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On QZ steps with perfect shifts and computing the index of a differential-algebraic equation
IMA Journal of Numerical Analysis ( IF 2.1 ) Pub Date : 2020-08-15 , DOI: 10.1093/imanum/draa049
Nicola Mastronardi 1 , Paul Van Dooren 2
Affiliation  

In this paper we revisit the problem of performing a |$QZ$| step with a so-called ‘perfect shift’, which is an ‘exact’ eigenvalue of a given regular pencil |$\lambda B-A$| in unreduced Hessenberg triangular form. In exact arithmetic, the |$QZ$| step moves that eigenvalue to the bottom of the pencil, while the rest of the pencil is maintained in Hessenberg triangular form, which then yields a deflation of the given eigenvalue. But in finite precision the |$QZ$| step gets ‘blurred’ and precludes the deflation of the given eigenvalue. In this paper we show that when we first compute the corresponding eigenvector to sufficient accuracy, then the |$QZ$| step can be constructed using this eigenvector, so that the deflation is also obtained in finite precision. An important application of this technique is the computation of the index of a system of differential algebraic equations, since an exact deflation of the infinite eigenvalues is needed to impose correctly the algebraic constraints of such differential equations.

中文翻译:

关于具有完美位移的QZ步长并计算微分代数方程的指数

在本文中,我们将回顾执行| $ QZ $ |的问题。进行所谓的“完美移位”,即给定常规铅笔| $ \ lambda BA $ |的“精确”特征值 呈未简化的Hessenberg三角形形式。用精确的算术,| $ QZ $ | 步骤将特征值移动到铅笔的底部,而铅笔的其余部分保持为Hessenberg三角形式,然后产生给定特征值的放气。但以有限的精度| $ QZ $ | 步骤“模糊”并排除给定特征值的放气。在本文中,我们证明了当我们首先以足够的精度计算相应的特征向量时,| $ QZ $ |可以使用该特征向量来构造步长,从而也可以以有限的精度获得放气。该技术的重要应用是计算微分代数方程组的索引,因为需要精确地对无穷特征值进行放气以正确施加此类微分方程的代数约束。
更新日期:2020-08-15
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