非正规Krylov子空间框架下对称线性方程组的求解
On solving symmetric systems of linear equations in an unnormalized Krylov subspace framework
摘要：
在求解对称线性方程组的非规范化Krylov子空间框架中，由Lanczos过程生成的正交向量不一定是梯度的形式。将每个正交向量与一个三元组相联系，仅利用三元组的三项递归，给出了对称线性方程组相容或不相容的条件.在兼容的情况下，给出解决方案，在不兼容的情况下，获得不兼容的证书。特别是，当矩阵是奇异的情况下，处理。我们还推导出一个最小残差方法的基础上，这个框架，并显示如何迭代可以更新显式的三元组的基础上，并在不相容的情况下，最小欧氏范数的最小残差的解决方案。
英文摘要：
In an unnormalized Krylov subspace framework for solving symmetric systems of linear equations, the orthogonal vectors that are generated by a Lanczos process are not necessarily on the form of gradients. Associating each orthogonal vector with a triple, and using only the three-term recurrences of the triples, we give conditions on whether a symmetric system of linear equations is compatible or incompatible. In the compatible case, a solution is given and in the incompatible case, a certificate of incompatibility is obtained. In particular, the case when the matrix is singular is handled. We also derive a minimum-residual method based on this framework and show how the iterates may be updated explicitly based on the triples, and in the incompatible case a minimum-residual solution of minimum Euclidean norm is obtained.
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