An Introduction to Scilab

Tsing Nam Kiu 丁南僑Department of Mathematics

The University of Hong Kong

2009 January 7

What is a Scilab?

Scilab is a mathematical software Similar software: Matlab, Mathematica, Octave,

Euler Math Toolbox, Maxima, …

What is special about Scilab: free, highly supported, powerful, many users, …

Home page of Scilab: www.scilab.org A short introduction of Scilab: http://hkumath.hku.hk~n

kt/Scilab/IntroToScilab.html

Using Scilab as a calculator

+, – , * (multiplication), / (division), ^ (power)

Examples:

> (12.34 + 0.03) / (2.8 – 1.2 * 3)

> 2^3 or 2*2*2

> 2^– 3

> 2^100

> ans^(1/100)

Using Scilab as a calculator (2)

Commonly used functions:

cos, sin, tan, acos, asin, atan, sqrt, exp, log, log10 Solving quadratic equation x^2 – x+1=0:

> a = 1, b = – 1, c = 1

> (– a + sqrt(b^2 – 4*a*c))/(2*a)

> (– a – sqrt(b^2 – 4*a*c))/(2*a) A smarter way to find roots of polynomials:

> p = poly([1 –1 1],"x","coeff")

> roots(p)

Using Scilab as a calculator (3)

special constants: %i, %pi, %e

> tan(%pi / 4)

> %e ( = exp(1) )

> (1+%i)*(1--%i) Learning how to use Scilab and getting help:

Click on “?” on menu

> help command

See documentation on Scilab website

Vectors and matrices in Scilab

Data types: (real or complex) numbers, vectors, matrices, polynomials, strings, functions, …

Vectors in Scilab:

> x = [0 1 2 – 3]

> y = [2; 4; 6; 8]

> z = [1 2 3 4] ’ ’ is conjugate transpose of a matrix

> 3*x, y+z, y–z

> x+y, x+1

Vectors and matrices in Scilab (2)

Matrices in Scilab:

> A = [0 1 0 1; 2 3 –4 0]

> B = A ’

> A * y, x * B, A * B, B * A, (B*A)^2 Special matrices (and vectors):

> ones(2,3), zeros(1,2), eye(3,3)

> rand, rand(3,2) Empty vector or matrix: > a = [ ] Building matrix by blocks:

> C = [A 2*A], x = [9 x 7], a = [a 1]

Solving linear equations

3 x1 + 2 x2 – x3 = 1

x1 + x3 = 2

2 x1 – 2 x2 + x3 = – 1 To solve the above system of linear equations:

> A = [3 2 – 1 ; 1 0 1; 2 – 2 1]

> b = [1 2 – 1]’

> x = inv(A)*b (inv is inverse of a matrix)

> x = A \ b Important remark: theoretically it does not make s

ense to divide something by a matrix!

The colon “:” operator

> 1:10, 1:100, xx = 1:100; Using “;” to suppress answer output > sum(xx) > 1:2:10, –3:3:11, 4:–1:1, 2:1:0, > t = 0: 0.1: 2*%pi

> y = sin(t)

> plot(t,y), plot(t,sin(t),t,cos(t)) Task 1: plot the straight lines

y = x +1 and y = exp(x) on the same graph, from x = – 2 to x = 2

Elements of vectors and matrices

Example

> v = rand(4,1)

> v(1), v(3), v([2 4]), v(4:-1:1), v($) “$” means the last entry Example

> A = [1 2 3 4 5; 6 7 8 9 10]

> A(2,3), A(1,:), A(:, 2), A(:, [4 2])

Exercises

Task 2: simulate tossing of a coin:

0 = head, 1 = tail.

functions to use: rand, round, …

Task 3: simulate tossing of 100 coins

Exercises (2)

Task 4: simulate throwing 3 dices, each dice has outcome from 1 to 6 with equal probabilities;

functions to use: rand, floor, ceil, …

Task 5 (challenging!): simulate tossing a coin 100 times and find the longest run of consecutive H’s or T’s in the resulting sequence;

functions to use: diff, find, max,

Programming in Scilab

Click on menu bar to open Scipad; then write your scilab function file.

Format of a function:function [out1, out2, ...] = name(in1, in2, ...) (body of function definition; may have many lines)endfunction

One file may contain more than one function. To use the functions, you must load the function fi

le by choosing File -> Execute the file from the menu.

Programming in Scilab (2)

A simple function to find the n-th term of the Fibonnaci sequence 0, 1, 1, 2, 3, 5, 8, 13, 21, …

function k = fibo(n) if n == 1, k = 0;elseif n==2, k = 1;else k = fibo(n-1) + fibo(n-2);end

endfunction Save the file as fibo.sci (or any other file name). Execute it from Scilab menu bar Try, say: > fibo(5), fibo(2), fibo(10), fibo(100)

Programming in Scilab (3)

An improved programme:function K = fibonacci(n)//function K = fibonacci(n)//Gives the n-th term of the Fibonacci sequence ,1,1,2,3,5,8,13,... if n==1,

K = 0; elseif n==2, K = 1;elseif n>2 & int(n)==n

// check if n is an integer greater than 2K = fibonacci(n-1) + fibonacci(n-2);

else disp('error! -- input is not a positive integer'); end

endfunction

Programming in Scilab (4)

Programming Task (challenging!): write a programme to automate Task 5, which is to perform the following experiment m times. The experiment is to simulate tossing a coin n times and find the longest run (k) of consecutive H’s or T’s in the resulting sequence.

For each time you do the experiment, you’ll get a number k. Therefore you should get m numbers k

1, k2, …, km at the end. Inputs of the function are m, n; output is a vector

k = [ k1 k2 … km].

Recap

We have discussed and learned the following: What Scilab is Basic usage of Scilab (as a calculator) Vectors and Matrices in Scilab Solving linear equations Simulation of some random events Basic Scilab programming