the 2nd and 3rd eigenvalues on triangles




Computer-assisted proof for the simplicity of the 2nd and 3rd dirichlet eigenvalues on triangular domains



  • DEMO

Guaranteed estimation of shape derivative for clustered eigenvalues over triangles

Ryoki ENDO

First version: 2023/09/07

Running environment

  • The codes run on latest MATLAB environment along with INTLAB toolbox (http://www.ti3.tu-harburg.de/rump/intlab/).
  • Since Octave does not supply ldl function for indefinite symmetric matrix, the rigorous evaluation is not available for Octave.


Edit my_intlab_mode_config.m to configure the computing environment.

%The path of INTLAB toolbox and initialization.
format compact short infsup;

%The path of the library of verified eigenvalue estimation for matrix.

%The path of the codes for switch between verified computing and approximate computing.

%The path of the library of FEMs on triangular domains

global INTERVAL_MODE; %The value can be 0 or 1. See readme.md

INTERVAL_MODE=1; for rigorous computing based on interval arithmetic.
%INTERVAL_MODE=0; for approximate computing with rounding error inside.

Remeber to run "my_intlab_mode_config" before other codes.

How to use it?

For the purpose of rigorous computation, please run the code in MATLAB along with the INTLAB toolbox.

The code has two running mode: approximate evaluation and verified evaluation.

To swith between each other, please set the value of INTERVAL_MODE.

  • INTERVAL_MODE=0: approximate computation mode.
  • INTERVAL_MODE=1: verified computation mode. It takes longer time that approximate computation mode. So be careful with the mesh size, which should not be too small.

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