Journal of the Korean Society for Industrial and Applied Mathematics

Korean Society for Industrial and Applied Mathematics (한국산업응용수학회)
권호 표지
DOI QR코드

DOI QR Code

A NEW UNDERSTANDING OF THE QR METHOD

  • Received : 2009.11.04
  • Accepted : 2010.03.08
  • Published : 2010.03.25
Download PDF
⟨ Previous Next ⟩

Abstract

The QR method is one of the most common methods for calculating the eigenvalues of a square matrix, however its understanding would require complicated and sophisticated mathematical logics. In this article, we present a simple way to understand QR method only with a minimal mathematical knowledge. A deflation technique is introduced, and its combination with the power iteration leads to extracting all the eigenvectors. The orthogonal iteration is then shown to be compatible with the combination of deflation and power iteration. The connection of QR method to orthogonal iteration is then briefly reviewed. Our presentation is unique and easy to understand among many accounts for the QR method by introducing the orthogonal iteration in terms of deflation and power iteration.

Keywords

References

  1. G. Ammar, D. Calvetti, W. Gragg, and L. Reichel. Polynomial zerofinders based on szego polynomials. J. Comput. Appl. Math., 127:1–16, 2001. https://doi.org/10.1016/S0377-0427(00)00491-X
  2. R. Burnside and P. Guest. A simple proof of the transposed qr algorithm. SIAM Review, 38:306–308, 1996. https://doi.org/10.1137/1038044
  3. D. Calvetti, S. Kim, and L. Reichel. The restarted qr-algorithm for eigenvalue computation of structured matrices. J. Comput. Appl. Math., 149:415–422, 2002. https://doi.org/10.1016/S0377-0427(02)00486-7
  4. J. Francis. The qr transformation i. Comput. J., 4:265–271, 1961. https://doi.org/10.1093/comjnl/4.3.265
  5. J. Francis. The qr transformation ii. Comput. J., 4:332–345, 1961.
  6. G. Golub and C. Loan. Matrix Computations. The John Hopkins University Press, 1989.
  7. B. Kagstrom and A. Ruhe. Matrix pencils. Springer-Verlag Lecture Notes in Mathematics, 973, 1983.
  8. V. Kublanovskaya. On some algorithms for the solution of the complete eigenvalue problem. USSR Comput. Math. Math. Phys., 3:637–657, 1961.
  9. L. Trefethen and D. Bau III. Numerical Linear Algebra. SIAM, 1997.
  10. D. Watkins. Understanding the qr algorithm. SIAM Review, 24:427–439, 1982. https://doi.org/10.1137/1024100
  11. D. Watkins. Qr-like algorithms for eigenvalue problems. J. Comput. Appl. Math., 123:67–83, 2000. https://doi.org/10.1016/S0377-0427(00)00402-7