Numerical Mathematics With MATLAB

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Numerical Mathem

atics with

MATLAB

Reza Abazari

2008

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2 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

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3 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 : [email protected] 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 points % fun_f(x) : functional value at x

% Copyright by Reza Abazari 2008-04-23 % Email : [email protected]

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')

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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)]), end 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' ); break 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 graphics

% f_name: definition of the equation to solve % a , c : end points of initial interval %

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 points % Copyright by Reza Abazari 2008-04-23

% Email : [email protected]

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' );

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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' ); break 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 endpoints % - 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 @ notation % 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 : [email protected] 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 fixed-point % - tol is the tolerance

% - max1 is the maximum number of iterations %Output- k is the number of iterations

% - p is the approximation to the fixed-point % - err is the error in the approximation % - P' contains the sequence {pn}

%If g is defined as an M-file function use the @ notation % 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 : [email protected] 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';

@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@ function [S,E,G]=golden(f,a,b,delta,epsilon)

%Input - f is the object function

% - a and b are the endpoints of the interval % - delta is the tolerance for the abscissas % - epsilon is the tolerance for the ordinates %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 contains [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 @ notation % 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 : [email protected] 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;

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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,epsilon,show)

%Input - F is the object function input as an M-file function % - G = -(1/norm(grad F))*grad F; the search direction % input as an M-file function

% - P0 is the initial starting point

% - max1 is the maximum number of iterations

% - 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 displayed %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.

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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;

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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

% Abstract : This simulation illustrates the convergence 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' limits for the plot based on the % given

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% '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 representing 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 upper and lower guesses') % This portion adds the text and math

subplot(2,1,1), text(0,0.8,['Check first if the lower 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-linear 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 in 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

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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 iterations') 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 of number of iterations') figure('Position',wind)

subplot(2,1,1), plot(n,Ea,'g','linewidth',2)

title('Absolute relative error as a function of number of iterations') subplot(2,1,2), plot(n,ea,'g','linewidth',2)

title('Absolute relative approximate error as a function of number of iterations')

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

@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@ % Copyright by Reza Abazari 2008-04-23

% Abstract : This simulation shows how the bisection method works % for finding roots of an equation f(x)=0

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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' limits 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 representing 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 initial guesses') hold off

%

% Iteration 1

figure('Position',wind) xr=(xu+xl)/2;

% This graphs the function and two lines representing 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)

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plot([xr,xr],[maxi,mini],'r','linewidth',2) plot([lxrange,uxrange],[0,0],'k','linewidth',1)

title('Entered function on given interval with upper and lower guesses') % This portion adds the text and math to the top part 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 the 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 between 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 representing 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 upper and lower… guesses') % This portion adds the text and math to the top part of the figure % window

text(0.2,.8,['xr = (xu + xl) / 2 = ',num2str(xr)])

hold off

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xl=xr;

else

xu=xr; end

% Calculate relative approximate error ea=abs((xr-xp)/xr)*100;

subplot(3,1,3), text(0,1,['Absolute relative approximate error']) text(0,.8,['ea = abs((xr - xp) / xr)*100 = ',num2str(ea),'%'])

text(0,.4,['Check the interval between which the root 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;

title('Entered function on given interval with upper and lower … guesses') % This portion adds the text and math to the top part of the figure % window

hold off

xl=xr; else

xu=xr; end

subplot(3,1,3), text(0,1,['Absolute relative approximate error']) text(0,.8,['ea = abs((xr - xp) / xr)*100 = ',num2str(ea),'%'])

text(0,.4,['Check the interval between which the root lies. Does it… lie in ( xl , xu ) or ( xr , xu )?'])

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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;

title('Entered function on given interval with upper and lower guesses') % This portion adds the text and math to the top part of the figure window subplot(3,1,1), text(0,1,['Iteration 4'])

text(0,.6,['Finding the value of the function at the lower and upper guesses and the estimated root'])

hold off

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,['Absolute relative approximate error'])

text(0,.8,['ea = abs((xr - xp) / xr)*100 = ',num2str(ea),'%'])

text(0,.4,['Check the interval between which the root 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 representing 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)

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plot([lxrange,uxrange],[0,0],'k','linewidth',1)

title('Entered function on given interval with upper and lower guesses') % This portion adds the text and math to the top part of the figure window subplot(3,1,1), text(0,1,['Iteration 5'])

text(0,.6,['Finding the value of the function at the lower and upper guesses and the estimated root'])

hold off

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,['Absolute relative approximate error'])

text(0,.8,['ea = abs((xr - xp) / xr)*100 = ',num2str(ea),'%'])

text(0,.4,['Check the interval between which the root 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)

