Theoretical analysis of the extended cyclic reduction algorithm

The extended cyclic reduction algorithm developed by Swarztrauber in 1974 was used to solve the block-tridiagonal linear system. The paper fills in the gap of theoretical results concerning the zeros of matrix polynomial $B_{i}^{(r)}$ with respect to a tridiagonal matrix which are computed by Newton's method in the extended cyclic reduction algorithm. Meanwhile, the forward error analysis of the extended cyclic reduction algorithm for solving the block-tridiagonal system is studied. To achieve the two aims, the critical point is to find out that the zeros of matrix polynomial $B_{i}^{(r)}$ are eigenvalues of a principal submatrix of the coefficient matrix.