Multiple root finder algorithm for Legendre and Chebyshev polynomials via Newton’s method

We exhibit a numerical technique based on Newton’s method for finding all the roots of Legendre and Chebyshev polynomials, which execute less iterations than the standard Newton’s method and whose results can be compared with those for Chebyshev polynomials roots, for which exists a well known analytical formula. Our algorithm guarantees at least nine decimal correct ciphers in the worst case, however, when comparing with Chebyshev roots given by its formula, even eighteen decimal correct ciphers are achieved in several roots, in the best case. As a comparison guide the results are collated with those gotten by MATLAB.