Direct least squares method and indirect least squares method for solving eigenvalue problems

For linear real eigenvalue problem A φ -λ.B λ =0, where A and B are symmetric linear operators, the accurate eigenvalues are real, but approximate eigenvalues by usual least squares method are complex. In this paper, direct least squares method and indirect least squares method which can avoid the contradiction are studied. The zero minimum principle of declination square functional is proved. The upper and lower bounds, the best upper and lower bounds, the second upper bound, the second lower bound of an eigenvalue are derived under the generalized orthogonality conditions (B λ 1 , B λ j )=0 (i=j). These two least squares methods can be applied to eigenvalue problems of differential equation, of system of differential equations, of matrix, etc..