Question

d. Use the solution found in part e as an initial guess with Newton's method and a = 0.1, to obtain a good estimate for the solution to the nonlinear system -200u1 + 100u2 = sin(0.1) + au ? 1001 - 20042 + 100u3 = sin(0.2) + auz? 100uz – 200uz + 100u4 = sin(0.3) + auz 100uz – 20014 + 100u5 = sin(0.4) + au42 10004 - 200us + 1000g = sin(0.5) + aus? 100ug - 2004+ 100u, = sin(0.6) + aus? 10006 - 2004, + 100ug = sin(0.7) + au 2 100u7 - 200ug + 100u, = sin(0.8) + aug? 100ug - 200u, = sin(0.9) + au, Note: You might have to do several iterations of Newton's method.

Answer 1

%This code made by Harish Chandra%

%choose initial guess accordingly  

%Main file%%%%%%%%%%%%%%%%%%

clc;
clear all;
format long;
iter=0;
error=1;
tol=1e-5;
x1=0.1;
x2=0.1;
x3=0.1;
x4=0.1;
x5=0.1;
x6=0.1;
x7=0.1;
x8=0.1;
x9=0.1;% initial guess.
% In case, MATLAB shows "Matrix is close to singular or badly scaled."
% then change the initial guess.
iter=0;
%while (error>tol)
for iter=1:10
x0=[x1;x2;x3;x4;x5;x6;x7;x8;x9];
[f J]=newtonraphson2(x0(1),x0(2),x0(3),x0(4),x0(5),x0(6),x0(7),x0(8),x0(9)); % subroutine file
if abs(det(J))<tol
error('newton-Jacobian is singular try for new x0')
break;
end
xn=x0-inv(J)*f;
[ff J]=newtonraphson2(xn(1),xn(2),xn(3),xn(4),xn(5),xn(6),xn(7),xn(8),xn(9));
error=max(abs(ff));
fprintf('\n error %i.',error)
if error<tol
sol=double(xn) % solution of the algebraic equation   
iter ; % print number of iterartion
x1=sol(1);x2=sol(2);x3=sol(3);x4=sol(4);x5=sol(5);x6=sol(6);x7=sol(7);
x8=sol(8);x9=sol(9);
end
x1=xn(1); x2=xn(2);x3=xn(3);x4=xn(4);x5=xn(5);x6=xn(6);x7=xn(7);x8=xn(8);
x9=xn(9);
iter=iter+1;
end

%J

%Function file%%%%%%%%%%%%%%%%

%--------------- Subroutine file -----------------------
function [f J]=newtonraphson2(z1,z2,z3,z4,z5,z6,z7,z8,z9)
syms x1 x2 x3 x4 x5 x6 x7 x8 x9
z=[z1 z2 z3 z4 z5 z6 z7 z8 z9];
%eqn1= 2*x1^2+x2^2-5*x1*x2+2*x1-2*x2+1;
%eqn1=4*x2^3-6*x2-12*x1+6;
%eqn2= x2^2-2*x1*x2-3*x1+x2+1;
%eqn2=-15*x2-3*x1^4-x1^2+10;
eqn1= -200*x1+100*x2-0.1*x1^2-sin(0.1);
eqn2=100*x1-200*x2+100*x3-0.1*x2^2-sin(0.2);
eqn3=100*x2-200*x3+100*x4-0.1*x3^2-sin(0.3);
eqn4=100*x3-200*x4+100*x5-0.1*x4^2-sin(0.4);
eqn5=100*x4-200*x5+100*x6-0.1*x5^2-sin(0.5);
eqn6=100*x5-200*x6+100*x7-0.1*x6^2-sin(0.6);
eqn7=100*x6-200*x7+100*x8-0.1*x7^2-sin(0.7);
eqn8=100*x7-200*x8+100*x9-0.1*x8^2-sin(0.8);
eqn9=100*x8-200*x9-0.1*x9^2-sin(0.9);
f1=subs(eqn1,{x1,x2,x3,x4,x5,x6,x7,x8,x9},z); f1=double(f1);
f2=subs(eqn2,{x1,x2,x3,x4,x5,x6,x7,x8,x9},z); f2=double(f2);
f3=subs(eqn3,{x1,x2,x3,x4,x5,x6,x7,x8,x9},z); f3=double(f3);
f4=subs(eqn4,{x1,x2,x3,x4,x5,x6,x7,x8,x9},z); f4=double(f4);
f5=subs(eqn5,{x1,x2,x3,x4,x5,x6,x7,x8,x9},z); f5=double(f5);
f6=subs(eqn6,{x1,x2,x3,x4,x5,x6,x7,x8,x9},z); f6=double(f6);
f7=subs(eqn7,{x1,x2,x3,x4,x5,x6,x7,x8,x9},z); f7=double(f7);
f8=subs(eqn8,{x1,x2,x3,x4,x5,x6,x7,x8,x9},z); f8=double(f8);
f9=subs(eqn9,{x1,x2,x3,x4,x5,x6,x7,x8,x9},z); f9=double(f9);
f=[f1;f2;f3;f4;f5;f6;f7;f8;f9];
D11=diff(eqn1,x1);D12=diff(eqn1,x2);D13=diff(eqn1,x3);D14=diff(eqn1,x4);D15=diff(eqn1,x5);
D16=diff(eqn1,x6);D17=diff(eqn1,x7);D18=diff(eqn1,x8);D19=diff(eqn1,x9);

