Dixon resultant’s solution of systems of geodetic polynomial equations

Abstract

The Dixon resultant is proposed as an alternative to Grobner basis or multipolynomial resultant approaches for solving systems of polynomial equations inherent in geodesy. Its smallness in size, high density (ratio on the number of nonzero elements to the number of all elements), speed, and robustness (insensitive to combinatorial sequence and monomial order, e.g., Grobner basis) makes it extremely attractive compared to its competitors. Using 3D-intersection and conformal C 7 datum transformation problems, we compare its performance to those of the Sturmfels’s resultant and Grobner basis. For the 3D-intersection problem, Sturmfels’s resultant needed 0.578 s to solve a 6 × 6 resultant matrix whose density was 0.639, the Dixon resultant on the other hand took 0.266 s to solve a 4 × 4 resultant matrix whose density was 0.870. For the conformal C 7 datum transformation problem, the Dixon resultant took 2.25 s to compute a quartic polynomial in scale parameter whereas the computaton of the Grobner basis fails. Using relative coordinates to compute the quartic polynomial in scale parameter, the Grobner basis needed 0.484 s, while the Dixon resultant took 0.016 s. This highlights the robustness of the Dixon resultant (i.e., the capability to use both absolute and relative coordinates with any order of variables) as opposed to Grobner basis, which only worked well with relative coordinates, and was sensitive to the combinatorial sequence and order of variables. Geodetic users uncomfortable with lengthy expressions of Grobner basis or multipolynomial resultants, and who aspire to optimize on the attractive features of Dixon resultant, may find it useful.