numerical mathematics with matlab

Numerical MathemNumerical MathemNumerical MathemNumerical Mathem atics withatics withatics withatics with

MATLABMATLABMATLABMATLAB

Reza Abazari

2008200820082008

Contents 1. Rootfinding for Nonlinear Equations ...… ......………….………………….1-38 2. Numerical Linear Algebra ……………… ……………….……..…………..40-76 3. Polynomial Interpolation ……………… ………………….….……………78-102 4. Numerical Integration ……………… ………..…………….…………..…104-137 5. Nonlinear Systems and Numerical Optimization ……..…. ………….139-169 6. Numerical solution of ODE ………………….…… ……….…………….171-199 7. Numerical solution of PDE ………… ………..……………..……………201-235

1. Rootfinding for Nonlinear Equations

function R = approot (f,X,epsilon) % Input - f is object function % - X is the vector of abscissas % - epsilon is the tolerance % Output - R is the vector of approximate locations for roots % If f is an M-file function call R = approot (@f,X ,epsilon). % If f is an anonymous function call R = approot (f ,X,epsilon). % Copyright by Reza Abazari 2008-04-23 % Email : abazari-r@uma.ac.ir Y=f(X); yrange = max(Y)-min(Y); epsilon2 = yrange*epsilon; n=length(X); m=0; X(n+1)=X(n); Y(n+1)=Y(n); for k=2:n if Y(k-1)*Y(k) <= 0, m=m+1; R(m)=(X(k-1)+X(k))/2; end s=(Y(k)-Y(k-1))*(Y(k+1)-Y(k)); if (abs(Y(k)) < epsilon2) & (s <= 0), m=m+1; R(m)=X(k); end end

@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@ % bisec_g(f_name, a,c, xmin, xmax, n_points) % Bisection method with graphics % a, c : end points of initial interval % xmin and xmax : limits on the graphic plot. % n_points: number of points used in plotting . % function bisec_g(f_name, a,c, xmin, xmax, n_points) % it_limit : limit of iteration number % Y_a, Y_c : y values of the current end poin ts % fun_f(x) : functional value at x % Copyright by Reza Abazari 2008-04-23 % Email : abazari-r@uma.ac.ir clg, hold off; hold on clear Y_a, clear Y_c wid_x = xmax - xmin; dx = (xmax- xmin)/n_points; xp=xmin:dx:xmax; yp=feval(f_name, xp); plot(xp,yp); xlabel('x');ylabel('f(x)'); title('Bisection Method') ymin=min(yp); ymax=max(yp);wid_y = ymax-ymin;

yp=0.*xp; plot(xp,yp) fprintf( 'Bisection Scheme\n\n' ); tolerance = 0.000001; it_limit = 30; fprintf( ' It. a b c f(a) '); fprintf( ' f(c) abs(f(c)-f(a))/2\n' ); it = 0; Y_a = feval(f_name, a ); Y_c = feval(f_name, c ); plot([a,a],[Y_a,0]); text(a,-0.1*wid_y,'x=a') plot([c,c],[Y_c,0]); text(c,-0.1*wid_y,'x=c') if ( Y_a*Y_c > 0 ) fprintf( ' f(a)f(c) > 0 \n' ) ; else while 1 it = it + 1; b = (a + c)/2; Y_b = feval(f_name, b ); plot([b,b],[Y_b,0],':'); plot(b,0,'o') if it<4, text(b, wid_y/20, [num2str(it)]), en d fprintf('%3.0f %10.6f, %10.6f', it, a, b ); fprintf('%10.6f, %10.6f, %10.6f', c, Y_a, Y_ c ); fprintf( ' %12.3e\n', abs((Y_c - Y_a)/2)); if ( abs(c-a)<=tolerance ) fprintf( ' Tolerance is satisfied. \n' ); break end if ( it>it_limit ) fprintf( 'Iteration limit exceeded.\n' ); b reak end if( Y_a*Y_b <= 0 ) c = b; Y_c = Y_b; else a = b; Y_a = Y_b; end end fprintf('Final result: Root = %12.6f \n', b ); end x=b; plot([x x],[0.05*wid_y 0.2*wid_y]) text( x, 0.25*wid_y, 'Final solution') plot([x (x-wid_x*0.004)],[0.05*wid_y 0.09*wid_y]) plot([x (x+wid_x*0.004)],[0.05*wid_y 0.09*wid_y])

@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@ % bisec_n(f_name, a,c): bisection method without gr aphics % f_name: definition of the equation to solve % a , c : end points of initial inter val % function bisec_n(f_name, a,c) f_name % tolerance : tolerance % it_limit : limit of iteration number % Y_a, Y_c : y values of the current end poin ts % Copyright by Reza Abazari 2008-04-23 % Email : abazari-r@uma.ac.ir fprintf( 'Bisection Scheme\n\n' ); tolerance = 0.000001; it_limit = 30; fprintf( ' It. a b c f(a) '); fprintf( ' f(b) f(c)\n' ); it = 0; Y_a = feval(f_name, a ); Y_c = feval(f_name, c ); if ( Y_a*Y_c > 0 ) fprintf( '\n\n Stopped because f(a)f(c) > 0 \n' ); fprintf( '\n Change a or b and run again.\n' ) ;

else while 1 it = it + 1; b = (a + c)/2; Y_b = feval(f_name, b ); fprintf('%3.0f %10.6f, %10.6f', it, a, b ); fprintf('%10.6f, %10.6f, %10.6f, %10.6f\n', c, Y_a, Y_b, Y_c ); if ( abs(c-a)<=tolerance ) fprintf( ' Tolerance is satisfied. \n' ); break end if ( it>it_limit ) fprintf( 'Iteration limit exceeded.\n' ); b reak end if( Y_a*Y_b <= 0 ) c = b; Y_c = Y_b; else a = b; Y_a = Y_b; end end fprintf('Final result: Root = %12.6f \n', b ); end.

@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@ function [c,err,yc]=bisect(f,a,b,delta) %Input - f is the function % - a and b are the left and right endpoin ts % - delta is the tolerance %Output - c is the zero % - yc= f(c) % - err is the error estimate for c %If f is defined as an M-file function use the @ no tation % call [c,err,yc]=bisect(@f,a,b,delta). %If f is defined as an anonymous function use the % call [c,err,yc]=bisect(f,a,b,delta). % Copyright by Reza Abazari 2008-04-23 % Email : abazari-r@uma.ac.ir ya=f(a); yb=f(b); if ya*yb > 0,return,end max1=1+round((log(b-a)-log(delta))/log(2)); for k=1:max1 c=(a+b)/2; yc=f(c); if yc==0 a=c; b=c; elseif yb*yc>0 b=c; yb=yc; else a=c; ya=yc; end if b-a < delta, break,end end c=(a+b)/2; err=abs(b-a); yc=f(c);

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function [k,p,err,P] = fixpt(g,p0,tol,max1) %Input - g is the iteration function % - p0 is the initial guess for the fixe d-point % - tol is the tolerance % - max1 is the maximum number of iterat ions %Output- k is the number of iterations % - p is the approximation to the fixedpoint % - err is the error in the approximation % - P' contains the sequence {pn} %If g is defined as an M-file function use the @ no tation % call [k,p,err,P]=fixedpoint(@g,p0,tol,max1). %If g is defined as an anonymous function use the % call [k,p,err,P]=fixedpoint(g,p0,tol,max1). % Copyright by Reza Abazari 2008-04-23 % Email : abazari-r@uma.ac.ir P(1)= p0; for k=2:max1 P(k)=g(P(k-1)); err=abs(P(k)-P(k-1)); relerr=err/(abs(P(k))+eps); p=P(k); if (err<tol) | (relerr<tol),break;end end P=P';

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function [S,E,G]=golden(f,a,b,delta,epsilon) %Input - f is the object function % - a and b are the endpoints of the interv al % - delta is the tolerance for the abscissa s % - epsilon is the tolerance for the ordina tes %Output - S=(p,yp) contains the abscissa p and % the ordinate yp of the minimum % - E=(dp,dy) contains the error bounds for p and yp % - G is an n x 4 matrix: the kth row conta ins [ak ck dk bk]; % the values of a, c, d, and b at the kth iteration %If f is defined as an M-file function use the @ no tation % call [S,E,G]=golden(@f,a,b,delta,epsilon). %If f is defined as an anonymous function use the % call [S,E,G]=golden(f,a,b,delta,epsilon). % Copyright by Reza Abazari 2008-04-23 % Email : abazari-r@uma.ac.ir r1=(sqrt(5)-1)/2; r2=r1^2; h=b-a; ya=f(a); yb=f(b); c=a+r2*h; d=a+r1*h;

yc=f(c); yd=f(d); k=1; A(k)=a; B(k)=b; C(k)=c; D(k)=d; while (abs(yb-ya)>epsilon)|(h>delta) k=k+1; if (yc<yd) b=d; yb=yd; d=c; yd=yc; h=b-a; c=a+r2*h; yc=f(c); else a=c; ya=yc; c=d; yc=yd; h=b-a; d=a+r1*h; yd=f(d); end A(k)=a; B(k)=b; C(k)=c; D(k)=d; end dp=abs(b-a); dy=abs(yb-ya); p=a; yp=ya; if (yb<ya) p=b; yp=yb; end G=[A' C' D' B']; S=[p yp]; E=[dp dy];

@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@ function [P0,y0,err,P]=grads(F,G,P0,max1,delta,epsi lon,show) %Input - F is the object function input as an M- file function % - G = -(1/norm(grad F))*grad F; the searc h direction % input as an M-file function % - P0 is the initial starting point % - max1 is the maximum number of iteration s % - delta is the tolerance for hmin in the single parameter % minimization in the search direction % - epsilon is the tolerance for the error in y0 % - show; if show==1 the iterations are dis played %Output - P0 is the point for the minimum. % - y0 is the function value F(P0) % - err is the error bound for y0 % - P is a vector containing the iterations %Use @F and @G to call the M-file functions F and G .

% Copyright by Reza Abazari 2008-04-23 % Email : abazari-r@uma.ac.ir if nargin==5, show = 0; end [mm n] = size(P0); maxj = 10; big = 1e8; h = 1; P=zeros(maxj,n+1); len = norm(P0); y0 = F(P0); if (len>1e4), h = len/1e4; end err = 1;cnt = 0;cond = 0; P(cnt+1,:)=[P0 y0]; while (cnt<max1 & cond~=5 & (h>delta | err>epsilon) ) %Compute search direction S = G(P0); %Start single parameter quadratic minimization P1 = P0 + h*S; P2 = P0 + 2*h*S; y1 = F(P1); y2 = F(P2); cond = 0; j = 0; while (j<maxj & cond==0) len = norm(P0); if (y0<y1) P2 = P1; y2 = y1; h = h/2; P1 = P0 + h*S; y1 = F(P1); else if (y2<y1) P1 = P2; y1 = y2; h = 2*h; P2 = P0 + 2*h*S; y2 = F(P2); else cond = -1; end end j = j+1; if (h<delta), cond=1; end if (abs(h)>big | len>big), cond=5; end end if (cond==5) Pmin = P1; ymin = y1; else d = 4*y1 - 2*y0 - 2*y2; if (d<0) hmin = h*(4*y1-3*y0-y2)/d; else cond = 4; hmin = h/3; end %Construct the next point Pmin = P0 + hmin*S;

ymin =F(Pmin); %Determine magnitude of next h h0 = abs(hmin); h1 = abs(hmin-h); h2 = abs(hmin-2*h); if (h0<h), h = h0; end if (h1<h), h = h1; end if (h2<h), h = h2; end if (h==0), h = hmin; end if (h<delta), cond=1; end %Termination test for minimization e0 = abs(y0-ymin); e1 = abs(y1-ymin); e2 = abs(y2-ymin); if (e0~=0 & e0<err), err = e0; end if (e1~=0 & e1<err), err = e1; end if (e2~=0 & e2<err), err = e2; end if (e0==0 & e1==0 & e2==0), err = 0; end if (err<epsilon), cond=2; end if (cond==2 & h<delta), cond=3; end end cnt = cnt+1; P(cnt+1,:)=[Pmin ymin]; P0 = Pmin; y0 = ymin; end if (show==1) disp(P); end

@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@ % Copyright by Reza Abazari 2008-04-23 % Email : abazari-r@uma.ac.ir % Abstract : This simulation illustrates the conv ergence of the % bisection method of finding roots of an equation f(x)=0. % % INPUTS: Enter the following % Function in f(x)=0 f = inline('x^3-0.165*x^2+3.993*10^(-4)'); % Lower initial guess xl = 0.0; % Upper intial guess xu = 0.11; % Lower bound of range of 'x' to be seen lxrange = -0.02; % Upper bound of range of 'x' to be seen uxrange = 0.12; % Maximum number of iterations nmax = 30; % Guess of root for Matlab root function xrguess=0.05; % % SOLUTION % The following finds the upper and lower 'y' limit s for the plot based on the % given

