Workshop of MATLAB. Section 3

Transcription

1 Workshop of MATLAB Section 3

2 Outlines Logic and Flow Control Loops Inline and temporary function definition Writing MATLAB Functions and Scripts Debugging Solving Differential Equations Symbolic Math

3 Logic and Flow Control (I) Name Description Symbol eq Equal == ne Not equal <> and ~= lt Less than < gt Greater than > le Less than or equal <= ge Greater than or equal >=» eq(a,b); means A==B» lt(a,b); means A<B» ne(a,b); means A~=B or A<>B or ~(A==B) Both of above expressions could be used in MATLAB. Relation Operation

4 Logic and Flow Control (II) Name Description Symbol Relop AND Relop OR and Circuit logical (Returns true (1) if both inputs evaluate to true, and false (0) if they do not.) Circuit logical (Returns true (1) if either input, or both, evaluate to true, and false (0) if they do not.) Wise logical (Returns true (1) if both inputs is nonzero, and false (0) if they do» or(a,b); xor means Logical EXCLUSIVE OR(Performs an exclusive OR operation on the A B corresponding elements of arrays A and B. The resulting element C(i,j,...) is logical» not(a); means ~Atrue (1) if A(i,j,...) or B(i,j,...), but not both, is nonzero.)» and(a,b); means A&B Both of above expressions could be used in MATLAB. not) or Wise logical (Returns true (0) if both inputs is zero, and false (1) if they do not) not Logical NOT(Returns true (0) if both inputs is nonzero, and false (1) if they do not) && & ~ xor(a,b) any True if any element of vector is nonzero any(a) all True if all elements of vector are nonzero all (A) Logical Operation

5 Logic and Flow Control (III) Logical variable is an array of true and false values» Logic=[true false false true true];» whos Logic Name/Attributes Size Bytes Class m 1x5 5 logical Logical Variables

6 Logic and Flow Control (IV) Symbol Description + Addition or unary plus. - Subtraction or unary minus. * Matrix multiplication..* Array multiplication. / Slash or matrix right division. B/A = B*inv(A)./ Array right division. A./B = A(i,j)/B(i,j) \ Backslash or matrix left division. A\B = inv(a)*b \. Array left division. A.\B = B(i,j)/A(i,j) ^ Matrix power..^ Array power. A.^B = A(i,j) to the B(i,j) power Matrix transpose. A' is the linear algebraic transpose of A. Array transpose. A.' is the array transpose of A Operator Precedence

7 Logic and Flow Control (V) When we want to check for equality between two variables, we might use: A == B This is valid MATLAB code, when A and B are scalars. When A and B are matrices, A == B does not test if they are equal. It tests where they are equal. The result is another matrix of 0 s and 1 s. If A and B are not the same size, then A == B is an error. Logical Indexing for Matrices

8 Logic and Flow Control (VI)» A = magic(4);» B = A;» B(1,1) = 0;» A == B ans = The proper way to check for equality between two variables is to use the isequal function. isequal returns 1 (representing true) or 0 (false).» isequal(a,b) ans = 0 Logical Indexing

9 Logic and Flow Control (VII) isequal: isequal True if arrays are numerically equal. isequal(a,b) returns logical 1 (TRUE) if arrays A and B are the same size and contain the same values, and logical 0 (FALSE) otherwise. If A or B contains a NaN element, isequal returns false because NaNs are not equal to each other isempty: isempty True for empty array. isempty(x) returns 1 if X is an empty array and 0 otherwise. An empty array has no elements Helpful Function if Expression

10 Logic and Flow Control (VIII) if Conditionally execute statements. The general form of the if statement is if expression1 statements1 elseif expression2 statements2 else statements3 End The statements are executed if the real part of the expression has all nonzero elements The ELSE and ELSEIF parts are optional. Zero or more ELSEIF parts can be used as well as nested IF's. If, elseif, else, end

11 Logic and Flow Control (IX) yournumber = input('enter a number: '); if yournumber < 0 disp('negative') elseif yournumber > 0 disp('positive') else disp('zero') end EXAMPLES if B > A 'greater' elseif A < B 'less' elseif A == B 'equal' else error('unexpected Situation') end If, elseif, else, end

12 Logic and Flow Control (X) Switch among several cases based on expression. The general form of the SWITCH statement is: switch switch_expr case case_expr, statement,..., statement case {case_expr1, case_expr2, case_expr3,...} statement,..., statement... otherwise, statement,..., statement end Switch & case

