The finite difference method based on staggered grids is applied to simulate the induction logging response in 3D media. The invariance property of the Krylov subspace is used to solve the resulting large sparse complex symmetric linear system. To improve iteration convergence the pseudo inverse of the coefficient matrix is employed when constructing the Krylov subspace instead of the coefficient matrix itself. In each iteration four 3D Poisson equations are computed to get a new Lanczos vector with the incomplete Cholesky decomposition conjugate method. Generally the desirable solution is achieved with the iteration less than 20 times. Also, a new material averaging formula is proposed to get a reasonable average of the conductivity, which guarantees the conservation of the electric current.