All roots spectral methods: Constraints, floating point arithmetic and root exclusion

Abstract

Abstract The nonlinear two-point boundary value problem (TPBVP for short) u x x + u 3 = 0 , u ( 0 ) = u ( 1 ) = 0 , offers several insights into spectral methods. First, it has been proved a priori that ? 0 1 u ( x ) d x = p / 2 . By building this constraint into the spectral approximation, the accuracy of N + 1 degrees of freedom is achieved from the work of solving a system with only N degrees of freedom. When N is small, generic polynomial system solvers, such as those in the computer algebra system Maple, can find all roots of the polynomial system, such as a spectral discretization of the TPBVP. Our second point is that floating point arithmetic in lieu of exact arithmetic can double the largest practical value of N . (Rational numbers with a huge number of digits are avoided, and eliminating M symbols like 2 and p reduces N + M -variate polynomials to polynomials in just the N unknowns.) Third, a disadvantage of an “all roots” approach is that the polynomial solver generates many roots – ( 3 N - 1 ) for our example – which are genuine solutions to the N -term discretization but spurious in the sense that they are not close to the spectral coefficients of a true solution to the TPBVP. We show here that a good tool for “root-exclusion” is calculating ? = ? n = 1 N b n 2 ; spurious roots have ? larger than that for the physical solution by at least an order of magnitude. The ? -criterion is suggestive rather than infallible, but root exclusion is very hard, and the best approach is to apply multiple tools with complementary failings.