Laguerre's method to obtain poly roots - matlab

Question

I have to write using Laguerre's method a piece of code to find real and complex roots of poly:
f(x)=a4x4+a3x3+a2x2+a1x+a0 , [a4 a3 a2 a1 a0]=[-2 5 5 2 1]
I got something:
function y = laguerre (x,coef)
tol = 10^(-10);
%polynomial order
n = length(coef);
i = 0;
while (abs(polyval(coef,x)) > tol)
i = i + 1
x
polyval(coef,x)
%second derivative
a=polyval(polyder(polyder(coef)), x);
%first derivative
b=polyval(polyder(coef), x);
%polynomial value
c=polyval(coef, x);
%square root
r = sqrt((n-1)*[(n-1)*b^2-n*a*c]);
d1 = b + r;
d2 = b - r;
%highest abs value denominator chosen
if (abs(d1) > abs(d2))
d = d1;
else
d = d2;
end
%next iteration value obtained
z = n*c/d;
x = x - z;
end
i = i + 1
x
polyval(coef,x)
y = x;
return
end
and I want to draw and/ or obtain my roots:
coef = [-2 5 5 2 1];
fh = #poly;
%diary on
%plot of the function
figure(1);
fplot(fh, [-500 500]);
hold on;
figure(2);
fplot(fh, [-1.5 3]);
hold on;
%roots obtained using MM2
%z1(1) = MM2(-500,coef);
%z1(2) = MM2(500,coef);
%z1(3) = MM2(1 + i, coef);
%z1(4) = MM2(0 - 2i, coef);
%roots obtained using Laguerre's
%z2(1) = laguerre(-500,coef);
z2(2) = laguerre(500,coef);
%z2(3) = laguerre(1 + i, coef);
%z2(4) = laguerre(0 - 2i, coef);
%additional points on plot
p = [z1(1) z2(2)];
plot(p,[0 0],'r*');
legend('function','zeros');
hold off;
figure(1);
plot(p,[0 0],'r*');
plot([-500 500],[0 0], 'b*');
legend('function','zeros','START points');
diary off;
hold off;
As you see, some part of code is commented because I really don't know how to move on. I got many warnings. I got one error in 1st part of the code. I am looking for some tips or direction what to do.

Answer 1

Your code works perfectly well, you should get the roots
x=-500 => x=-0.6694729, p(x)=-8.882D-16 in 9 steps
x= 500 => x=3.3489712, p(x)=1.621D-14 in 8 steps
x=1 + i => x=-0.0897491+i*0.4636332, p(x)=i*8.327D-17 in 4 steps
x=0 - 2i => x=-0.0897491-i*0.4636332, p(x)=i*5.551D-17 in 4 steps
To illustrate the bigger picture, the iteration map for the Laguerre method with initial values in [-4,4]+[-4,4]*i looks like
Every color shade represents one step. There is nothing nefarious in this polynomial preventing convergence of the method.
To compare with, the Halley iteration map is
and the Newton fractal