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Newton Method Non Linear System-Numerical Analysis-MATLAB Code, Exercises of Mathematical Methods for Numerical Analysis and Optimization

This is solution to one of problems in Numerical Analysis. This is matlab code. Its helpful to students of Computer Science, Electrical and Mechanical Engineering. This code also help to understand algorithm and logic behind the problem. This code includes: Newton, Method, Nonlinear, Systemns, Algorithm, Approximate, Solution, Initial, Iterations, Equations, Tolerance

Typology: Exercises

2011/2012

Uploaded on 07/31/2012

saripella
saripella 🇮🇳

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% NEWTON'S METHOD FOR SYSTEMS ALGORITHM 10.1
%
% To approximate the solution of the nonlinear system F(X)=0 given
% an initial approximation X:
%
% INPUT: Number n of equations and unknowns; initial approximation
% X=(X(1),...,X(n)); tolerance TOL; maximum number of
% iterations N.
%
% OUTPUT: Approximate solution X=(X(1),...,X(n)) or a message
% that the number of iterations was exceeded.
syms('OK', 'N', 'I', 'J', 'P', 'TOL', 'NN', 'X','ZZ');
syms('FLAG', 'NAME', 'OUP', 'K', 'A', 'R', 's', 'ss');
syms('K1', 'I1', 'Z1', 'IR1', 'IA1', 'J1', 'C1', 'L1', 'JA1');
TRUE = 1;
FALSE = 0;
fprintf(1,'This is the Newton Method for Nonlinear Systems.\n');
fprintf(1,'The functions could be input or defined in code.\n');
fprintf(1,'This code assumes input of functions - see \n');
fprintf(1,'comments in code for alternate version.\n');
fprintf(1,'This program also uses M-files JAC.M and FN.M \n');
fprintf(1,'If the number of equations exceeds 7 then JAC.M\n');
fprintf(1,'and FN.M must be changed.\n');
OK = FALSE;
while OK == FALSE
fprintf(1,'Input the number n of equations.\n');
N = input(' ');
if N >= 2
OK = TRUE;
else
fprintf(1,'N must be an integer greater than 1.\n');
end;
end;
A = zeros(N,N+1);
X = zeros(1,N);
Y = zeros(1,N);
% Define components of F as follows:
% s(1) = '3*y1-cos(y2*y3)-0.5';
% s(2) = 'y1^2-81*(y2+0.1)^2+sin(y3)+1.06';
% s(3) = 'exp(-y1*y2)+20*y3+(10*pi-3)/3';
for I = 1 : N
fprintf(1,'Input the function F_(%d) in terms of y1 ... y%d \n' ,I ,N);
s(I) = input(' ','s');
end;
for I = 1 : N
for J = 1 : N
fprintf(1,'Input the partial of F_(%d) with respect to x_%d \n',I,J);
fprintf(1,'in terms of y1, ..., y%d \n',N);
ss((I-1)*N+J) = input(' ','s');
end;
end;
% Define the entries of the Jacobian in row major ordering.
% ss(1) = '3';
% ss(2) = 'y3*sin(y2*y3)';
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% NEWTON'S METHOD FOR SYSTEMS ALGORITHM 10.