% 'x' range in the input section. maxi = f(lxrange); mini = f(lxrange); for i=lxrange:(uxrange-lxrange)/10:uxrange if f(i) > maxi maxi = f(i); end if f(i) < mini mini = f(i); end end tot=maxi-mini; mini=mini-0.1*tot; maxi=maxi+0.1*tot; % This calculates window size to be used in figures set(0,'Units','pixels') scnsize = get(0,'ScreenSize'); wid = round(scnsize(3)); hei = round(0.95*scnsize(4)); wind = [1, 1, wid, hei]; % This graphs the function and two lines representi ng the two guesses figure('Position',wind) clf subplot(2,1,2),fplot(f,[lxrange,uxrange]) hold on plot([xl,xl],[maxi,mini],'y','linewidth',2) plot([xu,xu],[maxi,mini],'g','linewidth',2) plot([lxrange,uxrange],[0,0],'k','linewidth',1) title('Entered function on given interval with uppe r and lower guesses') % This portion adds the text and math subplot(2,1,1), text(0,0.8,['Check first if the low er and upper guesses bracket

the root of the equation']) axis off text(0.2,0.6,['f(xl) = ',num2str(f(xl))]) text(0.2,0.4,['f(xu) = ',num2str(f(xu))]) text(0.2,0.2,['f(xu)*f(xl) = ',num2str(f(xu)*f(xl)) ]) hold off % True solution % This is how Matlab calculates the root of a non-l inear equation. xrtrue=fzero(f,xrguess); % Value of root by iterations % Here the bisection method algorithm is applied to generate the values of the

roots, true error, % absolute relative true error, approximate error , absolute relative

approximate error, and the % number of significant digits at least correct i n the estimated root as a

function of number of % iterations. for i=1:nmax xr(i)=(xl+xu)/2; if f(xu)*f(xr(i))<0 xl=xr(i); else xu=xr(i); end end

n=1:nmax; % Absolute true error for i=1:nmax Et(i)=abs(xrtrue-xr(i)); end % Absolute relative true error for i=1:nmax et(i)=abs(Et(i)/xrtrue)*100; end % Absolute approximate error for i=1:nmax if i==1 Ea(i)=0; else Ea(i)=abs(xr(i)-xr(i-1)); end end % Absolute relative approximate error for i=1:nmax ea(i)=abs(Ea(i)/xrtrue)*100; end % Significant digits at least correct for i=1:nmax if (ea(i)>5 | i==1) sigdigits(i)=0; else sigdigits(i)=floor((2-log10(ea(i)/0.5))); end end % The graphs figure('Position',wind) plot(n,xr,'r','linewidth',2) title('Estimated root as a function of number of it erations') figure('Position',wind) subplot(2,1,1), plot(n,Et,'b','linewidth',2) title('Absolute true error as a function of number of iterations') subplot(2,1,2), plot(n,et,'b','linewidth',2) title('Absolute relative true error as a function o f number of iterations') figure('Position',wind) subplot(2,1,1), plot(n,Ea,'g','linewidth',2) title('Absolute relative error as a function of num ber of iterations') subplot(2,1,2), plot(n,ea,'g','linewidth',2) title('Absolute relative approximate error as a fun ction of number of

iterations') figure('Position',wind) bar(sigdigits,'r')

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% Copyright by Reza Abazari 2008-04-23 % Email : abazari-r@uma.ac.ir % Abstract : This simulation shows how the bisect ion method works % for finding roots of an equation f(x )=0

clear all % INPUTS: Enter the following % Function in f(x)=0 f = inline('x^3-0.165*x^2+3.993*10^(-4)'); % Upper initial guess xu = 0.11; % Lower initial guess xl = 0.0; % Lower bound of range of 'x' to be seen lxrange = -0.02; % Upper bound of range of 'x' to be seen uxrange = 0.12; % % SOLUTION % The following finds the upper and lower 'y' limit s for the plot % based on the given % 'x' range in the input section. maxi = f(lxrange); mini = f(lxrange); for i=lxrange:(uxrange-lxrange)/10:uxrange if f(i) > maxi maxi = f(i); end if f(i) < mini mini = f(i); end end tot=maxi-mini; mini=mini-0.1*tot; maxi=maxi+0.1*tot; % This calculates window size to be used in figures set(0,'Units','pixels') scnsize = get(0,'ScreenSize'); wid = round(scnsize(3)); hei = round(0.95*scnsize(4)); wind = [1, 1, wid, hei]; % This graphs the function and two lines representi ng the two guesses figure('Position',wind) clf fplot(f,[lxrange,uxrange]) hold on plot([xl,xl],[maxi,mini],'y','linewidth',2) plot([xu,xu],[maxi,mini],'g','linewidth',2) plot([lxrange,uxrange],[0,0],'k','linewidth',1) title('Entered function on given interval with init ial guesses') hold off % % Iteration 1 figure('Position',wind) xr=(xu+xl)/2; % This graphs the function and two lines representi ng the two guesses clf subplot(3,1,2),fplot(f,[lxrange,uxrange]) hold on plot([xl,xl],[maxi,mini],'y','linewidth',2) plot([xu,xu],[maxi,mini],'g','linewidth',2)

plot([xr,xr],[maxi,mini],'r','linewidth',2) plot([lxrange,uxrange],[0,0],'k','linewidth',1) title('Entered function on given interval with uppe r and lower guesses') % This portion adds the text and math to the top pa rt of the figure % window subplot(3,1,1), text(0,1,['Iteration 1']) text(0.2,.8,['xr = (xu + xl)/2 = ',num2str(xr)]) text(0,.6,['Finding the value of the function at th e lower and upper… guesses

and the estimated root']) text(0.2,.4,['f(xl) = ',num2str(f(xl))]) text(0.2,.2,['f(xu) = ',num2str(f(xu))]) text(0.2,0,['f(xr) = ',num2str(f(xr))]) axis off hold off % Check the interval between which the root lies if f(xu)*f(xr)<0 xl=xr; else xu=xr; end % This portion adds the text and math to the bottom part of the % figure window subplot(3,1,3), text(0,1,['Check the interval betwe en which the root… lies. Does

it lie in ( xl , xu ) or ( xr , xu )?']) text(0,.8,['']) text(0.2,0.6,['xu = ',num2str(xu)]) text(0.2,0.4,['xl = ',num2str(xl)]) axis off xp=xr; % % Iteration 2 figure('Position',wind) xr=(xu+xl)/2; % This graphs the function and two lines representi ng the two guesses clf subplot(3,1,2),fplot(f,[lxrange,uxrange]) hold on plot([xl,xl],[maxi,mini],'y','linewidth',2) plot([xu,xu],[maxi,mini],'g','linewidth',2) plot([xr,xr],[maxi,mini],'r','linewidth',2) plot([lxrange,uxrange],[0,0],'k','linewidth',1) title('Entered function on given interval with uppe r and lower… guesses') % This portion adds the text and math to the top pa rt of the figure % window subplot(3,1,1), text(0,1,['Iteration 2']) text(0.2,.8,['xr = (xu + xl) / 2 = ',num2str(xr)]) text(0,.6,['Finding the value of the function at th e lower and upper… guesses

and the estimated root']) text(0.2,.4,['f(xl) = ',num2str(f(xl))]) text(0.2,.2,['f(xu) = ',num2str(f(xu))]) text(0.2,0,['f(xr) = ',num2str(f(xr))]) axis off hold off % Check the interval between which the root lies if f(xu)*f(xr)<0

xl=xr; else xu=xr; end % Calculate relative approximate error ea=abs((xr-xp)/xr)*100; % This portion adds the text and math to the bottom part of the % figure window subplot(3,1,3), text(0,1,['Absolute relative approx imate error']) text(0,.8,['ea = abs((xr - xp) / xr)*100 = ',num2st r(ea),'%']) text(0,.4,['Check the interval between which the ro ot lies. Does it… lie in ( xl

, xu ) or ( xr , xu )?']) text(0.2,0.2,['xu = ',num2str(xu)]) text(0.2,0,['xl = ',num2str(xl)]) axis off xp=xr; % % Iteration 3 figure('Position',wind) xr=(xu+xl)/2; % This graphs the function and two lines representi ng the two guesses clf subplot(3,1,2),fplot(f,[lxrange,uxrange]) hold on plot([xl,xl],[maxi,mini],'y','linewidth',2) plot([xu,xu],[maxi,mini],'g','linewidth',2) plot([xr,xr],[maxi,mini],'r','linewidth',2) plot([lxrange,uxrange],[0,0],'k','linewidth',1) title('Entered function on given interval with uppe r and lower … guesses') % This portion adds the text and math to the top pa rt of the figure % window subplot(3,1,1), text(0,1,['Iteration 3']) text(0.2,.8,['xr = (xu + xl) / 2 = ',num2str(xr)]) text(0,.6,['Finding the value of the function at th e lower and upper… guesses

and the estimated root']) text(0.2,.4,['f(xl) = ',num2str(f(xl))]) text(0.2,.2,['f(xu) = ',num2str(f(xu))]) text(0.2,0,['f(xr) = ',num2str(f(xr))]) axis off hold off % Check the interval between which the root lies if f(xu)*f(xr)<0 xl=xr; else xu=xr; end % Calculate relative approximate error ea=abs((xr-xp)/xr)*100; % This portion adds the text and math to the bottom part of the % figure window subplot(3,1,3), text(0,1,['Absolute relative approx imate error']) text(0,.8,['ea = abs((xr - xp) / xr)*100 = ',num2st r(ea),'%']) text(0,.4,['Check the interval between which the ro ot lies. Does it… lie in ( xl

, xu ) or ( xr , xu )?'])

text(0.2,0.2,['xu = ',num2str(xu)]) text(0.2,0,['xl = ',num2str(xl)]) axis off xp=xr; % % Iteration 4 figure('Position',wind) xr=(xu+xl)/2; % This graphs the function and two lines representi ng the two guesses clf subplot(3,1,2),fplot(f,[lxrange,uxrange]) hold on plot([xl,xl],[maxi,mini],'y','linewidth',2) plot([xu,xu],[maxi,mini],'g','linewidth',2) plot([xr,xr],[maxi,mini],'r','linewidth',2) plot([lxrange,uxrange],[0,0],'k','linewidth',1) title('Entered function on given interval with uppe r and lower guesses') % This portion adds the text and math to the top pa rt of the figure window subplot(3,1,1), text(0,1,['Iteration 4']) text(0.2,.8,['xr = (xu + xl) / 2 = ',num2str(xr)]) text(0,.6,['Finding the value of the function at th e lower and upper guesses and

the estimated root']) text(0.2,.4,['f(xl) = ',num2str(f(xl))]) text(0.2,.2,['f(xu) = ',num2str(f(xu))]) text(0.2,0,['f(xr) = ',num2str(f(xr))]) axis off hold off % Check the interval between which the root lies if f(xu)*f(xr)<0 xl=xr; else xu=xr; end % Calculate relative approximate error ea=abs((xr-xp)/xr)*100; % This portion adds the text and math to the bottom part of the figure window subplot(3,1,3), text(0,1,['Absolute relative approx imate error']) text(0,.8,['ea = abs((xr - xp) / xr)*100 = ',num2st r(ea),'%']) text(0,.4,['Check the interval between which the ro ot lies. Does it lie in ( xl

, xu ) or ( xr , xu )?']) text(0.2,0.2,['xu = ',num2str(xu)]) text(0.2,0,['xl = ',num2str(xl)]) axis off xp=xr; % % Iteration 5 figure('Position',wind) xr=(xu+xl)/2; % This graphs the function and two lines representi ng the two guesses clf subplot(3,1,2),fplot(f,[lxrange,uxrange]) hold on plot([xl,xl],[maxi,mini],'y','linewidth',2) plot([xu,xu],[maxi,mini],'g','linewidth',2) plot([xr,xr],[maxi,mini],'r','linewidth',2)

plot([lxrange,uxrange],[0,0],'k','linewidth',1) title('Entered function on given interval with uppe r and lower guesses') % This portion adds the text and math to the top pa rt of the figure window subplot(3,1,1), text(0,1,['Iteration 5']) text(0.2,.8,['xr = (xu + xl) / 2 = ',num2str(xr)]) text(0,.6,['Finding the value of the function at th e lower and upper guesses and

the estimated root']) text(0.2,.4,['f(xl) = ',num2str(f(xl))]) text(0.2,.2,['f(xu) = ',num2str(f(xu))]) text(0.2,0,['f(xr) = ',num2str(f(xr))]) axis off hold off % Check the interval between which the root lies if f(xu)*f(xr)<0 xl=xr; else xu=xr; end % Calculate relative approximate error ea=abs((xr-xp)/xr)*100; % This portion adds the text and math to the bottom part of the figure window subplot(3,1,3), text(0,1,['Absolute relative approx imate error']) text(0,.8,['ea = abs((xr - xp) / xr)*100 = ',num2st r(ea),'%']) text(0,.4,['Check the interval between which the ro ot lies. Does it lie in ( xl

, xu ) or ( xr , xu )?']) text(0.2,0.2,['xu = ',num2str(xu)]) text(0.2,0,['xl = ',num2str(xl)]) axis off xp=xr;

