on Notes Newton-Raphson A root-solving technique
1. Introduction to Newton-Raphson
(Updated 11/3/2010) (Dr. Shoane)
Based on geometric interpretation or Taylor series expansion of the function, f(x) at xi.
From the geometry of the situation seen in the graph, we have
f ( xi ) =
Hence
f ( xi ) 0 xi xi +1
xi+1 = xi
f ( xi ) f ( xi )
The Newton-Raphson algorithm attempts to minimize the difference between xi and xi+1 by iteratively updating the next xi value. This continues until xi and xi+1 are very close to each other. The critical point occurs when the function, f, crosses the zero line, which is where the equation f(x)=0 (e.g., x3+2x2+1=0) is satisfied. It can be shown that at this critical point, xi xi+1. Importantly, the value of x at this point is a root (or solution) of the equation f(x)=0.
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2. Examples Example 1 f ( x) = ( x 3) ( x 1) ( x 1) = x 3 5 x 2 + 7 x 3 = 0 f ( x) = 3 x 2 10 x + 7 xi +1 = xi xi3 5 xi2 + 7 xi 3 3xi2 10 xi + 7
Example 2 f ( x, y ) = x 2 + xy 10 = 0 f ( x, y ) xi +1 = xi df ( x, y ) / dy
3. Video http://www.youtube.com/watch?v=lFYzdOemDj8
4. Code for function newton http://math.fullerton.edu/mathews/n2003/newtonsmethod/New ton%27sMethodProg/Links/Newton%27sMethodProg_lnk_3.html
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Modified Program Demonstrating Newton-Raphson Technique (Dr. Shoane debugged the above program, and it should run as is, after cleaning up any comments that were moved out of position.) ******** Copy the entire code starting from here to the end of the program **********
function newton_test() % From http://math.fullerton.edu/mathews/n2003/newtonsmethod/Newton %27sMethodProg/Links/Newton%27sMethodProg_lnk_3.html % 1) Modified by Prof. George Shoane 11/2/2010: positions of functions, plus comments at each pause. % 2) The test functions have been put at the end of the program. % 3) It is suggested that the students learn how the function "newton" % works, and then write their own functions to solve any given problem. % This is probably more efficient than trying to input a complicated % nonlinear system equation with two variables to fit the format of the function "newton". % 4) % % % call % % If the nonlinear function consists of 2 variables where one of them is an indpendent variable, then one technique is to update the independent variable and set it as a constant at each iteration. Then the "newton" function can be used directly. The output after the function provides the estimated value of the dependent variable for that particular independent variable value. (Updated 11/3/2010)
% 5) Hint: Use symbolic representation to define the function and obtain the % derivative of the function. But do not use the symbolic variables in the % actual root solving program. (Updated 11/3/2010)
% THE FOLLOWING SCRIPT FILE WAS USED TO CALL THE NEWTON SUBROUTINE echo on;clc; %--------------------------------------------------------------------------% %A2_5 MATLAB script file for implementing Algorithm 2.5 % % NUMERICAL METHODS:MATLAB Programs,(c) John H.Mathews 1995 % To accompany the text % NUMERICAL METHODS for Mathematics,Science and Engineering,2nd Ed,1992 % Prentice Hall,Englewood Cliffs,New Jersey,07632,U.S.A % Prentice Hall,Inc.;USA,Canada,Mexico ISBN 0-13-624990-6 % Prentice Hall,International Editions:ISBN 0-13-625047-5 % This free software is compliments of the author % E-mail address: "[e-mail]" % % Algorithm 2.5 (Newton-Raphson Iteration) % Section 2.4,Newton-Raphson and Secant Methods,Page 84 %---------------------------------------------------------------------------% clear all; close all; format long;
