近似对称矩阵的直接解法及其应用

所谓近似对称矩阵是指矩阵中仅有极少一部分元素是非对称的，在将对角线以上的非对称元素用其对角线以下的对称元素替代后，该矩阵就变成了一个对称矩阵。在求解非线性连续介质力学问题时常常会遇到近似非对称矩阵。基于Sherman-Morrison定理，给出了一种新的近似非对称矩阵的分解算法。在确保数值稳定性的前提下，无论在求解效率还是在内存开销方面新算法都优于一般的非对称稀疏矩阵的求解器，且仅需对传统的基于LDLT分解的求解器略做修改，即可开发出适应于对称和非对称稀疏矩阵的求解器。最后用一个摩擦接触算例，显示了新算法的优越性。

Direct solution of near-symmetric matrices and its applications

By a near-symmetric matrix,we mean that only a very few of entries in the matrix are non-symmetric.If those non-symmetric entries above the diagonal are replaced with the corresponding entries below the diagonal,it will become symmetric.Such a matrix can be encountered in the analysis of nonlinear continuum problems.Based on Sherman-Morrison's formula,a new scheme for decomposing near-symmetric matrices is proposed,which is much more effective and less memory-used than those solvers for common sparse non-symmetric matrices under the condition that the numerical stability is assured.Moreover,the solver corresponding to the scheme,which is suitable for both symmetric and non-symmetric matrices,can be developed through slightly augmenting the solvers based on LDLT decomposition.With an example on a frictional contact problem,the advantages of the proposed scheme are illustrated.