The spectral Lanczos decomposition method

Thus, we arrive at the problem of computing a function of matrix B. This problem can be most effectively solved by the Spectral Lanczos Decomposition Method (SLDM), which involves partial tridiagonalization of the given matrix (Druskin and Knizhnerman, 1994 Golub, and Van Loan, 1996). 

Application of the spectral Lanczos decomposition method (SLDM) for solving the linear system of equations for discrete electromagnetic fields... 

The spectral Lanczos decomposition method is designed for effective calculation of the matrix functions. This method has found a useful application in the solution of linear equations in electromagnetic modeling (Druskin and Knizhnerman, 1994 Druskin et ah, 1999)). Let us consider, for example, the matrix equation (12.35). The formal solution of this equation has the form... 

The great success in solving boundary value problem of eqs. 12.7 and 12.9 was obtained by using the spectral Lanczos decomposition method (SLDM). SLDM was first introduced by Druskin and Knizhnerman to solve Maxwell s equations in both frequency and time domain (Druskin and Knizhnerman, 1988, 1994). 

It can be seen that the convergence of the truncated series depends on the value of w the convergence rate is slower when frequency oj is smaller. To overcome this problem the authors of SLDM proposed a new approach for solutions to Maxwell s equations — a preconditioned modification of SLDM for low frequencies. This new method is based on the standard SLDM but with Krylov subspaces generated from the inverse powers of the Maxwell operator. In Druskin et al. (1999) this method is referred to as spectral Lanczos decomposition method using inverse powers, or SLDMINV. 

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