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Chapter 1 Scientific Computing Approximation in Scientific Computing (1.2) January 12, 2010

Chapter 1 Scientific Computing Approximation in Scientific Computing (1.2) January 12, 2010. Absolute and Relative Errors. Example: Approximations. Model Earth as an ellipsoid?. Irrational number has infinite digits in decimal expansion. Floating-point number system.

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Chapter 1 Scientific Computing Approximation in Scientific Computing (1.2) January 12, 2010

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  1. Chapter 1 • Scientific Computing • Approximation in Scientific Computing (1.2) • January 12, 2010

  2. Absolute and Relative Errors

  3. Example: Approximations Model Earth as an ellipsoid? Irrational number has infinite digits in decimal expansion Floating-point number system

  4. General Strategy in Scientific Computing

  5. Sources of Approximation

  6. Computational and Data Errors

  7. Truncation and Rounding Errors

  8. Example: Finite Difference Approximation By Taylor Expansion Truncation Error

  9. Example: Finite Difference Approximation Rounding Error Minimizing mh/2 + 2epsilon /h

  10. Forward and Backward Errors

  11. Example (relative) backward error is about twice the forward error

  12. Example: Backward Error Analysis

  13. Example, cont. (relative) forward and backward errors are similar.

  14. Example -Sensitivity

  15. Sensitivity and Conditioning

  16. Condition Number

  17. Example

  18. Examples • What is the condition number of f (x) = sin(x) at x =0, pi/2 and pi? • cond# = | x cot (x) | • 2. What is the condition number of f (x) = x2 + 2x at x =0, 1 and 10? For sufficiently large x?

  19. Stability

  20. Accuracy

  21. Review Problems • Homework One is out and it is due next Thursday. (1.2) What are the approximate absolute and relative erros in approximating pi by a) 3 and b) 3.14? (1.5) Consider the function f(x, y) = x–y. Measure the size of the input (x, y) by | x | + | y |, and assuming that | x | + |y | ~ 1 and x – y ~ ε show that cond(f) ~ 1 / ε. What can you conclude about the sensitivity of substration

  22. (1.7) Let (b, p, U, L) be the four integers that characterize a floating-point number system. Given b= 10, what are the smallest values of p and U, and largest value of L such that both 2365.27 and 0.0000512 can be represented exactly in a normalized floating-point system? (1.17) Let x be a given nonzero floating-point number in a normalized system and let y be an adjacent floating-point number, also nonzero. a) What is the minimum possible spacing between x and y? b) What is the maximum possible spacing between x and y? (1.12) In floating-point arithmetic, which expressions can be evaluated more accurately? x2 –y2 or (x – y ) ( x + y) Example: x = 3469, y= 3451 b=10, p=3, chopping Exact value = 124560, and …

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