D21=diff(eqn2,x1);D22=diff(eqn2,x2);D23=diff(eqn2,x3);D24=diff(eqn2,x4);D25=diff(eqn2,x5);
D26=diff(eqn2,x6);D27=diff(eqn2,x7);D28=diff(eqn2,x8);D29=diff(eqn2,x9);

D31=diff(eqn3,x1);D32=diff(eqn3,x2);D33=diff(eqn3,x3);D34=diff(eqn3,x4);D35=diff(eqn3,x5);
D36=diff(eqn3,x6);D37=diff(eqn3,x7);D38=diff(eqn3,x8);D39=diff(eqn3,x9);

D41=diff(eqn4,x1);D42=diff(eqn4,x2);D43=diff(eqn4,x3);D44=diff(eqn4,x4);D45=diff(eqn4,x5);
D46=diff(eqn4,x6);D47=diff(eqn4,x7);D48=diff(eqn4,x8);D49=diff(eqn4,x9);

D51=diff(eqn5,x1);D52=diff(eqn5,x2);D53=diff(eqn5,x3);D54=diff(eqn5,x4);D55=diff(eqn5,x5);
D56=diff(eqn5,x6);D57=diff(eqn5,x7);D58=diff(eqn5,x8);D59=diff(eqn5,x9);

D61=diff(eqn6,x1);D62=diff(eqn6,x2);D63=diff(eqn6,x3);D64=diff(eqn6,x4);D65=diff(eqn6,x5);
D66=diff(eqn6,x6);D67=diff(eqn6,x7);D68=diff(eqn6,x8);D69=diff(eqn6,x9);

D71=diff(eqn7,x1);D72=diff(eqn7,x2);D73=diff(eqn7,x3);D74=diff(eqn7,x4);D75=diff(eqn7,x5);
D76=diff(eqn7,x6);D77=diff(eqn7,x7);D78=diff(eqn7,x8);D79=diff(eqn7,x9);

D81=diff(eqn8,x1);D82=diff(eqn8,x2);D83=diff(eqn8,x3);D84=diff(eqn8,x4);D85=diff(eqn8,x5);
D86=diff(eqn8,x6);D87=diff(eqn8,x7);D88=diff(eqn8,x8);D89=diff(eqn8,x9);

D91=diff(eqn9,x1);D92=diff(eqn9,x2);D93=diff(eqn9,x3);D94=diff(eqn9,x4);D95=diff(eqn9,x5);
D96=diff(eqn9,x6);D97=diff(eqn9,x7);D98=diff(eqn9,x8);D99=diff(eqn9,x9);

J11=subs(D11,{x1,x2,x3,x4,x5,x6,x7,x8,x9},z);
J12=subs(D12,{x1,x2,x3,x4,x5,x6,x7,x8,x9},z);
J13=subs(D13,{x1,x2,x3,x4,x5,x6,x7,x8,x9},z);
J14=subs(D14,{x1,x2,x3,x4,x5,x6,x7,x8,x9},z);
J15=subs(D15,{x1,x2,x3,x4,x5,x6,x7,x8,x9},z);
J16=subs(D16,{x1,x2,x3,x4,x5,x6,x7,x8,x9},z);
J17=subs(D17,{x1,x2,x3,x4,x5,x6,x7,x8,x9},z);
J18=subs(D18,{x1,x2,x3,x4,x5,x6,x7,x8,x9},z);
J19=subs(D19,{x1,x2,x3,x4,x5,x6,x7,x8,x9},z);