% - p0, p1, and p2 are the initial approximations % - delta is the tolerance for p0, p1, and p2

% - epsilon the the tolerance for the function values y % - max1 is the maximum number of iterations

%Output - p is the Muller approximation to the zero 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 @ notation % call [p,y,err]=muller(@f,p0,p1,p2,delta,epsilon,max1). %If f is defined as an anonymous function use the

% call [p,y,err]=muller(f,p0,p1,p2,delta,epsilon,max1). % Copyright by Reza Abazari 2008-04-23

% Email : [email protected] %Initalize the matrices P and Y P=[p0 p1 p2];

Y=f(P);

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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 to 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 @ notation % call [c,err,yc]=regula(@f,a,b,delta,epsilon,max1)

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% call [c,err,yc]=regula(f,a,b,delta,epsilon,max1) % Copyright by Reza Abazari 2008-04-23

% Email : [email protected] 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)

% - p0 and p1 are the initial approximations to a zero of f % - delta is the tolerance for p1

% - epsilon is the tolerance for the function values y % - max1 is the maximum number of iterations

%Output - p1 is the secant method approximation to 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 @ notation % 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).

for k=1:max1

p2=p1-f(p1)*(p1-p0)/(f(p1)-f(p0)); err=abs(p2-p1);

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

%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 the @ 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,max1).

% Copyright by Reza Abazari 2008-04-23 % Email : [email protected] 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),break,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 starting simplex

% - min1 & max1 are minimum and maximum number of iterations % - epsilon is the tolerance

% - show == 1 displays iterations (P and 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 minimum

% - P is a matrix containing the vertex iterations % - Q is an array containing the iterations for F(P) %Call [V0,y0,dV,dy]=nelder(@F,V,min1,max1,epsilon,show)

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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;

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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 given polynomial using

% newton raphson technique.this m-file calls the functions in the two % m-files named as syn_division.m and derivate.m.

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% polynomial.

% Copyright by Reza Abazari 2008-12-09 % Email : [email protected] 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_root_old^powers; powers=powers-1; end powers=order_der_fun-1; for index=1:order_der_fun dfx=dfx+coeff_der_function(index)*fun_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)<=error_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 function, the % limitation of

% this function is, it can calculate only the derivatives of % power(x,n)....

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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_function(index); der_order=der_order-1; end ++++++++++++++++++++++++++++++++++++++++++ % This m-file takes care of synthetic division.

% By giving one polynomial and one root this function returns

% the polynomial formed with the other roots of the given polynomial % excluding the given root.

% Copyright by Reza Abazari 2008-04-23 % Email : [email protected] function coeff_second=syn_division(coeff_function,fun_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 given the end of this m.file. % findzero.m

powers=ones(order+2,1)*([0:order]);% the powers of the polynomial % repeated in rows (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 from 1 in a column. % This is used when differntiation the polynomial

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 array

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 matrix elementwise

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M=[coeff_h coeff_E]; % the matrix of the LHS of the % linear system of equations

N= feval(fun,y); % the column vector of the RHS % of the linear system of equations

A=M\N; % solution of the linear system of equations, % first (order +1) element are the coefficients % of the polynomial. Last element is the value % of the error at these points

A1=A(1:end-1); % the coefficients only

A_der=A(2:end-1).*t; % the coeffcients of the derivative % of the polynomial

z(1)=interval(1); % z(1) is the start point 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 error 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 seek the point that % maximizes the magnitude of the error function. If there is an extreme % point (local maximum or local minimum) between such two points of the % z array then the derivative of the error function is zero at this % extreme point. We thus look for the extreme point by looking for the % root of the derivative of the error function between these two

% points. If the extreme point doesn't exist then 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 therefore 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 corressponding point % in the old array is less than a certain threshold then quit the loop

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if abs(y(ind)-y1(ind)) <2^-30 break;

end

% compare it also with the following point 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 root.

