Inexact Newton Method For Solving The Eigenvalue Problem Of A Large Symmetric Positive Definite Toeplitz Matrix

Inexact Newton method is an effective method to compute the eigenvalues of large sparsesymmetric matrices, which can reach superliner convergence in proper condition. For the similarspectral distribution between Toeplitz matrix and its approximate circulant matrix, the smallesteigenvalue of the circulant matrix can be used as the initial guess of inexact Newton method,combining with the fast Fourier transform, a new inexact Newton method for computing thesmallest eigenvalue of large symmetric positive definite Toeplitz matrix is presented. For clustereigenvalues, a preconditioned inexact Newton method is presented based on sine-transform tospeed up the convergence. Moreover, a block inexact Newton method is presented for computing afew of smallest eigenvalues of large symmetric Toeplitz matrix. Numerical results show that thepreconditioned inexact Newton method and the block inexact Newton method are efficient forcomputing the smallest eigenvalues of large symmetric Toeplitz matrix.