A very short introduction to Matlab and Octave

Transcription

1 A very short introduction to Matlab and Octave Claude Fuhrer 09 November 2016

2 Contents 1 Installation 3 2 Variables and literal values Variables Exercises Litteral values Numbers format Generating range of datas Operators Arithmetic operators Integer arithmetic Trigonometric function Examples of calculations An example: numerical integration The complete code

3 1 Installation The software GNU Octave is a mathematical software mainly devoted to simulation and numerical computing. It is a free software that is the source code for GNU Octave may be freely donwloaded and examined. There is an important community of user, sharing informations, pieces of code or even complete modules to solve some common or less common problem of engineering. To install GNU Octave, one can download it from the official repository which is at the adress The last stable version is version which was released on July 2, Clicking on the download button, at the right of the page, one can download version for Windows, MacOS or Linux (the official development platform). 2 Variables and literal values 2.1 Variables Matlab and Octave does not process variables like C does. For these two languages there is no need to declare a variable before one can use it. The type of a variable is automatically inferred by the program with its content. One can simply write: a=5 ans=0 a = 8 0 To be sure that a computation starts with a clean environment, one can use the clear all instruction. clear all a = 5 a ans=0 a = 5 a = 5 0 The list of all already variables (with optionally their size) can be found with the who or whos instruction. b = 3 a = 5 whos 3

4 b = 3 a = 5 Variables in the current scope: Attr Name Size Bytes Class ==== ==== ==== ===== ===== a 1x1 8 double b 1x1 8 double Total is 2 elements using 16 bytes The name MATLAB is an abbreviation for MATrix LABoratory. It means that the strength of MATLAB is mainly is the processing of vectorised data. This does not mean that all values processed by this way are vectors in the sense of linear algebra, but mostly that the same processing may be applied to many different data at one time. For that goal, Matlab provides a simple way to enter line or column vectors or two-dimensionnal arrays of data. x = [1, 2, 3] # x is a line vector y = [1 ; 2 ; 3] # y is a column vector z = [5 6 7] # z is another line vector 0; x = y = z = One can see that, for a line vector, the elements are separated by a comma or space. The use of the coma is highly recommended. For a column vector, the elements are separated by a semicolon. Two-dimensional arrays (or matrices) are simply entered by the combination of these two manner. A=[1,2,3 ; 4,5,6; 7,8,9] ans=0 4

5 A = More over, one should remark that the Matlab programming language is case sensitive. a = 4 A = 5 a A ans=0 a = 4 A = 5 a = 4 A = Exercises Try it by yourself: clear all m = 7 m+2 m m^2 m = m^2 clear m ans=0 clear all A=[7,8,9,10,11,12] A(2:3) B=[0:2:10] C = transpose(1:2:11) dot(b,c) D=rand(6,6) E=linspace(0,5,6) E * E E * E 5

6 1. Questions: a) What does the expression A(2:3)? b) What does the expression B=[0:2:10]? c) What does the function transpose()? d) What does the function rand()? e) What does the function linspace(0,5,6)? Is it possible to replace it with another already known expression? 2.2 Litteral values The use of types in Matlab is not so strict as for example in the C programming language. The language provides many types: double : the mostly used type single logical : a boolean value char : for a single character int8, int16, int32, int64 : signed integers on 8, 16, 32, or 64 bits uint8, uint16, uint32, uint64 : unsigned integers typeinfo(2) typeinfo(2.2) ans= 0; scalar scalar 2.3 Numbers format Matlab provides many format to enhance the lisibility of numbers. Theses formats are: Format Example format short format short e (+001) format long format long e (+001) format short g format long g format hex 403f6a7a e format rat 3550/113 format bank

7 2.4 Generating range of datas Within Octave it is always necessary to write explicitely a set of data, if these data are equally reparted over an interval. For example, on can write: 0:1:10 0:5:50 0:0.1:3 ans= Columns 1 through 7: Columns 8 through 14: Columns 15 through 21: Columns 22 through 28: Columns 29 through 31: The command x:y:z generates a set of values, starting at x, ending at z with a step y. There is a similar command, called linspace() which has almost the same effect. For example, one can type: 7

