Abstract
The nonlinear two-point boundary value problem (TPBVP for short) u_{xx}+u^{3}=0,\quad u(0)=u(1)=0,
First, it has been proved a priori that \int u(x)dx=\frac p{\sqrt{2}}.
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 \sqrt{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 \rho=\sqrt{\sum\limits_{n=1}^{N}b_{n}^{2}};
Authors
John P. Boyd
(Department of Climate & Space Sciences and Engineering, University of Michigan, United States)
Calin-Ioan Gheorghiu
(Tiberiu Popoviciu Institute of Numerical Analysis, Romanian Academy)
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Cite this paper as
J.P. Boyd, C.I. Gheorghiu, All roots spectral methods: Constraints, floating point arithmetic and root exclusion, Applied Mathematics Letters 67 (2017) 28–32
DOI: 10.1016/j.aml.2016.11.015
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