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

## Abstract

The nonlinear two-point boundary value problem (TPBVP for short) $u_{xx}+u^{3}=0,\quad u(0)=u(1)=0,$ offers several insights into spectral methods.

First, it has been proved a priori that $\int u(x)dx=\frac p{\sqrt{2}}.$ 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 $$\sqrt{2}$$ 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 $\rho=\sqrt{\sum\limits_{n=1}^{N}b_{n}^{2}};$ 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.

## 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)

## Keywords

Chebyshev polynomials; nonlinear ordinary differential equations; two-point boundary value problem; lemniscate elliptic function; computer algebra.

### References

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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) 28–3
DOI: 10.1016/j.aml.2016.11.015

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Applied Mathematics Letters

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0893-9659

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References

[1] J.P. Boyd, Tracing multiple solution branches for nonlinear ordinary differential equations: Chebyshev and Fourier spectral methods and a degree-increasing spectral homotopy [DISH], J. Sci. Comput. 19 (2016) 1113–1143.
[2] J.P. Boyd, Degree-increasing [N to N + 1] homotopy for Chebyshev and Fourier spectral methods, Appl. Math. Lett. 57 (2016) 77–81.
[3] C.I. Gheorghiu, D. Trif, The numerical approximation to positive solution for some reaction–diffusion problems, Pure Math. Appl.: Math. Optim. 11 (2001) 243–253.
[4] J.P. Boyd, Chebyshev and Fourier Spectral Methods, Dover, New York, 2001.
[5] C.-I. Gheorghiu, Spectral Methods for Differential Problems, Casa Cartii de Stiinta, Cluj-Napoca, Romania, 2007, 157 pp.