1 Getting started Math4414 Matlab Tutorial We start by defining the arithmetic operations in matlab in the following tables Arithmetic Operations * mu

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1 1 Getting started Math4414 Matlab Tutorial We start by defining the arithmetic operations in matlab in the following tables Arithmetic Operations * multiplication + addition - subtraction n left division / right division ^ power At the matlab prompt type >> format long >> orange = 3 orange= 3 >> orange = 3; >> orange orange = 3 >> apple = 13; >> orange + apple 13 >> c = apple*orange; >> c c = 39 >> b = c*orange/5 + apple^3 1

2 b = e+03 ( *10^3 ) >> d = sqrt(b) d = One may change the format of the output using the command 'format' as defined in the following table. format short format long format short e format long e format short g format long g format rational Let us change the format to see the effect >> format >> f = 2/3 5 digits 16 digits 5 digits scientific 16 digits scientific 5 digits 16 digits rational number f = >> format short >> g = 2/3 g = >> format short e >> 2/ e-01 >> format long e >> 2/ e-01 Vectors in Matlab 2

3 >>v = [1; 3; -3; 4] v = >> v = [ ] v = >> w = 2*v; >> z = w + v z = >> z' (change the orientation of the vector) >>x = 0:0.2:3; >> length(x) 16 >> x(8:12) >> y = cos(x); >> y y = Columns 1 through 7 3

4 Columns 8 through Columns 15 through >> y(1) ( will print y(1)) Operations of vectors and scalars if a =[a 1 ;a 2 ; a n ], b =[a 1 ;a 2 ; ;a n ],c is a scalar. a + c = [a 1 + c; a 2 + c; ;a n + c] a Λ c = [a 1 Λ c; a 2 Λ c; ;a n Λ c] a + b = [a 1 + b 1 ;a 2 + b 2 ; ;a n + c] a: Λ b = [a 1 Λ b 1 ;a 2 Λ b 2 ; ;a n Λ b n ] a:=b = [a 1 =b 1 ;a 2 =b 2 ; ;a n =b n ] a: b = [b 1 =a 1 ;b 2 =a 2 ; ;b n =a n ] a:^b = [a b1 1 ;ab2 2 ; ;abn n ] c:^b = [c a1 ;c a2 ; ;c an ] a:^c = [a c 1 ;ac 2 ; n] ;ac >> u = [w,v] (will app the two vectors) >> norm(v,p) ( computes the L^p norm, p>=1) 4

5 2 Buit-in functions sin cos tan artan acos asin exp log log10 log2 sinh cosh tanh sec abs sign max min sine function cosine function tangent inverse tangent inverse cosine inverse sine exponential natural logarithm decimal logarithm binary logarithm hyperbolic sine hyperbolic sine hyperbolic sine secant absolute value sign function maximum minimum These function may accept scalar, vector, and matrix arguments. 3 Plotting curves and surfaces The following table contains some important commands to plot curves and surfaces. figure(n) clf hold on plot xlabel ylabel zlabel xaxis yaxis zaxis subplot view surf contour meshgrid pcolor create or change figure n clears current figure hold existing plots in a figure plots a curve defines the label for x axis defines the label for y axis defines the label for z axis define range and tics on x axis define range and tics on y axis define range and tics on z axis define and switch subplots change view for surfaces plots surfaces plots contour for surfaces create coordinates for grid points plots solution in a plane 5

