This test matrix is from the analysis of the Ising model for ferromagnetic materials. The matrix A is the product of the two 2m × 2m matrices K and L,

where

It can be shown that the eigenvalues of A are the 2m numbers that are obtained by computing the eigenvalues of the m 2 by 2 matrices

for , where . Figure 1 shows the eigenvalue distribution of 100 by 100 Ising matrix with .

The following is the FORTRAN calling sequence for forming matrix-vector AX or A^{T}X: Fortran calling sequence.

This Ising model was proposed to explain properties of ferromagnets but since then it has found application to topics in chemistry and biology as well as physics. For any reader unfamiliar with the model an excellent introduction is [B. A. Cipra].

A numerical method for approximating the leading eigenvalues of 2D Ising models using a transfer matrix of order 2^{n} with n = 30 is reported in [B. Parlett and W. Heng].

We plan to include the transfer matrix in the future version of this collection.