Numerical computation of the characteristic polynomial of a complex matrix-猫眼课题宝

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2010

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作者：

Ilse C. F. Ipsen;Dean Lee;Rizwana Rehman

本文提出了复矩阵特征多项式的数值计算算法，并进行了灵敏度和稳定性分析。例如，在量子物理学中，需要特征多项式来计算费米子系统的热力学性质。一般的共识似乎是，计算特征多项式的数值方法是数值不准确和不稳定的。然而，为了判断方法的数值精度，需要研究特征多项式系数对矩阵扰动的敏感性。给出了n × n复矩阵特征多项式系数的前向误差界。这些界限由奇异值的初等对称函数组成。此外，我们还研究了两种特征多项式计算方法的数值稳定性。第一种方法由矩阵的特征值确定其特征多项式的系数。第二种方法需要将复矩阵A初步约化为其Hessenberg形式H。H的特征多项式是由H的首主子阵的特征多项式的逐次计算得到的。数值实验表明，第二种方法比由特征值确定特征多项式的方法更精确。

In this dissertation we present algorithms, and sensitivity and stability analyses for the numerical computation of characteristic polynomials of complex matrices. In Quantum Physics, for instance, characteristic polynomials are required to calculate thermodynamic properties of systems of fermions. The general consensus seems to be that numerical methods for computing characteristic polynomials are numerically inaccurate and unstable. However, in order to judge the numerical accuracy of a method, one needs to investigate the sensitivity of the coefficients of the characteristic polynomial to perturbations in the matrix. We derive forward error bounds for the coefficients of the characteristic polynomial of an n × n complex matrix. These bounds consist of elementary symmetric functions of singular values. Furthermore, we investigate the numerical stability of two methods for the computation of characteristic polynomials. The first method determines the coefficients of the characteristic polynomial of a matrix from its eigenvalues. The second method requires a preliminary reduction of a complex matrix A to its Hessenberg form H. The characteristic polynomial of H is obtained from successive computations of characteristic polynomials of leading principal submatrices of H. Our numerical experiments suggest that the second method is more accurate than the determination of the characteristic polynomial from eigenvalues.

Numerical computation of the characteristic polynomial of a complex matrix

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