6 For example the following program will plot the graph of y = cos(2x)+ exp(x), 0» x» 2 in dash-dot line and blue color >> x = 0:0.1:2; >>y = cos(2*x) + exp(x); >>plot(x,y,'-.b') You may set colors using b: blue, r:red, y:yellow and g:green. The type of lines is set by -:solid,..:dotted, -.:dash-dot, +:plus symbol: for other symbols and lines types consult the online tutorial. To plot two curves on the same figure use 'hold on'. >> x = 0:0.1:2; >>y = cos(2*x) + exp(x); >>z = tanh(x) ; >>plot(x,y,'-.b'); >>hold on >>plot(x,z,'-r'); Let us plot a 3D graph of z = cos(xy)+cos(x) >>figure(1) >> x = 0:0.1:2; >> y = 0:0.2:4; >> [X,Y] = meshgrid(x,y); >> Z = cos(x.*y) + cos(x); >>surf(x,y,z); >> View(30,65) >>title(' 3D Surface'); >>figure(2); >>pcolor(x,y,z); >>figure(3) >>contour(x,y,z,10); In order to create 4 plots in one figure. use >>subplot(2,2,1) %(to plot in the upper-left plot) >>subplot(2,2,2) %(to plot in the upper-right plot) >>subplot(2,2,3) %(to plot in the lower-left plot) >>subplot(2,2,4) %(to plot in the lower-right plot) On the figure window you may (i) left click on File menu and export your image as eps, tiff,. (ii) left click on theleft arrow and then on the plot and a window will appear where you can ajust the axes marks, labels, color and type of the plot. (iii) left click ona and left click on the plot to add text to your plot 6

7 4 Matrices >> A = [ 2 3 ; -7 9 ] A = >> C = [ 8 9 ; ] C = >> B = A + C B = >> D = A*B D = >> det(a) 39 >> inv(a) Matrix functions 7

8 det(a) A' eig(a) lu(a) qr(a) A n f norm(a,p) size(a) sparse full(as) hess(a) cond(a,p) luinc(a) trace(a) svd(a) ones(n,m) zeros(n,m) eye(n) spy(a) speye spdiag gmres(a,f) qmr(a,f) cgs(a,f) For more help determinant transpose of A eigenvalues of A lu factorization QR factorization solve Ax=f norm of A,in L p ; p =1; 2; inf, orfro: gives the number of rows and columns define a sparse matrix transform matrix from sparse format to full compute the Hessenberg form p =1; 2; inf; fro: computes the condition number incomplete lu factorization trace of A singular value decomposition an n m matrix containing all ones an n m matrix containing all zeros an n n identity matrix shows the sparse structure sparse identity matrix sparse diagonal matrix use gmres to solve Ax=f use qmr to solve Ax=f Preconditioned conjugate gradient 5 Programming in Matlab Relational Operators < less than <= less or equal > greater than >= greater or equal == equal!= not equal to Logical Operators & j e and or not Control Flow 8

9 One may write a set of instructions and save them in one file with extension :m Example of a script to illustrate the use of if if (a > 2) & a ( a < 11) else z = cos(a)^2; z = exp(a); A script that uses while i=1; while ( i < 14) x(i) = cos(pi*i/13); i = i+1; The following script computes the zero of your machine function epsilon = machinezero epsilon = 1; while (1+epsilon)> 1 epsilon = epsilon; epsilon = 2*epsilon Passing functions as arguments to other functions Consider the file mysum.m which contains the following function function S = mysum(x,y,myfun) S = feval(myfun,x) + feval(myfun,y); you would call it as >> x = 12; >>y=-13.8; >> d = mysum(x,y,'sin'); Timing your code: To time you code or parts of it you may use the cputime command as illustrated in the following example 9

10 >> t0 = cputime; >> x=12; >> y=13; >> s = mysum(x,y,'sin'); >>t1 = cputime >> cpu_time = t1-t0; One line help may be obtained by typing at the matlab prompt >>help or >> help norm %(to get help on norm) >>help lu % (help on how to use lu function) >>help format %(help on format) You may also check the online manual at 6 Solvin nonlinear algebraic equations The bisection method Let f be a continuous function on [a; b] such that f(a)f (b) < 0. Then there exists at least one root x Λ 2 (a; b) such that f(x Λ )=0. The bisection method consists in computing c = a+b, if the f(a)f(c) < 0 the 2 next interval will be [a; c], otherwise we use the interval [c; b]. We continue this process until we reach aninterval whose length and/or jf(c)j is less than a prescribed tolerance. function [a,b]=bisection(a1,b1,fun,eps,nmax) %This function uses the bisection method to approximate %roots of f(x)=0 %input: %[a1,b1] interval that contains a root %fun: the function f %eps: tolerance %nmax: maximum number of iterations allowed %output: [a,b] a final interval containing the root % fp1 = feval(fun,a1); fp2 = feval(fun,b1); if(fp1*fp2 > 0) % display(' This function might not have a root in this interval'); iter = 1; while (b1-a1) > eps & (iter <= nmax) 10

