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

作者：Boyd John P^*; Gheorghiu Calin Ioan

来源：Applied Mathematics Letters, 2017, 67: 28-32.

DOI：10.1016/j.aml.2016.11.015

• Chebyshev polynomials
• Nonlinear ordinary differential equations
• Two-point boundary value problem
• Lemniscate elliptic function
• Computer algebra

The nonlinear two -point boundary value problem (TPBVP for short)
u(xx) + u(3) = 0, u(0) = u(1) = 0,
offers several insights into spectral methods. First, it has been proved a priori that integral(1)(0) u(x)dx = pi/root 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 root 2 and pi 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 rho equivalent to root Sigma(N)(n=1) b(n)(2); spurious roots have rho larger than that for the physical solution by at least an order of magnitude. The rho-criterion is suggestive rather than infallible, but root exclusion is very hard, and the best approach is to apply multiple tools with complementary failings.

• 出版日期2017-5