@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@ function [p,y,err]=muller(f,p0,p1,p2,delta,epsilon, max1) %Input - f is the object function % - p0, p1, and p2 are the initial approximat ions % - delta is the tolerance for p0, p1, and p 2 % - epsilon the the tolerance for the functio n values y % - max1 is the maximum number of iterations %Output - p is the Muller approximation to the zer o of f % - y is the function value y = f(p) % - err is the error in the approximation of p. %If f is defined as an M-file function use the @ no tation % call [p,y,err]=muller(@f,p0,p1,p2,delta,epsilon,m ax1). %If f is defined as an anonymous function use the % call [p,y,err]=muller(f,p0,p1,p2,delta,epsilon,ma x1). % Copyright by Reza Abazari 2008-04-23 % Email : abazari-r@uma.ac.ir %Initalize the matrices P and Y P=[p0 p1 p2]; Y=f(P); %Calculate a and b in formula (15)

for k=1:max1 h0=P(1)-P(3);h1=P(2)-P(3);e0=Y(1)-Y(3);e1=Y(2)-Y (3);c=Y(3); denom=h1*h0^2-h0*h1^2; a=(e0*h1-e1*h0)/denom; b=(e1*h0^2-e0*h1^2)/denom; %Suppress any complex roots if b^2-4*a*c > 0 disc=sqrt(b^2-4*a*c); else disc=0; end %Find the smallest root of (17) if b < 0 disc=-disc; end z=-2*c/(b+disc); p=P(3)+z; %Sort the entries of P to find the two closest t o p if abs(p-P(2))<abs(p-P(1)) Q=[P(2) P(1) P(3)]; P=Q; Y=f(P); end if abs(p-P(3))<abs(p-P(2)) R=[P(1) P(3) P(2)]; P=R; Y=f(P); end %Replace the entry of P that was farthest from p with p P(3)=p; Y(3) = f(P(3)); y=Y(3); %Determine stopping criteria err=abs(z); relerr=err/(abs(p)+delta); if (err<delta)|(relerr<delta)|(abs(y)<epsilon) break end end

@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@ function [c,err,yc]=regula(f,a,b,delta,epsilon,max1 ) %Input - f is the function % - a and b are the left and right endpoints % - delta is the tolerance for the zero % - epsilon is the tolerance for the value of f at the zero % - max1 is the maximum number of iterations %Output - c is the zero % - yc=f(c) % - err is the error estimate for c %If f is defined as an M-file function use the @ no tation % call [c,err,yc]=regula(@f,a,b,delta,epsilon,max1) %If f is defined as an anonymous function use the

% call [c,err,yc]=regula(f,a,b,delta,epsilon,max1) % Copyright by Reza Abazari 2008-04-23 % Email : abazari-r@uma.ac.ir ya=f(a); yb=f(b); if ya*yb>0 disp('Note: f(a)*f(b) >0'), return, end for k=1:max1 dx=yb*(b-a)/(yb-ya); c=b-dx; ac=c-a; yc=f(c); if yc==0,break; elseif yb*yc>0 b=c; yb=yc; else a=c; ya=yc; end dx=min(abs(dx),ac); if abs(dx)<delta,break,end if abs(yc)<epsilon, break,end end c; err=abs(b-a)/2; yc=f(c);

@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@ function [p1,err,k,y]=secant(f,p0,p1,delta,epsilon, max1) %Input - f is the object function % - p0 and p1 are the initial approximation s to a zero of f % - delta is the tolerance for p1 % - epsilon is the tolerance for the functio n values y % - max1 is the maximum number of iterations %Output - p1 is the secant method approximation t o the zero % - err is the error estimate for p1 % - k is the number of iterations % - y is the function value f(p1) %If f is defined as an M-file function use the @ no tation % call [c,err,yc]=bisect(@f,a,b,delta). %If f is defined as an anonymous function use the % call [c,err,yc]=bisect(f,a,b,delta). % Copyright by Reza Abazari 2008-04-23 % Email : abazari-r@uma.ac.ir for k=1:max1 p2=p1-f(p1)*(p1-p0)/(f(p1)-f(p0)); err=abs(p2-p1); relerr=2*err/(abs(p2)+delta);

p0=p1; p1=p2; y=f(p1); if (err<delta)|(relerr<delta)|(abs(y)<epsilon),bre ak,end end

@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@ function [p0,err,k,y]=newton(f,df,p0,delta,epsilon, max1) %Input - f is the object function % - df is the derivative of f % - p0 is the initial approximation to a zero of f % - delta is the tolerance for p0 % - epsilon is the tolerance for the funct ion values y % - max1 is the maximum number of iteratio ns %Output - p0 is the Newton-Raphson approximation to the zero % - err is the error estimate for p0 % - k is the number of iterations % - y is the function value f(p0) %If f and df are defined as M-file functions use th e @ notation % call [p0,err,k,y]=newton(@f,@df,p0,delta,epsilon, max1). %If f and df are defined as anonymous functions use the % call [p0,err,k,y]=newton(f,df,p0,delta,epsilon,m ax1). % Copyright by Reza Abazari 2008-04-23 % Email : abazari-r@uma.ac.ir for k=1:max1 p1=p0-f(p0)/df(p0); err=abs(p1-p0); relerr=2*err/(abs(p1)+delta); p0=p1; y=f(p0); if (err<delta)|(relerr<delta)|(abs(y)<epsilon),bre ak,end end

@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@ function [V0,y0,dV,dy]=nelder(F,V,min1,max1,epsilon ,show) %Input - F is input as an M-file function % - V is a 3xn matrix containing starti ng simplex % - min1 & max1 are minimum and maximum number of iterations % - epsilon is the tolerance % - show == 1 displays iterations (P an d Q) %Output - V0 is the vertex forthe minimum % - y0 is the function value F(V0) % - dV is the size of the final simplex % - dy is the error bound for the minim um % - P is a matrix containing the vertex iterations % - Q is an array containing the iterat ions for F(P) %Call [V0,y0,dV,dy]=nelder(@F,V,min1,max1,epsilon, show) % Copyright by Reza Abazari 2008-04-23 % Email : abazari-r@uma.ac.ir

if nargin==5, show=0; end [mm n]=size(V); % Order the vertices for j=1:n+1 Z=V(j,1:n); Y(j)=F(Z); end [mm lo]=min(Y); [mm hi]=max(Y); li=hi; ho=lo; for j=1:n+1 if(j~=lo&j~=hi&Y(j)<=Y(li)) li=j; end if (j~=hi&j~=lo&Y(j)>=Y(ho)) ho=j; end end cnt=0; % Start of Nelder-Mead algorithm while (Y(hi)>Y(lo)+epsilon&cnt<max1)|cnt<min1 S=zeros(1:n); for j=1:n+1 S=S+V(j,1:n); end M=(S-V(hi,1:n))/n; R=2*M-V(hi,1:n); yR=F(R); if (yR<Y(ho)) if (Y(li)<yR) V(hi,1:n)=R; Y(hi)=yR; else E=2*R-M; yE=F(E); if (yE<Y(li)) V(hi,1:n)=E; Y(hi)=yE; else V(hi,1:n)=R; Y(hi)=yR; end end else if (yR<Y(hi)) V(hi,1:n)=R; Y(hi)=yR; end C=(V(hi,1:n)+M)/2; yC=F(C); C2=(M+R)/2;

yC2=F(C2); if (yC2<yC) C=C2; yC=yC2; end if (yC<Y(hi)) V(hi,1:n)=C; Y(hi)=yC; else for j=1:n+1 if (j~=lo) V(j,1:n)=(V(j,1:n)+V(lo,1:n))/2; Z=V(j,1:n); Y(j)=F(Z); end end end end [mm lo]=min(Y); [mm hi]=max(Y); li=hi; ho=lo; for j=1:n+1 if (j~=lo&j~=hi&Y(j)<=Y(li)) li=j; end if (j~=hi&j~=lo&Y(j)>=Y(ho)) ho=j; end end cnt=cnt+1; P(cnt,:)=V(lo,:); Q(cnt)=Y(lo); end % End of Nelder-Mead algorithm %Determine size of simplex snorm=0; for j=1:n+1 s=norm(V(j)-V(lo)); if(s>=snorm) snorm=s; end end Q=Q'; V0=V(lo,1:n); y0=Y(lo); dV=snorm; dy=abs(Y(hi)-Y(lo)); if (show==1) disp([P Q]) end

@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@ % this m-file calculates the real roots of the give n polynomial using % newton raphson technique.this m-file calls the fu nctions in the two % m-files named as syn_division.m and derivate.m. % coeff_function is the 1xn matrix conatining the c oeff of the

% polynomial. % Copyright by Reza Abazari 2008-12-09 % Email : abazari-r@uma.ac.ir function [final_roots,functionvalue] =

newton(coeff_function,initial_guess,error_tolerance ,max_iterations) iterations=0; max_no_roots=size(coeff_function,2); final_roots=0; functionvalue=0; for no_roots=1:max_no_roots-1 fun_root_new=initial_guess; flag=1; coeff_der_function=derivate(coeff_function); order_fun=size(coeff_function,2); order_der_fun=size(coeff_der_function,2); while flag==1 fun_root_old=fun_root_new; fx=0; dfx=0; nonzero=1; while nonzero==1 powers=order_fun-1; for index=1:order_fun fx=fx+coeff_function(index)*fun_roo t_old^powers; powers=powers-1; end powers=order_der_fun-1; for index=1:order_der_fun dfx=dfx+coeff_der_function(index)*f un_root_old^powers; powers=powers-1; end if dfx==0 fun_root_old=fun_root_old+1; else nonzero=0; end end iterations = iterations + 1; fun_root_new = fun_root_old - fx/dfx; if iterations >= max_iterations flag=0; elseif abs(fun_root_new-fun_root_old)<=err or_tolerance flag=0; final_roots(no_roots)=fun_root_new; functionvalue(no_roots)=fx; end end coeff_function=syn_division(coeff_function,fun_ root_new); end

++++++++++++++++++++++++++++++++++++++++++

% This m-file calculates the derivative of the func tion, the % limitation of % this function is, it can calculate only the deriv atives of % power(x,n).... % Copyright by Reza Abazari 2008-04-23 % Email : abazari-r@uma.ac.ir

function coeff_derivative=derivate(coeff_function) der_order=size((coeff_function),2)-1; coeff_derivative=0; for index=1:size((coeff_function),2)-1 coeff_derivative(index)=der_order*coeff_functio n(index); der_order=der_order-1; end

++++++++++++++++++++++++++++++++++++++++++ % This m-file takes care of synthetic division. % By giving one polynomial and one root this functi on returns % the polynomial formed with the other roots of the given polynomial % excluding the given root. % Copyright by Reza Abazari 2008-04-23 % Email : abazari-r@uma.ac.ir function coeff_second=syn_division(coeff_function,f un_root_new) order_fun=size((coeff_function),2); coeff_second=0; for index=1:size((coeff_function),2)-1 if index==1 coeff_second(index)=coeff_function(index); else

coeff_second(index)=coeff_function(index)+fun_root_ new*coeff_second(index-1); end end

@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@ function A=remez(fun, fun_der,interval,order) %This M.file used findzero.m and err.m that’s giv en the end of this m.file. % findzero.m % Copyright by Reza Abazari 2008-12-09 % Email : abazari-r@uma.ac.ir powers=ones(order+2,1)*([0:order]);% the powers of the polynomial % repeated in ro ws (order +2) % times coeff_E =(-1).^[1:order+2]; coeff_E=coeff_E(:); % the coefficients of the E as a column array t=1:order; t=t(:); % the powers of the polynomial starting f rom 1 in a column. % This is used when differntiation the po lynomial y=linspace(interval(1),interval(2),order+2); % the first choice of % the (order+2) points for i=1:10 y=y(:); % make the points array a column arr ay h=(y-interval(1))*ones(1,order+1); % repeat the points column % minus the start of the % interval ( order +1) times coeff_h=h.^powers; % raise the h matrix by the % power matr ix elementwise

M=[coeff_h coeff_E]; % the matrix of the LHS of the % linear sys tem of equations N= feval(fun,y); % the column vecto r of the RHS % of the linear sy stem of equations A=M\N; % solution of the linear syst em of equations, % first (order +1) element ar e the coefficients % of the polynomial. Last ele ment is the value % of the error at these point s A1=A(1:end-1); % the coefficients only A_der=A(2:end-1).*t; % the coeffcients o f the derivative % of the polynomial z(1)=interval(1); % z(1) is the start poi nt of the interval z(order+3)=interval(2); % z(order+3) is the end point of the % interval % in between we fill in with the roots of the e rror function for k=1: order+1 z(k+1)=findzero(@err,y(k),y(k+1),fun,A1 ,interval(1)); end % between every two points in the array z, we s eek the point that % maximizes the magnitude of the error function . If there is an extreme % point (local maximum or local minimum) betwee n such two points of the % z array then the derivative of the error func tion is zero at this % extreme point. We thus look for the extreme p oint by looking for the % root of the derivative of the error function between these two % points. If the extreme point doesn't exist th en we check the value of % the error function at the two current points of z and pick the one % that gives maximum magnitude. for k=1:order+2 if sign(err(z(k),fun_der,A_der,interval(1) ))~=... sign(err(z(k+1),fun_der,A_der,interval(1 ))) % check for a change in sign y1(k)=findzero(@err,z(k),z(k+1),fun_der,A _der,interval(1)); % the extreme point that we seek v(k)=abs(err(y1(k),fun,A1,interval(1))); % the value of the error function at the extreme point else % if there is no change in sign therefor e there is no extreme % point and we compare the endpoints of the sub-interval v1=abs(err(z(k),fun,A1,interval(1))); % magnitude of the error function at the start of the sub-interval v2=abs(err(z(k+1),fun,A1,interval(1))); % magnitude of the error function at the end of the sub-interval % pick the larger of the two if v1>v2 y1(k)=z(k); v(k)=v1; else y1(k)=z(k+1); v(k)=v2; end end end [mx ind]=max(v); % search for the point in the extreme points array that gives maximum % magnitude for the error function % if the difference between this point and the corr essponding point % in the old array is less than a certain threshold then quit the loop

if abs(y(ind)-y1(ind)) <2^-30 break; end % compare it also with the following poi nt if it is not % the last point if ind<length(y) & abs(y(ind+1)-y1(ind)) < 2^- 30 break end % replace the old points with the new points y=y1; end