% To approximate the solution of the nonlinear system F(X)=0 given % an initial approximation X: % % INPUT: Number n of equations and unknowns; initial approximation % X=(X(1),...,X(n)); tolerance TOL; maximum number of % iterations N. % % OUTPUT: Approximate solution X=(X(1),...,X(n)) or a message % that the number of iterations was exceeded. syms('OK', 'N', 'I', 'J', 'P', 'TOL', 'NN', 'X','ZZ'); syms('FLAG', 'NAME', 'OUP', 'K', 'A', 'R', 's', 'ss'); syms('K1', 'I1', 'Z1', 'IR1', 'IA1', 'J1', 'C1', 'L1', 'JA1'); TRUE = 1; FALSE = 0; fprintf(1,'This is the Newton Method for Nonlinear Systems.\n'); fprintf(1,'The functions could be input or defined in code.\n'); fprintf(1,'This code assumes input of functions - see \n'); fprintf(1,'comments in code for alternate version.\n'); fprintf(1,'This program also uses M-files JAC.M and FN.M \n'); fprintf(1,'If the number of equations exceeds 7 then JAC.M\n'); fprintf(1,'and FN.M must be changed.\n'); OK = FALSE; while OK == FALSE fprintf(1,'Input the number n of equations.\n'); N = input(' '); if N >= 2 OK = TRUE; else fprintf(1,'N must be an integer greater than 1.\n'); end; end; A = zeros(N,N+1); X = zeros(1,N); Y = zeros(1,N); % Define components of F as follows: % s(1) = '3y1-cos(y2y3)-0.5'; % s(2) = 'y1^2-81(y2+0.1)^2+sin(y3)+1.06'; % s(3) = 'exp(-y1y2)+20y3+(10pi-3)/3'; for I = 1 : N fprintf(1,'Input the function F_(%d) in terms of y1 ... y%d \n' ,I ,N); s(I) = input(' ','s'); end; for I = 1 : N for J = 1 : N fprintf(1,'Input the partial of F_(%d) with respect to x_%d \n',I,J); fprintf(1,'in terms of y1, ..., y%d \n',N); ss((I-1)N+J) = input(' ','s'); end; end; % Define the entries of the Jacobian in row major ordering. % ss(1) = '3'; % ss(2) = 'y3sin(y2*y3)';

% ss(3) = 'y2sin(y2y3)'; % ss(4) = '2y1'; % ss(5) = '-162(y2+0.1)'; % ss(6) = 'cos(y3)'; % ss(7) = '-y2exp(-y1y2)'; % ss(8) = '-y1exp(-y1y2)'; % ss(9) = '20'; OK = FALSE; while OK == FALSE fprintf(1,'Input the Tolerance.\n'); TOL = input(' '); if TOL > 0 OK = TRUE; else fprintf(1,'Tolerance must be positive.\n'); end; end; OK = FALSE; while OK == FALSE fprintf(1,'Input the maximum number of iterations.\n'); NN = input(' '); if NN > 0 OK = TRUE; else fprintf(1,'Must be a positive integer.\n'); end; end; for I = 1 : N fprintf(1,'Input initial approximation X(%d).\n', I); X(I) = input(' '); end; fprintf(1,'Select output destination\n'); fprintf(1,'1. Screen\n'); fprintf(1,'2. Text file\n'); fprintf(1,'Enter 1 or 2\n'); FLAG = input(' '); if FLAG == 2 fprintf(1,'Input the file name in the form - drive\:name.ext\n'); fprintf(1,'for example A:\OUTPUT.DTA\n'); NAME = input(' ','s'); OUP = fopen(NAME,'wt'); else OUP = 1; end; fprintf(1,'Select amount of output\n'); fprintf(1,'1. Answer only\n'); fprintf(1,'2. All intermediate approximations\n'); fprintf(1,'Enter 1 or 2\n'); FLAG = input(' '); fprintf(OUP, 'NEWTONS METHOD FOR NONLINEAR SYSTEMS\n\n'); if FLAG == 2 fprintf(OUP, 'Iteration, Approximation, Error\n'); end; % STEP 1

J1 = N-I1;

JA1 = J1+1;

C1 = A(J1,N+1);

for L1 = JA1 : N C1 = C1-A(J1,L1)*Y(L1); end; Y(J1) = C1/A(J1,J1); end; end; end; if OK == FALSE fprintf(1,'Linear system is singular\n'); end; if OK == TRUE % STEP 5 R = 0; for I = 1 : N if abs(Y(I)) > R R = abs(Y(I)); end; X(I) = X(I)+Y(I); end; if FLAG == 2 fprintf(OUP, ' %2d', K); for I = 1 : N fprintf(OUP, ' %11.8f', X(I)); end; fprintf(OUP, '\n%12.6e\n', R); end; % STEP 6 if R < TOL OK = FALSE; fprintf(OUP, 'Iteration %d gives solution:\n\n', K); for I = 1 : N fprintf(OUP, ' %11.8f', X(I)); end; fprintf(OUP, '\n\nto within tolerance %.10e\n', TOL); % STEP 7 else K = K+1; end; end; end; if K > NN % STEP 8 fprintf(OUP, 'Procedure does not converge in %d iterations\n', NN); end; if OUP ~= 1 fclose(OUP); fprintf(1,'Output file %s created successfully \n',NAME); end;