The Lanczos algorithm and complex Gauss quadrature
The Lanczos algorithm and complex Gauss quadrature
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DOI：
10.1553/etna_vol50s1
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发表时间：
2018
期刊：
影响因子：
--
通讯作者：
S. Pozza;M. Pranic;Z. Strakoš
中科院分区：
文献类型：
--
作者：
S. Pozza;M. Pranic;Z. Strakoš
.高斯求积可以自然地推广以近似准定线性泛函，其中与（形式）正交多项式，（复）雅可比矩阵和Lanczos算法的互连类似于正定情况。在这次调查中，我们回顾这些关系，并参考文献，介绍他们在几个相关的背景下。特别地，n -权（复）高斯求积的存在对应于成功地执行Lanczos算法的前n步，以生成两个相关Krylov子空间的双正交基。（复）Jacobi矩阵的Jordan分解可以用高斯求积节点和权重以及相关的正交多项式来明确表示。由于只要输入是真实的，Lanczos算法的输出就可以是真实的，所以只要准定线性泛函的所有相关矩都是真实的，高斯求积的值就是真实的数。
. Gauss quadrature can be naturally generalized in order to approximate quasi-definite linear functionals, where the interconnections with (formal) orthogonal polynomials, (complex) Jacobi matrices, and the Lanczos algorithm are analogous to those in the positive definite case. In this survey we review these relationships with giving references to the literature that presents them in several related contexts. In particular, the existence of the n -weight (complex) Gauss quadrature corresponds to successfully performing the first n steps of the Lanczos algorithm for generating biorthogonal bases of the two associated Krylov subspaces. The Jordan decomposition of the (complex) Jacobi matrix can be explicitly expressed in terms of the Gauss quadrature nodes and weights and the associated orthogonal polynomials. Since the output of the Lanczos algorithm can be made real whenever the input is real, the value of the Gauss quadrature is a real number whenever all relevant moments of the quasi-definite linear functional are real.