Numerical Methods in Scientific Computation

Transcription

1 Numerical Methods in Scientific Computation Programming and Software Introduction to error analysis 1

2 Packages vs. Programming Packages MATLAB Excel Mathematica Maple Packages do the work for you Most offer an interactive environment 2

3 MATLAB Software product from The MathWorks, Inc. Originally focused on matrix manipulations Interactive tool to do numerical functions, and visualization Commands can be saved into user scripts call and m-file Graphics and graphical user interfaces (GUI) are built into program 3

4 MATLAB Base product is extended with a variety of toolboxes Optimization Statistics Curve fitting Image processing 4

5 MATLAB Example: Find roots of >> p = [ ] >> r = roots (p) >> r = x 3 6x 2 72x 27 5

6 MATLAB Advantages: Large library of functions to evaluate a wide variety of numerical problems Interactive tool allows immediate evaluation of results Disadvantages Expensive Not as fast as C/C++/FORTRAN code 6

7 Programming language concepts Interpreted Languages MATLAB Usually integrated with an interactive GUI Allows quick evaluation of algorithms Ideal for debugging and one-time analysis and in cases where high speed is not critical 7

8 Programming language concepts Compiled Languages FORTRAN, C, C++ Source code is created with an editor as a plain text file Programs then needs to be compiled (converted from source code to machine instructions with relative addresses and undefined external routines still needed). The compiled routines (called object modules) need then to be linked with system libraries. Faster execution than interpreted languages 8

9 Programming concepts Data representation Constants, variables, type declarations Data organization and structures Arrays, lists, trees, etc. 9

10 Programming concepts Mathematical formulas Assignment, priority rules Input/Output Stdin / Stdout, files, GUI 10

11 Programming concepts Structured programming Set of rules that prescribe good programming style Single entry point and exit point to all control structures Flexible enough to allow creativity and personal expression Easy to debut, test and update 11

12 Structured programming Control structures Sequence Selection Repetition Computer code is clearer and easier to follow 12

13 Algorithm expression Flowcharts Visual representation Useful in planning Pseudocode Simple representation of an algorithm Basic program constructs without being language-specific Easy to modify and share with others 13

14 Flowcharts 14

15 Control structures Sequence Implement one instruction at a time 15

16 Control structures Selection Branching IF/THEN/ELSE CASE/ELSE 16

17 Control structures 17

18 Control structures 18

19 Control structures Repetition DOEXIT loops (break loop) Similar to C while loop DOFOR loop (count-controlled) 19

20 Control structures pretest posttest midtest 20

21 Control structures 21

22 Modular programming Break complicated tasks into manageable parts Independent and self-contained Well-defined modules with a single entry point and single exit point Always a good idea to return a value for status or error code 22

23 Modular programming Makes logic easier to understand and digest Allows black-box development of modules Requires well-defined interfaces with no side effects Isolates errors Allows for reuse of modules and development of libraries 23

24 Software engineering approach Specification: A clear statement of the problem, the requirements, and any special parameters of operation Algorithm/Design: A flow chart or pseudo code representation of how exactly how will the problem be solved. 24

25 Software engineering approach Implementation: Breaking the algorithm into manageable pieces that can be coded into the language of choice, and putting all the pieces together to solve the problem. Verification: Checking that the implementation solves the original specification. In numerical problems, this is difficult because most of the time you don t know what the correct answer is. 25

26 Example case How do you solve the sum of integers problem? S = n i= 0 i a) Simplest sum = N Coded specifically for specific values of N. 26

27 Possible solutions b) Intermediate solution Does not require much thought, takes advantage of looping ability of most languages: C/C++: MATLAB 27

28 Possible solutions c) Analytical solution Requires some thought about the sequence remember back to one of your math classes. sum = n* ( n + 1) 2 28

29 Verification of algorithm What can fail in the above algorithms. (To know all the possible failure modes requires knowledge of how computers work). Some examples of failures are: For (b): This algorithm is fairly robust. But, when N is large execution will be slow compared to (c) 29

30 Algorithm (c) This is most common algorithm for this type of problem, and it has many potential failure modes. For example: (c.1) What if N is less than zero? Still returns an answer but not what would be expected. (What happens in (b) if N is negative?). 30

31 Algorithm (c) (c.2) In which order will the multiplication and division be done. For all values of N, either N or N+1 will be even and therefore N*(N+1) will always be even but what if the division is done first? Algorithm will work half of the time. If division is done first and N+1 is odd, the algorithm will return a result but it will not be correct. This is a bug. For verification, it means the algorithm works sometimes but not always. 31

32 Algorithm (c) (c.3) What if N is very large? What is the maximum value N*(N+1) can have? There are maximum values that integers can be represented in a computer and we may overflow. What happens then? Can we code this to handle large values of N? 32

33 Verification By breaking the program into small modules, each of which can be checked, the sum of the parts is likely to be correct but not always.. Problems can be that the program only works some of the time or it may not work on all platforms. The critical issue to realize all possible cases that might occur. 33

34 Introduction to error analysis Error is the difference between the exact solution and a numerical method solution In most cases, you don t know what the exact solution is, so how do you calculate the error Error analysis is the process of predicting what the error will be even if you don t know what the exact solution Errors can also be introduced by the fact that the numerical algorithm has been implemented on a computer 34

35 Significant digits Can a number be used with confidence? How accurate is the number? How many digits of the number do we trust? 35

36 Significant digits Can the speed be estimated to one decimal place? 36

37 Significant digits The significant digits of a number are those that can be used with confidence The digits that are known with certainty plus one estimated digit zeros are not always significant digits , , x10 4, 4.530x10 4, x10 4 Exact numbers vs. measured numbers Exact numbers have an infinite number of significant digits π is an exact number but usually subject to round-off 37

38 Significant digits 38

39 Accuracy and precision Accuracy is how close a computed or measured value is to the true value Precision is how close individual computed or measured values agree with each other Reproducibility Inaccuracy/Bias vs. Imprecision/Uncertainty Inaccuracy: systematic deviation from the truth imprecision: magnitude of the scatter 39

40 Accuracy and precision 40

41 Accuracy and precision The level of accuracy and precision required depend on the problem Error to represent both the inaccuracy and imprecision of predictions 41

42 Error definitions Two general types of errors Truncation errors due to approximations of exact mathematical functions Round-off errors due to limited significant digit representation of exact numbers E t = true value approximation 42

43 Error definitions E t does not capture the order of magnitude of error 1V error probably doesn t matter if you re measuring line voltage, but it does matter if you re measuring the voltage supply to a VLSI chip Therefore, its better to normalize the error relative to the value 43

44 Error definitions Example: Line voltage Chip supply voltage 44

45 Error definitions What if we don t know the true value? Use an approximation of the true value a How do we calculate the approximate error? Use an iterative approach Approximate error = current approximation previous approximation Assumes that the iteration will converge 45

46 Error Definitions For most problems, we are interested in keeping the error less than specified error tolerance How many iterations do you do, before you re satisfied that the result is correct to at least n significant digits? 46

47 Example Infinite series expansion of e x As we add terms to the expansion, the expression becomes more exact Using this series expansion, can we calculate e 0.5 to three significant digits? 47

48 Example Calculate the error tolerance First approximation Second approximation Error approximation 48

49 Example Third approximation Error approximation 49

50 Example 50

51 Next class HW1, due 9/8 Chapra & Canale 6 th Edition 1.8 (typo: dy/dx -> dy/dt), 1.12, 2.5 (choose order n=6), 2.14 Next class Continue on error analysis 51