In this thesis, the conjugate gradient (CG) technique is applied to dense multilevel block-Toeplitz matrix equations from the mixed potential integral equation (MPIE). To significantly reduce the memory requirement and computational cost, the conjugate gradient (CG) method combined with the fast Fourier transformation (FFT), which is often referred to as the CG-FFT method is adapted. When FFT technique is used, the vector-vector multiplication in spectral domain can replace the Topelitz matrix-vectormultiplication in spatial domain. Therefore, the computational complexity of O(N2)is reduced to O(N log N) per iteration.However, the FFT technique can’t reduce the iteration number of CG method, which is largely depends on the spectral properties of the integral operator or the matrices of discrete linear systems. The preconditioning is simply a means of transforming the original linear system into one which has the same solution, but which is likely to be easier to solve by reducing the condition number of the operator equations.The banded diagonal matrix, symmetric successive overrelaxation (SSOR), block diagonal matrix, sparse approximate invert and wavelet based sparse approximate inverse preconditioning technique are applied to CG method in this thesis. Our numerical calculations show that the PCG-FFT algorithms with these preconditioners converge much faster than the conventional one to the MPIE for microwave circuits. Some typical microstrip discontinuities are analyzed and the good results demonstrate the validity of all these proposed algorithms.