8 linspace(0,10,11) linspace(0,50,11) linspace(0,3,31) Columns 1 through 8: Columns 9 through 16: Columns 17 through 24: Columns 25 through 31: Naturally, one can use every value as limit to generates these datas series. exemple: 0:pi/5: 2 * pi linspace(0, 2 * pi, 11) For Columns 1 through 7: Columns 8 through 11: 8

9 Columns 1 through 8: Columns 9 through 11: Operators 3.1 Arithmetic operators The arithmetical operators defined by Matlab are the standard operators, that is: + : addition, e.g : substraction e.g. 4-2 * : multiplication / : real division e.g. 4 / 3 \ : inverse division : e.g. 4 \ 3 is the same as 3 / 4. Moreover, as Matlab provides tools to vectorize operation, all these operations may be preceeded by a dot to apply them element by elements. x = [1, 2, 3] y = [2, 4, 6] x * y # A line vector cannot be multiplied by another line vector disp("result of element by element multiplication") x. * y # Multiplication element by element disp("result of : x * y") x * y # A quote means "transpose" disp("result of : x * y ") x * y # This computes the standard scalar product 0; 9

10 x = y = Result of element by element multiplication Result of : x * y Result of : x * y Integer arithmetic As previously seen the standard aritmetical operators produce double values. If one need to do integer arithmetic (mainly a division), one could use the following functions: clear all x = 17 y = 5 idivide(x,y) mod(x,y) x = 17 y = Trigonometric function As a scientific computing tool, Octave provides a whole set of trigonometric function. Normally these functions use angle in radians (the mathematical way to define 10

11 angles), but one can also find their equivalent for angles given in degree. The direct functions are: Function ~sin() ~cos() ~tan() ~cot() sec() csc() Description Sine of an angle in radians Cosine of an angle in radians Tangent of an angle in radians Cotangent of an angle in radians Secant of an angle in radians Cosecant of an angle in radians The inverse functions are: Function asin() acos() atan() acot() asec() acsc() Description The angle in radians for the given sine value The angle in radians for the given cosine value The angle in radians for the given tangent (in the first quadrant) The angle in radian for the given cotangent The angle in radians for the secant value The angle in radians for the given cosecant Adding the suffix d to these functions gives the equivalent functions but computing angles in degrees. For exemple sind(), cosd(), asind(), acosd(), etc Examples of calculations As for (almost) every other operators and functions defined by Octave, the trigonometric function may compute many values with a simple instruction. For example, one can write: clear all angle = 0:10:180 sind(angle) angle = Columns 1 through 13: Columns 14 through 19:

12 Columns 1 through 8: Columns 9 through 16: Columns 17 through 19: An example: numerical integration Suppose that one have the function f (x) = sin(x) + sin( 2 x ) between the limits 0 and π. The graph of this function is given by: One now would be able to compute the area between the graph of this function and the x -axis. For that goal one can use the integration operation, but for the current example, one will compute an approximation using the rectangle method. That is, for every computed value of the function (f (x))) one compute the area of a rectangle of height f (x) and width x i = x i+1 x i. One can show this by: 12

13 The area of one rectangle is then area = f (x i ) x i. One can compute the area of all rectangles defined by all x i and store these area into one rectangle. In our case, the value of x i is the same for all x i, but one will ignore this property and explicitely compute all the x i value The complete code # Example 1 : Numerical integration of the function # f(x) = sin(x) + sin(x/2) # over the interval [0:2pi] # frc1 22 novembre 2016, v1.0 # Step 1 : generate N values equally spaced over the intervall [0:2pi] N=10; x=linspace(0,2 * pi,10); # Step 2: compute the function f(x) for each of the values of the # vector x y=sin(x) + sin(x/2); # Step 3: compute the vector deltax which contains the difference # between the value x(i+1) and x(i) for each value of the # vector x deltax = diff(x) # REMARK : The number of elements of vector deltax is one less # than the number of element of the vector y # Step 4: now compute the surface of each rectangle over the # interval [0:2pi]. # This surface is given by the formula # rectsurf = y(i) * deltax(i), # but, due to fact that the vector deltax has not the same # number of elements this computing should be slightly adatpted rect = y(1:end-1). * deltax; # Step 5: The surface under the curve of the function can be computed by # summing the surface of all rectangles. sum(rect) 13