Abstract: In order to solve the singular linear saddle point problem, the positive definite symmetric block diagonal preconditioner is used for the singular saddle point linear system (1). The eigenvalue distribution of the matrix of the block diagonal preconditioned singular linear system is discussed. The distribution range of the nonzero eigenvalues of the preconditioned system matrix and the algebraic multiplicity of the eigenvalues 1 and 0 are obtained. We find that the preconditioner satisfies the proper splitting, and further obtain the convergence of the preconditioned GMRES algorithm and the preconditioned TFQMR algorithm for solving this block diagonal preconditioned singular linear system. Numerical experiments show that the preconditioned GMRES algorithm and the preconditioned TFQMR algorithm have obvious advantages over the GMRES algorithm and TFQMR algorithm without preconditioning.

Key words: block diagonal preconditioner, singular saddle point problem, eigenvalue