# Comparing Eigenvalues between Different Levels

## Explanation of Notation for the Data in Folder 116.119

Here we calculate the outer approximations of the same twisted SG for different levels. We know that they should decrease as we go up higher levels. We observed this for the large eigenvalues, but the very beginning of the spectrum increases and does not satisfy this. In the excel file the numbers 116 through 119 indicate different SGs. For each SG the first column is the ratio of eigenvalues of level 0.025 over level 0.05, whereas the second column is level 0.05 over 0.1. By level 0.1 we mean that we draw the triangles which has a contraction ratio bigger than 0.1.

Corresponding contraction ratios for T1, T2, T3,T4:

T1=(0.5462 0.5462 0.4034) (triangle 116)

T2=(0.5345 0.5345 0.4286) (triangle 117)

T3=(0.5229 0.5229 0.4532) (triangle 118)

T4=( 0.5114 0.5114 0.4770) (triangle 119)

The explanation of the names of the files:(only for one of them, the others are the same)

matlabT1: level 0.025 , the eigenvalues for the zero mesh refinement, triangle 116.

matlab05_T1: level 0.05, the eigenvalues for the zero mesh refinement, triangle 116.

matlab01_T1: level 0.1, the eigenvalues for the zero mesh refinement, triangle 116.

## Explanation of Notation for r_r_2 Folder

We compare the eigenvalues for different levels of r, r^2, r^2 lattice case. The excel file contains the data for the ratios of eigenvalues for different levels. The first 3 columns are level 0.025 over level 0.05, the second three columns are level 0.05 over level 0.1 for zero, first and second mesh refinements.

Matlab Files:

matlab025_l2: the eigenvalues for the second mesh refinement of level 0.025

matlab05_l1: the eigenvalues for the first mesh refinement of level 0.05

matlab01_l3: the eigenvalues for the third mesh refinement of level 0.1

The decrease of the eigenvalues from level 0.1 to level 0.05 can be seen at the bottom of the spectrum.