of the error on the result and the error on the data ▫ Absolute condition kA and relative condition kR
  10. 10. Condition • If F(g) is a differentiable function:
  11. 11. Overview • • • • • • • • Description of a numerical problem Condition Condition : example Numerical stability Numerical stability : forward stability Numerical stability : weak stability Numerical stability : backward stability Numerical stability : example
  12. 12. Condition : example • What is the condition of the evaluation of the function f : • Using the formula from the previous section:
  13. 13. Condition : example • What can we conclude? ▫ De denominator approaches zero for values {x1 = –1; x2 = 3/2} ▫ For these values the function is ill-conditioned, as the relative error becomes very large.
  14. 14. Overview • • • • • • • • Description of a numerical problem Condition Condition : example Numerical stability Numerical stability : forward stability Numerical stability : weak stability Numerical stability : backward stability Numerical stability : example
  15. 15. Numerical stability • Implementing an exact relation F is usually not feasable: ▫ Discretization ▫ Rounding error F* • Definition: • “numerical stability measures the deviation of F* (the approximation) from F (the exact result).”
  16. 16. Numerical stability : forward stability • Given by:
  17. 17. Numerical stability : forward stability
  18. 18. Numerical stability : weak stability
  19. 19. Numerical stability : weak stability
  20. 20. Numerical stability : backward stability • The idea is the following: ▫ Consider the result r* = F(g) to be the exact result ▫ Find data g*’ corresponding to r* ▫ Measure the stability with the following:
  21. 21. Numerical stability : backward stability
  22. 22. Overview • • • • • • • • Description of a numerical problem Condition Condition : example Numerical stability Numerical stability : forward stability Numerical stability : weak stability Numerical stability : backward stability Numerical stability : example
  23. 23. Numerical stability : example • Investigate the stability of algorithms A and B for the function f:
  24. 24. Numerical stability : example
  25. 25. Numerical stability : example • Resulting formula: for x1 = 0, the relative error is large, and the condition is small: Unstable but for -3/2 the problem is also ill-conditioned Stability is weak
  26. 26. Numerical stability : example • Can you evaluate algorithm B?
  27. 27. Sources • “Inleiding tot de numerieke wiskunde”, A. Bultheel, 2007, Acco • http://en.wikipedia.org/wiki/Numerical_analysis • http://en.wikipedia.org/wiki/Condition_number By knowledgedriver, 2012.

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