矩阵B^TA^(-1)B的特征值估计及预条件处理

Eigenvalue Estimation and Preconditioning for B^TA^(-1)B

在矩阵A为对称正定和矩阵B为列满秩的假设下,研究矩阵BTA-1B的特征值上下界估计,进而给出了BTA-1B的谱条件数的估计·基于以上论述,论证了当矩阵A的条件较好时矩阵Q=BTB可作为矩阵BTA-1B的预条件矩阵·在数值实验中,采用预条件共轭梯度算法(PCG)对Stokes方程求解,实验结果表明Q=BTB确实是一类有效的预条件矩阵·这一结果也和其他文献的数值结果相吻合·

The estimation for the bounds of the eigenvalues of matrix B^TA^(-1)B is studied on the assumption that the matrix A is symmetric and positive definite(SPD) and B has full column rank. Furthermore, the estimation for the spectral condition number of B^TA^(-1)B is obtained. Based on what was concluded above, it is proved that Q=B^TB is a good preconditioner for matrix B^TA^(-1)B provided A is well conditioned. In the numerical experiment, the Stokes equation is solved by preconditioned conjugate gradient method (PCG). The experimental result demonstrates that Q=B^TB is indeed an effective preconditioner. This result is consistent to the numerical results of some references.