CS227-Scientific Computing. Lecture 3-MATLAB Programming

Transcription

1 CS227-Scientific Computing Lecture 3-MATLAB Programming

2 Contents of this lecture Relational operators The MATLAB while Function M-files vs script M-files The MATLAB for Logical Operators The MATLAB if Random number generation

3 Altering the flow of control M-files seen so far are just sequences of commands, executed one after the other, and bundled as a single command Real programs allow you to alter this straight-line execution: blocks of commands can be executed repeatedly, or skipped deping on the current value of variables

4 Example: Computing Machine Epsilon Here is an algorithm for computing this value: Set x = 1, and continue dividing x by 2 until 1+x evaluates to 1 Machine epsilon is the next-to-last value of x obtained in this process

5 Example: Computing Machine Epsilon set x = 1 is 1+x the same as 1? yes done no set x = x/2

6 Example: Computing Machine Epsilon How do we code this? set x = 1 is 1+x the same as 1? yes done no set x = x/2

7 while while condition Execute the block of s as long as the expression evaluates to true

8 while while condition How do we encode the condition?

9 Relational Operators These are <,<=,>,>==,== (is equal to), ~= (is not equal to) The value of an expression using one of these operators is either 0 (false) or 1 (true) >> 7<3 ans =0 >> 3<=7 ans =1 >> 3==7 ans =0 >> 3=7??? 3=7 Error: The expression to the left of the equals sign is not a valid target for an assignment

10 The Machine Epsilon Script %computation of machine %epsilon x=1; while 1+x/2~=1 x=x/2; x

11 The Machine Epsilon Script %computation of machine epsilon x=1; while 1+x/2~=1 x=x/2; x Note that the original algorithm was off by one a common problem with loop structure--- and had machine epsilon as the next-to-last value of x This is why we write the condition as 1+x/2~=1 instead of 1+x~=1

12 Function M-files function s = powersin(x) %POWERSIN Power series for sin(x) % y = POWERSIN(x) tries to compute %sin(x) from its power series s = 0; t = x; n = 1; while s+t ~= s; s = s + t; t = -x^2/((n+1)*(n+2))*t; n = n + 2; This example from Moller s book is a MATLAB function that evaluates the power series for sin x at its input argument and returns the value of the sum in its output argument

13 You invoke the function just as you would any built-in MATLAB function

14 A MATLAB function has its own private space for variable names The variables x,y,s and n in the Command Window are completely different from the variables with the same names in the function powersin

15 A MATLAB function has its own private space for variable names The function communicates with the user exclusively through the value assigned to its input argument and the value returned in its output argument Typically a function reads no input from the keyboard, and prints no output to the screen

16 Here is the computation of machine epsilon implemented as a function function x = whatseps2( ) %WHATSEPS2 returns machine epsilon as a value x = 1; while 1+x/2~=1 x=x/2; Note that there is no input argument The second, closing the keyword function, is optional The closing while is not

17 Here is the computation of machine epsilon implemented as a function If a function has no input arguments, it can be invoked with or without parentheses In fact the built-in eps in MATLAB is a function with no arguments

18 Functions can have more than one output argument function [x,n] = whatseps3( ) %WHATSEPS3 returns machine epsilon as a value %and the corresponding power of two as a second %output argument x = 1; n=0; while 1+x/2~=1 x=x/2; n=n+1; n=-n; In this version of whatseps, the second output argument contains the value n such that eps = 2^n

19 Functions can have more than one output argument When you invoke a function with more than one output argument, you have to explicitly grab the output values after the first, or they will be ignored

20 The for Syntax for variable = startvalue:increment:value 1 2 Semantics: variable is set in turn to each of the values startvalue, startvalue + increment, etc For each of these values, the sequence of s is executed

21 Example This program prints the sum of the first million terms of the harmonic series, both forward and backward forward_sum=0; backward_sum=0; for i=1: forward_sum=forward_sum+1/i; backward_sum=backward_sum+1/( i+1); forward_sum backward_sum

22 Warning! MATLAB is different from other programming languages, in that it is optimized to perform vectorized operations rapidly The use of for loops is sometimes unavoidable, but in this case it is much faster to sum the harmonic series by with the sum(1/1: ) See Vectorizing Loops under Techniques for Improving Performance in MATLAB Help

23 Random number generation The built-in MATLAB function rand() returns a value uniformly distributed between 0 and 1 >> rand() ans = >> rand() ans = >> rand() ans = We ll have a LOT more to say about random numbers later in the course!

