Origin-shifted algorithm for matrix eigenvalues

  • Authors:

  • Y. Y. Nie;Z. Li;J. D. Han

  • Affiliations:

  • Robotics Laboratory, Shenyang Institute of Automation, Academia Sinica No.114, Nanta Avenue, Shenyang 110016, P.R. China;School of Sciences, Northeastern University, Shenyang 110004, P.R. China;Robotics Laboratory, Shenyang Institute of Automation, Academia Sinica No.114, Nanta Avenue, Shenyang 110016, P.R. China

  • Venue:
  • International Journal of Computer Mathematics
  • Year:
  • 2008

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Abstract

In this paper an origin-shifted algorithm for matrix eigenvalues based on Frobenius-like form of matrix and the quasi-Routh array for polynomial stability is given. First, using Householder's transformations, a general matrix A is reduced to upper Hessenberg form. Secondly, with scaling strategy, the origin-shifted Hessenberg matrices are reduced to the Frobenius-like forms. Thirdly, using quasi-Routh array, the Frobenius-like matrices are determined whether they are stable. Finally, we get the approximate eigenvalues of A with the largest real-part. All the eigenvalues of A are obtained with matrix deflation. The algorithm is numerically stable. In the algorithm, we describe the errors of eigenvalues using two quantities, shifted-accuracy and satisfactory-threshold. The results of numerical tests compared with QR algorithm show that the origin-shifted algorithm is fiducial and efficient for all the eigenvalues of general matrix or for all the roots of polynomial.