++++++++++++++++++++++++++++++++++++++++++ function y=findzero(fun,x0,x1,varargin) % fun is the function that we need to compute its r oot. % x0 and x1 are two arguments to the function such that the root that % we seek lie between them and the function has dif ferent sign at % these two points varargin are the other arguments of the function % the value of the function at the first point f0=feval(fun,x0,varargin{:}); %the value of the function at the second point f1=feval(fun,x1,varargin{:}); % check that the sign of the function at x0 and x1 is different. % Otherwise report an error % Copyright by Reza Abazari 2008-04-23 % Email : abazari-r@uma.ac.ir if sign(f0)==sign(f1) error('the function at the two endpoints must b e of opposite... signs'); end % find a closer point to the root using the method of chords. In the % method of chords we simply connect the two points (x0, f(x0)) and %(x1, f(x1))using a straight line and compute the i ntersection point % with this line and the horizontal axis. This new point is closer to % the desired root x=x0 - f0 * ((x1-x0)/(f1-f0)); %evaluate the function at this new point f=feval(fun,x,varargin{:}); % enter this root as long as the difference between the two points % that sandwitch the desired root is larger than a certain threshold while abs(f)>2^-52 % we keep one of the two old points that has a different sign than the % new point and we overwrite the other old poin t with the new point if sign(f)==sign(f0) x0=x; f0=f; else x1=x; f1=f; end x=x0 - f0 * ((x1-x0)/(f1-f0)); f=feval(fun,x,varargin{:}); end

% at the end of the loop we reach the root with the desired precision and % it is given by x y=x;

++++++++++++++++++++++++++++++++++++++++++ function e= err(x,fun, A, first) % Copyright by Reza Abazari 2008-04-23 % Email : abazari-r@uma.ac.ir % the polynomial coefficients array , make it a col umn array A=A(:); % the argument array , make it a column array x=x(:); % order of the polynomial is equal to the number of coefficients % minus one order=length(A)-1; % the powers out in a row and repeated for each arg ument to form a % matrix for example if the order is 2 and we have 3 arguments in x % then % [0 1 2] % powers= [0 1 2] % [0 1 2] powers=ones(length(x),1)*[0:order]; % each argument is repeated a number of times equal to the number of % coefficients to form a row then each element of t he resulting row % is raised with the corresponding power in the pow ers matrix temp=((x-first)*ones(1,order+1)).^powers; % multiply the resulting matrix with the coefficien ts table in order % to obtain a column array. Each element of the res ulting array is % equal to the polynomial evaluated at the distance between the % corresponding argument and the start of the inter val temp=temp*A; % the error vector is then given as the difference between the % function evaluated at the argument array and the polynomial % evaluated at the argument array e=feval(fun,x)-temp; Test:

some test for this m.file: %Testing Script% % Example 1 fun=inline('exp(x)'); fun_der= inline('exp(x)'); interval=[0, 2^(-10)]; order =2; A= remez(fun, fun_der, interval, order); A1=A(1:end-1); % the 3 coefficients of the second order polynomial E=A(end); % the maximum approximation error

% plotting the error of the whole interval x=0: 2^-15:2^-10 ; e=err(x,fun, A1, interval(1)); plot(x,e) xlabel('x') ylabel('e(x)=f(x)-p(x)') title('Error function for when approximating exp(x) ') % Example 2 fun=inline('sin(x)'); fun_der= inline('cos(x)'); interval=[0, 2^(-10)]; order =2; A= remez(fun, fun_der, interval, order); A1=A(1:end-1); % the 3 coefficients of the second order polynomial E=A(end); % the maximum approximation error % plotting the error of the whole interval x=0: 2^-15:2^-10 ; e=err(x,fun, A1, interval(1)); figure; plot(x,e) xlabel('x') ylabel('e(x)=f(x)-p(x)') title('Error function for when approximating sin(x) ')

@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@

function r = bisect(fun,xb,xtol,ftol,verbose) % Copyright by Reza Abazari 2008-04-23 % Email : abazari-r@uma.ac.ir % bisect Use bisection to find a root of the scala r equation f(x) = 0 % % Synopsis: r = bisect(fun,xb) % r = bisect(fun,xb,xtol) % r = bisect(fun,xb,xtol,ftol) % r = bisect(fun,xb,xtol,ftol,verbose) % % Input: fun = (string) name of function for wh ich roots are sought % xb = vector of bracket endpoints. xle ft = xb(1), xright = xb(2) % xtol = (optional) relative x tolerance. Default: xtol=5*eps % ftol = (optional) relative f(x) toleran ce. Default: ftol=5*eps % verbose = (optional) print switch. Default : verbose=0, no printing % % Output: r = root (or singularity) of the functio n in xb(1) <= x <= xb(2) if size(xb,1)>1, warning('Only first row of xb is u sed as bracket'); end if nargin < 3, xtol = 5*eps; end if nargin < 4, ftol = 5*eps; end if nargin < 5, verbose = 0; end xeps = max(xtol,5*eps); % Smallest toleran ces are 5*eps feps = max(ftol,5*eps); a = xb(1,1); b = xb(1,2); % Use first row if xb is a matrix xref = abs(b - a); % Use initial brac ket in convergence test fa = feval(fun,a); fb = feval(fun,b); fref = max([abs(fa) abs(fb)]); % Use max f in con vergence test

if sign(fa)==sign(fb) % Verify sign chan ge in the interval error(sprintf('Root not bracketed by [%f, %f]',a, b)); end if verbose fprintf('\nBisection iterations for %s.m\n',fun); fprintf(' k xm fm\n'); end k = 0; maxit = 50; % Current and max number of iterations while k < maxit k = k + 1; dx = b - a; xm = a + 0.5*dx; % Minimize roundoff in co mputing the midpoint fm = feval(fun,xm); if verbose, fprintf('%4d %12.4e %12.4e\n',k,xm, fm); end if (abs(fm)/fref < feps) | (abs(dx)/xref < xeps) % True when root is found r = xm; return; end if sign(fm)==sign(fa) a = xm; fa = fm; % Root lies in interval [ xm,b], replace a and fa else b = xm; fb = fm; % Root lies in interval [ a,xm], replace b and fb end end warning(sprintf('root not within tolerance after %d iterations\n',k)); @@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@ function Xb = brackPlot(fun,xmin,xmax,nx) % Copyright by Reza Abazari 2008-04-23 % Email : abazari-r@uma.ac.ir % brackPlot Find subintervals on x that contain si gn changes of f(x) % % Synopsis: Xb = brackPlot(fun,xmin,xmax) % Xb = brackPlot(fun,xmin,xmax,nx) % % Input: fun = (string) name of mfile function that evaluates f(x) % xmin,xmax = endpoints of interval to s ubdivide into brackets. % nx = (optional) number of samples alon g x axis used to test for % brackets. The interval xmin <= x <= xmax is divided into % nx-1 subintervals. Default: nx = 20. % % Output: Xb = two column matrix of bracket limi ts. Xb(k,1) is the left % (lower x value) bracket and Xb(k, 2) is the right bracket for % the k^th potential root. If no b rackets are found, Xb = []. if nargin<4, nx=20; end % --- Create data for a plot of f(x) on interval xm in <= x <= xmax xp = linspace(xmin,xmax); yp = feval(fun,xp); % --- Save data used to draw boxes that indicate br ackets ytop = max(yp); ybot = min(yp); % y coord inates of the box ybox = 0.05*[ybot ytop ytop ybot ybot]; % around a bracket c = [0.7 0.7 0.7]; % RGB col or used to fill the box % --- Begin search for brackets x = linspace(xmin,xmax,nx); % Vector of potenti al bracket limits f = feval(fun,x); % Vector of f(x) va lues at potential brackets

nb = 0; Xb = []; % Xb is null unless brackets are found for k = 1:length(f)-1 if sign(f(k))~=sign(f(k+1)) % True if sign of f (x) changes in the interval nb = nb + 1; Xb(nb,:) = [x(k) x(k+1)]; % Save left and rig ht ends of the bracket hold on; fill([x(k) x(k) x(k+1) x(k+1) x(k)],y box,c); % Add filled box end end if isempty(Xb) % Free advice warning('No brackets found. Check [xmin,xmax] or increase nx'); return; % return without drawing a plot end % --- Add plot here so that curve is on top of boxe s used to indicate brackets plot(xp,yp,[xmin xmax],[0 0]); grid on; xlabel('x'); if isa(fun,'inline') ylabel(sprintf('Roots of f(x) = %s',formula(fun)) ); % label is formul in f(x) else ylabel(sprintf('Roots of f(x) defined in %s',fun) ); % label is name of m-file end hold off @@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@ function y = bisection ( f, a, b, TOL ) %BISECTION bisection method for locating the root of a nonlinear function % calling sequences: % y = bisection ( 'f', a, b, TOL ) % bisection ( 'f', a, b, TOL ) % inputs: % f string containing name of m-f ile defining function % whose root is to be located % a,b left and right endpoints, res pectively, of interval % known to contain root of f % TOL absolute error convergence to lerance % (iterations will be performed until the size of % enclosing interval is smaller than 2*TOL) % output: % y approximate value of root % % NOTE: % if BISECTION is called with no output arguments, the iteration % number, the current enclosing interva l and the current % approximation to the root are display ed % % Copyright by Reza Abazari 2008-12-09 % Email : abazari-r@uma.ac.ir sfa = sign(feval(f,a)); Nmax = floor ( log((b-a)/TOL) / log(2.0) ) + 1 for i = 1 : Nmax p = ( a + b ) / 2.0; sfp = sign(feval(f,p)); if ( nargout == 0 ) disp (sprintf('\t\t %3d \t (%.10f,%.10f) \t %.1 0f \n', i, a, b, p)) end if ( (b-a)<2*TOL | fp == 0 ) if ( nargout == 1 )

y = p; end return elseif ( sfa * sfp < 0 ) b = p; else a = p; sfa = sfp; end end @@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@ function y = false_pos ( f, a, b, TOL, Nmax ) %FALSE_POS uses method of false position (regula falsi) to locate the root % of a nonlinear function % % calling sequences: % y = false_pos ( 'f', a, b, TOL, Nmax ) % false_pos ( 'f', a, b, TOL, Nmax ) % % inputs: % f string containing name of m-f ile defining function % whose root is to be located % a,b left and right endpoints, res pectively, of interval % known to contain root of f % TOL absolute error convergence to lerance % Nmax maximum number of iterations to be performed % % output: % y approximate value of root % % NOTE: % if FALSE_POS is called with no output arguments, the iteration % number, the current enclosing interva l and the current % approximation to the root are display ed % % if the maximum number of iterations i s exceeded, a message % to this effect will be displayed and the most recent % approximation will be returned in the output value % % Copyright by Reza Abazari 2008-04-23 % Email : abazari-r@uma.ac.ir old = b; fa = feval(f,a); fb = feval(f,b); for i = 1 : Nmax new = b - fb * ( b - a ) / ( fb - fa ); fnew = feval(f,new); if ( nargout == 0 ) disp(sprintf('\t\t %3d \t (%.10f,%.10f) \t %.10 f \n', i, a, b, new)) end if ( abs(new-old) < TOL ) if ( nargout == 1 ) y = new; end

return elseif ( fa * fnew < 0 ) b = new; fb = fnew; else a = new; fa = fnew; end old = new; end disp('Maximum number of iterations exceeded') if ( nargout == 1 ) y = new; end @@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@ function y = false_pos_aitken ( f, a, b, TOL, Nmax ) %FALSE_POS_AITKEN uses method of false position ( regula falsi) with Aitken % extrapolation to locate the roo t of a nonlinear function % % calling sequences: % y = false_pos_aitken ( 'f', a, b, TOL , Nmax ) % false_pos_aitken ( 'f', a, b, TOL, Nm ax ) % % inputs: % f string containing name of m-f ile defining function % whose root is to be located % a,b left and right endpoints, res pectively, of interval % known to contain root of f % TOL absolute error convergence to lerance % (applied to extrapolated sequ ence of approximations) % Nmax maximum number of iterations to be performed % % output: % y approximate value of root % % NOTE: % if FALSE_POS_AITKEN is called with no output arguments, % the iteration number, the current fal se position % approximation to the root and the cur rent extrapolated % approximation to the root are display ed. % % if the maximum number of iterations i s exceeded, a message % to this effect will be displayed and the most recent % approximation will be returned in the output value % % Copyright by Reza Abazari 2008-04-23 % Email : abazari-r@uma.ac.ir old = b; phatold = b; fa = feval(f,a); fb = feval(f,b); for i = 1 : Nmax new = b - fb * ( b - a ) / ( fb - fa ); fnew = feval(f,new); if ( i == 1 | i == 2 )

if ( nargout == 0 ) disp ( sprintf ( '\t\t %3d \t %.10f \n', i, new ) ) end else phat = new - ( new - old ) ^ 2 / ( new - 2 * ol d + older ); if ( nargout == 0 ) disp ( sprintf ( '\t\t %3d \t %.10f \t %.10f \n', i, new, phat ) ) end if ( abs(phat-phatold) < TOL ) if ( nargout == 1 ) y = phat; end return else phatold = phat; end end if ( sign(fa) * sign(fnew) < 0 ) b = new; fb = fnew; else a = new; fa = fnew; end older = old; old = new; end disp('Maximum number of iterations exceeded') if ( nargout == 1 ) y = phat; end @@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@ function y = fixed_point ( f, x0, TOL, Nmax ) %FIXED_POINT locate the fixed point of an arbitra ry function using general % fixed_point iteration % % calling sequences: % y = fixed_point ( 'g', x0, TOL, Nmax ) % fixed_point ( 'g', x0, TOL, Nmax ) % % inputs: % g string containing name of m-f ile defining the % iteration function % x0 initial approximation to loca tion of fixed point % TOL absolute error convergence to lerance % Nmax maximum number of iterations to be performed % % output: % y approximate value of fixed po int % % NOTE: % if FIXED_POINT is called with no outp ut arguments, the % iteration number and the current appr oximation to the % fixed point are displayed % % if the maximum number of iterations i s exceeded, a message

