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

The nonlinear two-point boundary value problem (TPBVP for short) uxx+u3=0,u(0)=u(1)=0,offers several insights into spectral methods. First, it has been proved a priori that ?01u(x)dx=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 - (3N-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=1Nbn2; 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.