Block conjugate gradient type methods for the approximation of bilinear form CHA−1BCHA−1B

In this paper, we consider computing the approximation of block bilinear form CHA−1BCHA−1B, where square matrix AA is large sparse, and BB and CC are rectangular matrices with the same size. We propose block conjugate gradient (CG) type methods for the approximation of CHA−1BCHA−1B, in which approximation XkXk of linear systems AX=BAX=B does not need to be computed explicitly and only the approximation ηkηk of CHA−1BCHA−1B, which is mathematically equivalent to CHXkCHXk, will be calculated instead. Numerical results show the effectiveness of our proposed methods.