  In this work we discuss the possibility to reduce the computational
complexity of modal methods, i.e. methods based on eigenmodes expansion, from
the third power to the second power of the number of eigenmodes. The proposed
approach is based on the calculation of the eigenmodes part by part by using
shift-and-invert iterative technique and by applying the iterative approach to
solve linear equations to compute eigenmodes expansion coefficients. As
practical implementation, the iterative modal methods based on polynomials and
trigonometric functions as well as on finite-difference scheme are developed.
Alternatives to the scattering matrix (S-matrix) technique which are based on
pure iterative or mixed direct-iteractive approaches allowing to markedly
reduce the number of required numerical operations are discussed. Additionally,
the possibility of diminishing the memory demand of the whole algorithm from
second to first power of the number of modes by implementing the iterative
approach is demonstrated. This allows to carry out calculations up to hundreds
of thousands eigenmodes without using a supercomputer.