13 Logic and Flow Control (XI) The switch_expr can be a scalar or a string. A scalar switch_expr matches a case_expr if switch_expr == case_expr. A string switch_expr matches a case_expr if strcmp(switch_expr,case_expr) returns 1 (true). If switch_expr matches the case_expr the statements following the case will be executed. Switch & case

14 Logic and Flow Control (XII) When the case expression is a cell array, the case_expr matches if any of the elements of the cell array match the switch expression. If none of the case expressions match the switch expression then the otherwise case is executed. Only one case is executed and execution resumes with the statement after the end. Only the statements between the matching case and the next case, otherwise, or end are executed. Switch & case

15 Logic and Flow Control (XIII) Example: To execute a certain block of code based on what the string, method, is set to, method = 'Bilinear'; switch lower(method) case {'linear','bilinear'} disp('method is linear') case 'cubic disp('method is cubic') case 'nearest disp('method is nearest') otherwise disp('unknown method.') end Switch & case

16 Logic and Flow Control (Exercise) 1. Make MATLAB write something depending on which test our month "passes". [hint: month = input('give month number (1-12): ' ); ] 2. Use the command rem (remainder). The syntax is rem(x,y) and returns the remainder after the division of the two integers x and y. [hint : number = input('give an integer: ' );remainder2 = rem(number,2); remainder3 = rem(number,3); ] 3. Make MATLAB show the number of day that user key in. [Hint : [daynum, daystring] = weekday(date, 'long', 'en_us'); e.g User key in Tuesday, Matlab display Day2. ] Exercise

17 Loops (I) FOR Repeat statements a specific number of times. The general form of a FOR statement is: for variable = expr statement statement end The columns of the expression are stored one at a time in the variable and then the following statements, up to the end, are executed for

18 Loops (II) The expression is often of the form X:Y, in which case its columns are simply scalars. for R = 1:20 for C = 1:20 end end A(R,C) = 1/(R+C-1); Long loops are more memory efficient when the colon expression appears in the for statement since the index vector is never created. for

19 Loops (III) Step S with increments of :0.1 for S = 1.0: -0.1: 0.0 S end Set E to the unit 5:vectors for E = eye(5) E end Set D to a defined vectors V = [ ]; for D = V D end for

20 Loops (IV) It is recommended that you do not assign to the loop control variable while in the body of a loop. for k=1:2 disp(sprintf('at the start of the loop, k = %d', k)) k = 10; disp(sprintf('following the assignment, k = %d\n', k)) end At the start of the loop, k = 1 Following the assignment, k = 10 At the start of the loop, k = 2 Following the assignment, k = 10 for

21 Loops (V) The WHILE loop repeats a group of statements an indefinite number of times. under control of a logical condition. A matching end delineates the statements. The general form of a while statement is: while expression statements end The statements are executed while the real part of the expression has all nonzero elements. while

22 While loop Loops (VI) >> h = 0.001; >> x = [0:h:2]; >> y = 0*x; >> y(1) = 1; >> i = 1; >> size(x) >> max(size(x)) >> while(i<max(size(x))) y(i+1) = y(i) + h*(x(i)-abs(y(i))); i = i + 1; end >> plot(x,y,'go') >> plot(x,y) while

23 Loops (VII) Continue The continue statement passes control to the next iteration of the for loop or while loop in which it appears. Skipping any remaining statements in the body of the loop. Break break terminates the execution of for and while loops. In nested loops, break exits from the innermost loop only. break is not defined outside of a for or while loop. Return return terminates the current sequence of commands and returns control to the invoking function or to the keyboard. return is also used to terminate keyboard mode. jumper

24 Loops (VIII) Continue % The loop below will calculate and print values of k^2-50 % for all values of the requested k % for which k^2-50 is positive. for k=-10:1:10 if (k^2-50<0) continue; end val=k^2-50; fprintf( \n k=%g val=%g,k,val) end jumper

25 Loops (IX) Break % The loop below will calculate values of k^2-50 % for all values of the requested k % until it turns negative for k=-10:1:10 if (k^2-50<0) break; end val=k^2-50; fprintf( \n k=%g val=%g,k,val) end jumper

26 Loops (X) Return % The loop below will calculate values of k^2-50 % for all values of the requested k % terminate once the condition not fullfill for k=-10:1:10 if (k^2-50<0) return; end val=k^2-50; fprintf( \n k=%g val=%g,k,val) end jumper

