我们研究作用于具有简单谱的Jacobi矩阵的Wilkinson移位迭代的收敛速率。我们证明，对于无AP谱（即不含三项算术级数的简单谱），收敛是三次的。在阶数3上，存在一个三对角对称矩阵P_0，它是Wilkinson迭代序列的极限，且所有收敛到P_0的迭代都是严格二次的。在P_0附近的三对角矩阵中，收敛到P_0的初始条件集合X相当薄：它是若干不相交弧X_s的并集，在P_0处相交，其中s遍历符号序列s: N -> {1,-1}的Cantor集。Wilkinson步将X_s映射到X_{s'}，其中s'是s的左移位。在与P_0共轭的三对角矩阵中，P_0附近但不在X中的初始条件以三次速率收敛。
英文摘要：
We study the rate of convergence of Wilkinson's shift iteration acting on Jacobi matrices with simple spectrum. We show that for AP-free spectra (i.e., simple spectra containing no arithmetic progression with 3 terms), convergence is cubic. In order 3, there exists a tridiagonal symmetric matrix P_0 which is the limit of a sequence of a Wilkinson iteration, with the additional property that all iterations converging to P_0 are strictly quadratic. Among tridiagonal matrices near P_0, the set X of initial conditions with convergence to P_0 is rather thin: it is a union of disjoint arcs X_s meeting at P_0, where s ranges over the Cantor set of sign sequences s: N -> {1,-1}. Wilkinson's step takes X_s to X_{s'}, where s' is the left shift of s. Among tridiagonal matrices conjugate to P_0, initial conditions near P_0 but not in X converge at a cubic rate.
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Positive semidefinite and positive semidefinite splitting iteration methods for solving nonsingular positive semidefinite systems
Minab Higher Education Center, University of Hormozgan（霍尔木甘大学米纳布高等教育学院）；Department of Mathematics, University of Hormozgan（霍尔木甘大学数学系）；Faculty of Mathematical Sciences, University of Guilan（吉兰大学数理学院）
How long does it take to compute the eigenvalues of a random symmetric matrix?
J.P. Morgan Securities LLC（摩根大通证券）；Courant Institute for Mathematical Sciences, New York University（纽约大学柯朗数学科学研究所）；Division of Applied Mathematics, Brown University（布朗大学应用数学系）
We present a comprehensive convergence analysis for Self-Consistent Field (SCF) iteration to solve a class of nonlinear eigenvalue problems with eigenvector-dependency (NEPv). Using a tangent-angle matrix as an intermediate measure for approximation error, we establish new formulas for two fundamental quantities that optimally characterize the local convergence of the plain SCF: the local contraction factor and the local average contraction factor. In comparison with previously established results, new convergence rate estimates provide much sharper bounds on the convergence speed. As an appli…
The rational QZ method generalizes the QZ method by implicitly supporting rational subspace iteration. In this paper we extend the rational QZ method by introducing shifts and poles of higher multiplicity in the Hessenberg pencil, which is a pencil consisting of two Hessenberg matrices. The result is a multishift, multipole iteration on block Hessenberg pencils which allows one to stick to real arithmetic for a real input pencil. In combination with optimally packed shifts and aggressive early deflation as an advanced deflation technique we obtain an efficient method for the dense generalized …
