MATLAB QUICK START TUTORIAL

Transcription

1 MATLAB QUICK START TUTORIAL This tutorial is a brief introduction to MATLAB which is considered one of the most powerful languages of technical computing. In the following sections, the basic knowledge with the most common commands and functions are to be introduced. Command Window After starting MATLAB, the command window is opened. All commands, statements and functions should be typed in the command window just after the prompt (>>) then ENTER should be pressed: MATLAB shows the result of the entered statement or functions directly after pressing ENTER. To stop showing the result, semi-colon (;) should be typed at the end of the line: If more than one statement is to be given in one line, they should be separated by commas (,) or semi-colons (;): or >> x= x = >> >> x=; >> >> x=, y=6, z=x+y x = y = 6 z = 7 >> x=; y=6; z=x+y z = 7 New statements and commands should always be typed at the last prompt (>>) in the command window. The command window is cleared (i.e. the old commands are erased) by typing clc. Finally, the command window can be used as a simple calculator: >>.*5^ 78.5 >> 7.5*ans Variables The variable names in MATLAB must start with a letter (a-z, A-Z), and must not contain any signs (+ - * / ^?! & % ). >> Temp = ; % Correct >> 7up = 777; % Wrong >> &r+s = 8.8; % Wrong >> R- = ; % Wrong >> R_ = ; % Correct Note: MATLAB is case sensitive. So, any different letter (small/capital) can cause errors: >> TEMP = ; Temp = 6; >> TEMP, Temp TEMP = Temp = 6 Murad Elarbi Jan. 8

2 The variables in MATLAB can be scalar (one value) or arrays (vectors and matrices). They can be integer, real or complex. These types will be discussed in the following sections. Scalar Arithmetic It means the mathematical operations and calculations of scalar values and variables. The precedence of the operations is shown in next table: Operators Operation ( ) Parenthesis contents are calculated first ^ Power * / \ Multiplication, right division and left division + - Addition and subtraction >> *^ - - 8/* *5 55 >> (*)^ - - 8/(*) + (8 + )*5 86 >> a = 6 \ a =.5 Evaluation of the functions, like sin(.*/8), is done first. Predefined Constants Some common constants are defined in MATLAB as follows: Constant Meaning pi Inf, -Inf, - (infinity) NaN / (Not a Number) eps the smallest number that is not rounded to zero if added to. i,j (the imaginary unit) >> A = * pi * 5 A = >> b = * (5 + 7i) % a complex number b =. +8.i >> t = /Inf % division by infinity t = If a variable is given a name of predefined constant, MATLAB works with the new values of that variable: >> pi = pi = >> A = * pi * 5 % pi has a new value here. A = 6 Built-in Functions MATLAB has a very large library of functions for very wide range of scientific and technical applications. Starting with the most common mathematical and trigonometric functions in this section, functions of other applications will be discussed as needed later Function Meaning Function Meaning Function Meaning Function Meaning sqrt(x) log(x) log(x) exp(x) x lnx logx e x sin(x) cos(x) tan(x) asin(x) sin x cos x tan x sin - x / 9 acos(x) atan(x) sinh(x) cosh(x) cos - x tan - x sinh x cosh x tanh(x) asinh(x) acosh(x) atanh(x) tanh x sinh - x cosh - x tanh - x

3 Arrays MATLAB was initially designed to solve problems of arrays and matrices, so, it is very powerful in handling arrays of different dimensions. One-dimensional arrays, which known as vectors, can be created by typing elements between two brackets ( [ ] ) and separated by commas (,) or space: >> x = [ 5] x = 5 >> y = [,5,7,9] y = If the elements of the array are in equal increment, they can be defined using colon ( : ) operator: >> y = ::9 y = One-dimensional arrays can be added, subtracted and multiplied by scalar values: >> z = x + y, z = y - x, z = 5 * y z = 5 8 z = z = They cannot be multiplied by other arrays if the rules of matrix multiplications are not satisfied: >> z = x * y??? Error using ==> mtimes Inner matrix dimensions must agree. Matrices Matrix creation and operations are done very easily. Matrix is created by typing the elements of columns separated by commas (,) or spaces, and semi-colon (;) or ENTER between rows: >> M =[ 5; 6] M = 5 6 >> n =[ ] n = The rules of matrix operations must be considered, otherwise, error messages will appear: >> M + n??? Error using ==> plus Matrix dimensions must agree. But, in case of matrix multiplication: >> M*n >> n*m / 9

4 By-element operations MATLAB provides operators that perform mathematical operations on each element individually or on equivalent elements of two arrays and matrices having same orders. The operators are: Operators Operation.^ Power.*./.\ Multiplication, right division and left division >> x.*y >> M = [ ; 5; 5 6] M = >> M.^ >> r = [6 ]; p = [ ]; >> r.^p Note: M^ gives the same result as M*M. Matrix transpose, determinant and inverse Matrix, or vector, transpose is obtained in MATLAB by apostrophe (') (single quotation mark) operator: >> a = [ ]; >> a' >> b = [ ; 6; 8] b = 6 8 >> b' 6 8 Square matrix determinant is obtained by the function det( ): >> M = [ ; 5; ]; >> det(m) - The inverse of a matrix is obtained by the function inv( ): >> inv(m) >> inv(m) * M % should give unity matrix.... / 9