% to this effect will be displayed and the most recent % approximation will be returned in the output value % % Copyright by Reza Abazari 2008-04-23 % Email : abazari-r@uma.ac.ir old = x0 for i = 1 : Nmax new = feval(f,old); if ( nargout == 0 ) disp ( sprintf ( '\t\t %3d \t %.10f \n', i, new ) ) end if ( abs(new-old) < TOL ) if ( nargout == 1 ) y = new; end return else old = new; end end disp('Maximum number of iterations exceeded') if ( nargout == 1 ) y = new; end @@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@

function y = fixed_point_aitken ( f, x0, TOL, Nmax ) %FIXED_POINT_AITKEN uses general fixed point iter ation with Aitken % extrapolation to locate the r oot of a nonlinear function % % calling sequences: % y = fixed_point_aitken ( 'g', x0, TOL , Nmax ) % fixed_point_aitken ( 'g', x0, TOL, Nm ax ) % % inputs: % g string containing name of m-f ile defining the % iteration function % x0 initial approximation to loca tion of fixed point % TOL absolute error convergence to lerance % (applied to extrapolated sequ ence of approximations) % Nmax maximum number of iterations to be performed % % output: % y approximate value of fixed po int % % NOTE: % if FIXED_POINT_AITKEN is called with no output arguments, % the iteration number, the current fun ctional iteration % approximation to the fixed point and the current extrapolated % approximation to the fixed point are displayed. % % if the maximum number of iterations i s exceeded, a message % to this effect will be displayed and the most recent

% approximation will be returned in the output value % % Copyright by Reza Abazari 2008-04-23 % Email : abazari-r@uma.ac.ir old = x0; phatold = x0; for i = 1 : Nmax new = feval(f,old); if ( i == 1 | i == 2 ) if ( nargout == 0 ) disp ( sprintf ( '\t\t %3d \t %.10f \n', i, new ) ) end else phat = new - ( new - old ) ^ 2 / ( new - 2 * ol d + older ); if ( nargout == 0 ) disp ( sprintf ( '\t\t %3d \t %.10f \t %.10f \n', i, new, phat ) ) end if ( abs(phat-phatold) < TOL ) if ( nargout == 1 ) y = phat; end return else phatold = phat; end end older = old; old = new; end disp('Maximum number of iterations exceeded') if ( nargout == 1 ) y = phat; end @@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@ function r = polyroots ( poly, Nmax, TOL ) %POLYROOTS locate the roots of an arbitrary pol ynomial using Laguerre's % method % % calling sequences: % r = polyroots ( poly, Nmax, TOL ) % polyroots ( poly, Nmax, TOL ) % % inputs: % poly vector containing the coeffic ients of the polynomial % whose roots are to be compute d. the first entry in % the vector must be the leadin g coefficient of the % polynomial, and the final ent ry must be the constant % term. zero coefficients must be explicitly provided. % Nmax maximum number of iterations to be performed to % compute each root

% TOL absolute error convergence to lerance % % output: % r vector containing the roots o f the polynomial % % NOTE: % if POLYROOTS is called with no output arguments, the % roots of the polynomial and the numbe r of iterations % required to compute each root are dis played % % if the maximum number of iterations i s exceeded, a message % to this effect will be displayed and the roots for which % convergence had been achieved will be returned % % Copyright by Reza Abazari 2008-04-23 % Email : abazari-r@uma.ac.ir q = poly; degree = length(poly)-1; r = zeros ( 1, degree ); its = zeros ( 1, degree ); found = 0; while ( found < degree - 2 ) [new, done] = laguerre ( q, 0.0, TOL, Nmax ); if ( done == 0 ) disp('Maximum number of iterations exceeded') return end; if ( abs(imag(new)) == TOL ) found = found + 1; r(found) = real(new); its(found) = done; q = deconv ( q, [ 1 -real(new) ] ); else r(found+1) = new; r(found+2) = conj(new); its(found+1) = done; its(found+2) = done; found = found + 2; q = deconv ( q, conv ( [1 -new], [1 -conj(new)] ) ); end; q = real(q); end; if ( found == degree - 2 ) if ( q(2) == 0 ) r(degree-1) = sqrt ( -q(3)/q(1) ); r(degree) = -r(degree-1); else r(degree-1)=2*q(3)/(-q(2)-sign(q(2))*sqrt(q(2 )*q(2)-4*q(1)*q(3))); r(degree) = q(3) / ( r(degree-1) * q(1) ); end; else r(degree) = -q(2)/q(1); end; if ( nargout == 0 )

disp ( [r' its'] ) end; @@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@ function [P,iter]= seidel(G,P,delta, max1) %Input - G is the nonlinear fixed-point system % saved as an M-file function % - P is the initial guess at the solut ion % - delta is the error bound % - max1 is the number of iterations %Output - P is the seidel approximation to the solu tion % - iter is the number of iterations required %Use the @ notation call [P,iter]=seidel(@G, P, del ta, max1). % Copyright by Reza Abazari 2008-04-23 % Email : abazari-r@uma.ac.ir N=length(P); for k=1:max1 X=P; % X is the kth approximation to the solution for j=1:N A=G(X); % Update the terms of X as they are calculate d X(j)=A(j); end err=abs(norm(X-P)); relerr=err/(norm(X)+eps); P=X; iter=k; if (err<delta)|(relerr<delta) break end end @@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@ function [P,iter,err]=newdim(F,JF,P,delta,epsilon,m ax1) %Input -F is the system saved as the M-file F.m % -JF is the Jacobian of F saved as the M-file JF.M % -P is the inital approximation to the solution % -delta is the tolerance for P % -epsilon is the tolerance for F(P) % -max1 is the maximum number of iterati ons %Output -P is the approximation to the solution % -iter is the number of iterations requ ired % -err is the error estimate for P %Use the @ notation call %[P,iter,err]=newdim(@F, @JF, P, delta, epsilon, ma x1). % Copyright by Reza Abazari 2008-04-23 % Email : abazari-r@uma.ac.ir

Y=F(P); for k=1:max1 J=JF(P); Q=P-(J\Y')'; Z=F(Q); err=norm(Q-P); relerr=err/(norm(Q)+eps); P=Q; Y=Z; iter=k; if (err<delta)|(relerr<delta)|(abs(Y)<epsilon) break end end @@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@ function [y, done] = laguerre ( f, x0, TOL, Nmax ) % Copyright by Reza Abazari 2008-04-23 % Email : abazari-r@uma.ac.ir done = 0; n = length ( f ) - 1; fp = polyder ( f ); fp2 = polyder ( fp ); for i = 1 : Nmax fx = polyval ( f, x0 ); fpx = polyval ( fp, x0 ); fp2x = polyval ( fp2, x0 ); if ( abs(fx) < TOL ) y = x0; done = i; return end; gx = fpx / fx; g2 = gx * gx; hx = g2 - fp2x / fx; disc = sqrt ( (n-1) * ( n * hx - g2 ) ); if ( abs(gx-disc) < abs(gx+disc) ) denom = gx+disc; else denom = gx-disc; end dx = n / denom; x0 = x0 - dx; if ( abs(dx) < TOL ) y = x0; done = i; return end end y = x0; @@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@

function y = steffensen ( f, x0, TOL, Nmax ) %STEFFENSEN locate the fixed point of an arbitra ry function using % Steffensen's method % % calling sequences: % y = steffensen ( 'g', x0, TOL, Nmax ) % steffensen ( 'g', x0, TOL, Nmax ) % % inputs: % g string containing name of m-f ile defining the % iteration function % x0 initial approximation to loca tion of fixed point % TOL absolute error convergence to lerance % Nmax maximum number of iterations to be performed % % output: % y approximate value of fixed po int % % NOTE: % if STEFFENSEN is called with no outpu t arguments, the % iteration number and the current appr oximation to the % fixed point are displayed % % if the maximum number of iterations i s exceeded, a message % to this effect will be displayed and the most recent % approximation will be returned in the output value % % Copyright by Reza Abazari 2008-04-23 % Email : abazari-r@uma.ac.ir p0 = x0; phatold = x0; for i = 1 : Nmax p1 = feval(f,p0); p2 = feval(f,p1); phat = p2 - ( p2 - p1 ) ^ 2 / ( p2 - 2 * p1 + p0 ) ; if ( nargout == 0 ) disp ( sprintf ( '\t\t %3d \t %.10f \n', i, ph at ) ) end if ( abs(phat-phatold) < TOL ) if ( nargout == 1 ) y = phat; end return else phatold = phat; p0 = phat; end end disp('Maximum number of iterations exceeded') if ( nargout == 1 ) y = phat; end @@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@

2. Numerical Linear Algebra function [P,iter]= seidel(G,P,delta, max1) %Input - G is the nonlinear fixed-point system % saved as an M-file function % - P is the initial guess at the solut ion % - delta is the error bound % - max1 is the number of iterations %Output - P is the seidel approximation to the solu tion % - iter is the number of iterations required %Use the @ notation call [P,iter]=seidel(@G, P, del ta, max1). % Copyright by Reza Abazari 2008-04-23 % Email : abazari-r@uma.ac.ir N=length(P); for k=1:max1 X=P; % X is the kth approximation to the solution for j=1:N A=G(X); % Update the terms of X as they are calculate d X(j)=A(j); end err=abs(norm(X-P)); relerr=err/(norm(X)+eps); P=X; iter=k; if (err<delta)|(relerr<delta) break end end

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function [p,Q]=steff(f,df,p0,delta,epsilon,max1) %Input - f is the object function % - df is the derivative of f input as a str ing 'df' % - p0 is the initial approximation to a ze ro of f % - delta is the tolerance for p0 % - epsilon is the tolerance for the functio n values y % - max1 is the maximum number of iterations %Output - p is the Steffensen approximation to th e zero % - Q is the matrix containing the Steffens en sequence %If f and df are defined as M-file functions use th e @ notation % call [p,Q]=steff(@f,@df,p0,delta,epsilon,max1). %If f and df are defined as anonymous functions use the % call [p,Q]=steff(f,df,p0,delta,epsilon,max1). %Initialize the matrix R

R=zeros(max1,3); R(1,1)=p0; for k=1:max1 for j=2:3 %Denominator in Newton-Raphson method is calcul ated nrdenom=df(R(k,j-1)); %Conditional calculates Newton-Raphson approxim ations if nrdenom==0 'division by zero in Newton-Raphson method' break else R(k,j)=R(k,j-1)-f(R(k,j-1))/nrdenom; end %Denominator in Aitkens Acceleration process is calculated aadenom=R(k,3)-2*R(k,2)+R(k,1); %Conditional calculates Aitkens Acceleration ap proximations if aadenom==0 'division by zero in Aitkens Acceleration' break else R(k+1,1)=R(k,1)-(R(k,2)-R(k,1))^2/aadenom; end end %Breaks out and ends program if division by zero occured if (nrdenom==0)|(aadenom==0) break end %Stopping criteria are evaluated; p and the matr ix Q are determined err=abs(R(k,1)-R(k+1,1)); relerr=err/(abs(R(k+1,1))+delta); y=f(R(k+1,1)); if (err<delta)|(relerr<delta)|(y<epsilon) p=R(k+1,1); Q=R(1:k+1,:); break end end @@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@ function [P,iter,err]=newdim(F,JF,P,delta,epsilon,m ax1) %Input -F is the system saved as the M-file F.m % -JF is the Jacobian of F saved as the M-file JF.M % -P is the inital approximation to the solution % -delta is the tolerance for P % -epsilon is the tolerance for F(P) % -max1 is the maximum number of iterati ons %Output -P is the approximation to the solution % -iter is the number of iterations requ ired % -err is the error estimate for P %Use the @ notation call %[P,iter,err]=newdim(@F, @JF, P, delta, epsilon, ma x1).

