Recently, there has been interest in high precision approximations of the first eigenvalue of the Laplace--Beltrami operator on spherical triangles for combinatorial purposes. We compute improved and certified enclosures to these eigenvalues. This is achieved by applying the method of particular solutions in high precision, the enclosure being obtained by a combination of interval arithmetic and Taylor models. The index of the eigenvalue is certified by exploiting the monotonicity of the eigenvalue with respect to the domain. The classically troublesome case of singular corners is handled by combining expansions at all corners and an expansion from an interior point. In particular, this allows us to compute 100 digits of the fundamental eigenvalue for the three-dimensional Kreweras model that has been the object of previous efforts.


  1. Laplacian eigenvalue
  2. interval analysis
  3. lattice walks

MSC codes

  1. 65N35
  2. 65N25
  3. 65G20
  4. 05A15
  5. 05A16

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Information & Authors


Published In

cover image SIAM Journal on Scientific Computing
SIAM Journal on Scientific Computing
Pages: A3210 - A3232
ISSN (online): 1095-7197


Submitted: 20 March 2020
Accepted: 27 July 2020
Published online: 19 October 2020


  1. Laplacian eigenvalue
  2. interval analysis
  3. lattice walks

MSC codes

  1. 65N35
  2. 65N25
  3. 65G20
  4. 05A15
  5. 05A16



Funding Information

Agence Nationale de la Recherche https://doi.org/10.13039/501100001665 : ANR-19-CE40-0018

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