Robust plotting of generalized lemniscates Original Research Article

A root neighborhood (or pseudozero set) of a degree-n polynomial p(z) is the set of roots of all polynomials View the MathML source whose coefficients differ from those of p(z), under a specified norm in View the MathML source, by no more than a given amount ε. In the case of a weighted infinity norm, root neighborhoods are bounded by generalized lemniscates (algebraic curves of degree 4n). A simple description of generalized lemniscates may be cast in terms of multipolar coordinates: namely, the product of the distances r1,…,rn of a variable point z from the n roots z1,…,zn of p(z) is equal to a degree-n polynomial in the distance r of z from the origin (in the case of “ordinary” lemniscates, this polynomial specializes to a constant). The ability to efficiently and faithfully graph lemniscates is important in numerical analysis, control theory, and other scientific/engineering applications. By constructing the tensor-product Bernstein form over a bounding rectangle, we develop a robust adaptive algorithm to plot lemniscates, employing quadtree subdivision to efficiently achieve any desired resolution. Moreover, singular points of the lemniscate (if any) may be explicitly identified and incorporated into the subdivision.