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%-------------------------% % This program implements the Newton-Raphson method.% % % Define and store the functions f(x) and f'(x) % in the M-files f.m and df.m respectively disp('(1) Display the functions f and df. Press any key'); pause % Press any key to continue % GS clc; %.......................................................... % Begin a section which enters the function(s) necessary for the example % into M-file(s) by executing the diary command in this script file % The preferred programming method is not to use these steps % One should enter the function(s) into the M-file(s) with an editor delete output delete f.m %diary f.m;disp('function y=f(x)');... disp('y=x.^3-x-3;');... diary off; delete df.m %diary df.m;disp('function y1=df(x)');... disp('y1=3*x.^2-1;');... diary off; % Remark.f.m,df.m and newton.m are used for Algorithm 2.5 f(0); df(0); % Test for files f.m,df.m disp('(2) Plot y=f(x) versus x.; Set max iterations to 50. Press any key'); pause % Press any key to see the graph y=f(x).clc; %~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~% % Prepare graphics arrays to plot y=f(x) %~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~% a=-3.0; b=3.0; h=(b-a)/150; X=a:h:b; Y=f(X); % GS clc; figure(1); % GS clf; %~~~~~~~~~~~~~~~~~~~~~~~% % Begin graphics section %~~~~~~~~~~~~~~~~~~~~~~~% a=-3.0; b=3.0; c=-10; d=10; whitebg('w'); plot([a b],[0 0],'b',[0 0],[c d],'b'); axis([a b c d]); axis(axis); hold on; plot(X,Y,'-g');
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xlabel('x'); ylabel('y'); title('Graph of y=f(x).'); grid; hold off; disp('(3) Prepare Initial Iterations. Press any key'); figure(gcf);pause % Press any key to perform Newton-Raphson iteration % GS clc; %------------------------------% % Example,page 79 Use Newton-Raphson iteration for finding % a zero of the function f(x)=x^3-x-3. % % Enter the starting value in p0 % Enter the abscissa tolerance in delta % Enter the ordinate tolerance epsilon in % Enter the maximum number of iterations in max1 p0=2.0; delta=1e-12; epsilon=1e-12; max1=50; [p0,y0,err,P]=newton('f','df',p0,delta,epsilon,max1); disp('(4) Show Graphical Anayisis for Newton method. pause % Press any key for the list of iterations % GS clc; %~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~% % Prepare arrays to graph and print results %~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~% a=1.6; b=2.05; h=(b-a)/150; X=a:h:b; Y=f(X); max1=length(P); clear Vx Vy for i=1:max1, k1=2*i-1; k2=2*i; Vx(k1)=P(i); Vy(k1)=0; Vx(k2)=P(i); Vy(k2)=f(P(i)); end Z0=zeros(1,length(P)); % GS clc; figure(2); % GS clf; %~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~% % Begin graphics section for the results %~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~% a=1.6; Press any key');
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b=2.05; c=-0.5; d=3.5; whitebg('w'); plot([a b],[0 0],'b',[0 0],[c d],'b'); axis([a b c d]); axis(axis); hold on; plot(X,Y,'-g',Vx,Vy,'-r',P,Z0,'or'); xlabel('x'); ylabel('y'); title('Graphical analysis for Newton`s method.'); grid; hold off; figure(gcf); disp('(5) Print results. Press any key'); pause % Press any key to continue %.......... % Prepare results %.......... J=1:max1; Yp=f(P); points=[J;P';Yp']; % GS clc; %............................................ % Begin section to print the results % Diary commands are included which write all % the results to the Matlab textfile output %............................................ Mx1='Iterations for Newton`s method.'; Mx2=' k p(k) f(p(k))'; Mx3='The solution is:'; Mx4='The error estimate for p is'; % GS clc, echo off,diary output,... disp(''),disp(Mx1),disp(''),disp(Mx2),disp(points'),... disp('Iteration converged quadratically to the root.'),... disp(''),disp(Mx3),disp(''),disp('p='),... disp(p0),disp('f(p)='),disp(y0),... disp([Mx4,num2str(err)]),diary off,echo on disp('(6) This time set max iterations to 12. Press any key'); pause % Press any key to perform Newton-Raphson iteration.clc; %----------------------------------------------------% % Example,page 79 Use Newton-Raphson iteration for finding % a zero of the function f(x)=x^3-x-3. % % Enter the starting value in p0 % Enter the abscissa tolerance in delta % Enter the ordinate tolerance in epsilon % Enter the maximum number of iterations in max1 p0=0.0;