% x0 and x1 are two arguments to the function such that the root that % we seek lie between them and the function has different 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

if sign(f0)==sign(f1)

error('the function at the two endpoints must be 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 intersection 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 point 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

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% 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)

% the polynomial coefficients array , make it a column 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 argument 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 the resulting row % is raised with the corresponding power in the powers matrix

temp=((x-first)*ones(1,order+1)).^powers;

% multiply the resulting matrix with the coefficients table in order % to obtain a column array. Each element of the resulting array is % equal to the polynomial evaluated at the distance between the % corresponding argument and the start of the interval

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

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% 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)

% bisect Use bisection to find a root of the scalar 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 which roots are sought

% xb = vector of bracket endpoints. xleft = xb(1), xright = xb(2) % xtol = (optional) relative x tolerance. Default: xtol=5*eps % ftol = (optional) relative f(x) tolerance. Default: ftol=5*eps % verbose = (optional) print switch. Default: verbose=0, no printing %

% Output: r = root (or singularity) of the function in xb(1) <= x <= xb(2) if size(xb,1)>1, warning('Only first row of xb is used 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 tolerances 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 bracket in convergence test fa = feval(fun,a); fb = feval(fun,b);

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if sign(fa)==sign(fb) % Verify sign change 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 computing 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)

% brackPlot Find subintervals on x that contain sign 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 subdivide into brackets. % nx = (optional) number of samples along 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 limits. 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 brackets are found, Xb = []. if nargin<4, nx=20; end

% --- Create data for a plot of f(x) on interval xmin <= x <= xmax xp = linspace(xmin,xmax); yp = feval(fun,xp);

% --- Save data used to draw boxes that indicate brackets

ytop = max(yp); ybot = min(yp); % y coordinates of the box ybox = 0.05*[ybot ytop ytop ybot ybot]; % around a bracket

c = [0.7 0.7 0.7]; % RGB color used to fill the box % --- Begin search for brackets

x = linspace(xmin,xmax,nx); % Vector of potential bracket limits

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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 right ends of the bracket hold on; fill([x(k) x(k) x(k+1) x(k+1) x(k)],ybox,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 boxes 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-file defining function % whose root is to be located

% a,b left and right endpoints, respectively, of interval % known to contain root of f

% TOL absolute error convergence tolerance

% (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 interval and the current

% approximation to the root are displayed %

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 %.10f \n', i, a, b, p)) end if ( (b-a)<2*TOL | fp == 0 ) if ( nargout == 1 )

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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:

% Nmax maximum number of iterations to be performed %

% output:

% NOTE:

% if FALSE_POS is called with no output arguments, the iteration % number, the current enclosing interval and the current

% approximation to the root are displayed %

% if the maximum number of iterations is 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 : [email protected] 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 %.10f \n', i, a, b, new)) end if ( abs(new-old) < TOL ) if ( nargout == 1 ) y = new; end

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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 root of a nonlinear function %

% y = false_pos_aitken ( 'f', a, b, TOL, Nmax ) % false_pos_aitken ( 'f', a, b, TOL, Nmax ) %

% inputs:

% (applied to extrapolated sequence of approximations) % Nmax maximum number of iterations to be performed

%

% output:

% NOTE:

% if FALSE_POS_AITKEN is called with no output arguments, % the iteration number, the current false position

% approximation to the root and the current extrapolated % approximation to the root are displayed.

%

% Copyright by Reza Abazari 2008-04-23 % Email : [email protected] 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 )

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if ( nargout == 0 )

disp ( sprintf ( '\t\t %3d \t %.10f \n', i, new ) ) end

else

phat = new - ( new - old ) ^ 2 / ( new - 2 * old + 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 arbitrary function using general % fixed_point iteration

%

% y = fixed_point ( 'g', x0, TOL, Nmax ) % fixed_point ( 'g', x0, TOL, Nmax ) %

% inputs:

% g string containing name of m-file defining the % iteration function

% x0 initial approximation to location of fixed point % TOL absolute error convergence tolerance

% output:

% y approximate value of fixed point %

% NOTE:

% if FIXED_POINT is called with no output arguments, the % iteration number and the current approximation to the % fixed point are displayed

%

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% to this effect will be displayed and the most recent % approximation will be returned in the output value %

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

y = new; end

@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@ function y = fixed_point_aitken ( f, x0, TOL, Nmax )

%FIXED_POINT_AITKEN uses general fixed point iteration with Aitken

% extrapolation to locate the root of a nonlinear function %

% y = fixed_point_aitken ( 'g', x0, TOL, Nmax ) % fixed_point_aitken ( 'g', x0, TOL, Nmax ) %

% inputs:

% (applied to extrapolated sequence of approximations) % Nmax maximum number of iterations to be performed

%

% output:

% NOTE:

% if FIXED_POINT_AITKEN is called with no output arguments, % the iteration number, the current functional iteration

% approximation to the fixed point and the current extrapolated % approximation to the fixed point are displayed.

