Telemark University College Department of Electrical Engineering, Information Technology and Cybernetics. MathScript

Transcription

1 Telemark University College Department of Electrical Engineering, Information Technology and Cybernetics Solutions So You Think You Can HANS-PETTER HALVORSEN, MathScript Part I: Introduction to MathScript Faculty of Technology, Postboks 203, Kjølnes ring 56, N-3901 Porsgrunn, Norway. Tel: Fax:

2 Table of Contents Table of Contents...ii 1 Introduction to MathScript Matrices and Vectors Solving Linear Equations Plotting Functions Calling functions in MathScript User-Defined Functions in MathScript Scripts Flow Control Additional Tasks ii

3 1 Introduction to MathScript Task 1: Math function Given the following function: ( ) Find ( ) and ( ) We find ( ): In order to do that, we type the following in the Command Window: First we need to define x: x=2 Then we can define the function: y = (4*x + 2)/x Then MathScript respond with the following answer: y = 5 We find ( ): We define a new value for x: x=5 Then we execute the function once more: y = (4*x + 2)/x 3

4 4 Introduction to MathScript Then MathScript respond with the following answer: y = 4.4 Task 2: Math function Create the following mathematical expression in MathScript: ( ) is a function of and, i.e. ( ). Use MathScript to find ( ). Tip! Use the following built-in functionality: x^2 sqrt(x) ( ) log(x) exp(x) MathScript code: x=2; y=2; z=3*x^2 + sqrt(x^2+y^2)+exp(log(x)); 1.1 Matrices and Vectors Task 3: Vectors and Matrices Type the following vector in MathScript in the Command window: [ ]

5 5 Introduction to MathScript Type the following matrix in MathScript in the Command window: * + Type the following matrix in MathScript in the Command window: [ ] MathScript Code: x=[1 2 3] A=[0 1; -2-3] C=[-1 2 0; ; 1 0 6] Use Use MathScript to find the value in the second row and the third column of matrix C. C(2,3) This gives: ans = -2 Use MathScript to find the second row of matrix C. C(2,:) This gives: ans = Use MathScript to find the third column of matrix C. C(:,3) This gives:

6 6 Introduction to MathScript ans = Task 4: Linear Algebra (You may want to skip this task if you haven t learned about Linear Algebra yet) Given the matrix A, B and C: * + * + * + Solve the following basic matrix operations using MathScript: ( ) ( ) ( ) ( ) ( ) ( ) where eig = Eigenvalues, diag = Diagonal, det = Determinant A=[0 1;-2-3] B=[1 0; 3-2] C=[1-1; -2 2] A+B A-B A' inv(a) diag(a) diag(b) det(a) det(b) det(a*b) eig(a)

7 7 Introduction to MathScript Task 5: Matrices (You may want to skip this task if you haven t learned about Linear Algebra yet) Use MathScript to prove the following: ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) where is the unit matrix Since we haven t learned about if yet, you can simple do like this: A*B This gives the following answer: ans =

8 8 Introduction to MathScript B*A This gives the following answer: ans = We see that the answers is not the same, and therefore 1.2 Solving Linear Equations Task 6: Equations (You may want to skip this task if you haven t learned about Linear Algebra yet) Given the equations: Set the equations on the following form: Find A and b. Solve the equations, i.e., find, using MathScript. A=[1 2; 3 4] b=[5;6] x=inv(a)*b The solution is displayed in the Output window:

9 9 Introduction to MathScript

10 2 Plotting Task 7: Plotting In the Command window in MathScript window input the time from t=0 seconds to t=10 seconds in increments of 0.1 seconds as follows: >>t=[0:0.1:10]; Then, compute the output y as follows: >>y=cos(t); Use the Plot command: >>plot(t,y) Task 8: Plotting Plot ( ) and ( ) for in the same plot. Make sure to add labels and a legend, and use different line styles and colors for the plots. Note! We cannot use Greek letters in MathScript, but use, e.g., theta, x or another name for your variable. Using the hold on command: The code becomes: x=0:0.01:2*pi; clf plot(x, sin(x), 'c+:') hold on 10

11 11 Plotting plot(x, cos(x), 'r:') hold off legend('sin', 'cos') xlabel('x') ylabel('f(x)') The plot becomes: You may also use the plot command like this instead of using the hold on command: plot(x, sin(x), x, cos(x)) For line colors and line-styles we have the following properties we can use for the plot function: Line Styles:

12 12 Plotting Marker specifiers: Colors:

13 13 Plotting Task 9: Plot of dynamic system Given the following differential equation: where,where is the time constant The solution for the differential equation is: ( ) Set and the initial condition ( ) Plot the solution ( ) in the time interval Add Grid, and proper Title and Axis Labels to the plot. We define a script (m-file): %Define Variabes T=5; a=-1/t; %Start Condition, etc