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delta=1e-12; epsilon=1e-12; max1=12; [p0,y0,err,P]=newton('f','df',p0,delta,epsilon,max1); disp('(7) Plot graph. Press any key'); pause % Press any key for the list of iterations.clc; %~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~% % Prepare arrays to graph and print results %~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~% a=-3.5; b=0.5; h=(b-a)/100; X=a:h:b; Y=f(X); max1=length(P); clear Vx Vy for i=1:max1,k1=2*i-1; k2=2*i; Vx(k1)=P(i); Vy(k1)=0; Vx(k2)=P(i); Vy(k2)=f(P(i)); end Z0=zeros(1,length(P)); % clc; figure(3); % clf; %~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~% % Begin graphics section for the results %~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~% a=-3.5; b=0.5; c=-30; d=5; whitebg('w'); plot([a b],[0 0],'b',[0 0],[c d],'b'); axis([a b c d]); axis(axis); hold on; plot(X,Y,'-g',Vx,Vy,'-r',P,Z0,'or'); xlabel('x'); ylabel('y'); title('Graphical analysis for Newton`s method.'); grid; hold off; figure(gcf); disp('(8) Print results. Press any key'); pause % Press any key to continue %..............
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% Prepare results %.............. J=1:max1; Yp=f(P); points=[J;P';Yp']; % GS clc; %............................................ % Begin section to print the results % Diary commands are included which write all % the results to the Matlab textfile output %............................................ Mx1='Iterations for Newton`s method.'; Mx2=' k p(k) f(p(k))'; % GS clc, echo off,diary output,... disp(''),disp(Mx1),disp(''),disp(Mx2),disp(points'),... disp('Iteration did not occur.This is a case of "cycling."'),... diary off,echo on
%(c) John H. Mathews 2004 %end
function[p0,y0,err,P]=newton(f,df,p0,delta,epsilon,max1) %--------------------------------------------------------% %NEWTON Newton's method is used to locate a root % Sample calls %[p0,y0,err]=newton('f','df',p0,delta,epsilon,max1) %[p0,y0,err,P]=newton('f','df',p0,delta,epsilon,max1) % Inputs % f name of the function % df name of the function's derivative input % p0 starting value % delta convergence tolerance for p0 % epsilon convergence tolerance for y0 % max1 maximum number of iterations % Return % p0 solution:the root % y0 solution:the function value % err error estimate in the solution p0 % P History vector of the iterations % % NUMERICAL METHODS:MATLAB Programs,(c) John H.Mathews 1995 % To accompany the text % NUMERICAL METHODS for Mathematics,Science and Engineering, 2nd Ed,1992 % Prentice Hall,Englewood Cliffs,New Jersey,07632,U.S.A % Prentice Hall,Inc.;USA,Canada,Mexico ISBN 0-13-624990-6 % Prentice Hall,International Editions:ISBN 0-13-625047-5 % This free software is compliments of the author % E-mail address: "[e-mail]" % % Algorithm 2.5 (Newton-Raphson Iteration) % Section 2.4,Newton-Raphson and Secant Methods,Page 84
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%--------------------------------------------------------% P(1)=p0; y0=feval(f,p0); for k=1:max1,df0=feval(df,p0); %if df0&Equal;0,dp=0; if df0 == 0 % GS 11/2/2010 dp=0; % GS 11/2/2010 else dp=y0/df0; end p1=p0-dp; y1=feval(f,p1); err=abs(dp); relerr=err/(abs(p1)+eps); p0=p1; y0=y1; P=[P;p1]; if (err<delta)|(relerr<delta)|(abs(y1)<epsilon),break,end end function y=f(x) y=x.^3-x-3; function y1=df(x) y1=3*x.^2-1;
**************************** End of Program ****************************
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