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MatLAB Lesson 5&6: Symbolic Mathematics & File I/O

MatLAB Lesson 5&6: Symbolic Mathematics & File I/O. Edward Cheung email: icec@polyu.edu.hk Room W311g. 2008. Polar Plots. In polar coordinates, the position of a point is given by an angle  and the radius. Polar plot is a plot of angles vs magnitudes for points of a function

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MatLAB Lesson 5&6: Symbolic Mathematics & File I/O

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  1. MatLABLesson 5&6: Symbolic Mathematics & File I/O Edward Cheung email: icec@polyu.edu.hk Room W311g 2008

  2. Polar Plots • In polar coordinates, the position of a point is given by an angle  and the radius. • Polar plot is a plot of angles vs magnitudes for points of a function • Compass plot is a plot of vector emanating from the origin • The polar command can be used to create polar plots. • Usage:- polar(theta, radius, ‘line specifiers’) • Example:- Plot >>t=linspace(0,2*pi,200); r=3*cos(0.5*t).^2+t; Polar(t,r)

  3. Equation Solving using the Left Division Operator • By the left division ‘\’ operator • Designed for numerical computation & faster than using the inverse; a preference method in MatLab • Works fine with systems having a unique solution • Example:- >> A=[6 12 4;7 -2 3;2 8 -9]; B=[70;5;64]; xyz=A\B The solution is [3 5 -2]

  4. Matrix Plotting • Matlab will take each column as a line/axis. A = 6 12 4 7 -2 3 2 8 -9 >> plot(A) >> grid

  5. Colormap • A colormap is an m-by-3 matrix of real numbers between 0.0 and 1.0. Each row is an RGB vector that defines one colour. The kth row of the colormap defines the k-th colour, where map(k,:) = [r(k) g(k) b(k)] specifies the intensity of red, green, and blue. • colormap(map) sets the colormap to the matrix map. colormap('default') sets the current colormap to the default colormap. • Example:- MATLAB 's default colormap contains 64 colors and the 57th color is red. >>cm = colormap; cm(57,:) ans = 1 0 0

  6. Colormap (cont.) • M-files in ..\toolbox\matlab\graph3d generate a number of colormaps. Each M-file accepts the colormap size as an argument. • For example, >>colormap(cool(128)) % creates an hsv colormap with 128 colors. If you do not specify a size, MATLAB creates a colormap the same size as the current colormap. % Demonstration of changing and rotating image colormaps >>imageext

  7. Example: cool.m

  8. Colormap demonstration • Example:- MatLab has a sample image call mandrill ..\toolbox\matlab\demos\mandrill.mat >>load mandrill % noted the map & X >>image(X) colormap map colormap default % or colormap jet resume to normal • You can try to change the colormap as in page 43 of the workbook.

  9. colorbar • The colorbar function displays the current colormap, either vertically or horizontally, in the figure window along with the graph. • Example:- page 44 of the workbook [x,y] = meshgrid([-2:0.2:2]); z = x.*exp(-x.^2-y.^2); surf(x,y,z,gradient(z)) %colour is gradient(z) colorbar colorbar horiz

  10. Formatted Output • The display command is used to display the elements of a variable without displaying the name of the variable and to display text string. • disp(variable_name) or disp(‘string’) • The fprintf command is used for formatted output • fprintf(‘formated text & data’) • does not automatically append LF for flexibility in formatting • Same formatting syntax as in C

  11. Example – Formatted Output >> p=0.052;n=100;fname='output.txt'; fprintf(1,'The sample size is %d\nwith a %5.1f%% defective rate.\n',n,100*p); The sample size is 100 with a 5.2% defective rate. >> fprintf(1,'The data is stored in %s.\n',fname); The data is stored in output.txt. Example:- > s=sprintf('%f',pi) % write pi to string s s = 3.141593 > sprintf('%s','hello') % print string

  12. Output Data to File • Writing the output to a file require 3 steps:- • Open a file using fopen • Write the output to the file using fprintf • Close the file using fclose >>fid=fopen(‘filename’,’permission’) fid = A scaler value created for file identification fclose(fid)

  13. Example on Writing Output to a File • Create a text file called ‘sin.txt’ containing a short table of the sine function: >>x=0:360; y=[x;sin(x*pi/180)]; fid=fopen('sin.txt','wt'); fprintf(fid,'%6.2f%12.8f\n',y); fclose(fid); >> type sin.txt % checking

