Linear Algebra and MATLAB

Transcription

1 Linear Algebra and MATLAB Pre-Semester Course Mathematics Christian Seckinger

2 MATLAB 1 1 This script is based on a MATLAB course of Alexey Cherepnev and Tobias Nagel at University of Mainz.

3 References Judd, K.L. (1998): Numerical Methods in economics. The MIT Press. Miranda, M.J. and P.L.Fackler (2002): Applied computational economics and nance. The MIT Press. Otto, S.R. and J.P. Denier (2005): An introduction to Programming and Numerical Methods in MATLAB. Springer, London. Wooldridge, J.M. (2002): Introductory Econometrics - A modern Approach (2nd edition). ltps Thomson Learning.

4 References cont. There are also many scripts concerning MATLAB online, e.g.: Griths, D.F. (2005): An introduction to Matlab: Version 2.3. University of Dundee, scriptum. or search for 'Matlab Primer'

5 What is MATLAB? MATLAB: MATrix LABoratory Environment for numerical computation developed by MathWorks: 'MATLAB is a high-level language and interactive environment that enables you to perform computationally intensive tasks faster than with traditional programming languages such as C, C++, and Fortran.' ( Many built-in functions Also possibility for development of own routines High graphical capability, e.g. 2D and 3D graphics

6 What is MATLAB? cont. Alternatives: Oktave: OpenSource project ( Freeware, mostly compatible with MATLAB. Scilab: Free software ( For special problems also possible: C, C++, Microsoft Excel,...

7 Workspace

8 Workspace cont. Parts of the workspace (Note: You can individualize your workspace): Command window Editor: Main part, where you can do everything indicates MATLAB interpreter As the name suggest: an editor (with syntax highlighting). Write/Save your own code here (e.g. m-les). You could also use any editor you prefer. Command history: Workspace: Shows everything you typed into the Command Window Shows all current dened variables with value and class Double click on name opens Array Editor

9 Scilab Free software Workspace: Console: Equivalent to MATLAB Command Window Editor: Equivalent to MATLAB Editor Variable Browser: Equivalent to MATLAB Workspace (Console Applications Variable Browser) Most of the commands are the same There exist commands to convert Scilab-les.sci to MATLAB-les.m e.g. mfile2sci (opens a GUI where you can select the les you want to convert)

10 Notation in this script Anytime code is included along the slides, the typewriter-font will be used, esp.: xyz: Commands that can be used directly in MATLAB [xyz...]: Optional information <xyz>: Generic name that you should replace when using the command

11 Basics to keep in mind MATLAB is case-sensitive All commands have to be typed behind (the MATLAB interpreter) The output is shown in the command window, but can be suppressed if your statement ends with ; Last result is saved in ans by default Without own assignments you will lose earlier results There is a very good help le, which can accessed by Drop-down menu help <name> MATLAB allows tab-completion Arrow keys will go through command history

12 Useful commands clc clear [var1...] who whos edit <filename> CTRL + C ;, %... clears command window (no variables deleted) deletes selected variables shows all actual dened variables like who, with additional information on Size, Bytes and Class of variables opens the le lename.m in the editor aborts actual computation separates commands and suppresses output separates commands, all output will be shown comments allows line-breaks for readability

13 Easy calculations arithmetic operations:+, -,, *, / assignment with = Example»2 + 2 ans = 4 Example» a = ans * 2 a = 8 Precedence and rules for brackets as usual Example»12 / 2 * 3 ans = 18

14 Built-in functions Arguments have to be in round brackets sin cos tan sine-function (arguments should be in radians) cosine-function (arguments should be in radians) tangent-function (arguments should be in radians) Example» x = cos(pi/4) x = sqrt square root exp exponential function log natural logarithm log10 logarithm to basis 10

15 Built-in functions cont. gcd(a,b) lcm(a,b) abs fix floor ceil round greatest common divisor least common multiple absolute value round towards zero round towards nearest integer towards minus innity round towards nearest integer towards plus innity round towards nearest integer Many more built-in functions

16 Numbers and Formats Several kind of numbers Type Example Integer 1232, Real , Rational 1/3, 1/9 Complex i (i= 1) Inf Innity (results of dividing by 0) NaN Not a Number (0/0) Very large or very small numbers are represented by 2.334e-03= = You can inuence how MATLAB prints numbers, e.g. format long: xed point format with 15 digits All possibilities: help format All computations are done with double precision (no matter of format)