J21=subs(D21,{x1,x2,x3,x4,x5,x6,x7,x8,x9},z);
J22=subs(D22,{x1,x2,x3,x4,x5,x6,x7,x8,x9},z);
J23=subs(D23,{x1,x2,x3,x4,x5,x6,x7,x8,x9},z);
J24=subs(D24,{x1,x2,x3,x4,x5,x6,x7,x8,x9},z);
J25=subs(D25,{x1,x2,x3,x4,x5,x6,x7,x8,x9},z);
J26=subs(D26,{x1,x2,x3,x4,x5,x6,x7,x8,x9},z);
J27=subs(D27,{x1,x2,x3,x4,x5,x6,x7,x8,x9},z);
J28=subs(D28,{x1,x2,x3,x4,x5,x6,x7,x8,x9},z);
J29=subs(D29,{x1,x2,x3,x4,x5,x6,x7,x8,x9},z);

J31=subs(D31,{x1,x2,x3,x4,x5,x6,x7,x8,x9},z);
J32=subs(D32,{x1,x2,x3,x4,x5,x6,x7,x8,x9},z);
J33=subs(D33,{x1,x2,x3,x4,x5,x6,x7,x8,x9},z);
J34=subs(D34,{x1,x2,x3,x4,x5,x6,x7,x8,x9},z);
J35=subs(D35,{x1,x2,x3,x4,x5,x6,x7,x8,x9},z);
J36=subs(D36,{x1,x2,x3,x4,x5,x6,x7,x8,x9},z);
J37=subs(D37,{x1,x2,x3,x4,x5,x6,x7,x8,x9},z);
J38=subs(D38,{x1,x2,x3,x4,x5,x6,x7,x8,x9},z);
J39=subs(D39,{x1,x2,x3,x4,x5,x6,x7,x8,x9},z);

J41=subs(D41,{x1,x2,x3,x4,x5,x6,x7,x8,x9},z);
J42=subs(D42,{x1,x2,x3,x4,x5,x6,x7,x8,x9},z);
J43=subs(D43,{x1,x2,x3,x4,x5,x6,x7,x8,x9},z);
J44=subs(D44,{x1,x2,x3,x4,x5,x6,x7,x8,x9},z);
J45=subs(D45,{x1,x2,x3,x4,x5,x6,x7,x8,x9},z);
J46=subs(D46,{x1,x2,x3,x4,x5,x6,x7,x8,x9},z);
J47=subs(D47,{x1,x2,x3,x4,x5,x6,x7,x8,x9},z);
J48=subs(D48,{x1,x2,x3,x4,x5,x6,x7,x8,x9},z);
J49=subs(D49,{x1,x2,x3,x4,x5,x6,x7,x8,x9},z);

J51=subs(D51,{x1,x2,x3,x4,x5,x6,x7,x8,x9},z);
J52=subs(D52,{x1,x2,x3,x4,x5,x6,x7,x8,x9},z);
J53=subs(D53,{x1,x2,x3,x4,x5,x6,x7,x8,x9},z);
J54=subs(D54,{x1,x2,x3,x4,x5,x6,x7,x8,x9},z);
J55=subs(D55,{x1,x2,x3,x4,x5,x6,x7,x8,x9},z);
J56=subs(D56,{x1,x2,x3,x4,x5,x6,x7,x8,x9},z);
J57=subs(D57,{x1,x2,x3,x4,x5,x6,x7,x8,x9},z);
J58=subs(D58,{x1,x2,x3,x4,x5,x6,x7,x8,x9},z);
J59=subs(D59,{x1,x2,x3,x4,x5,x6,x7,x8,x9},z);

J61=subs(D61,{x1,x2,x3,x4,x5,x6,x7,x8,x9},z);
J62=subs(D62,{x1,x2,x3,x4,x5,x6,x7,x8,x9},z);
J63=subs(D63,{x1,x2,x3,x4,x5,x6,x7,x8,x9},z);
J64=subs(D64,{x1,x2,x3,x4,x5,x6,x7,x8,x9},z);
J65=subs(D65,{x1,x2,x3,x4,x5,x6,x7,x8,x9},z);
J66=subs(D66,{x1,x2,x3,x4,x5,x6,x7,x8,x9},z);
J67=subs(D67,{x1,x2,x3,x4,x5,x6,x7,x8,x9},z);
J68=subs(D68,{x1,x2,x3,x4,x5,x6,x7,x8,x9},z);
J69=subs(D69,{x1,x2,x3,x4,x5,x6,x7,x8,x9},z);

J71=subs(D71,{x1,x2,x3,x4,x5,x6,x7,x8,x9},z);
J72=subs(D72,{x1,x2,x3,x4,x5,x6,x7,x8,x9},z);
J73=subs(D73,{x1,x2,x3,x4,x5,x6,x7,x8,x9},z);
J74=subs(D74,{x1,x2,x3,x4,x5,x6,x7,x8,x9},z);
J75=subs(D75,{x1,x2,x3,x4,x5,x6,x7,x8,x9},z);
J76=subs(D76,{x1,x2,x3,x4,x5,x6,x7,x8,x9},z);
J77=subs(D77,{x1,x2,x3,x4,x5,x6,x7,x8,x9},z);
J78=subs(D78,{x1,x2,x3,x4,x5,x6,x7,x8,x9},z);
J79=subs(D79,{x1,x2,x3,x4,x5,x6,x7,x8,x9},z);