  14. DataInput from String • sscanf reads data from string variable s, convert it according to the specified format string and return a matrix >>s = '2.7183 3.1416'; % s is a string A = sscanf(s,'%f') : % A is a vector A = 2.7183 3.1416 >> A(1) ans = 2.7183

  15. Formatted Input from File • fscanf • Read formatted data from file • Usage:- • A = fscanf(fid,format) • [A,count] = fscanf(fid,format,size) • Size=n %Read n elements into a column vector. • Size=inf %Read to the end of the file, resulting in a column vector containing the same number of elements as are in the file. • Size=[m,n] % Read enough elements to fill an m-by-n matrix, filling the matrix in column order. n can be inf, but not m.

  16. Example on Reading Formatted Data from File >>fid=fopen('earth.txt','r'); s=fscanf(fid,'%f %f',[2 inf]); fclose(fid); plot(s(1,:),s(2,:),'.'); • Rewrite for a count of data read:- • >> fid=fopen('earth.txt','r'); • [s,count]=fscanf(fid,'%f %f',[2 inf]); • count • count = • 13840 • ginput enables you to select points from the figure using the mouse for cursor positioning. e.g. [x,y] = ginput(10)

  17. Other File Formats • MatLab has many file I/O functions, the following are examples, you can call help to explore them. >> help iofun • Popular file formats dlmread / dlmwrite - Read & write ASCII delimited file. • A wizard uiimport is available for import data. xlsread / xlswrite - data and text from a spreadsheet in an Excel workbook. • Internet functions:- urlread / urlwrite - Read & Write URL as a string. • From Ports Serial – construct a serial port object for fopen, etc.

  18. Symbolic Processing • A symbolic expression is stored as a character string and it is one of the data types supported by MatLab • Based on Maple from The University of Waterloo • http://www.maplesoft.com • Symbolic toolbox contains functions in calculus, linear algebra, simplification and solution of equations, variable precision arithmetic, transforms (derivatives, integrals) • User can manipulate symbolic expressions for simplification, solve and evaluate equations in symbolic form, do Calculus, do linear algebra, Laplace & Fourier transforms • Symbolic object can be created using sym() function or syms • Syms is a short-hand notation to sym() • Use findsym() to identify the variables in an expression • Given a function f(x), MatLab can provide the derivative and the integral in symbolic form using symbolic math toolbox

  19. Symbolic Variables To create symbolic variable >>syms x y % shortcut to create sym variables >>z=5*x % z is sym if x is sym variable >>symx % ans=sym(‘x’) (x undefined) >>x=sym('x') % create sym variable x using sym() >>v=sym('u*sin(pi*t/2)') % create fun from string v = u*sin(pi*t/2) % u, t are absent in the workspace window because they are not explicitly defined % MatLab does not indent symbolic variables like numerical variables (ans does not indented) • The memory size of symbolic variable is variable • Convert symbolic matrix to numeric form:- • Usage:- r = double(S)

  20. Example on Symbolic Toolbox Applications >> x=y^2 % define function x >> ztrans(x) % get z transform of x ans = z*(z+1)/(z-1)^3 >> laplace(x) % get Laplace transform of x ans = 2/s^3 >> u=cos(x) % we can also do substitution u =cos(y^2) >> laplace(u) % get Laplace transform of u ans =1/2*2^(1/2)*pi^(1/2)*sin(1/4*s^2)*(1/2-FresnelC(1/2*s*2^(1/2)/pi^(1/2)))+1/2*2^(1/2)*pi^(1/2)*cos(1/4*s^2)*(1/2-FresnelS(1/2*s*2^(1/2)/pi^(1/2)))

  21. Symbolic Toolbox Application (cont. example) • FresnelC is the Fresnel Cosine Integral defined as:- • Over fifty special functions are available from the Symbolic Toolbox and can be evaluated using mfun(‘function’, variable) >> help mfun • Function list is available with >>mfunlist

  22. Special Function Evaluation - Example • Let’s try to plot the Fresnel Cosine Integral >> z=0:0.01:10; >> y=mfun('FresnelC',z); >> plot(z,y)