%

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% Copyright by Reza Abazari 2008-04-23 % Email : [email protected] 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 * old + 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

y = phat; end

@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@ function r = polyroots ( poly, Nmax, TOL )

%POLYROOTS locate the roots of an arbitrary polynomial using Laguerre's % method

%

% r = polyroots ( poly, Nmax, TOL ) % polyroots ( poly, Nmax, TOL ) %

% inputs:

% poly vector containing the coefficients of the polynomial % whose roots are to be computed. the first entry in % the vector must be the leading coefficient of the % polynomial, and the final entry must be the constant % term. zero coefficients must be explicitly provided. % Nmax maximum number of iterations to be performed to % compute each root

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% TOL absolute error convergence tolerance %

% output:

% r vector containing the roots of the polynomial %

% NOTE:

% if POLYROOTS is called with no output arguments, the % roots of the polynomial and the number of iterations % required to compute each root are displayed

%

% if the maximum number of iterations is exceeded, a message % to this effect will be displayed and the roots for which % convergence had been achieved will be returned

%

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 )

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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 solution % - delta is the error bound

% - max1 is the number of iterations

%Output - P is the seidel approximation to the solution % - iter is the number of iterations required %Use the @ notation call [P,iter]=seidel(@G, P, delta, max1). % Copyright by Reza Abazari 2008-04-23

% Email : [email protected] 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 calculated 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,max1)

%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 iterations %Output -P is the approximation to the solution

% -iter is the number of iterations required % -err is the error estimate for P

%Use the @ notation call

%[P,iter,err]=newdim(@F, @JF, P, delta, epsilon, max1). % Copyright by Reza Abazari 2008-04-23

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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 : [email protected] 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; @@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@

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function y = steffensen ( f, x0, TOL, Nmax )

%STEFFENSEN locate the fixed point of an arbitrary function using % Steffensen's method

%

% y = steffensen ( 'g', x0, TOL, Nmax ) % steffensen ( 'g', x0, TOL, Nmax ) %

% inputs:

% output:

% NOTE:

% if STEFFENSEN is called with no output arguments, the % iteration number and the current approximation to the % fixed point are displayed

%

% Copyright by Reza Abazari 2008-04-23 % Email : [email protected] 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, phat ) ) end if ( abs(phat-phatold) < TOL ) if ( nargout == 1 ) y = phat; end return else phatold = phat; p0 = phat; end end

y = phat; end

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40 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 solution % - delta is the error bound

% - max1 is the number of iterations

%Output - P is the seidel approximation to the solution % - iter is the number of iterations required %Use the @ notation call [P,iter]=seidel(@G, P, delta, max1). % Copyright by Reza Abazari 2008-04-23

% Email : [email protected] 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 calculated 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,Q]=steff(f,df,p0,delta,epsilon,max1)

% - df is the derivative of f input as a string 'df' % - p0 is the initial approximation to a zero of f % - delta is the tolerance for p0

%Output - p is the Steffensen approximation to the zero % - Q is the matrix containing the Steffensen sequence %If f and df are defined as M-file functions use the @ 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).

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R=zeros(max1,3);

R(1,1)=p0; for k=1:max1 for j=2:3

%Denominator in Newton-Raphson method is calculated nrdenom=df(R(k,j-1));

%Conditional calculates Newton-Raphson approximations 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 approximations 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 matrix 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,max1)

%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 iterations %Output -P is the approximation to the solution

% -iter is the number of iterations required % -err is the error estimate for P

%Use the @ notation call

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% Copyright by Reza Abazari 2008-04-23 % Email : [email protected] 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 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 routine - no pivoting % strategy is implemented to reduce the effects of roundoff % error

%

[nrow ncol] = size ( A ); if ( nrow ~= ncol )

disp ( 'gauss_elim error: Square coefficient matrix required' ); return;

end;

nb = length ( b ); if ( nrow ~= nb )

disp('gauss_elim error: Size of b-vector not compatible with matrix dimension') return;

end;

x = zeros ( 1, nrow ); % Gaussian elimination

Numerical Mathematics With MATLAB