% Copyright by Reza Abazari 2008-04-23 % Email : abazari-r@uma.ac.ir Y=F(P); for k=1:max1 J=JF(P); Q=P-(J\Y')'; Z=F(Q); err=norm(Q-P); relerr=err/(norm(Q)+eps); P=Q; Y=Z; iter=k; if (err<delta)|(relerr<delta)|(abs(Y)<epsilon) break end end

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function x = gauss_elim ( A, b ) %GAUSS_ELIM solve the linear system Ax = b using Gaussian elimination % with back substitution % % calling sequences: % x = gauss_elim ( A, b ) % gauss_elim ( A, b ) % % inputs: % A coefficient matrix for linear system % (matrix must be square) % b right-hand side vector % % output: % x solution vector (i.e., vector for which Ax = b) % % NOTE: % this is intended as a demonstration r outine - no pivoting % strategy is implemented to reduce the effects of roundoff % error % % Copyright by Reza Abazari 2008-04-23 % Email : abazari-r@uma.ac.ir [nrow ncol] = size ( A ); if ( nrow ~= ncol ) disp ( 'gauss_elim error: Square coefficient mat rix required' ); return; end; nb = length ( b ); if ( nrow ~= nb ) disp('gauss_elim error: Size of b-vector not com patible with matrix dimension') return; end; x = zeros ( 1, nrow ); % Gaussian elimination

for i = 1 : nrow - 1 if ( A(i,i) == 0 ) t = min ( find ( A(i+1:nrow,i) ~= 0 ) + i ); if ( isempty(t) ) disp ( 'gauss_elim error: A matrix is singul ar' ); return end; temp = A(i,:); tb = b(i); A(i,:) = A(t,:); b(i) = b(t); A(t,:) = temp; b(t) = tb; end; for j = i+1 : nrow m = -A(j,i) / A(i,i); A(j,i) = 0; A(j, i+1:nrow) = A(j, i+1:nrow) + m * A(i, i+1:nr ow); b(j) = b(j) + m * b(i); end; end; % % back substitution % x(nrow) = b(nrow) / A(nrow, nrow); for i = nrow - 1 : -1 : 1 x(i) = ( b(i) - sum ( x(i+1:nrow) .* A(i, i+1:n row) ) ) / A(i,i); end @@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@ function x = gauss_seidel ( A, b, xold, TOL, Nmax ) %GAUSS_SEIDEL approximate the solution of the linea r system Ax = b by applying % the Gauss-Seidel method (successive r elaxation) % % calling sequences: % x = gauss_seidel ( A, b, xold, TOL, N max ) % gauss_seidel ( A, b, xold, TOL, Nmax ) % % inputs: % A coefficient matrix for linear system - must be a % square matrix % b right-hand side vector for li near system % xold vector containing initial gue ss for solution of % linear system % TOL convergence tolerance - appli ed to maximum norm of % difference between successive approximations % NMax maximum number of iterations to be performed % % output: % x approximate solution of linea r system % % NOTE: % if GAUSS_SEIDEL is called with no out put arguments, the % iteration number and the current appr oximation to the % solution are displayed % % if the maximum number of iterations i s exceeded, a meesage % to this effect will be displayed and the current approximation % will be returned in the output value %

% Copyright by Reza Abazari 2008-04-23 % Email : abazari-r@uma.ac.ir n = length ( b ); [r c] = size ( A ); if ( c ~= n ) disp ( 'gauss_seidel error: matrix dimensions an d vector dimension not compatible' ) return end; xnew = zeros ( 1, n ); if ( nargout == 0 ) s = sprintf ( '%3d \t %10f ', 0, xold(1) ); for j = 2 : n s = sprintf ( '%s%10f ', s, xold(j) ); end; disp ( s ); end; for its = 1 : Nmax xnew(1) = ( b(1) - sum ( A(1,2:n) .* xold(2:n) ) ) / A(1,1); for i = 2 : n-1 xnew(i) = ( b(i) - sum( A(i,1:i-1) .* xnew(1 :i-1)) - ... sum(A(i,i+1:n).*xold(i+1:n))) / A(i,i); end; xnew(n) = ( b(n) - sum ( A(n,1:n-1) .* xnew(1:n-1) ) ) / A(n,n); if ( nargout == 0 ) s = sprintf ( '%3d \t %10f ', its, xnew(1) ); for j = 2 : n s = sprintf ( '%s%10f ', s, xnew(j) ); end; disp ( s ); end; conv = max ( abs ( xnew - xold ) ); if ( conv < TOL ) x = xnew; return else xold = xnew; end; end; disp ( 'gauss_seidel error: maximum number of itera tions exceeded' ); if ( nargout == 1 ) x = xnew; end @@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@ function y = broyden ( F, J, x0, TOL, Nmax ) %BROYDEN solve the system of nonlinear equatio ns F(x) = 0 using % Broyden's method % % calling sequences: % y = broyden ( F, J, x0, TOL, Nmax ) % broyden ( F, J, x0, TOL, Nmax ) % % inputs:

% F vector-valued function of a v ector argument which % defines the system of equatio ns to be solved % J matrix-valued function which computes the Jacobian % associated with the function F % x0 vector containing initial gue ss for solution of % nonlinear system % TOL convergence tolerance - appli ed to maximum norm of % difference between successive approximations % NMax maximum number of iterations to be performed % % output: % y approximate solution of nonli near system % % % NOTE: % if BROYDEN is called with no output a rguments, each % approximation to the solution is disp layed % % if the maximum number of iterations i s exceeded, a meesage % to this effect will be displayed and the current approximation % will be returned in the output value % % Copyright by Reza Abazari 2008-04-23 % Email : abazari-r@uma.ac.ir Fold = feval(F,x0)'; Jold = feval(J,x0); A0 = inv ( Jold ); dx = -A0 * Fold; x0 = x0 + dx; if ( nargout == 0 ) disp ( x0' ) end for i = 2 : Nmax Fnew = feval(F,x0)'; dy = Fnew - Fold; u = A0 * dy; v = dx' * A0; denom = dx' * u; A0 = A0 + ( dx - u ) * v / denom; dx = -A0 * Fnew; x0 = x0 + dx; if ( nargout == 0 ) disp ( x0' ) end if ( max(abs(dx)) < TOL ) if ( nargout == 1 ) y = x0; end return else Fold = Fnew; end end disp('broyden error: Maximum number of iterations e xceeded');

if ( nargout == 1 ) y = x0; end @@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@ function x = conj_grad ( A, b, x, TOL, Nmax ) %CONJ_GRAD approximate the solution of the linea r system Ax = b by applying % the conjugate gradient method % % calling sequences: % x = conj_grad ( A, b, x, TOL, Nmax ) % conj_grad ( A, b, x, TOL, Nmax ) % % inputs: % A coefficient matrix for linear system - must be a % square matrix % b right-hand side vector for li near system - must be % a column vector % x column vector containing init ial guess for solution of % linear system % TOL convergence tolerance - appli ed to Euclidean norm of % residual vector % NMax maximum number of iterations to be performed % % output: % x approximate solution of linea r system % % NOTE: % if CONJ_GRAD is called with no output arguments, the % iteration number and the current appr oximation to the % solution are displayed % % if the maximum number of iterations i s exceeded, a meesage % to this effect will be displayed and the current approximation % will be returned in the output value % % Copyright by Reza Abazari 2008-04-23 % Email : abazari-r@uma.ac.ir n = length ( b ); [r c] = size ( A ); if ( c ~= n ) disp('conj_grad error: matrix dimensions and vec tor dimension not compatible') return end; if ( nargout == 0 ) s = sprintf ( '%3d \t %10f ', 0, x(1) ); for j = 2 : n s = sprintf ( '%s%10f ', s, x(j) ); end; disp ( s ); end; r = A * x - b; delta0 = r' * r; d = -r; for its = 1 : Nmax

h = A * d; lambda = delta0 / ( d' * h ); x = x + lambda * d; r = r + lambda * h; delta1 = r' * r; if ( nargout == 0 ) s = sprintf ( '%3d \t %10f ', its, x(1) ); for j = 2 : n s = sprintf ( '%s%10f ', s, x(j) ); end; disp ( s ); end; if ( sqrt(delta1) < TOL ) return else alpha = delta1 / delta0; delta0 = delta1; d = -r + alpha * d; end; end; disp ( 'conj_grad error: maximum number of iteratio ns exceeded' ); @@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@ function x = jacobi ( A, b, xold, TOL, Nmax ) %JACOBI approximate the solution of the linea r system Ax = b by applying % the Jacobi method (simultaneous relax ation) % % calling sequences: % x = jacobi ( A, b, xold, TOL, Nmax ) % jacobi ( A, b, xold, TOL, Nmax ) % % inputs: % A coefficient matrix for linear system - must be a % square matrix % b right-hand side vector for li near system % xold vector containing initial gue ss for solution of % linear system % TOL convergence tolerance - appli ed to maximum norm of % difference between successive approximations % NMax maximum number of iterations to be performed % % output: % x approximate solution of linea r system % % NOTE: % if JACOBI is called with no output ar guments, the % iteration number and the current appr oximation to the % solution are displayed % % if the maximum number of iterations i s exceeded, a meesage % to this effect will be displayed and the current approximation % will be returned in the output value % % Copyright by Reza Abazari 2008-04-23 % Email : abazari-r@uma.ac.ir n = length ( b );

[r c] = size ( A ); if ( c ~= n ) disp('jacobi error: matrix dimensions and vector dimension not compatible') return end; xnew = zeros ( 1, n ); if ( nargout == 0 ) s = sprintf ( '%3d \t %10f ', 0, xold(1) ); for j = 2 : n s = sprintf ( '%s%10f ', s, xold(j) ); end; disp ( s ); end; for its = 1 : Nmax xnew(1) = ( b(1) - sum ( A(1,2:n) .* xold(2:n) ) ) / A(1,1); for i = 2 : n-1 xnew(i) = ( b(i) - sum ( A(i,1:i-1) .* xold(1: i-1) ) -... sum ( A(i,i+1:n) .* xold(i+1:n) ) ) / A(i,i); end; xnew(n) = ( b(n) - sum ( A(n,1:n-1) .* xold(1:n-1) ) ) / A(n,n); if ( nargout == 0 ) s = sprintf ( '%3d \t %10f ', its, xnew(1) ); for j = 2 : n s = sprintf ( '%s%10f ', s, xnew(j) ); end; disp ( s ); end; conv = max ( abs ( xnew - xold ) ); if ( conv < TOL ) x = xnew; return else xold = xnew; end; end; disp ( 'jacobi error: maximum number of iterations exceeded' ); if ( nargout == 1 ) x = xnew; end. @@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@ function [lu, pvt] = LUfactor ( A ) %LUFACTOR compute the LU decomposition for the matrix A % % calling sequence: % [lu, pvt] = LUfactor ( A ) % input: % A coefficient matrix for linear system % (matrix must be square) % outputs: % lu matrix containing LU decompos ition of input matrix % A - the L matrix in the decom position consists of % 1's along the main diagonal t ogether with the % strictly lower triangular por tion of the matrix lu; % the U matrix in the decomposi tion is the upper

% triangular portion of the mat rix lu % pvt vector which indicates the pe rmutation of the rows % performed during the decompos ition process % NOTE: % this routine performs partial pivotin g during the % decomposition process % % the system Ax = b can be solved by fi rst applying LUfactor % to the coefficient matrix A and then applying the companion % routine, LUsolve, for each right-hand side vector b % % Copyright by Reza Abazari 2008-04-23 % Email : abazari-r@uma.ac.ir [nrow ncol] = size ( A ); if ( nrow ~= ncol ) disp ( 'LUfactor error: Square coefficient matri x required' ); return; end % % initialize row pointers % for i=1:nrow pvt(i) = i; end for i = 1 : nrow - 1 % % partial pivoting % t =min(find(abs(A(pvt(i:nrow),i)) == max(abs(A(pvt (i:nrow),i))))+ i-1 ); if ( t ~= i ) temp = pvt(i); pvt(i) = pvt(t); pvt(t) = temp; end % % terminate if matrix is singular % if ( A(pvt(i),i) == 0 ) disp ( 'LUfactor error: coefficient matrix is s ingular' ); lu = A; return end % % elimination steps % for j = i+1 : nrow m = -A(pvt(j),i) / A(pvt(i),i); A(pvt(j),i) = -m; A(pvt(j), i+1:nrow) = A(pvt(j), i+1:nrow) + m * A (pvt(i), i+1:nrow); end end lu = A;

@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@ function x = LUsolve ( lu, b, pvt ) %LUSOLVE perform forward and backward substitu tion to obtain the % solution to the linear system Ax = b, where the LU % decomposition of the coefficient matr ix has previously % been determined % % calling sequence: % x = LUsolve ( lu, b, pvt ) % LUsolve ( lu, b, pvt ) % % inputs: % lu matrix containing LU decompos ition of coefficient % matrix for the linear system - the L matrix in the % decomposition must consists o f 1's along the main % diagonal together with the st rictly lower triangular % portion of the matrix lu; the U matrix in the % decomposition must be the upp er triangular portion % of the matrix lu % b right-hand side vector for li near system % pvt vector which indicates the pe rmutation of the rows % performed during the LU decom position of the % coefficient matrix % % output: % x solution vector % % NOTE: % the system Ax = b can be solved by fi rst applying LUfactor % to the coefficient matrix A and then applying LUsolve once % for each right-hand side vector b % % Copyright by Reza Abazari 2008-04-23 % Email : abazari-r@uma.ac.ir [nrow ncol] = size ( lu ); % % forward substitution % z(1) = b(pvt(1)); for i = 2 : nrow z(i) = b(pvt(i)) - sum ( z(1:i-1) .* lu(pvt(i), 1:i-1) ); end % % back substitution % x(nrow) = z(nrow) / lu(pvt(nrow), nrow); for i = nrow - 1 : -1 : 1 x(i) = ( z(i) - sum ( x(i+1:nrow) .* lu(pvt(i), i+1:nrow) ) ) / lu(pvt(i),i); end @@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@ function y = newton_sys ( F, J, x0, TOL, Nmax )

