A Note on Simultaneous Rootfinding for Algebraic, Exponential, and Trigonometric Polynomials

Abstract A common derivation and convergence analysis is presented for two methods for the simultaneous computation of all the zeros of an algebraic, exponential, or trigonometric polynomial. The analysis is performed considering a certain class of generalized polynomials which allow a factorization in terms of translations of a function q(t) such that Lagrangian—like interpolation is exact. The methods were already known for algebraic polynomials (q(t) = t) but new for exponential ( (q(t) = sinh( t 2 )) and trigonometric polynomials (q(t) = sin( t 2 )) . Using the properties q(0) = 0, q′(0) ≠ 0, and q″(0) = 0, we prove the local convergence of the methods presented in this note. The algorithms only require the evaluation of the generalized polynomial at certain points but no derivatives or coefficients. Numerical examples are included.