A COMPLETED THEORY OF THE UNSYMMETRIC LANCZOS PROCESS AND RELATED ALGORITHMS .1.
A COMPLETED THEORY OF THE UNSYMMETRIC LANCZOS PROCESS AND RELATED ALGORITHMS .1.
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DOI：
10.1137/0613037
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发表时间：
1992-04-01
影响因子：
1.5
通讯作者：
GUTKNECHT, MH
中科院分区：
文献类型：
--
作者：
GUTKNECHT, MH
理论的“非对称”Lanczos双正交化（BO）算法，这到目前为止一直被限制到一个基本上通用的情况下（其特征在于相关的矩矩阵的领导主子矩阵的非奇异性或存在一个完整的正规形式的正交多项式）扩展到非通用的情况。 可以克服由于出现两个正交的右和左迭代向量x（n）和y（n）而引起的“严重”故障。 对于有限秩N的运算符，非一般BO算法推广了Parlett、Taylor和Liu的前瞻Lanczos算法[Math. Comp.，44（1985），pp. 105-124]，在最多N步中有规律地终止，除非发生取决于初始向量的非常特殊的情况;但即使这样，该算法在最多N个步骤中产生块三对角矩阵，其块是小的或稀疏的，并且其特征多项式是该算子对不变子空间的限制的最小多项式。相应的线性方程求解器BIORES（简称BIORTHORES或Lanczos/ORTHORES）。 整个理论的发展作为一个后果已知的相应结果正式正交多项式和帕德逼近，其中许多新的和更简单的推导。
The theory of the "unsymmetric" Lanczos biorthogonalization (BO) algorithm, which has so far been restricted to an essentially generic situation (characterized by the nonsingularity of the leading principal submatrices of the associated moment matrix or by the existence of a full set of regular formal orthogonal polynomials) is extended to the nongeneric case. The "serious" breakdowns due to the occurrence of two orthogonal right and left iteration vectors x(n) and y(n) can be overcome. For an operator of finite rank N the nongeneric BO algorithm, which generalizes the look-ahead Lanczos algorithm of Parlett, Taylor, and Liu [Math. Comp., 44 (1985), pp. 105-124], terminates regularly in at most N steps, except when a very special situation depending on the initial vectors occurs; but even then the algorithm produces in at most N steps a block tridiagonal matrix whose blocks are either small or sparse and whose characteristic polynomial is the minimal polynomial of the restriction of the operator to an invariant subspace.Formulas are also derived for a nongeneric version of the corresponding linear equation solver BIORES (brief for BIORTHORES or Lanczos/ORTHORES). The whole theory is developed as a consequence of known corresponding results on formal orthogonal polynomials and Pade approximants, for many of which new and simpler derivations are given.