%NEWTON_SYS solve the system of nonlinear equatio ns F(x) = 0 using % Newton's method % % calling sequences: % y = newton_sys ( F, J, x0, TOL, Nmax ) % newton_sys ( F, J, x0, TOL, Nmax ) % % inputs: % F vector-valued function of a v ector argument which % defines the system of equatio ns to be solved % J matrix-valued function which computes the Jacobian % associated with the function F % x0 vector containing initial gue ss for solution of % nonlinear system % TOL convergence tolerance - appli ed to maximum norm of % difference between successive approximations % NMax maximum number of iterations to be performed % % output: % y approximate solution of nonli near system % % dependencies: % this routine uses both LUfactor and L Usolve % % NOTE: % if NEWTON_SYS is called with no outpu t arguments, each % approximation to the solution is disp layed % % if the maximum number of iterations i s exceeded, a meesage % to this effect will be displayed and the current approximation % will be returned in the output value % % Copyright by Reza Abazari 2008-04-23 % Email : abazari-r@uma.ac.ir old = x0; for i = 1 : Nmax Fold = feval(F,old); Jold = feval(J,old); [lu pvt] = LUfactor ( Jold ); dx = LUsolve ( lu, -Fold, pvt ); new = old + dx; if ( nargout == 0 ) disp ( new ) end if ( max(abs(dx)) < TOL ) if ( nargout == 1 ) y = new; end return else old = new; end end disp('newton_sys error: Maximum number of iteration s exceeded'); if ( nargout == 1 )

y = new; end @@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@ function y = tridiagonal ( c, a, b, r ) %TRIDIAGONAL solve a linear system with a tridiago nal coefficient matrix % % calling sequence: % x = tridiagonal ( c, a, b, r ) % tridiagonal ( c, a, b, r ) % % inputs: % c vector containing the entries along lower diagonal % of the coefficient matrix % a vector containing the entries along main diagonal % of the coefficient matrix % b vector containing the entries along upper diagonal % of the coefficient matrix % r right-hand side vector % % output: % x solution vector % % NOTE: % the entries in the vectors c, a and b are assumed to be % numbered as follows: % % | a(1) b(1) | % | c(1) a(2) b(2) | % | c(2) a(3) b(3) | % | . . . | % | . . . | % | . . b(n-1) | % | c(n-1) a(n) | % % Copyright by Reza Abazari 2008-04-23 % Email : abazari-r@uma.ac.ir n = length ( a ); % % factorization step % for i = 1 : n-1 b(i) = b(i) / a(i); a(i+1) = a(i+1) - c(i) * b(i); end % % forward substitution % r(1) = r(1) / a(1); for i = 2 : n r(i) = ( r(i) - c(i-1) * r(i-1) ) / a(i); end

% % back substitution % for i = n-1 : -1 : 1 r(i) = r(i) - r(i+1) * b(i); end if ( nargout == 0 ) disp ( r ) else y = r; end @@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@ function x = sor ( A, b, xold, omega, TOL, Nmax ) %SOR approximate the solution of the linea r system Ax = b by applying % the SOR method (successive over-relax ation) % % calling sequences: % x = sor ( A, b, xold, omega, TOL, Nma x ) % sor ( A, b, xold, omega, TOL, Nmax ) % % inputs: % A coefficient matrix for linear system - must be a % square matrix % b right-hand side vector for li near system % xold vector containing initial gue ss for solution of % linear system % omega relaxation parameter % TOL convergence tolerance - appli ed to maximum norm of % difference between successive approximations % NMax maximum number of iterations to be performed % % output: % x approximate solution of linea r system % % NOTE: % if SOR is called with no output argum ents, the % iteration number and the current appr oximation to the % solution are displayed % % if the maximum number of iterations i s exceeded, a meesage % to this effect will be displayed and the current approximation % will be returned in the output value % % Copyright by Reza Abazari 2008-04-23 % Email : abazari-r@uma.ac.ir n = length ( b ); [r c] = size ( A ); if ( c ~= n ) disp ( 'sor error: matrix dimensions and vector dimension not compatible' ) return end xnew = zeros ( 1, n ); if ( nargout == 0 )

s = sprintf ( '%3d \t %10f ', 0, xold(1) ); for j = 2 : n s = sprintf ( '%s%10f ', s, xold(j) ); end disp ( s ); end for its = 1 : Nmax xnew(1) = ( 1 - omega ) * xold(1) + omega *... ( b(1) - sum ( A(1,2:n) .* xold(2:n) ) ) / A(1,1); for i = 2 : n-1 xnew(i) = ( 1 - omega ) * xold(i) + omega * ( b(i) -... sum ( A(i,1:i-1) .* xnew(1:i-1) ) -... sum ( A(i,i+1:n) .* xold(i+1:n) ) ) / A(i,i); end xnew(n) = ( 1 - omega ) * xold(n) + omega *... ( b(n) - sum ( A(n,1:n-1) .* xnew(1 :n-1) ) ) / A(n,n); if ( nargout == 0 ) s = sprintf ( '%3d \t %10f ', its, xnew(1) ); for j = 2 : n s = sprintf ( '%s%10f ', s, xnew(j) ); end disp ( s ); end conv = max ( abs ( xnew - xold ) ); if ( conv < TOL ) x = xnew; return else xold = xnew; end end disp ( 'sor error: maximum number of iterations exc eeded' ); if ( nargout == 1 ) x = xnew; end @@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@ function [lambda, v] = sympower ( A, x, TOL, Nmax ) %SYMPOWER approximate the dominant eigenvalue and an associated % eigenvector for a symmetric matrix using the power % method % % calling sequences: % [lambda, v] = sympower ( A, x, TOL, N max ) % lambda = sympower ( A, x, TOL, Nmax ) % sympower ( A, x, TOL, Nmax ) % % inputs: % A square symmetric matrix whose dominant eigenvalue % is to be approximated % x initial approximation to eige nvector corresponding % to the dominant eigenvalue % TOL absolute error convergence to lerance % (convergence is measured in t erms of the Euclidean % norm of the difference betwee n successive terms % in the eigenvector seqeunce)

% Nmax maximum number of iterations to be performed % % outputs: % lambda approximation to dominant eig envalue of A % v an eigenvector of A correspon ding to the eigenvalue % lambda - vector will be norma lized to unit length % in the Euclidean norm % % NOTE: % if SYMPOWER is called with no output arguments, the % iteration number, the current eigenve ctor approximation, % the current eigenvalue approximation and an estimate of % the rate of convergence of the eigenv alue sequence will % be displayed % % if the maximum number of iterations i s exceeded, a message % to this effect will be displayed and the most recent % approximations to the dominant eigenv alue and its corresponding % eigenvector will be returned in the o utput values % % Copyright by Reza Abazari 2008-04-23 % Email : abazari-r@uma.ac.ir [r c] = size ( A ); [rx rc] = size ( x ); if ( rx == 1 ) x = x'; rx = rc; end; if ( r ~= c ) disp ( 'sympower error: matrix must be square' ) ; return; elseif ( r ~= rx ) disp ( 'sympower error: dimensions of matrix and vector are not compatible' ); return; end x = x / sqrt ( sum ( x .* x ) ); mu_old = 0; if ( nargout == 0 ) s = sprintf ( '%3d \t %10f ', 0, x(1) ); for j = 2 : rx s = sprintf ( '%s%10f ', s, x(j) ); end disp ( s ); end for i = 1 : Nmax xnew = A * x; mu = sum ( x .* xnew ); xnew = xnew / sqrt ( sum ( xnew .* xnew ) ); if ( nargout == 0 ) s = sprintf ( '%3d \t %10f ', i, xnew(1) ); for j = 2 : rx s = sprintf ( '%s%10f ', s, xnew(j) ); end s = sprintf ( '%s \t %10f', s, mu ); if ( i >= 3 ) s = sprintf ( '%s \t \t %10f', s, abs((m u-mu_old)/(mu_old-mu_older)) );

end disp ( s ); end delta = x - sign(mu) * xnew; err = sqrt ( sum ( delta .* delta ) ); if ( err < TOL ) if ( nargout >= 1 ) lambda = mu; end; if ( nargout >= 2 ) v = xnew; end; return; else x = xnew; mu_older = mu_old; mu_old = mu; end end disp ( 'sympower error: Maximum number of iteration exceeded' ); if ( nargout >= 1 ) lambda = mu; end if ( nargout >= 2 ) v = xnew; end @@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@ function [lambda, vects] = hotelling ( A, num_pairs , TOL, Nmax ) %HOTELLING apply Hotelling deflation to a symm etric matrix to % approximate a specified number of e igenpairs associated % with the largest eigenvalues of the matrix - the routine % SYMPOWER is used to approximate eac h eigenpair % % calling sequences: % [lambda, vects] = hotelling ( A, num_ pairs, TOL, Nmax ) % lambda = hotelling ( A, num_pairs, TO L, Nmax ) % hotelling ( A, num_pairs, TOL, Nmax ) % % inputs: % A square symmetric matrix whose eigenpairs are to be % approximated % num_pairs % number of eigenpairs to appro ximate % TOL absolute error convergence to lerance applied during % each call to SYMPOWER % (convergence is measured in t erms of the Euclidean % norm of the difference betwee n successive terms % in the eigenvector seqeunce) % Nmax maximum number of iterations to be performed during % each call to SYMPOWER % % outputs: % lambda vector containing the largest 'num_pairs' eigenvalues % of the matrix A % vects matrix containing eigenvector s corresponding to the % entries in the output vector 'lambda' % - the i-th column of this mat rix is an eigenvector, % normalized to unit length i n the Euclidean norm, % corresponding to the i-th e ntry in the vector

% 'lambda' % % dependencies: % this routine makes use of the routine SYMPOWER % % NOTE: % if the maximum number of iterations i s exceeded during any % call to SYMPOWER, a message to this e ffect will be displayed % % Copyright by Reza Abazari 2008-04-23 % Email : abazari-r@uma.ac.ir [r c] = size ( A ); if ( r ~= c ) disp ( 'hotelling error: matrix must be square' ); return; end n = r; l = zeros ( 1, num_pairs ); v = zeros ( n, num_pairs ); [l(1) v(:,1)] = sympower ( A, rand(n,1), TOL, Nmax ); for i = 2:num_pairs A = hd ( A, l(i-1), v(:,i-1) ); [l(i) v(:,i)] = sympower ( A, rand(n,1), TOL, N max ); end lambda = l; vects = v; return; %++++++++++++++++++++++++++++++++++++++++++++++++++ ++++++++++++++++++++++++++++++++ function B = hd ( A, l, v ) B = A - ( l / dot ( v, v ) ) * v * v'; return; @@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@ function [lambda, v] = inv_power ( A, q, x, TOL, Nm ax ) %INV_POWER approximate the eigenvalue nearest to the number q, % and an associated eigenvector, for an arbitrary matrix % using the inverse power method % % calling sequences: % [lambda, v] = inv_power ( A, q, x, TO L, Nmax ) % lambda = inv_power ( A, q, x, TOL, Nm ax ) % inv_power ( A, q, x, TOL, Nmax ) % % inputs: % A square matrix whose eigenvalu e nearest to the value % q is to be approximated % q approximation to an eigenvalu e of A % x initial approximation to eige nvector corresponding % to the eigenvalue nearest to q % TOL absolute error convergence to lerance % (convergence is measured in t erms of the maximum % norm of the difference betwee n successive terms % in the eigenvector seqeunce)

% Nmax maximum number of iterations to be performed % % outputs: % lambda approximation to dominant eig envalue of A % v an eigenvector of A correspon ding to the eigenvalue % lambda - vector will be norma lized to unit length % in the maximum norm % % dependencies: % this routine makes use of both LUfact or and LUsolve from % "Systems of Equations" library % % NOTE: % if INV_POWER is called with no output arguments, the % iteration number, the current eigenve ctor approximation, % the current eigenvalue approximation and an estimate of % the rate of convergence of the eigenv alue sequence will % be displayed % % if the maximum number of iterations i s exceeded, a message % to this effect will be displayed and the most recent % approximations to the eigenvalue near est to q and its % corresponding eigenvector will be ret urned in the output values % % Copyright by Reza Abazari 2008-04-23 % Email : abazari-r@uma.ac.ir [r c] = size ( A ); [rx rc] = size ( x ); if ( rc == 1 ) x = x'; rc = rx; end; if ( r ~= c ) disp ( 'inv_power error: matrix must be square' ); return; elseif ( r ~= rc ) disp ( 'inv_power error: dimensions of matrix an d vector are not compatible' ); return; end; p = min ( find ( abs(x) == max(abs(x)) ) ); x = x / x(p); mu_old = 0; if ( nargout == 0 ) s = sprintf ( '%3d \t %10f ', 0, x(1) ); for j = 2 : rc s = sprintf ( '%s%10f ', s, x(j) ); end; disp ( s ); end; [lu pvt] = LUfactor ( A - q*eye(rc) ); for i = 1 : Nmax xnew = LUsolve ( lu, x, pvt ); mu = xnew(p); p = min ( find ( abs(xnew) == max(abs(xnew)) ) ); xnew = xnew / xnew(p);

if ( nargout == 0 ) s = sprintf ( '%3d \t %10f ', i, xnew(1) ); for j = 2 : rc s = sprintf ( '%s%10f ', s, xnew(j) ); end s = sprintf ( '%s \t %10f', s, 1/mu+q ); if ( i >= 2 ) s = sprintf ( '%s \t \t %10f', s, abs(( mu-mu_old)/(mu_old-mu_older)) ); end disp ( s ); end err = max ( abs ( x - xnew ) ); if ( err < TOL ) if ( nargout >= 1 ) lambda = 1/mu+q; end; if ( nargout >= 2 ) v = xnew'; end; return; else x = xnew; mu_older = mu_old; mu_old = mu; end end disp ( 'inv_power error: Maximum number of iteratio n exceeded' ); if ( nargout >= 1 ) lambda = 1/mu+q; end if ( nargout >= 2 ) v = xnew; end @@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@ function [lambda, vects] = wielandt ( A, num_pairs, TOL, Nmax ) %WIELANDT apply Wielandt deflation to an arbi trary matrix to % approximate a specified number of e igenpairs associated % with the either largest or the smal lest eigenvalues of % the matrix - the routine POWER_METH OD is used to approximate % eigenpairs associated with the larg est eigenvalues, the % routine INV_POWER is used to approx imate eigenpairs % associated with the smallest eigenv alues % % calling sequences: % [lambda, vects] = wielandt ( A, num_p airs, TOL, Nmax ) % lambda = wielandt ( A, num_pairs, TOL , Nmax ) % wielandt ( A, num_pairs, TOL, Nmax ) % % inputs: % A square symmetric matrix whose eigenpairs are to be % approximated % num_pairs % number of eigenpairs to appro ximate % if num_pairs > 0 then approxi mate largest eigenvalues % if num_pairs < 0 then approxi mate smallest eigenvalues % TOL absolute error convergence to lerance applied during % each call to either POWER_MET HOD or INV_POWER % (convergence is measured in t erms of the maximum % norm of the difference betwee n successive terms