27 Loops (Exercise) 1. Create a vector x with the elements with FOR loop a. 2, 4, 6, 8,... b. 10, 8, 6, 4, 2, 0, -2, -4 c. 1, 1/2, 1/3, 1/4, 1/5, Create a vector with the elements with WHILE loop a. 2, 4, 6, 8,... b. 10, 8, 6, 4, 2, 0, -2, -4 c. 1, 1/2, 1/3, 1/4, 1/5, Construct a vector containing the squares of the integers 2 through Compute and display the even powers of 2 less than or equal to val. 5. Create an Hilbert matrix, H. [Hint : H(i,j) = 1/(i+j-1)] 6. Produces a vector with the same elements as x, but they are arranged in the reverse order. 7. The program prompts the user to enter any number. If the number is less than 0, the break statement terminates the execution of the loop. If the number is greater than 10, the continue statement skips the value and jumps to the do while loop without terminating the loop. Otherwise, it will print the entered number. Exercise

28 Inline and temporary function definition (I) Inline functions are like function M-files, i.e. they accept (usually numerical) input and return output. inline( expr,arg, n ) In the first command (expr) the formula is edited as a string, i.e. it is quoted by apostrophes. In the second command (arg) the string is evaluated and converted to an inline function. With the third argument (n) we fix the parameter of the inline function. Inline and Temporary Function Definition

29 Inline and temporary function definition(ii) Example: Let's defined the function y(x) introduced symbolically in the previous section as inline function: y= (1+x) 2 /(1+x 2 )» '(1+x).^2./(1+x.^2) ans = (1+x).^2./(1+x.^2)» y= y=inline(ans,'x ) y = Inline function: y(x) = (1+x).^2./(1+x.^2) This function is also array smart w.r.t. x:» y(1),y([1 2 3]) ans = 2 ans = fplot(y,[-1 1]) Inline and Temporary Function Definition

30 Inline and temporary function definition» argnames(y),formula(y) ans = 'x' (III) ans = (1+x).^2./(1+x.^2) Inline and Temporary Function Definition

31 Inline and temporary function definition (Exercise) 1. Define the function y(x)=(1+x) 2 /(1+x 2 ). 2. Define as inline function. Evaluate this function at 4 and plot it on the interval [-5,5]. [tips : fplot function] 3. Make an inline function g(x)=x+cos(x 2 ). Plot it using vector x=-5:.1:5 and y=g(x). Exercise

32 Writing MATLAB Functions and Scripts (I) Inline Function f = inline('sin(x)','x'); Operation and evaluation takes place in the current workspace Script Function function y = f(x) y = sin(x) Operate in separate workspaces Comparison

33 Writing MATLAB Functions and Scripts (II) Projectile Modeling Assume a projectile by initial velocity V=10 m/s and A=60 degree as its angel of throwing calculate time and distance of first contact with earth also and max height in second and meter respectively.» V = 20;» A = 60;» g = 9.8;» Ar = pi*a/180;» Vv = V*sin(Ar);» Vh = V*cos(Ar);» T = 2*Vv/g;» t = 0:.01:T;» H = Vv*t g/2*(t.^2);» D = Vh*t;» plot(d,h);» Hmax = max(h)» Dend = max(d) A Model Example

34 Writing MATLAB Functions and Scripts (III) We already knows command history and its functionality. One of command history capabilities is its ability to save a list of command as an M-file. This M-file could be executed to reproduce our desired result later from command window. A Command History

35 Writing MATLAB Functions and Scripts (IV) Click on third line of modeling assignment (gravity) line. Hold shift key and click on bottom most line of command history. Right-click on selected area and click on Create M-file. Save your M-file as ProjectileModel.m. Close M-file Editor. Creating Script Files

36 Writing MATLAB Functions and Scripts (V) When we save an M-file in current directory we will be able to run it by typing its name in Command Window. Close figure. Type clc to clear Command Window. Type ProjectileModel in command prompt. Press Enter to run your first M-file Running Scripts

37 Writing MATLAB Functions and Scripts (VI) M-files often have a natural structure consisting of multiple sections. For larger files, We typically focus efforts on a single section at a time, working with the code in just that section. Similarly, when conveying information about our M-files to others, often we describe the sections of the code. To facilitate these processes, we can use M-file cells, where cell means a section of code. Specifically, MATLAB software uses cells for: Rapid code iteration in the Editor. This makes the experimental phase of our work with M-file scripts easier. Publishing M-files. This allows us to include code and results in a presentation format such as HTML. Cells

38 Writing MATLAB Functions and Scripts (VII) This is the overall process of using cells for rapid code iteration: In the MATLAB Editor, enable cell mode by selecting Cell > Enable Cell Mode. Define the boundaries of the cells in an M-file script using cell features. Once you define the cells, use cell features to navigate quickly from cell to cell in your file, evaluate the code in a cell in the base workspace, and view the results. Rapid Code Interaction by Cells

