1. Basis of Mathematical Modeling LECTURE 3 Numerical Analysis with MATLAB Dr. N.K. Sakhnenko, PhD, Professor Associate

2. Outline Finding roots with MATLAB Interpolation Numerical Integration Optimization Problems: Minimizing Functions of One Variable; Minimizing Functions of Several Variables; Linear Programming

3. Root finding Computing roots of the polynomials Polynomials are represented in MATLAB by their coefficients in the descending order of powers. For instance, the cubic polynomial p(x) = 3x 3 + 2x 2 + 1 is represented as p = [3 2 0 1]. Its roots can be found using function roots. >> format short >> r = roots(p) r = 0.1667 + 0.5528i 0.1667 - 0.5528i

4. Let now f be an one variable function f=f(x). MATLAB function fzero computes a zero of the function f using user supplied initial guess. In the following example let f(x) = cos(x) – x. First we define a function y = f 1 (x) function y = f1(x) y = cos(x) - x; To compute its zero we use MATLAB function fzero >> format long >>r = fzero('f1', 0.5) r =0.73908513321516 Name of the function whose zero is computed is entered as a string. Second argument of function fzero is the initial approximation of r. Finding zeros using MATLAB function fzero f=0

5. Function fzero A user can enter a two-element vector that designates a starting interval. In our example we choose [ 0 1] as a starting interval to obtain r = fzero('f1', [0 1]) r = 0.73908513321516 One can check last result using function feval err = feval('f1', r) err =0 By adding the third input parameter tol you can force MATLAB to compute the zero of a function with the relative error tolerance tol. In our example we let tol = 10 -3 to obtain rt = fzero('f1',.5, 1e-3) rt = 0.73886572291538

6. Function fzero function f = myf(x) f = sin(x) - х.^2.*cos(x); >> x1=fzero ('myf', -5) x1 = -4.7566 >> x2=fzero ('myf', 0.1) x2 = 7.9161e-019 >> x3=fzero ('myf', 5) x3 = 4.6665 f=0

7. Systems of nonlinear equations MATLAB function fsolve computes solution of this system as well. For more information help fsolve

8. Systems of nonlinear equations function F=mysys(x) F(1)=x(1)*(2-x(2))-cos(x(1))*exp(x(2)); F(2)=2+x(1)-x(2)-cos(x(1))-exp(x(2)); >> [x,f]=fsolve('mysys', [0 0]) x = 0.7391 0.4429 f = 1.0e-011 * -0.4702 -0.6404

9. Interpolation Given set of n+1 points x k, y k, k=(0…n), with x 0 < x 1 < … < x n, find a function f(x) whose graph interpolates the data points, i.e., f(x k ) = y k, for k = 0, 1, …, n. The general form of the function interp1 is yi = interp1(x, y, xi, method), where the vectors x and y are the vectors holding the x- and the y- coordinates of points to be interpolated, respectively, x i is a vector holding points of evaluation, i.e., y i = f(x i ) and method is an optional string specifying an interpolation method. The following methods work with the function interp1:  Nearest neighbor interpolation, method = 'nearest'. Produces a locally piecewise constant interpolant.  Linear interpolation method = 'linear'. Produces a piecewise linear interpolant.  Cubic spline interpolation, method = 'spline'. Produces a cubic spline interpolant.  Cubic interpolation, method = 'cubic'. Produces a piecewise cubic polynomial.

10. Interpolation In this example, the following points (x k, y k ) = (k/5, sin(2k)), k = 0, 1, …, 5, x = 0:pi/5:pi; y = sin(2.*x); are interpolated using the 'cubic' method of interpolation. The interpolant is evaluated at the following points xi = 0:pi/100:pi; yi = interp1(x, y, xi, 'cubic'); Points of interpolation together with the resulting interpolant are displayed below plot(x, y, 'o', xi, yi), title('Cubic interpolant of y = sin(2x)')

11. Numeric Integration Two MATLAB functions quad('f ', a, b, tol) and quad8('f ', a, b, tol) are designed for numerical integration of the univariate functions. The input parameter 'f' is a string containing the name of the function to be integrated from a to b. The fourth input parameter tol is optional and specifies user's chosen relative error in the computed integral. >> quad('sin',0,pi/2) ans = 0.99999999787312 >> quad('sin',0,pi/2,0.0000001) ans = 0.99999999946896 The default accuracy is 0.001 function f=fint(x) f=exp(-x.^2); >> quad('fint',0,1) ans = 0.74682418072642

12. Numeric Integration of the bivariate functions Function dblquad computes a numerical approximation of the double integral where D = {(x, y): a < x< b, c< y <d} is the domain of integration. Syntax of the function is dblquad (fun, a, b, c, d, tol), where the parameter tol has the same meaning as in the function quad.