11 a=a1; b=b1; p = (a1+b1)/2; fp = feval(fun,p); if (fp*fp1 <0) else b1 = p; fp2 = fp; a1 = p; fp1 = fp; iter = iter +1; Advantages: 1- one function evaluation per iteration 2- converges to a root for all [a; b] such that f(a)f(b) < 0 Disadvantages: Linear convergence Newton-Raphson method We assume f 2 C 2, select an initial guess x 0 and approxmiate f with the tangent line to f at x 0 as f(x) ß f 0 (x 0 )(x x 0 )+f(x 0 )= ~ f(x) and solve ~ f 0 (x) = 0 to define x 1 x 1 = x 0 f(x 0) f 0 (x 0 ) We continue this process from x 1 to get x 2 as follows x n = x n 1 f(x n 1) f 0 (x n 1) ;n 1 Advantages: Quadratic convergence, i.e., jx n x Λ j lim n!1 jx n 1 x Λ = j ratio > o 2 Disadvantages: Newton method requires 1. the initial x 0 to be close enough to x Λ for convergence 2. two function evaluations per iteration 3. both f and f 0 11

12 function [x1,iter,fail] = newton(x0,tol,nmax,f,fd) %This routine find a root of f starting from %the initial guess x0 % %input parameters %f function f %fd derivative of f %x0: initial guess %tol: tolerance %Nmax: maximum number of iterations allowed % err = 10; iter = 0; while (i<= Nmax) & (abs(err) > tol) err = feval(f,x0)/feval(fd,x0); x1 = x0 - err; if (abs(err) < tol) & (iter ==Nmax) fail = 1; x0=x1; iter = iter + 1; 25 Newton s method 20 x 2 4 x + 1 = x 0 0 x 0 = 2.5 x 1 x 2 x 2 x Figure 1: Geometric illustration of Newton method for x 2 4x+1 = 0, x 0 = 2:5 12

13 Results for Newton's method applied to x^2-4 *x + 1 =0, with x0 = 0 and x0=5 n x_n f(x_n) x_n - ( 2 -sqrt(3)) e e e e e e e e e e e e e-16 n x_n f(x_n ) x_n -- ( 2 +sqrt(3)) e e e e e e e e e e e e e e e e Secant Method In order to avoid the use of derivative f 0 we use the secant method which consists in approximating f by a linear function ~ f that passes through two points (x 0 ;f(x 0 ) and (x 1 ;f(x 1 )) as f(x) ß f(x 1) f(x 0 ) (x x 0 )+f(x 0 )= f(x) ~ x 1 x 0 We solve ~ f(x) = 0 to obtain x 2 as In general we have x n+1 = x n x 2 = x 1 Advantages: 1. One function evaluation per iteration 2. Derivative is not required 3. Superlinear convergence x 1 x 0 f(x 1 ) f(x 0 ) f(x 0) x n x n 1 f(x n ) f(x n 1) f(x n 1); n 1: jx n x Λ j lim n!1 jx n 1 x Λ = ratio > o j1:6 function [p,iter]=secant(a1,b1,fun,eps,nmax) %This function computes approximation to roots of f(x)=0 %using the secant method %input: %a1,a2 are two intial guesses %fun : function f 13

14 %eps : tolerance %nmax: maximum number of iterations %output: %p : approximation to the root of f(x)=0 % fp1 = feval(fun,a1); % iter = 1; while abs(b1-a1) > eps & (iter <= nmax) fp2 = feval(fun,b1); p= a1 - fp1*(b1 -a1)/(fp2-fp1); a1 = b1; fp1 = fp2; b1 = p ; iter = iter +1; [a1,b1,abs(b1-a1),iter] 25 Secant method 20 for x 2 4x + 1 = 0 x 0 = 2 x 1 = x 2 x 1 x 4 0 x 0 x Figure 2: Geometric illustration of the secant method for x 2 4x +1 = 0, x 0 = 2 and x 1 =3. Results for the secant method applied to x^2-4x +1 =0, with x0 = -2, x1 = 3 14

15 n x_n x_n - (2+sqrt(3)) ratio e e e e e e e e e e e e e e e e e e e e e e e e e e e e e