14 14 Plotting x0=1; t=[0:1:25] %Define the function x=exp(a*t)*x0; %Plotting plot(t,x); grid title('simulation of x'' = ax') xlabel('time') ylabel('x') The result becomes: Task 10: Sub-plots Plot Sin(x) and Cos(x) in 2 different subplots. Add Titles and Labels. We define a script: % Define x-values

15 15 Plotting x=0:0.01:2*pi; % subplot 1 subplot(2,1,1) plot(x, sin(x)) title('plotting sin(x)') xlabel('x') ylabel('sin(x)') % Subplot 2 subplot(2,1,2) plot(x, cos(x)) title('plotting cos(x)') xlabel('x') ylabel('cos(x)') This gives the following plot:

16 3 Functions 3.1 Calling functions in MathScript Task 11: Statistics Given the vector: >>x=[ ] Find the mean value of the vector x. Find the minimum value of the vector x. Find the maximum value of the vector x. x=[ ] mean(x) min(x) max(x) 3.2 User-Defined Functions in MathScript Task 12: User-defined function Create a function calc_average that finds the average of two numbers. Test the function afterwards as follows: >>x=2; >>y=4; >>z=calc_average(x,y) 16

17 17 Functions Make sure to enter search path for the function: The function becomes: 3.3 Scripts Task 13: Script Create a new Script where you use the function you created in the previous task. Call it several times with different inputs.

18 18 Functions Task 14: Conversion It is quite easy to convert from radians to degrees or from degrees to radians. We have that: This gives: [ ] [ ] [ ] [ ] ( ) [ ] [ ] ( ) Create two functions that convert from radians to degrees (r2d(x)) and from degrees to radians (d2r(x)) respectively. Test the functions to make sure that they work as expected. The functions are as follows: function d = r2d(r) d=r*180/pi; and function r = d2r(d) r=d*pi/180; Testing the functions: >> r2d(2*pi) ans = 360 >> d2r(180) ans = Or: x = 0:0.1:360; x2 = d2r(x) y = sin(x2) plot(x,y)

19 19 Functions From the tests, we see the functions are correct.

20 4 Flow Control Task 15: For Loop Extend your calc_average function from a previous task so it can calculate the average of a vector with random elements. Use a For loop to iterate through the values in the vector and find sum in each iteration, e.g.: mysum = mysum + x(i); Test the function in the Command window Function: function av = calc_average2(x) %This function calculates the average of a vector x N=length(x); tot=0; for i=1:n tot=tot+x(i); end av=tot; Calling the function: x=rand(100) calc_average2(x) Task 16: If-else Depending on a variable a, different functions should be executed and plotted. ( ) ( ) ( ) Create a script that plot based on the value of the input. 20

21 21 Flow Control Code: clear clc a=1; %Try with different values here! x = 0:0.1:2*pi; if a == 1 y=sin(x) elseif a == 2; y=cos(x) elseif a == 3 y=tan(x); else y=x; end plot(x,y) Task 17: Fibonacci Numbers In mathematics, Fibonacci numbers are the numbers in the following sequence: 0, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, By definition, the first two Fibonacci numbers are 0 and 1, and each subsequent number is the sum of the previous two. Some sources omit the initial 0, instead beginning the sequence with two 1s. In mathematical terms, the sequence of Fibonacci numbers is defined by the recurrence relation: with seed values: Write a function in MathScript that calculates the N first Fibonacci numbers, e.g., >> N=10; >> fibonacci(n) ans =

22 22 Flow Control Use a For loop to solve the problem. Fibonacci numbers are used in the analysis of financial markets, in strategies such as Fibonacci retracement, and are used in computer algorithms such as the Fibonacci search technique and the Fibonacci heap data structure. They also appear in biological settings, such as branching in trees, arrangement of leaves on a stem, the fruitlets of a pineapple, the flowering of artichoke, an uncurling fern and the arrangement of a pine cone. function f = fibonacci(n) f=zeros(n,1); f(1)=0; f(2)=1; for k=3:n f(k)=f(k-1)+f(k-2); end

23 5 Additional Tasks Task 18: If-else Given the second order algebraic equation: The solution (roots) is as follows: where - there is no solution, - any complex number is a solution { Create a function that finds the solution for based on different input values for a, b and c, e.g., function x = solveeq(a,b,c) Use if-else statements to solve the problems Test the function from the Command window to make sure it works as expected, e.g., >> a=0, b=2, c=1 >> solveeq(a,b,c) You may define the function like this: function x = solveeq(a,b,c) if a~=0 x = zeros(2,1); x(1,1)=(-b+sqrt(b^2-4*a*c))/(2*a); x(2,1)=(-b-sqrt(b^2-4*a*c))/(2*a); elseif b~=0 23

