%SHOWNEWTSYS Displays the interesting behavior of problem 1.8 #12

format long, clear pta ptb

s1='[x*y^2-2*z+sin(x)+2; 2*x^4+3*x*y^2+y*z-1;';
s2='2*x*y+cos(x*y)+2*z^3-3]';
F=inline([s1 s2],'x','y','z')

s1='[y^2+cos(x), 2*x*y, -2; 8*x^3+3*y^2, 6*x*y+z, y;';
s2='2*y-sin(x*y)*y, 2*x-sin(x*y)*x, 8*z^2]';
J=inline([s1 s2],'x','y','z')

disp('FIRST TRY A GOOD STARTING POINT:')
x=[-0.1; 0.5; 1.1]; x'
j=1;
while 1
    pta(j,:)=x';  j=j+1;
    dx=-J(x(1),x(2),x(3))\F(x(1),x(2),x(3));
    if norm(dx)<1e-14, break, end
    x=x+dx; x'
end

disp('NEXT A BAD STARTING POINT:')
x=[-2; -2; 2]; x'
j=1;
while 1
    ptb(j,:)=x';  j=j+1;
    dx=-J(x(1),x(2),x(3))\F(x(1),x(2),x(3));
    if norm(dx)<1e-14, break, end
    x=x+dx; x'
end

plot3(pta(:,1),pta(:,2),pta(:,3),'o',...
      ptb(:,1),ptb(:,2),ptb(:,3),'*')
grid on