Eigenvalue problems solved by reorthogonalization Lanczos method for the large non-orthonormal sparse matrix

Abstract

Using shell model to calculate the nuclear systems in a large model space is an important method in the field of nuclear physics.On the basis of the nuclear shell model,a large symmetric non-orthonormal sparse Hamiltonian matrix is generated when adopting the generalized seniority method to truncate the many-body space.Calculating the energy eigenvalues and energy eigenvectors of the large symmetric non-orthonormal sparse Hamiltonian matrix is of indispensable steps before energies of nucleus are further calculated.In the mean time,some low-lying energy eigenvalues are always the focus of attention on the occasion of large scale shell model calculation.In this paper,by combining reorthogonalization Lanczos method with Cholesky decomposition method and Elementary transformation method,converting the generalized eigenvalue problems into the standard eigenvalue problems,and transforming the large standard eigenvalue problems into the small standard eigenvalue problems,we successfully calculate the eigenvalues and eigenvectors of large non-orthonormal sparse matrices with the help of computers with limited memory.The values obtained by using this method to calculate the small matrix agree with the exact values,which demonstrates that this method is accurate and can be used to calculate the energy eigenvalues and energy eigenvectors of large symmetric nonorthonormal sparse matrix.We take 116Sn (s=8,the number of unpaired particles,namely the generalized seniority) as an example in which there are active valence neutrons but inert protons at the magic number,and calculate ten of its lowest energy eigenvalues.Through calculation,we find that among these low-lying energy eigenvalues,the lowest energy eigenvalue converges fastest.A comparison between the calculation values and the experiment values shows that the difference between the calculated high-lying energy eigenvalue and its corresponding experimental one arrives at hundreds of keV,while for the low-lying energy eigenvalue,its calculation value can reach an accuracy of a few tens of keV.The results demonstrate that the Lanczos method is feasible in Matlab programming and shell model calculations. The significance of this research lies in the fact that this method will not only greatly help to calculate and obtain the low-lying energy eigenvalues of some medium-mass and heavy nuclei,but also possess great importance in calculating partial eigenvalues involved in large matrices in other theoretical researches and engineering designs.