13. Numeric Integration of the bivariate functions The M-file esin is used to evaluate function f: function z = esin(x,y); z = exp(-x.*y).*sin(x.*y); >>result = dblquad('esin', -1, 1, 0, 1) result = -0.22176646183245

14. Optimization The problem of optimality criterion finding is in a base of most managing protocols in modern telecom technologies. The optimization problem is the problem of finding the best solution from all feasible solutions (from Wikipedia, the free encyclopedia) This may correspond to maximizing profit in a company, or minimizing loss in a conflict. For Telecom, for example, it is necessary to minimize middle delivery of packages in a network; to maximize the productivity of telecom system; to minimize the cost of the using of resources of networks; to maximize a profit of the sale of telecommunications services.

15. Minimizing Functions of One Variable For a given mathematical function of a single variable coded in an M-file, you can use the fminbnd function to find a local minimum of the function in a given interval. If one needs a maximum he should to put minus before the function. [x,fval] = fminbnd(@myfun,x1,x2) returns value of x in which the function myfun(x) has a minimum on the interval x1<x<x2 and returns fval, that is the value of the function in the minimum. function f = myfun(x) f =... % the function under consideration For minimizing one can use the following functions [x,fval] = fminsearch(@myfun,x 0 ) [x,fval] = fminunc(@myfun, x 0 ) where x 0 is initial guess.

16. Minimizing Functions of One Variable Find the minimum of the function f(x)=x 3 -2x-5 on the interval from 0 to 2 clear all; fplot('fun1',[0,2]); grid on; title('fun1'); xlabel('x'); ylabel('f(x)'); [x,fval]=fminbnd(@fun1,0,2) % fun1.m function f = fun1(x) f=x^3-2*x-5 Solution: x=0.8165; fval=-6.0887

17. Minimizing Functions of One Variable For minimizing problem one can use the following functions [x,fval] = fminsearch(@myfun,x0) [x,fval] = fminunc(@myfun, x0) where x0 is initial guess. E.g., Find the minimum of the function f(x)=arctg(x 3 -2x-5) on the interval from -2 to 2 clear all; fplot('fun2',[-2,2]); grid on; title('fun2'); xlabel('x'); ylabel('f(x)'); [x,fval]=fminunc(@fun2,0) Solution: x=0.8165; fval=-1.4080 [x,fval]=fminunc(@fun2,-1) Solution: x =-2; fval=-1.5708 % fun2.m function f=fun2(x) f=atan(x^3-2*x-5);

18. Minimizing Functions of Several Variables The fminsearch function minimizes a function of several variables. Syntax x = fminsearch(fun,x0) x = fminsearch(fun,x0,options) [x,fval] = fminsearch(...) Description fminsearch finds the minimum of a scalar function of several variables, starting at an initial estimate. This is generally referred to as unconstrained nonlinear optimization. x = fminsearch(fun,x0) starts at the point x 0 and finds a local minimum x of the function described in fun. x 0 can be a scalar, vector, or matrix.

19. Minimizing Functions of Several Variables Find the minimum of the function f(x)= 3x 2 + 2xy + y 2 +5 near the point [20, -20] clear all; x0 = [20,-20]; [x,fval] = fminunc(@fun3,x0) % fun3.m function f = fun3(x) f = 3*x(1)^2 + 2*x(1)*x(2) + x(2)^2+5;

20. Linear Programming A linear programming problem (or LP in brief) is any decision problem of the form

21. Linear Programming First, enter the coefficients f = [-5; -4; -6] A = [1 -1 1 3 2 4 3 2 0]; b = [20; 42; 30]; lb = zeros(3,1); Next, call a linear programming routine: [x,fval] = linprog(f,A,b,[],[],lb) Optimization terminated successfully. x = 0.0000 15.0000 3.0000 fval = -78.0000