% in the eigenvector seqeunce) % Nmax maximum number of iterations to be performed during % each call to either POWER_MET HOD or INV_POWER % % outputs: % lambda vector containing the largest /smallest 'num_pairs' % eigenvalues of the matrix A % vects matrix containing eigenvector s corresponding to the % entries in the output vector 'lambda' % - the i-th column of this mat rix is an eigenvector, % normalized to unit length i n the maximum norm, % corresponding to the i-th e ntry in the vector % 'lambda' % % dependencies: % this routine makes use of the routine s POWER_METHOD and % INV_POWER % % NOTE: % if the maximum number of iterations i s exceeded during any % call to either POWER_METHOD or INV_PO WER, a message to this % effect will be displayed % % Copyright by Reza Abazari 2008-04-23 % Email : abazari-r@uma.ac.ir [r c] = size ( A ); if ( r ~= c ) disp ( 'wielandt error: matrix must be square' ) ; return; end; n = r; if ( num_pairs < 0 ) num_pairs = abs(num_pairs); small = 1; else small = 0; end; l = zeros ( 1, num_pairs ); v = zeros ( n, num_pairs ); j = zeros ( 1, num_pairs ); x = zeros ( num_pairs, n ); if ( small == 0 ) [l(1) v(:,1)] = power_method ( A, rand(n,1), TOL , Nmax ); else [l(1) v(:,1)] = inv_power ( A, 0, rand(1,n), TOL , Nmax ); end; for i = 2:num_pairs j(i) = min ( find ( abs(v(:,i-1)) == max(abs(v (:,i-1))) ) ); x(i,1:n+2-i) = A(j(i),:); A = wd ( A, n+2-i, j(i), v(:,i-1) ); if ( small == 0 ) [l(i) v(1:n-i+1,i)] = power_method ( A, rand(n- i+1,1), TOL, Nmax ); else [l(i) v(1:n-i+1,i)] = inv_power ( A, 0, rand(1, n-i+1), TOL, Nmax ); end;

end; for i=num_pairs:-1:2 for k=i:-1:2 temp = [v(1:j(k)-1, i); 0; v(j(k):n-k+1, i)]; v(1:n-k+2,i) = (l(i) - l(k-1))*temp +(x(k,1:n+2-k )*temp)*v(1:n-k+2,k-1)/v(j(k),k-1); end; p = min ( find ( abs(v(:,i)) == max(abs(v(:,i) )) ) ); v(:,i) = v(:,i) / v(p,i); end; lambda = l; vects = v; return; %++++++++++++++++++++++++++++++++++++++++++++++++++ ++++++++++++++++++++++++++++++++ function B = wd ( A, n, j, v ) temp = ( v / v(j) ) * A(j,:); B = [A(1:j-1,1:j-1)-temp(1:j-1,1:j-1) A(1:j-1,j+1:n ) - temp(1:j-1,j+1:n); ... A(j+1:n,1:j-1)-temp(j+1:n,1:j-1) A(j+1:n,j+1:n ) - temp(j+1:n,j+1:n)]; return; @@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@ function [lambda, v] = power_method ( A, x, TOL, Nm ax ) %POWER_METHOD approximate the dominant eigenvalue and an associated % eigenvector for an arbitrary matrix using the power % method % % calling sequences: % [lambda, v] = power_method ( A, x, TO L, Nmax ) % lambda = power_method ( A, x, TOL, Nm ax ) % power_method ( A, x, TOL, Nmax ) % % inputs: % A square matrix whose dominant eigenvalue is to be % approximated % x initial approximation to eige nvector corresponding % to the dominant eigenvalue % TOL absolute error convergence to lerance % (convergence is measured in t erms of the maximum % norm of the difference betwee n successive terms % in the eigenvector seqeunce) % Nmax maximum number of iterations to be performed % % outputs: % lambda approximation to dominant eig envalue of A % v an eigenvector of A correspon ding to the eigenvalue % lambda - vector will be norma lized to unit length % in the maximum norm % % NOTE: % if POWER_METHOD is called with no out put arguments, the % iteration number, the current eigenve ctor approximation, % the current eigenvalue approximation and an estimate of % the rate of convergence of the eigenv alue sequence will % be displayed %

% if the maximum number of iterations i s exceeded, a message % to this effect will be displayed and the most recent % approximations to the dominant eigenv alue and its corresponding % eigenvector will be returned in the o utput values % % Copyright by Reza Abazari 2008-04-23 % Email : abazari-r@uma.ac.ir [r c] = size ( A ); [rx rc] = size ( x ); if ( rx == 1 ) x = x'; rx = rc; end; if ( r ~= c ) disp ( 'power_method error: matrix must be squar e' ); return; elseif ( r ~= rx ) disp ( 'power_method error: dimensions of matrix and vector are not compatible' ); return; end; p = min ( find ( abs(x) == max(abs(x)) ) ); x = x / x(p); mu_old = 0; if ( nargout == 0 ) s = sprintf ( '%3d \t %10f ', 0, x(1) ); for j = 2 : rx s = sprintf ( '%s%10f ', s, x(j) ); end; disp ( s ); end; for i = 1 : Nmax xnew = A * x; mu = xnew(p); p = min ( find ( abs(xnew) == max(abs(xnew)) ) ); xnew = xnew / xnew(p); if ( nargout == 0 ) s = sprintf ( '%3d \t %10f ', i, xnew(1) ); for j = 2 : rx s = sprintf ( '%s%10f ', s, xnew(j) ); end; s = sprintf ( '%s \t %10f', s, mu ); if ( i >= 2 ) s = sprintf ( '%s \t \t %10f', s, abs((m u-mu_old)/(mu_old-mu_older)) ); end; disp ( s ); end; err = max ( abs ( x - xnew ) ); if ( err < TOL ) if ( nargout >= 1 ) lambda = mu; end; if ( nargout >= 2 ) v = xnew; end; return; else x = xnew; mu_older = mu_old; mu_old = mu;

end; end; disp ( 'power_method error: Maximum number of itera tion exceeded' ); if ( nargout >= 1 ) lambda = mu; end; if ( nargout >= 2 ) v = xnew; end. @@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@ function [lambda, v] = qrst ( a, b, TOL, Nmax, vect s ) %QRST determine all of the eigenvalues (a nd optionally all of the % eigenvectors) of a symmetric tridia gonal matrix using the % QR algorithm with Wilkinson shift % % calling sequences: % [lambda, v] = qrst ( a, b, TOL, Nmax, vects ) % [lambda, v] = qrst ( a, b, TOL, Nmax ) % lambda = qrst ( a, b, TOL, Nmax ) % qrst ( a, b, TOL, Nmax, vects ) % qrst ( a, b, TOL, Nmax ) % % inputs: % a vector containing elements al ong the main diagonal % of the symmetric tridiagonal matrix whose eigenvalues % are to be determined % b vector containing elements al ong the off diagonal % of the symmetric tridiagonal matrix whose eigenvalues % are to be determined % TOL convergence tolerance % Nmax maximum number of iterations % vects optional input argument % matrix containing eigenvector information produced % during the reduction of the o riginal symmetric % matrix to symmetric tridiagon al form % - this input is needed only i f computation of the % eigenvectors is requested ( by including the second % output argument) and the or iginal matrix was not in % symmetric tridiagonal form % % output: % lambda vector containing the eigenva lues of the symmetric % tridiagonal matrix determined by the vectors a and b % v optional output argument % matrix containing the eigenve ctors of the symmetric % tridiagonal matrix determined by the vectors a and b % - the i-th column of this mat rix is an eigenvector % which corresponds to the i- th eigenvalue in the % vector lambda % - eigenvectors will be not co mputed if this second % output argument is omitted % % NOTE: % if the maximum number of iterations i s exceeded, a message % to this effect will be displayed, alo ng with the number of % eigenvalues which had been determined - these eigenvalues % will be returned in the last entries of the output vector % lambda % % Copyright by Reza Abazari 2008-04-23

% Email : abazari-r@uma.ac.ir n = length(a); if ( length(b) == n-1 ) b(2:n) = b(1:n-1); end; c = zeros ( 1, n ); s = zeros ( 1, n ); shift = 0; togo = n; if ( nargout >= 2 ) if ( nargin >= 5 ) v = vects; else v = eye(n); end; end; for its = 1 : Nmax if ( togo == 1 ) lambda(1) = a(1) + shift; disp ( its ); return; end; trace = a(togo-1) + a(togo); det = a(togo-1)*a(togo) - b(togo)*b(togo); disc = sqrt ( trace*trace - 4*det ); mu1 = (1/2) * ( trace + disc ); mu2 = (1/2) * ( trace - disc ); if ( abs ( mu1 - a(togo) ) < abs ( mu2 - a(togo) ) ) s = mu1; else s = mu2; end; shift = shift + s; for i = 1:togo a(i) = a(i) - s; end; oldb = b(2); for i = 2:togo j = i-1; r = sqrt ( a(j)^2 + oldb^2 ); c(i) = a(j) / r; s(i) = oldb / r; a(j) = r; temp1 = c(i)*b(i) + s(i)*a(i); temp2 = -s(i)*b(i) + c(i)*a(i); b(i) = temp1; a(i) = temp2; if ( i ~= togo ) oldb = b(i+1); b(i+1) = c(i)* b(i+1); end; end; a(1) = c(2)*a(1) + s(2)*b(2); b(2) = s(2)*a(2); for i = 2:togo-1 a(i) = s(i+1)*b(i+1) + c(i)*c(i+1)*a(i);

b(i+1) = s(i+1)*a(i+1); end; a(togo) = c(togo)*a(togo); if ( nargout >= 2 ) for i = 2 : togo col1 = v(:,i-1) * c(i) + v(:,i) * s(i); v(:,i) = -s(i) * v(:,i-1) + c(i) * v(:,i); v(:,i-1) = col1; end; end; if ( abs(b(togo)) < TOL ) lambda(togo) = a(togo) + shift; disp([lambda(togo) its]); togo = togo - 1; end; end; disp ( 'qrst error: Maximum number of iterations ex ceeded' ); disp ( sprintf ( '%d eigenvalues determined \n', n- togo ) ); @@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@ function [a, b, V] = red2st ( A ) %RED2ST perform similarity transformations to reduce the symmetric % matrix A to symmetric tridiagonal f orm % % calling sequences: % [a, b, V] = red2st ( A ) % [a, b] = red2st ( A ) % red2st ( A ) % % input: % A square symmetric matrix to be reduced to symmetric % tridiagonal form % % outputs: % a vector containing elements al ong the main diagonal % of the symmetric tridiagonal from of A % b vector containing elements al ong the off diagonal % of the symmetric tridiagonal from of A % V optional output argument % matrix containing eigenvector information for the % matrix A % % Copyright by Reza Abazari 2008-04-23 % Email : abazari-r@uma.ac.ir [nrow ncol] = size ( A ); if ( nrow ~= ncol ) disp ( 'red2st error: square matrix required' ); return; end; n = nrow; if ( nargout >= 3 ) V = eye(n); end;

for i = 1 : n-2 w = zeros ( n, 1 ); x = A(:,n-i+1); alpha = - sign(x(n-i)) * norm ( x(1:n-i) ); if ( alpha ~= 0 ) w(n-i) = sqrt ( (1/2) * ( 1 - x(n-i)/alpha ) ); w(1:n-i-1) = -(1/2) * x(1:n-i-1) / ( alpha * w( n-i) ); u = A * w; K = dot ( w, u ); q = u - K * w; A = A - 2*w*q' - 2*q*w'; if ( nargout >= 3 ) V = V - 2*V*w*w'; V; end; end; end; a = diag(A); b = zeros ( n, 1 ); for i = 2:n b(i) = A(i,i-1); end; @@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@

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