The partition function for one-dimensional nearest neighbour Ising models is estimated by summing all the energy terms in the Hamiltonian for N sites. The algebraic expression for the partition function is then employed to deduce the eigenvalues of the basic $2 \times 2$ matrix and the corresponding Hermitian Toeplitz matrix is derived using the Discrete Fourier Transform. A new recurrence relation pertaining to the partition function for two-dimensional Ising models in zero magnetic field is also proposed.

The partition function for two-dimensional nearest neighbour Ising model in a non-zero magnetic field have been derived for a finite square lattice of 16, 25, 36 and 64 sites with the help of