Use the bairstows method to find the roots of the following

Use the bairstow\'s method to find the roots of the following polynomial.

x⁵ - 9x⁴ +25x³ -5x² -26x + 24 = 0

Make sure to determine the roots correct to at least four significant figures.

Solution

Roots are :  

-4.36376519662558

1.39848742554330-j1.8100778592698

1.39848742554330+j1.8100778592698

-0.820722036190821

1.38750880631095  

Since the equation will take about 16 iterations to solve, It can implemented only in software. I used MATLAB-2013a to solve it. But can use any other software to solve it.

I am here by providing the basic algorithm update that you will find usefull. Having need to get complete code feel free to contact.

Algorithm :

Solution to Bairstow method problem given by:

x^5-9*x^(4)+25*x^(3)-5*x^(2)-26*x^(1)+24.

Coefficent is denoted by a

The accuracy to which the polynomial is satisfied is given by 10^-5 denoted by tol.

The solutions are obtained according to the following algorithm:

while degree of the polynomial is greater than>2

Initialise the loop via (you can use other initialisation):

u=1; v=1; st=1;

while st>tol

    b(1)=a(1)-u; b(2)=a(2)-b(1)*u-v;

    for k=3:n

      b(k)=a(k)-b(k-1)*u-b(k-2)*v;

    end;

    c(1)=b(1)-u; c(2)=b(2)-c(1)*u-v;

    for k=3:n-1

      c(k)=b(k)-c(k-1)*u-c(k-2)*v;

    end;

calculate change in u and v

    c1=c(n-1); b1=b(n); cb=c(n-1)*b(n-1);

    c2=c(n-2)*c(n-2); bc=b(n-1)*c(n-2);

    if n>3, c1=c1*c(n-3); b1=b1*c(n-3); end;

    dn=c1-c2;

    du=(b1-bc)/dn; dv=(cb-c(n-2)*b(n))/dn;

    u=u+du; v=v+dv;

    st=norm([du dv]); it=it+1;

end;

Solve the quadratic equation parameterized by u,v,n,a that will provide you real and imaginary part of roots given by r1,r2,im1,im2

rts(n,1:2)=[r1 im1]; rts(n-1,1:2)=[r2 im2];

n=n-2;

a(1:n)=b(1:n);

end;

u=a(1); v=a(2);

Again,Solve the quadratic equation parameterized by u,v,n,a that will provide you real and imaginary part of roots given by r1,r2,im1,im2

rts(n,1:2)=[r1 im1];

if n==2

rts(n-1,1:2)=[r2 im2];

end;