The QR Algorithm for Real Hessenberg Matrices

  • R. S. Martin
  • G. Peters
  • J. H. Wilkinson
Part of the Die Grundlehren der mathematischen Wissenschaften book series (GL, volume 186)


The QR algorithm of Francis [1] and Kublanovskaya [4] with shifts of origin is described by the relations
$$ \matrix{ {{Q_s}({A_s} - {k_s}I) = {R_s},} & {{A_{s + 1}} = {R_s}Q_s^T + {k_s}I,} & {giving} \cr } \matrix{ {{A_{s + 1}} = } \hfill \cr } {Q_s}{A_s}Q_s^T, $$
where Q s is orthogonal, R s is upper triangular and k s is the shift of origin. When the initial matrix A 1 is of upper Hessenberg form then it is easy to show that this is true of all A s . The volume of work involved in a QR step is far less if the matrix is of Hessenberg form, and since there are several stable ways of reducing a general matrix to this form [3,5, 8], the QR algorithm is invariably used after such a reduction.


National Physical Laboratory Matrix Iteration Principal Submatrix Hessenberg Matrix Current Matrix 
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© Springer-Verlag Berlin · Heidelberg 1971

Authors and Affiliations

  • R. S. Martin
  • G. Peters
  • J. H. Wilkinson

There are no affiliations available

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