Chapter 1 Introduction

1. Chapter 1Introduction Prof. Chuan-Ming Liu MCSE Lab, NTUT TAWIAN MCSE Lab, NTUT

2. Outline • Scientific Computing • Computational Errors • Error Propagation • Error Propagation in Basic Arithmetic Operations MCSE Lab, NTUT

3. Scientific Computing Scientific computing The interface of Computer Science, Engineering and Applied Mathematics. MCSE Lab, NTUT

4. Scientific Computing • People want the methods to be fast and accurate. • Applications of Scientific Computing • Fluid dynamics • Material science • Semiconductor device simulation • Weather modeling • Computational biology • Econometrics and computational finance • Computer graphics MCSE Lab, NTUT

5. Scientific Computing • In this course, we will consider the time complexity and accuracy for the solutions to some problems. • Time complexity relates to the algorithm analysis. • Accuracy relates to mathematic analysis. MCSE Lab, NTUT

6. Scientific Computing • Why do we need numerical analysis (methods)? • It is all about the reality. • Example: consider a 3D solid if revolution can be created by revolving the curve y = r(x) about the x-axis. MCSE Lab, NTUT

7. y y = r(x) x z Scientific Computing • If r(x) is simple, e.g. r(x) = ex , then close form solution MCSE Lab, NTUT

8. Scientific Computing • If r(x) is not simple, for example it is not possible to get the close form: MCSE Lab, NTUT

9. Scientific Computing • Numerical analysis then comes in. • To obtain the (approximated) value of V, we can subdivide the interval [0, 1] into n equal subintervals as MCSE Lab, NTUT

10. Scientific Computing • Rewrite the integration into the sum MCSE Lab, NTUT

11. constructing solving Physicalphenomena Mathematical models Mathematical solutions explanation or prediction Scientific Computing • Note: what we did here is the rectangular rule, we will discuss this rule later in this course. • In practice, many problems do not have a close form solution. MCSE Lab, NTUT

12. L θ m mgsinθ mg Scientific Computing • Example: motion of a pendulum L: string length θ: the angle, a function of time t. Using Newton’s second law, we can get the following model MCSE Lab, NTUT

13. Scientific Computing • Initial-value problems MCSE Lab, NTUT

14. Scientific Computing • The general solution MCSE Lab, NTUT

15. Scientific Computing MCSE Lab, NTUT

16. Scientific Computing • Note: • Note: This close form is inaccurate when θdoes not approach to 0. MCSE Lab, NTUT

17. Scientific Computing • Discussion on the first example (Integration) • discretization • discretization error • how the error is OK • efficiency • The scientific computing is used to denote the use of numerical methods in a complex hardware and software environment to solve models from various scientific disciplines. MCSE Lab, NTUT

18. 3 Ω 5 Ω 1 2 100 v 10 Ω a 2 Ω 6 Ω 1 Ω 3 b 1 Ω 7 Ω 0 v 6 5 4 Scientific Computing • Example: consider the electrical network obtain the voltages at nodes 1 through 6. MCSE Lab, NTUT

19. Scientific Computing MCSE Lab, NTUT

20. Scientific Computing MCSE Lab, NTUT

21. Scientific Computing MCSE Lab, NTUT

22. Scientific Computing MCSE Lab, NTUT

23. Scientific Computing • In this course, we will learn many numerical methods as well as how to tell a method is good or not. First of all, we start with the “computational error”. MCSE Lab, NTUT

24. Computational Errors • The computational error comes from: • Machine representation • Number representation • Round off errors • Rounding • Chopping • Rounding up • Rounding down • Formula for the problem • Truncation error MCSE Lab, NTUT

25. 0 ∞ -∞ real number Computational Errors • Roundoff errors • finite precision numbers present infinite precision numbers • The result of an operation on two machine-representable numbers is not necessarily a machine-representable number. MCSE Lab, NTUT

26. Computational Errors 11 digits e.g. : a = 9.99*104 and b = 9.99*10-4 a + b = (9.99+9.99*10-8)*10-4 =9.9900000999*104. Suppose the machine only computes to 6-digits.Then, a + b = 9.99*104 =>b has no effect !! (why ?) However, a/b = 108 a*b = 99.8001 are still valid (why ?) The reason will be discussed later 6-digit representation is OK with the result MCSE Lab, NTUT

27. Truncation error Computational Errors • Truncation error • Occurs when applying a formula MCSE Lab, NTUT

28. Please review how to calculate the derivatives for some common functions e.g. ex, sinx, logx Taylor Theorem MCSE Lab, NTUT

29. Taylor Theorem • Pn(x) is the nth Taylor polynomial for f about x0 • Rn(x) is the remainder term (or truncation error) associated with Pn(x). • The infinite series obtained by taking n ∞ is called the Taylor series for x about x0 • When x0 = 0, the Taylor polynomial is often called a Maclaurin polynomialand the Taylor series is called a Maclaurin series. MCSE Lab, NTUT

30. Taylor Theorem -Example • Determine (a) the second and (b) the third Taylor polynomials for f(x) = cosx about x0 = 0 and use these polynomials to approximate cos(0.01) • (a) MCSE Lab, NTUT

31. Taylor Theorem - Example (b) exercise. MCSE Lab, NTUT

32. Computational Errors approximation of derivative of function f. Let f be a function. Obtain the derivative at a where f is cont. on [a-h, a+h]. MCSE Lab, NTUT

33. Computational Errors MCSE Lab, NTUT

34. actual f ’(a) approximate f ’(a) a-h a a+h MCSE Lab, NTUT

35. Computational Errors • The more terms use to estimate, the more accurate the answer is MCSE Lab, NTUT

36. Computational Errors • Absolute v.s. relativeerrors • If P* is an approximation to P, • |P*-P| is the absolute error. • | P*-P |/|P| where |P|≠0 is the relative error. • Error tolerance • A computation is acceptable if the error (absolute or relative) is no more than a specific error tolerance. • |computed solution – exact solution| < tolerance i.e. | P*-P | < tol or |P*-P|/|P| < tol. MCSE Lab, NTUT

37. Computational Errors • Error tolerances should be part of the specifications of an algorithm. • High accuracy usually needs high cost! • Accuracy appr. MCSE Lab, NTUT

38. Computational Errors • The number of decimal places of accuracy, d, for fixed error, |error| = 10-dis given by d = -log10|error|, i.e. the approximation has d accurate decimal places. • example MCSE Lab, NTUT

39. Computational Errors MCSE Lab, NTUT

40. Computational Errors • example MCSE Lab, NTUT

41. Error Propagation • Error Propagation (overall) MCSE Lab, NTUT

42. Error Propagation MCSE Lab, NTUT

43. Error Propagation • Example: c = ? MCSE Lab, NTUT

44. Error Propagation • Sol. MCSE Lab, NTUT

45. Error Propagation • The sensitivity of a problem is indicated by the relative dependence of output error to input error. • Definition: MCSE Lab, NTUT

46. Error Propagation in Basic Arithmetic Operations • Error Propagation in Arithmetic Operations MCSE Lab, NTUT

47. Error Propagation in Basic Arithmetic Operations MCSE Lab, NTUT

48. (1+x)-1 = 1-x+x2+… Error Propagation in Basic Arithmetic Operations MCSE Lab, NTUT

49. Error Propagation in Basic Arithmetic Operations MCSE Lab, NTUT

50. Error Propagation in Basic Arithmetic Operations • Avoiding cancellation errors • Rationalization • Taylor expansion (two nearly equal numbers) MCSE Lab, NTUT