  23. Example - Factorization of Large Integer >> n=sym('224') % To factorize integer 224 >> factor(n) ans = (2)^5*(7) >> 2^5*7 ans = 224 % Try a 40-digit number >>n1=sym('1234567890123456789012345678901234567890') >> factor(n1) ans = (2)*(3)^2*(5)*(73)*(101)*(137)*(1676321)*(5964848081)*(3803)*(3607)*(27961)*(3541) >> isprime(1676321) % check prime factor ans = 1 % 0=false 1=true

  24. Factorization & Simplification with Symbolic Toobox >> syms x y % define x, y symbolic variables >> y=2*(x+3)^2/(x^2+6*x+9); % define function y >> [num,denum]=numden(y) % separate num & denum num =2*(x+3)^2 denum =x^2+6*x+9 >> expand(num) ans =2*x^2+12*x+18 >> factor(num) ans =2*(x+3)^2 >> factor(denum) ans =(x+3)^2 >> simplify(y) ans =2

  25. Numerical Calculations with Symbolic Expressions • Symbolic substitution subs(s,old,new) can be used to subsititute numerical value to evaluate symbolic expression • Usage:- Y=subs(Expression, {var1,var2},{n1,n2}) >> syms x m b % create symbolic variables y=m*x+b; % define eqn >> subs(y,{x,b,m},{1,10,2}) % subst values into eqn ans = 12 >> diff(y) ans = m

  26. Expansion using Collect() Symbolic variables are stored as strings. Collect() can be used to expand a polynomial >> y=x^2*(x^3+x)*(x^2*3*x+3)*(x-4); >> collect(y) ans = 3*x^9-12*x^8+3*x^7-9*x^6-12*x^5+3*x^4-12*x^3

  27. Simplify • The simplify function uses operations such as addition, multiplication, rules of fractions, powers, logarithms, special functions and trigonometric identities to simply the expression. >> eqn1=sym('fx=x^4-2*x^3-33*x^2+18*x+216'); simplify(eqn1) ans = fx = x^4-2*x^3-33*x^2+18*x+216 • As the expression is being placed in quotes, x is not defined as symbolic variable. This is not recommended as it will cause error if x is being called later by other expressions. >>syms x; eqn2=cos(x)^2+sin(x)^2; simplify(eqn2) ans =1

  28. Example on the Simple Function • The simple function simplify a symbolic expression using different methods and report the shortest result. >> syms x; num=x^2+4*x+4; den=x+2; fx=num/den simple(fx) simplify: x+2 radsimp: x+2 combine(trig): x+2 factor: x+2 expand: 1/(x+2)*x^2+4/(x+2)*x+4/(x+2) combine: (x^2+4*x+4)/(x+2) convert(exp): (x^2+4*x+4)/(x+2) convert(sincos): (x^2+4*x+4)/(x+2) convert(tan): (x^2+4*x+4)/(x+2) collect(x): (x^2+4*x+4)/(x+2) ans = x+2 >>pretty(fx) % display the function in typeset form

  29. Further Example on Simple Function • Rework eqn1 in previous slide:- >> eqn1=sym('fx=x^4-2*x^3-33*x^2+18*x+216'); simplify(eqn1) simple(eqn1) ans = fx = (x-3)*(x-6)*(x+4)*(x+3) • Alternative usage for simple(). • In the following example, the shortest form is assigned to g and the name of the simplification method used is assigned to how as a character string >> [g how]=simple(eqn1) fx = (x-3)*(x-6)*(x+4)*(x+3) how = factor

  30. Symbolic Expression Plotting >> syms x y y=x^2; ezplot(y) grid on ezplot(y,[0,3]) grid on

  31. Polynomial & Symbolic Expressions • poly2sym() & sym2poly() convert vectors or polynomials into symbolic expressions or vice versa >>p=[1 0 -7 6]; p_sym=poly2sym(p) p_sym = x^3-7*x+6 >> pretty(p_sym) >> p1=sym2poly(p_sym) p1 = 1 0 -7 6 • horner() transpose a symbolic polynomial into its nested (Horner) representation >> horner(y^2+y) % ans = (1+y)*y >> px=x^4+2*x^3+x+10 horner(px) ans = 10+(1+(x+2)*x^2)*x

