Laguerre's Method catastrophic cancellation

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Hunter Thurmond 2014-4-17

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此问题的直接链接

https://ww2.mathworks.cn/matlabcentral/answers/126060-laguerre-s-method-catastrophic-cancellation

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I am trying to find the roots of f(x) = x^5 - 9x^4 -x^3 +17x^2 -8x -8 , using Laguerre's method.

I wrote a general code to run different fixed point methods,

%%%%%%%%%%%

function [c, k] = fixd(g,a, tol)
    k = 1;
    x = zeros(1,1,'double');
    x = a;
    next=zeros(1,1,'double');
    MAX = 10000;
    next = g(x);
    while k < MAX && abs(next - x) > tol
        x = next;
        next = g(x);
        k = k + 1;
    end
      c = x;
  end

%%%%%%%%%%%

and codes for f(x), f'(x), and f''(x) :

%%% f(x) %%%

function val = fun3(x)
    val = x*(x*(x*(x*(x - 9)-1)+17)-8)-8;
end

%%%%%%%%%%%%%

%%% f'(x) %%%

function val = fun3prime(x)
val = x*(x*(x*(5*x-36)-3)+34)-8;
end

%%%%%%%%%%%%%

%%% f''(x) %%%

function val = fun3dprime(x)
    val = x*(x*(20*x - 108) - 6) + 34 ;
end

%%%%%%%%%%%%%%

And here is the code I wrote for calculating the value for a in Laguerre's method:

%%% Laguerre's iterative function %%%

function [c] = laguerrefunct(x)
a=zeros(1,1,'double');
G=zeros(1,1,'double');
H=zeros(1,1,'double');
c=zeros(1,1,'double');
G = fun3prime(x) / fun3(x) ;
H = (G*G) - (fun3dprime(x) / fun3(x)) ;
if G + sqrt(4*(5*H - G*G)) > G - sqrt(4*(5*H - G*G))
    a = G + sqrt(4*(5*H - G*G));
else
    a = G - sqrt(4*(5*H - G*G));
end
c = x - a;
end