BTTB Preconditioner For Block Toeplitz Systems Based On Trigonometric Polynomial Approximation

This thesis consists of two chapters. The main contents are as follows.In chapter 1, we give some introductory materials. We will introduce Toeplitzmatrix, block-Toeplitz-Toeplitz-block (BTTB) matrix and their generating func-tions. Some related symbols, conceptions and theorems are given. Two iterationmethods, the conjugate gradient (CG) method and preconditioned conjugategradient (PCG) method are also introduced. In this chapter, we will also intro-duce some background of this thesis and list some important preconditionerswhich are related to this problem.In chapter 2, we will use PCG method to solve the BTTB system T_m,nx=b,where T_m,n is a BTTB matrix. The PCG method is efficient for solving thissystems. The main problem is how to choose suitable preconditioner to speed upthe convergence of the iteration. In this thesis, we use trigonometric polynomialapproximations to generate our preconditioner. In this chapter, we prove thatour preconditioner matrix is a efficient preconditioner. Numerical results showthat our preconditioners are more efficient than circulant preconditioners.