  32. Solve Equations • The solve(eqn) sets the expression to 0 and solve for roots >> y=sym('3*x-9'); solve(y) ans = 3 • If the equation has one variable, the solution is numerical. If the equation has more than one variables, a solution will be obtained for any of the variables in terms of others • Usage: y=solve(eqn,var) >> eqn=sym('y=x^2+2*t*x+30') solve(eqn) ans = [ -t+(t^2+y-30)^(1/2)] [ -t-(t^2+y-30)^(1/2)] syms t x solve(eqn,t) ans =-1/2*(-y+x^2+30)/x

  33. Solving System of Equations • Usage:- soln= solve(eqn1, eqn2,….,eqnN) • soln is the name of the structure in the form • Structure_name.field_name • Example >> eqn1=sym('5*x+6*y=20'); eqn2=sym('6*x+7*y=34'); soln=solve(eqn1,eqn2) soln = x: [1x1 sym] % name of solution structure y: [1x1 sym] >> soln.x % display the answer ans = 64 >> soln.y ans = -50

  34. Differentiation • Diff() can be used to determine the derivative of function • Usage:- diff(function) or diff(function, variable) • Example:- >> sym(‘x’); p_sym = x^3-7*x+6; pretty(diff(p_sym)) 2 3 x - 7 pretty(diff(p_sym,2)) % double differentiation 6 x

  35. Partial Differentiation >> syms r t x >> r=6*x+8*cos(3*t) r = 6*x+8*cos(3*t) >> diff(r,t) ans = -24*sin(3*t) >> diff(r,x) ans = 6

  36. Integration >> syms x >> y=cos(x) y = cos(x) >> int(y) ans = sin(x) >> int(y,0,pi) ans = 0 >> int(y,0,pi/2) ans = 1

  37. Integration with respect to a variable >> syms n x >> eqn=x^n; >> int(eqn,n) ans = 1/log(x)*x^n

  38. Example on Integration • Integral sometimes does not exist >> syms x eqn=int(1/(x-1)) eqn = log(x-1) >> eqn=int(1/(x-1),0,2) Warning: Explicit integral could not be found. >> eqn=int(1/(x-1),2,3) eqn = log(2) >> double(eqn) %conversion ans = 0.6931 The indefinite integral exist but the definite integral does not exist because a singularity at x=1

  39. Ordinary Differential Equation • First order differential equation • t is an independent variable • x is a function of t • Solution is in a form of x=g(t) + C • C can be determined when applying boundary conditions • x=x(t1) when t=t1 • If the condition obtained at t=0, it is called the initial condition • Using symbolic toolbox, we can obtain an analytical solution rather than numerical solution

  40. Solution to ODE using dsolve >> help dsolve • Usage:- dsolve('eqn1','eqn2', ...) • By default, the independent variable is 't'. It can be changed by including a different variable at the last input argument. • The letter 'D' denotes differentiation • e.g., D2y denotes the second derivative of y w.r.t. t • Due to this syntax, the names of symbolic variables cannot contain the letter "D". • Initial conditions are specified by equations like 'y(a)=b' or 'Dy(a) = b‘ • dsolve accepts up to 12 input arguments • dsolve returns a warning message, if it cannot find an analytic solution – may try numerical solution

  41. First Order ODE Solution >> dsolve('Dx+2*x=10') ans = 5+exp(-2*t)*C1 % note C1 is the constant % with initial condition that e.g. x(0)=0 >> y=dsolve('Dx+2*x=10','x(0)=0') y = 5-5*exp(-2*t) >> ezplot(y)

  42. Solution of Set of Equations >>[x,y]=dsolve('Dx=3*x+4*y','Dy=-4*x+3*y') x = exp(3*t)*(cos(4*t)*C1+sin(4*t)*C2) y = exp(3*t)*(-sin(4*t)*C1+cos(4*t)*C2) • Example:- • Solution:-

  43. Second Order ODE with Boundary Conditions • Example:- • Boundary conditions:- >>y=dsolve('D2y+9*y=0','y(0)=1','Dy(pi)=2') y = -2/3*sin(3*t)+cos(3*t)

  44. Solution to First Order Nonlinear Differential Equations >> dsolve('Dy=4-y^2','y(0)=1') ans = (2*exp(4*t+log(-3))+2)/(-1+exp(4*t+log(-3))) simple(ans) ans = (-6*exp(4*t)+2)/(-1-3*exp(4*t)) simple(ans) ans = (6*exp(4*t)-2)/(1+3*exp(4*t)) • Sometimes, more than one application of simple function is needed to get a simplified answer

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