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

2. Absolute and Relative Errors

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

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 Minimizing mh/2 + 2epsilon /h Rounding Error

11. Forward and Backward Errors

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

13. Example: Backward Error Analysis

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

15. Example -Sensitivity

16. Sensitivity and Conditioning

17. Condition Number

18. Example

19. Examples 1.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) = x 2 + 2x at x =0, 1 and 10? For sufficiently large x?

20. Stability

21. Accuracy

22. 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

23. (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? x 2 –y 2 or (x – y ) ( x + y) Example: x = 3469, y= 3451 b=10, p=3, chopping Exact value = 124560, and …