J81=subs(D81,{x1,x2,x3,x4,x5,x6,x7,x8,x9},z);
J82=subs(D82,{x1,x2,x3,x4,x5,x6,x7,x8,x9},z);
J83=subs(D83,{x1,x2,x3,x4,x5,x6,x7,x8,x9},z);
J84=subs(D84,{x1,x2,x3,x4,x5,x6,x7,x8,x9},z);
J85=subs(D85,{x1,x2,x3,x4,x5,x6,x7,x8,x9},z);
J86=subs(D86,{x1,x2,x3,x4,x5,x6,x7,x8,x9},z);
J87=subs(D87,{x1,x2,x3,x4,x5,x6,x7,x8,x9},z);
J88=subs(D88,{x1,x2,x3,x4,x5,x6,x7,x8,x9},z);
J89=subs(D89,{x1,x2,x3,x4,x5,x6,x7,x8,x9},z);

J91=subs(D91,{x1,x2,x3,x4,x5,x6,x7,x8,x9},z);
J92=subs(D92,{x1,x2,x3,x4,x5,x6,x7,x8,x9},z);
J93=subs(D93,{x1,x2,x3,x4,x5,x6,x7,x8,x9},z);
J94=subs(D94,{x1,x2,x3,x4,x5,x6,x7,x8,x9},z);
J95=subs(D95,{x1,x2,x3,x4,x5,x6,x7,x8,x9},z);
J96=subs(D96,{x1,x2,x3,x4,x5,x6,x7,x8,x9},z);
J97=subs(D97,{x1,x2,x3,x4,x5,x6,x7,x8,x9},z);
J98=subs(D98,{x1,x2,x3,x4,x5,x6,x7,x8,x9},z);
J99=subs(D99,{x1,x2,x3,x4,x5,x6,x7,x8,x9},z);

J11=double(J11);J12=double(J12);J13=double(J13);J14=double(J14);J15=double(J15);
J16=double(J16);J17=double(J17);J18=double(J18);J19=double(J19);

J21=double(J21);J22=double(J22);J23=double(J23);J24=double(J24);J25=double(J25);
J26=double(J26);J27=double(J27);J28=double(J28);J29=double(J29);

J31=double(J31);J32=double(J32);J33=double(J33);J34=double(J34);J35=double(J35);
J36=double(J36);J37=double(J37);J38=double(J38);J39=double(J39);

J41=double(J41);J42=double(J42);J43=double(J43);J44=double(J44);J45=double(J45);
J46=double(J46);J47=double(J47);J48=double(J48);J49=double(J49);

J51=double(J51);J52=double(J52);J53=double(J53);J54=double(J54);J55=double(J55);
J56=double(J56);J57=double(J57);J58=double(J58);J59=double(J59);

J61=double(J61);J62=double(J62);J63=double(J63);J64=double(J64);J65=double(J65);
J66=double(J66);J67=double(J67);J68=double(J68);J69=double(J69);

J71=double(J71);J72=double(J72);J73=double(J73);J74=double(J74);J75=double(J75);
J76=double(J76);J77=double(J77);J78=double(J78);J79=double(J79);

J81=double(J81);J82=double(J82);J83=double(J83);J84=double(J84);J85=double(J85);
J86=double(J86);J87=double(J87);J88=double(J88);J89=double(J89);

J91=double(J91);J92=double(J92);J93=double(J93);J94=double(J94);J95=double(J95);
J96=double(J96);J97=double(J97);J98=double(J98);J99=double(J99);

J=[J11 J12 J13 J14 J15 J16 J17 J18 J19;J21 J22 J23 J24 J25 J26 J27 J28 J29;...
J31 J32 J33 J34 J35 J36 J37 J38 J39;J41 J42 J43 J44 J45 J46 J47 J48 J49;...
J51 J52 J53 J54 J55 J56 J57 J58 J59;J61 J62 J63 J64 J65 J66 J67 J68 J69;...
J71 J72 J73 J74 J75 J76 J77 J78 J79;J81 J82 J83 J84 J85 J86 J87 J88 J89;...
J91 J92 J93 J94 J95 J96 J97 J98 J99;]; % Jacobian
end

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%Answer is also given bellow

%sol =

-0.015707875470060
-0.030417170036299
-0.043138846090355
-0.052903459117756
-0.058771089946084
-0.059841011347356
-0.055260927068038
-0.044235612146284
-0.026034779526152