39 Writing MATLAB Functions and Scripts (VIII) Cells

40 Writing MATLAB Functions and Scripts (IX) Cells

41 Writing MATLAB Functions and Scripts (Exercise) 1. A very simple model of this would be that we have 100 units in the reservoir and remove 20% each time step (in other words, 20% of the snow melts each day). We can make a script file to calculate the amount of melting and the amount remaining after each day by making the script file melting.m. [hint : need array to store Initial amount in the Reservoir, empty array to hold the Reservoir values] 2. A simple model might be that if the reservoir is less than 30, the melting rate is 30% per day, otherwise it is 20% per day. Plot of the Reservoir and the Melting rate as a function of time by making the script file meltplot.m. [hint : need array to store Initial amount in the Reservoir, empty array to hold the Reservoir values] 3. Write a script file to draw a pretty circle [hint : angle = linspace(0, 2*pi, 360);x = cos(angle);y = sin(angle);] 4. Write a script to sums the first n squares up to n=10 and save in file sumsquares. Exercise

42 Debugging (I) Function echo disp sprintf fprintf whos size keyboard Description Display function or script code as it executes. Display specified values or messages. Display formatted data of different types. List variables in the workspace. Show array dimensions. Interrupt program execution and allow input from keyboard. return warning MException Resume execution following a keyboard interruption. Display specified warning message. Access information on the cause of an error. Debugging

43 Debugging (II) Function dbstop dbclear dbcont dbdown dbmex dbstack dbstatus dbstep dbtype dbup dbquit Description Set breakpoints Clear breakpoints Resume execution Reverse workspace shift performed by dbup, while in debug mode Enable MEX-file debugging (on UNIX platforms) Function call stack List breakpoints Execute one or more lines from current breakpoint List text file with line numbers Shift current workspace to workspace of caller, while in debug mode Quit debug mode Debugging

44 Debugging (III) test.m function a = test(b) c = sqrt(b)*cos(b); a = test1(b,c); >>dbtype test [list the line number] >>dbstop in test [set breaking point] >> test(magic(3)) >> dbstack [display the function calls that led to the breakpoint] >> dbcont [Continue execution of the function] Debugging

45 Debugging (Exercise) Create a vector with the elements with FOR loop 0, 1/2, 2/3, 3/4, 4/5,... for d1=0:4 end for d2=1:5 rats(d3) d3=d1./d2 Try debug Exercise

46 Solving Differential Equations (I) MATLAB has two functions, ode23 and ode45, which are capable of numerically solving differential equations. Solver Problem Type When to Use ode45 Nonstiff Most of the time. This should be the first solver you try. ode23 Nonstiff For problems with crude error tolerances or for solving moderately stiff problems. Differential Equation Solution

47 Solving Differential Equations (II) The syntax for actually solving a differential equation with these functions is:» [T,Y] = ode45(functionname,t0,tf,y0,options); Argument Name functionname T0 and tf y0 >>help odeset Description the name of a function that we write that describes our system of differential equations. the initial and final times that we want a solution for It returns a vector of times, T, and a matrix Y that has the values of each variable in our system. Each column of Y is a different variable. vector of the initial values of the variables in our system of equations >> option = odeset('reltol', 10^-4); options Optional integration argument created using the odeset function. to tighten the default relative The tolerance odeset function from 10^-3 lets to you 10^-4. adjust the integration parameters of the particular ODE solvers. Differential Equation Solution

48 Solving Differential Equations (III) Solve y -2y-1, y(0)=1, for 0<t<10 [T,Y] = ode45( ex1, [0,10], 1); function yprime=ex1(t,y) yprime=2*y-1; Nonlinear Solver

49 Solving Differential Equations (IV) Example (1 st order) dv/dt= -g+4/15*v 2 /m function rk=f(t,y) mass=80; g=9.81; rk=-g+4/15*y^2/mass; Save as f.m >> timerange=[0 30]; >> initialvelocity=0; >> [t,y]=ode45( f,timerange,initialvelocity) >> plot(t,y(:,1)) Type in MATLAB command window Differential Equation Solution

50 Solving Differential Equations (V) Example (2 nd order) Step1: Convert into state space form Let V=y 1 and dv/dt=y 2 Therefore, (d/dt)v= y 2 (d 2 /dt 2 ) y 1 =-2t(y 2 )-9 y 1 Differential Equation Solution