5 Solution of system of linear equations by MATLAB A system of linear equations can be solved by using the backslash (\) operator. Let's try to solve the following system: x x 5x.5x.5 x x x x x x.5x.5x.8 x x >> a = [ ;.5; -.5; - - -]; % matrix of coefficients >> b = [.5;.8; ; 5]; % vector of the constant terms >> x = a\b x = Access to elements of arrays Elements of arrays and matrices can be accessed (picked) by using the array index: A(i), M(i,j): >> a(,) % Element from matrix a at row and column >> b() % Element from vector b at row.8 >> a(,:) % Elements of row and column from to Polynomials MATLAB deals with polynomials by handling the array of coefficients. The polynomial f ( x) x x x can be written as >> f = [ - ] f = - The roots of the polynomial are obtained by the function roots( ): >> roots(f) i i i i or simply: >> roots([ - ]) i i i i The value of the polynomial f(x) for a given x can be obtained by using polyval( ): >> polyval(f,) 8 >> polyval(f,-) 8 5 / 9

6 If the roots are known, the function poly( ) gives the coefficients of the polynomial: >> poly([, +5i, -5i]) -7 - Operations on polynomials Two polynomials or more can be added and subtracted in same way of one-dimensional arrays. Elements of coefficient arrays must be of identical terms. Considering the following polynomials f ( x) 9x 5x x 7 and g ( x) 6x x The sum of them is >> f = [9, -5,, 7]; g = [, 6, -, ]; % Zero term must be added to g >> sum = f + g sum = 9 9 Multiplication and division of polynomials are called convolution and deconvolution, respectively. The operations f ( x) g( x) and f ( x) g( x) can be done as: >> f = [9, -5,, 7]; g = [6, -, ]; >> prod = conv(f,g) prod = >> [q, r]=deconv(f,g) q = r = This means: f ( x) g( x) 5x 9x x 9x x and f ( x).5x. 58 with remainder equals to.58x g( x) Curve fitting with Polynomials Polynomials of different degrees can be used to fit sets of data. The set of data should be put in two equivalent arrays, then the function polyfit( ) is applied. Let's fit the following set of data: X Y >> X = [ ]; >> Y = [ ]; >> p = polyfit(x,y,5) p = This means that the 5 th degree polynomial that fits the given set of data is 5 p ( x).689x.967 x 9.59x 8.886x.9x.57 Now, to get the value of Y at X=.5, for example, polyval( ) is used: >> polyval(p,.5) % which means p(.5) 9.7 To check the accuracy of the curve fitting, compute values of p(x) at each value of X in the set above: >> polyval(p,x) >> err = Y - ans err = / 9

7 Calculus with MATLAB MATLAB has commands for most of the computations of basic calculus in its Symbolic Math Toolbox. This Toolbox enables the user to work with symbols (x, y, t, ) to do differentiation, integration, etc. So, the variables to be used in equations should be declared as symbols by syms first. Differentiation The function diff( )is used to differentiate the symbolic functions. Let's find d ( x ) : dx >> syms x; >> diff(x^) *x^ Let's find f (t) and f (t ) for f ( t) t cost : >> syms t >> f - cos(t) ; f = Inline function: f(t) = *t^ - cos(t) >> diff(f(t)) *t^+sin(t) >> diff(f(t),) 6*t^+cos(t) Integration MATLAB can compute definite and indefinite integrals by using the function int( ): >> syms x >> int(x^) /*x^ As an example of definite integration, let's find the integral sin xdx : >> syms x >> int(asin(x),, ) /*pi- sin x Double integration is possible too. Let's find x y >> syms x y >> int(int(x^+y^, y,, sin(x)), x,, pi) pi^-/9 dydx Taylor series Taylos polynomial expansion of a specific order for a function can be generated by taylor( ): >> syms x >> taylor(sin(x), x, ) x-/6*x^+/*x^5-/5*x^7+/688*x^9 7 / 9

8 Sums and products Finite numerical sums and products can be computed easily using the vector capabilities of MATLAB and the commands sum( ) and prod( ): >> X = :7 X = >> sum(x) % Computes X()+X()+ +X(7) 8 >> prod(x) % Computes X()*X()* *X(7) 5 Finite and infinite symbolic sums can be done by using symsum( ). Let's find evaluate n n >> syms k n; >> symsum(/k-/(+k),, n) -/(n+)+ >> symsum(/n^,, Inf) /6*pi^ n k k k and Plotting functions and data with MATLAB The graphics are one of the most powerful tools of MATLAB. It provides easy methods with very wide range of types for plotting functions and data. In this context, only two-dimensional plotting functions are discussed. First, let's see the function fplot( ): >> fplot(@(x)x^ + x +, [-, ]) The graph appears in a new figure window. The array [-, ] defines the domain of x values. The command plot( ) is more common in technical use. It plots (x,y) arrays and provides many options to format the plotted data points and curves. Let's see how to plot sin(x) for x= to : >> x=:.:*pi; % x from to pi by step. >> y=sin(x); >> plot(x,y) 8 / 9

9 Plot command can be used to show two or more curves in one graph. Let's plot cos(x) and tan(x) wih the sin(x) for the same domain: >> x=:.:*pi; % formation of x array >> y=sin(x); y=cos(x); y=tan(x); % calculation of the arrays y, y and y >> plot(x,y, x,y,'r--', x,y,'k-o') % r-- : red dashed line, k-o : black circles >> axis([ *pi - ]) >> xlabel('x'), ylabel('y, y, y') >> title('trigonometric Functions') >> legend('sin x', 'cos x', 'tan x') % the order of curves is the same as in plot() To understand the new commands in the example, check their effects by entering them at the command window one by one. Help on MATLAB The user can find the definition of all MATLAB commands and functions with examples and demos in the MATLAB Help. The command help gives quick access to information about any command or function. For example, the command >> help plot shows a summary about the function and its attributes. More information with examples can be found in the documentation that comes with MATLAB. Very useful information and pdf files can also be found in the website www. mathworks.com. 9 / 9