17 Numerical results Numerical analysis is approximative Example 1» 9 * ans = » format long» 9 * ans = Example 2» sin(pi) ans = e-016

18 How to use variables Assignment of value to a variable by = Variable names: Any combination of letters, underscores and digits, starting with a letter Use names that reect the values they represent Avoid special names like: pi, i, j, eps,... You can overwrite those variables, but this will very likely lead to mistakes in ongoing program. Use type <program_name> or which <variable_name> to see whether a program name or variable name is already in use. These rules also hold for le-names and are part of a good programming practice. A style guide by R. Johnson (2002):

19 Classes of variables Arrays: A (multidimensional) list of elements Examples: Vectors and Matrices (we will come to this later on) Number of entries is called length Entries are called elements or components Example» x = [ ] x = Entries must be enclosed in square brackets To access elements use round brackets (e.g. x(2))

20 Classes of variables cont. Strings: Variables containing text: Strings (a vector of characters) Text has to be marked by ' : my_name = A String Specic commands like: num2str, int2str, str2num,... Numbers: Dierent types: integer, rational, real, complex Typical class: double Dierent representation through system Boolean: Logical data structure Two states: true & false 1 & 0

21 m-les Equivalent to programs, functions, subroutines or procedures in other programming languages ASCII (text) les, that contain MATLAB commands Specic le-extensions is.m Distinguish between script-les and function-les

22 script-les Commands of script-le are read by interpreter like in command-window use ; to avoid unintended prompt. Start with a command hello_world.m %hello_world.m %prints string hello world to command window var = Hello world! ; disp(var); Now you can use this script in the command window:» hello_world Hello world! Caution: Your current directory must be the one where you saved the le

23 function-les Combination of script-le and mathematical function le-name and function-name must be equal times2.m function out1 = times2(x) %Doubles value of x %INPUT: x - number %OUTPUT: out1 - number out1 = 2*x;

24 function-les cont. command window» times2(4) ans = 8» % option 2» feval( times2,4) ans = 8 Use comments at beginning and through your program

25 function-les cont. Types of functions Primary functions First function dened in.m-le File name has to be identical to primary function name Subfunctions 2nd, 3rd, 4th,...function dened in.m-le Can be called from primary function or other subfunctions in the.m-le Nested functions Function within another function Can be called from enclosing function or from other nested function in the same enclosing function Private functions Primary function saved into subfolder named private Can be called only by functions in the parent folder of the private subfolder You can only call primary functions from MATLAB command window or other scripts

26 function-les cont.

27 function-les cont. Search order to nd functions: MATLAB interpreter looks for given function-name 1. Is it a variable? 2. Is it a nested function inside the current function? 3. Is it a sub-function? 4. Is the le in directory private? 5. Is the le in the current directory? 6. Is the le in any directors in its search path? (Use: path, addpath <directory>) 7. If all this failed, MATLAB displays error message

28 Loop structure Toolbox: for-loop: if-statements switch-statements while-loop

29 for-loop Basic idea: Repeat one (or several) operations for all elements of a vector. Syntax for <name> = <expression> <commands> end Example for n = 1:3 disp([int2str(n), x2=,int2str(2 n)]) end Output 1x2=2 2x2=4 3x2=6

30 Nested loops Many algorithms require to use loops within loops Nested loops Nested loops increase computation time, so try to use as few Example x = as possible 3 m=1 n=1 3 n m sum=0; for m = 1:3 for n = 1:3 sum = sum + n m; end x=sum;

31 Conditional statements Basic idea: Commands are executed according to a fulllment of an expression. Syntax if <expression> <commands> end Example if x == 0 disp( x equals 0! ) end Commands are executed if the expression (<expression>) is true, otherwise the program continues beyond end.

