Composition-Restricted Polynomial Approximations to the Matrix Sign Function

Determine, as a function of m and δ, the minimal extra approximation cost incurred when restricting to degree-2 compositions p_t(x) = a_t x + b_t x^3 in building polynomials for approximating the sign function on I = [−1,−δ] ∪ [δ,1]; specifically, compare the best achievable uniform error ε_{2T}^{comp} over compositions of length T with the optimal ε_m^* over all polynomials computable with m matrix–matrix multiplications.

Background

Approximating the matrix sign function using a limited number of matrix–matrix multiplications is critical for divide-and-conquer eigensolvers and purification methods in electronic structure.

Compositions of cubic polynomials are attractive due to low multiplication counts, but the gap between the best composition-based approximation and the unrestricted best-in-class approximation is unknown.