论文标题
Hessenberg的全局收敛QR II：数值稳定性
Global Convergence of Hessenberg Shifted QR II: Numerical Stability
论文作者
论文摘要
我们开发了一个框架，以证明在有限的算术中使用丽思值作为移位的移动QR算法的快速收敛。 Our key contribution is a dichotomy result which addresses the known forward-instability issues surrounding the shifted QR iteration [Parlett and Le 1993]: we give a procedure which provably either computes a set of approximate Ritz values of a Hessenberg matrix with good forward stability properties, or leads to early decoupling of the matrix via a small number of QR steps. 使用此框架，我们表明，本系列（银行，Garza-vargas和Srivastava 2021）中引入的转移策略在有限算术中迅速收敛，并在需要精确的polygarithmit上绑定了精确的位数，当时需要在受控的eigenvector条件和minemenvecter numine and Minemenv eigiemenv gap gap上调用。
We develop a framework for proving rapid convergence of shifted QR algorithms which use Ritz values as shifts, in finite arithmetic. Our key contribution is a dichotomy result which addresses the known forward-instability issues surrounding the shifted QR iteration [Parlett and Le 1993]: we give a procedure which provably either computes a set of approximate Ritz values of a Hessenberg matrix with good forward stability properties, or leads to early decoupling of the matrix via a small number of QR steps. Using this framework, we show that the shifting strategy introduced in Part I of this series [Banks, Garza-Vargas, and Srivastava 2021] converges rapidly in finite arithmetic with a polylogarithmic bound on the number of bits of precision required, when invoked on matrices of controlled eigenvector condition number and minimum eigenvalue gap.