24 Example: A random walk This program returns the result of a random walk At each step, a random angle between 0 and 2π is generated, and a step of length 1 is taken in that direction function x = randomwalk( numsteps ) %RANDOMWALK A random walk in the plane % Each step is of length 1 in a random direction % numsteps is the total number of steps % x is a 2 x numsteps matrix giving the coordinates of %the points visited in the walk x=zeros(2,numsteps); for step=2:numsteps theta=2*pi*rand(); x(:,step)=x(:,step-1)+[cos(theta);sin(theta)];

25 Example: A random walk Plots from two sample runs of the program:

26 Example: Nested for s This script computes and prints Pascal s triangle: The triangular array of binomial coefficients C(n,k),where 0<=k<=n These are computed by the formulas C(n,0)=C(n,n)=1, C(n+1,k+1)=C(n,k+1)+C(n,k) vec=zeros(11,11); vec(1,1)=1; fprintf('%d\n',vec(1,1)) for row=2:11 vec(row,1)=1; fprintf( %d\t,vec(row,1)); for col=2:row vec(row,col)=vec(row-1,col)+vec(row-1,col-1); fprintf('%d\t',vec(row,col)); fprintf('\n');

27 Example: Nested for s A run of the program >> pascaltriangle >>

28 Logical Operators Expressions with value true or false can be combined using && (and), or ( ) and not (~) For instance, the expression below is true if and only if n is a leap year (divisible by 4, and not divisible by 100 unless it is divisible by 400) Note the use of rem(n,k), which returns the remainder of n upon division by k (rem(n,4)==1)&&((rem(n,100)~=0) (rem(n,400)==0))

29 The if if condition if condition else if condition elseif condition elseif condition else

30 Example: A version of the random walk in which every step is horizontal or vertical, with each of the four directions equally probable function x = randomwalk2( numsteps ) %RANDOMWALK A random walk in the plane % Each step is 1 step north, south, east or west % numsteps is the total number of steps % x is a 2 x numsteps matrix giving the coordinates of %the points visited in the walk x=zeros(2,numsteps); for step=2:numsteps r=rand(); if r<025 x(:,step)=x(:,step-1)+[1;0]; elseif r<05 x(:,step)=x(:,step-1)+[-1;0]; elseif r<075 x(:,step)=x(:,step-1)+[0;1]; else x(:,step)=x(:,step-1)+[0;-1];

31 A plot of the output is peculiar-looking!

32 Example: This executes the random walk until it wanders a specified distance from the origin function [ X n ] = randomwalk3( radius ) %RANDOMWALK3 A random walk with each step of length 1 in a random %direction The walk continues until it wanders a distance of %radius from the origin % X is a 2-row matrix giving the coordinates of the successive % points on the walk n is the total number of steps taken %Note that we build the matrix X one column at a time X=[0;0]; n=1; while(x(1,n)^2+x(2,n)^2<= radius^2) theta = 2*pi*rand(); y = X(:,n)+[cos(theta);sin(theta)]; X = [X y]; n=n+1;

33 Example: In this version, the random walk proceeds until either the required distance from the origin is reached, or the maximum allowable number of steps are taken function [ X n ] = randomwalk4( radius, maxsteps ) %RANDOMWALK3 A random walk with each step of length 1 in a random %direction The walk continues until it wanders a distance of %radius from the origin, or until it takes maxsteps steps, whichever %comes first % X is a 2-row matrix giving the coordinates of the successive % points on the walk n is the total number of steps taken % Note that we pre-allocate a matrix X large enough to hold all the % steps, and then trim it to the correct size X=zeros(2,maxsteps); n=1; while((x(1,n)^2+x(2,n)^2<= radius^2)&&(n<=maxsteps)) theta = 2*pi*rand(); X(:,n+1) = X(:,n)+[cos(theta);sin(theta)]; n=n+1; %trim unused columns from the matrix X=X(:,1:n);