32 Logical expressions Closer look at <expression>: Comparison: a<b True if a is less than b a<=b True if a is less than or equal to b a>(=)b True if a is greater (or equal) to b a==b True if a is equal to b a =b True if a is not equal to b Compound statements: and(a,b) a& b: logical AND or(a,b) a b: logical OR not(a) a: logical NOT xor(a,b) logical exclusive OR a&&b Short-circuiting capability AND a b Short-circuiting capability OR

33 Logical expressions cont. Truth tables: AND false true false false false true false true OR false true false false true true true true XOR false true false false true true true false

34 Nested if statements Two branches: Syntax if <expression> <commands 1> else <commands 2> end Example if x == 0 error( x equals 0! ); else erg = 1/x; end

35 Nested if statements cont. If <expression> is true, <commands 1> will be executed. If <expression> is false, <commands 2> will be executed instead. More branches: Syntax if <expression 1> <commands 1> elseif <expression 2> <commands 2>... [else <commands 3>] end

36 Nested if statements cont. Example if a == b erg = 0; elseif a < b erg = a; else erg = b; end The else-part is optional Expressions are tested from top to bottom until the rst true. The corresponding commands are executed and the program continues beyond end.

37 switch-case Basic idea: Possibility to execute commands on dierent cases of an expression. Syntax switch <switch-expression> case <case-expr1> <commands 1> case <case-expr2> <commands 2>... otherwise <commands> end

38 Example Example option = 3; switch option case 1 disp( option 1 ); case 2 disp( option 2 ); otherwise disp( wrong argument ) end <switch-expression> can be a scalar or a string scalar: commands executed if <switch-expr>==<case-expr> string: commands executed if strcmp(switch-expr, case-expr) returns 1 (true)

39 Conditional loops Basic idea: Repeat a loop while a certain condition is satised. Syntax while <condition> <commands> end As long as <condition> is true, <commands> are executed Caution: Maybe <condition> is never fullled

40 Conditional loops cont. Example x = 1; count = 0; while x<10 disp(xˆ2); x = x + 1; count = count + 1; end disp([int2str(x), repetitions ])

41 The break command Sometimes it is necessary to stop a while or for loop (e.g. if a condition is reached but it is unknown how many repetitions are necessary) break: Stops execution of commands and the code beyond end is executed. In nested loops, break exits from the innermost loop only Example x = 1; while 1 == 1 %Caution: infinite loop x = x + 1; if x > 10 break; end end

42 Keeping a record Save your work you are doing in your command window diary <mysession>: all subsequent text appearing in command window will be saved to the le <mysession> diary off: stops recording <mysession> is saved at current folder you are working in Save current values of variables save <thissession>: All data saved to <thissession.mat> save <varfile> var1 [var2...]: Saves specic variables to <varle.mat> load <thissession>: reload the data

43 Reading data les If you have external data (e.g. Excel spreadsheet), save this data as comma-separated le (.csv or.txt) Separators don't matter as long as you know what it is How to import this le: Import data using graphical user interface (GUI) File import Data...(follow assistant) Use MATLAB command importdata

44 Reading data les cont. Example» A = importdata(mappe1.txt) A = mappe1.txt 1;10 2;20 3;30

45 Reading data les cont. There is a more sophisticated way of le management for formatted les:» fid=fopen( mappe1.txt, r ); % 1» a = fscanf(fid, %1d;%2d ); % 2» fclose(fid); % 3 Commands in detail: Our le is opened for reading ('r'). The variable fid (le-identier) is assigned a unique integer which identies the le used Read pairs of gures with 1 digit followed by an ';' and a 2 digit gure close le The result is a column vector a Use vector function a = reshape(a,2,3)

46 Writing data Nothing comparable to importdata Make use of le management for formatted les A short table of the exponential function» x = 0:.2:1; y = [x; exp(x)];» fid = fopen( exp.txt, wt );» fprintf(fid, %6.2f %12.8f\n,y);» fclose(fid); Open le for writing ('wt') Write pairs of gures of width 6 and 12 resp. (denoted by %) (Matlab takes the gures column-wise) Write the next pair into a new line (\n) Write these pair of gures with precision 2 and 8 resp. Specication of what to write (y)

47 Writing data cont. exp.txt For a detailed description of all possibilities for formatted les: cf.

48 Input during runtime Another helpful command is: in = input( Value of variable in = ) Prompts Value of variable in = and waits for user input from keyboard Any valid MATLAB expression is accepted instr = input( text, s ): typed input is not evaluated but saved as string