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NEWTON-RAPSON METHOD-8th degree Legendre polynomial


## Newton-Rapson Method to the smallest non negative root
## of the 8th degree Legendre Polynomial
## P8(x)=(1/128)(6435x^8-12012x^6+6930x^4-1260x^2+35)
## where -1<=x<=1.
## for the smallest non negative root, we can ignore
## all the terms except the last two by truncated
## the function to be zero and find
## x=0.167 as the initial smallest non negative
## root.
##Constants and initializations
x=[]; ## Empty array for the iterated x roots
x(1)=0.16700000; ## Initial guess to begin the iteration for the
## smallest non-negative root.
L8=[]; ## Empty array for the Legendre polynomial
L8p=[]; ## Empty array for the derivative of the Legendre polynomial
for i=1:100
##The value of the function at x
L8(i)=(1/128)*(6435*x(i)^8-12012*x(i)^6+6930*x(i)^4-1260*x(i)^2+35);
##The value of the derivative of the function at x
L8p(i)=(1/128)*(6435*8*x(i)^7-12012*6*x(i)^5+6930*4*x(i)^3-1260*2*x(i));
x(i+1)=x(i)-L8(i)/L8p(i); ## the iteration
endfor
## For plot let's define a new variable t instead of x and P8(t) instead of L8
t=-1:0.01:1;
P8=(1/128)*(6435*t.^8-12012*t.^6+6930*t.^4-1260*t.^2+35);
plot(t,P8);
title('8th degree Legendre polynomial vs x');
xlabel('x');
ylabel('P8(x)');
legend('P8(x)');
printf('The smallest non-negative root of the 8th Legendre polynomial is=%f',x(i))
save -text NEWRAPLEGENDRE.dat
print('-dpsc','NEWRAPLEGENDRE.ps');

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