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

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• 作者：John P. Boyd^a ; ^{jpboyd@umich.edu} ; Călin-Ioan Gheorghiu^b ; ^{ghcalin@ictp.acad.ro}
• 关键词：Chebyshev polynomials ; Nonlinear ordinary differential equations ; Two-point boundary value problem ; Lemniscate elliptic function ; Computer algebra
• 刊名：Applied Mathematics Letters
• 出版年：2017
• 出版时间：May 2017
• 年：2017
• 卷：67
• 期：Complete
• 页码：28-32
• 全文大小：810 K
• 卷排序：67

文摘

The nonlinear two-point boundary value problem (TPBVP for short) uxx+u3=0,u(0)=u(1)=0,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=π∕2. By building this constraint into the spectral approximation, the accuracy of N+1N+1 degrees of freedom is achieved from the work of solving a system with only NN degrees of freedom. When NN 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 NN. (Rational numbers with a huge number of digits are avoided, and eliminating MM symbols like 2 and ππ reduces N+MN+M-variate polynomials to polynomials in just the NN unknowns.) Third, a disadvantage of an “all roots” approach is that the polynomial solver generates many roots – (3N−1)(3N−1) for our example – which are genuine solutions to the NN-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 ρ≡∑n=1Nbn2; spurious roots have ρρ larger than that for the physical solution by at least an order of magnitude. The ρρ-criterion is suggestive rather than infallible, but root exclusion is very hard, and the best approach is to apply multiple tools with complementary failings.