一个有理QZ方法
A rational QZ method
AI总结：
本文提出了一种有理QZ方法，用于求解密集非对称广义特征值问题。该方法通过隐式处理Hessenberg-Hessenberg铅笔而非Hessenberg-三角铅笔，允许通过有理函数驱动子空间迭代，从而能够选择极点，进而证明了方法的唯一性和收敛性，并通过数值实验展示了其在速度和精度上的竞争力。
摘要：
我们提出了一种有理QZ方法，用于求解密集、非对称的广义特征值问题。该方法是对经典QZ方法的扩展，隐式地在Hessenberg-Hessenberg铅笔上操作，而非在Hessenberg-三角铅笔上。与QZ方法通过多项式驱动嵌套子空间迭代不同，有理QZ方法允许通过有理函数驱动嵌套子空间迭代，从而提供了选择极点的额外自由度。本文研究了Hessenberg-Hessenberg铅笔，将其与有理Krylov子空间联系起来，提出了一种直接减少到此类铅笔的方法，并介绍了隐式有理QZ步骤。与有理Krylov子空间的联系使我们能够证明有理QZ迭代的本质唯一性（隐式Q定理）以及所提出方法的收敛性。在证明过程中，我们直接在铅笔上操作，而不是将其全部重新表述为单个矩阵。数值实验用于展示其在速度和精度上与经典方法的竞争力。另外两种实验展示了新的可能性：首先，我们证明良好的极点选择可以在减少阶段用于消去原始问题；其次，我们使用有理QZ方法在迭代方法中隐式过滤有理Krylov子空间。
英文摘要：
We propose a rational QZ method for the solution of the dense, unsymmetric generalized eigenvalue problem. This generalization of the classical QZ method operates implicitly on a Hessenberg, Hessenberg pencil instead of on a Hessenberg, triangular pencil. Whereas the QZ method performs nested subspace iteration driven by a polynomial, the rational QZ method allows for nested subspace iteration driven by a rational function, this creates the additional freedom of selecting poles. In this article we study Hessenberg, Hessenberg pencils, link them to rational Krylov subspaces, propose a direct reduction method to such a pencil, and introduce the implicit rational QZ step. The link with rational Krylov subspaces allows us to prove essential uniqueness (implicit Q theorem) of the rational QZ iterates as well as convergence of the proposed method. In the proofs, we operate directly on the pencil instead of rephrasing it all in terms of a single matrix. Numerical experiments are included to illustrate competitiveness in terms of speed and accuracy with the classical approach. Two other types of experiments exemplify new possibilities. First we illustrate that good pole selection can be used to deflate the original problem during the reduction phase, and second we use the rational QZ method to implicitly filter a rational Krylov subspace in an iterative method.
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