MathematicsRoots, Differentiation and Integration

Prof. Muhammad Saeed

1. r = roots(p)2. r = fzero(func,x0), 3. r = fzero(func,[x1 x2])

a) r = fzero('3*x^3+2*x^2-5*x+7',5)b) r = fzero(@myfun,x0)c) r = fzero(@(x) exp(x)*sin(x),x0)d) Hfnc = @(x) x^2*cos(2*x)*sin(x*x)

r = fzero(Hfnc, [x0 x1])4. a= 1.5; r = fzero(@(x) myfun(x,a),0.1)5. options = optimset('Display','iter','TolFun',1e-8)

opts=optimset(options,'TolX',1e-4)r = fzero(fun,x0,opts)

2Mathematical modeling & Simulations

6. [r,fval] = fzero(...)7. [r,fval,exitflag] = fzero(...)8. [r,fval,exitflag,output] = fzero(...)

output.algorithm : Algorithm usedoutput.funcCount Number of function evaluationsoutput.intervaliterations: Number of iterations taken to find an

intervaloutput.iterations: Number of zero-finding iterationsoutput.message: Exit message

ExitFlags1 Function converged to a solution x.

-1 Algorithm was terminated by the output function. -3 NaN or Inf function value was encountered during search for

an interval containing a sign change. -4 Complex function value was encountered during search for an

interval containing a sign change. -5 Algorithm might have converged to a singular point.9. [….. ] =fminbnd(…)

3Mathematical modeling & Simulations

4Mathematical modeling & Simulations

[r,p,k] = residue(b,a)[b,a] = residue(r,p,k)

9732

1847

)(

)(23

23

SSS

SSS

sa

sb

1. Symbolica. syms x t z alpha; # int(-2*x/(1+x^2)^2)

# int(x/(1+z^2),z) # int(x*log(1+x),0,1)# int(2*x, sin(t), 1)

5Mathematical modeling & Simulations

2. Numerical# Z = trapz(Y)# Z = trapz(X,Y)

Example: IntegralTrapz.m# Z = quad(hfun,a,b)# Z = quad(hfun,a,b,tol)# [Z,fcnt] = quad(...)# Z= quad(@fun,a,b)# [Z, fcnt]=quad(……)# Z=quad(fun,a,b,tol,trace)# Z=quadl(……..)“The quad function may be most efficient for low accuracies with nonsmooth integrands. The quadl function may be more efficient than quad at higher accuracies with smooth integrands.”

q = quadgk(fun,a,b)[q,errbnd] = quadgk(fun,a,b,tol)[q,errbnd] = quadgk(fun,a,b,param1,val1,param2,val2,...)[q,errbnd] = quadgk(@(x)x.^5.*exp(-x).*sin(x),0,inf,

'RelTol',1e-8,'AbsTol',1e-12) “The ‘quadgk’ function may be most efficient for high accuracies and oscillatory integrands. It supports infinite intervals and can handle moderate singularities at the endpoints. It also supports contour integration along piecewise linear paths.”

q = dblquad(fun,xmin,xmax,ymin,ymax)q = dblquad(fun,xmin,xmax,ymin,ymax,tol)q = dblquad(fun,xmin,xmax,ymin,ymax,tol,method)q = dblquad(@(x,y)sqrt(max(1-(x.^2+y.^2),0)), -1, 1, -1, 1)

triplequad(fun,xmin,xmax,ymin,ymax,zmin,zmax)triplequad(fun,xmin,xmax,ymin,ymax,zmin,zmax,tol)triplequad(fun,xmin,xmax,ymin,ymax,zmin,zmax,tol,method)F = @(x,y,z)y*sin(x)+z*cos(x); Q = triplequad(F,0,pi,0,1,-1,1);

1. Symbolic• syms x

f = sin(5*x)g = exp(x)*cos(x); diff(g); diff(g,2)

• syms s t f = sin(s*t) ; diff(f,t) ; diff(f,s);diff(f,t,2);

2. Numerical• diff(x) ; diff(y)

z=diff(y)./diff(x) z=diff(y,2)./diff(x,2)• polyder(p)• polyder(a,b)

7Mathematical modeling & Simulations

8Mathematical modeling & Simulations

END

9Mathematical modeling & Simulations