To find roots using the Newton-Raphson method, the initial guess is very important otherwise it may take several iterations to give the value of roots. For the given Legendre polynomial ($ P _ 8 $), what should be the starting point? $$ P _ 8 ( x ) = \frac { 6435 x ^ 8 - 12012 x ^ 6 + 6930 x ^ 4 - 1260 x ^ 2 + 35 } { 128 } $$
My Attempt: For polynomials, the initial guess doesn't matter but if explicitly asked to make a good guess we will be required to plot the curve (plot attached) once but that defeats the purpose of finding solutions using NR method.
Kindly suggest if there is an alternative approach for this problem.
You are very lucky with this problem since Francesco Tricomi gave a very nice asymptotic approximation of the zero's of Legendre polynomials.
Letting $x_{n,k}$ be the $k^{\text{th}}$ root of $P_n(x)$, ordered in decreasing order, they are given by
$$x_{n,k}\sim\left(1-\frac{1}{8 n^2}+\frac{1}{8 n^3}+O\left(\frac{1}{n^4}\right)\right)\cos\left(\frac{4k-1}{4n+2}\pi\right)$$
For the case of $P_8(x)$, the results $$\left( \begin{array}{ccc} 1 & +0.960182 & +0.960290 \\ 2 & +0.796653 & +0.796666 \\ 3 & +0.525532 & +0.525532 \\ 4 & +0.183435 & +0.183435 \\ 5 & -0.183435 & -0.183435 \\ 6 & -0.525532 & -0.525532 \\ 7 & -0.796653 & -0.796666 \\ 8 & -0.960182 & -0.960290 \end{array} \right)$$
With such estimates, Newton method will converge very fast.