51 Solving Differential Equations (VI) Example (2 nd order) Step2: Create f2.m file function rk=f2(t,y) rk=[y(2); -2*t*y(2)-9 *y(1)]; (d/dt)v= y 2 (d 2 /dt 2 ) y 1 =-2t(y 2 )-9 y 1 Differential Equation Solution

52 Solving Differential Equations (VII) Example (2 nd order) Step3: Run in Matlab command window >> timerange=[0 5]; >> initialvelocity=[0;1]; >> [t,y]=ode45( f2,timerange,initialvelocity) >> plot(t,y) Differential Equation Solution

53 Solving Differential Equations (Exercise_1) 1. Plot the solution to y =2y-1, y(0)=1 and tspan = [0 10]. 2. Plot the solution to y =xy 2 +y with y(0)=1 and tspan = [0 10]. 3. Plot the solution to y +8y +2y=0 with y(0)=0, (d/dt)y(0)=1 and tspan = [0 10]. 4. Plot the solution to (d 2 /dt 2 )y+3*(d/dt)y + 2y=4exp(- 2t) -5 where y(0)=2, (d/dt)y(0)=-1and and tspan = [0 10]. 5. Solve and plot the solution to x +y -2x = 3 where x =0, y =1, x =0 and tspan = [0 10]. Exercise

54 Solving Differential Equations 6. Plot the solution to (Exercise_2) where σ = 10, β = 8/3, and ρ = 28, 1. x(0) = 8, y(0) = 8, and z(0) = 27 and tspan = [0 10]. 7. Plot the solution to Where y 1 (0) =600 and y 2 (0)= 0 Exercise

55 Symbolic Math (I) MATLAB is capable of doing fairly simple symbolic math analysis. It is designed for numerical work, rather than symbolic work, and in order to do symbolic math MATLAB actually uses the engine for another math software product, MAPLE. Every time we're trying to do in MATLAB's symbolic math toolkit involves mostly calling the "maple" command (which simulates Maple). Symbolic Math

56 Symbolic Math (II) For basics, though, the symbolic math toolkit is useful. A symbolic equation in MATLAB is represented as a string, such as» 'x + 2 = 5' The basic symbolic math operations all take string equations as arguments, and return string equations. MATLAB decides which variable in that string is the symbolic variable.» x = sym('x','real'); Could create a symbolic object known as x. Symbolic Math

57 Symbolic Math (III) To clear the symbolic objects x of 'real status, type» syms x clear Basic symbolic functions all work as we would expect, too.» g = ('x^2 + 2*x + 3') MATLAB also has symbolic functions such as int & diff for integrate and differentiate.» int('x^2') - int(expr,var) computes the indefinite integral of expr with respect to the symbolic scalar variable var» diff(g) simplify to simplify an expression.» f = 3*x *x» simplify(f) Symbolic Math

58 Symbolic Math (IV) taylor to generate a taylor series expansion.» syms x;» f = sin(x)/x;» t = taylor(f)» 1-1/6*x^2+1/120*x^4 We can also use solve to solve algebraic equations. solve(x + 1, x) -> 1 dsolve to solve a differential equation. dsolve('dy=y+1','x') -> -1+exp(x)*C1 Symbolic Math

59 Symbolic Math (V) The function numeric('symbolic expression') lets us convert a symbolic representation to numbers wherever possible, which also helps in simplification. Symbolic Math

60 Symbolic Math (VI) The function ezplot('symbolic function ) takes a symbolic equation and plots it as a function of x. We can specify a range or let it use the default. To specify a range, use ezplot('symbolic function',[xmin xmax]'). For more details on the symbolic toolkit by:» help sym. Symbolic Math

61 Symbolic Math (VII) Symbolic Math

62 Symbolic Math (Exercise_1) 1. Create the 3-by-4 symbolic matrix A with the auto-generated elements A1_1,..., A3_4 2. Find the determinant and the trace of the matrix A. [hint : det function and trace function] ] 3. Simplify z=sin 2 (α)+cos 2 (α). 4. Take the derivative of function f(x)=x 3 -cos(x) 5. Integrate function f(x,y).. [hint : int function] Exercise

63 Symbolic Math (Exercise_2) 6. Find the roots of this polynomial, f(x)=2x 3 +4x-8. [hint: solve roots function] 7. Symbolic variable S containing the expression x^2-y^2. Let's factor this expression. [hint : factor function] 8. We want to enter f(x,y)=(4x 2-1)e -x2-y2 as a symbolic expression and compute f(1,2). [hint : subs function] 9. Solve xy +y=2e 2x [hint :Dy= (2/x)*exp(2*x)- y/x] Exercise