24 24 Additional Tasks x=-c/b; elseif c~=0 disp('no solution') else disp('any complex number is a solution') end We test the function: >> a=0, b=2,c=1 a = 0 b = 2 c = 1 >> solveeq(a,b,c) ans = Or: >> a=1;, b=2;,c=1; >> solveeq(a,b,c) ans = -1 0 Or: >> a=0;, b=0;,c=1; >> solveeq(a,b,c) No solution Or: >> a=0;, b=0;,c=0; >> solveeq(a,b,c) Any complex number is a solution Task 19: Basic Math function Create a function that calculates the following mathematical expression: ( )

25 25 Additional Tasks The function becomes: function z=calcexpression(x,y) z=3*x^2 + sqrt(x^2+y^2)+exp(log(x)); Testing the function gives: >> x=2; y=2; >> calcexpression(x,y) ans = Task 20: User-defined function Create a function that uses Pythagoras to calculate the hypotenuse of a right-angled triangle, e.g.: function h = pyt(a,b) %.. h = Pythagoras theorem is as follows: Note! The function should handle that a and b could be vectors. The function may be written like this: function h = pyt(a,b) % This function calculates the hypotenuse of a right-angled triangle h=sqrt(a^2 + b^2);

26 26 Additional Tasks Testing the function in the Command window gives: >> pyt(2,3) ans = What if a and b are vectors? Let s try: >> a=rand(10,1)*10; >> b=rand(10,1)*10; >> pyt(a,b)??? Error using ==> mpower Matrix must be square. Error in ==> pyt at 4 h=sqrt(a^2 + b^2); As you see this gives an error. Let s modify our function so it possible for a and b to be vectors: function h = pyt(a,b) % This function calculates the hypotenuse of a right-angled triangle h=sqrt(a.^2 + b.^2); Testing the function gives: >> a=rand(10,1)*10; >> b=rand(10,1)*10; >> pyt(a,b) ans = As you see it now works with arrays to, thanks to a.^2 and b.^2. Task 21: Cylinder surface area Create a function that finds the surface area of a cylinder based on the height (h) and the radius (r) of the cylinder.

27 27 Additional Tasks We do like this: The function becomes: function A = cylindar_surface(h,r) % This function calculates the surface of a sylindar A = 2*pi*r^2 +2*pi*r*h; Testing the function: >> h=8 h = 8 >> r=3 r = 3 >> cylindar_surface(h,r) ans =

28 28 Additional Tasks Task 22: Create advanced expressions in MathScript Create the following expression in MathScript: ( ) ( ) ( ) ( )( ) Given Find ( ) (The answer should be ( ) ) Tip! You should split the expressions into different parts, such as: poly = num = den =. f = This makes the expression simpler to read and understand, and you minimize the risk of making an error while typing the expression in MathScript. The Script (advexpression.m) becomes: % Define the variables: a = 1; b = 3; c = 5; x = 9; % Split functions into parts for easy maintenance poly = a*x^2 + b*x + c; num = log(poly)-sin(poly); den = 4*pi*x^2 +cos(x-2)*poly; f = num/den Testing the Script: >> advexpression f =

29 29 Additional Tasks Task 23: Solving Equations Find the solution(s) for the given equations: Set the equations on the following form: The solution is: Lets find A and b: [ ] [ ] Code: >> A=[1 2; 3 4; 7 8] A = >> b=[5;6;9] b = >> x=inv(a)*b??? Error using ==> inv Matrix must be square. We get an error (as expected!) because A must be square in order to find the inverse of A. In this case we have to do the following: >> x=a\b

30 30 Additional Tasks x = Task 24: Script Given the famous equation from Albert Einstein: The sun radiates of energy per day. Calculate how much of the mass on the sun is used to create this energy per day. How many years will it take to convert all the mass of the sun completely? Do we need to worry if the sun will be used up in our generation or the next? The mass of the sun is The speed of light is: The mass on the sun is needed to create this energy per day: where: Energy per day is: This gives: Where ( )

31 31 Additional Tasks This gives: Times to it takes to convert all the mass of the sun completely: This gives: The script becomes as follows: clc %Define variables c = 3e8; %[m/s] E = 385e24; %[J/s] %Need to convert: E = E*60*60*24; %[J] %Mass of the sun used to create this amount of energy m = E/c^2 %[kg] %Total mass of sun: m_sun = 2e30; %[kg] %Number of days before sun is gone days = m_sun/m; %[days] %Number of years before sun is gone years = days/365 %[years] The Answers is: m = e+014 years = e+013 The mass on the sun is needed to create this energy per day:

32 32 Additional Tasks Numbers of years it will take to convert all the mass of the sun completely:

33 Telemark University College Faculty of Technology Kjølnes Ring 56 N-3914 Porsgrunn, Norway Hans-Petter Halvorsen, M.Sc. Telemark University College Department of Electrical Engineering, Information Technology and Cybernetics Phone: Blog: Room: B-237a