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LANCZOS ALGORITHM ON THE GRASSMANN MANIFOLD

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8th. World Congress on Computational Mechanics (WCCM8) 5th. European Congress on Computational Methods in Applied Sciences and Engineering (ECCOMAS 2008) June 30 – July 5, 2008

Venice, Italy

LANCZOS ALGORITHM ON THE GRASSMANN MANIFOLD

* Ana Paula Lopes¹, Ana Júlia Viamonte

²

and António José Pascoal³

¹ ISCAP - IPP Porto - Portugal [email protected]

²

Universidade Portucalense Porto - Portugal

[email protected]

³ Universidade Portucalense Porto - Portugal

[email protected]

Key Words: Lanczos Method, Grassmann Manifold, Stiefel Manifold, Invariant Subspaces, Eigenvalues and Eigenvectors.

ABSTRACT

The problem of computing eigenvalues, eigenvectors and invariant subspaces is always present in areas as diverse as Engineering, Physics, Computer Science and Mathematics. Considering the importance of these problems in many practical applications, it is not surprising that has been and continues to be the subject of intense research.

We developed a new Lanczos algorithm on the Grassmann manifold. This work comes in the wake of the article by A. Edelman, T. A. Arias and S. T. Smith, The geometry of algorithms with orthogonality constraints, where they presented a new conjugate gradient algorithm on the Grassmann and Stiefel manifolds. These manifolds which are based on orthogonality constraints, yields penetrating insight into many numerical algorithms of linear algebra. They have developed an approach to numerical algorithms involving orthogonality constraints. As the Lanczos method and the method of conjugate gradients are closely related, and one of the main problems of the Lanczos method is the loss of orthogonality, arose the idea of checking whether it would be possible to get a Lanczos algorithm on the Grassmann manifold.

REFERENCES

[1] P.-A Absil, R. Sepulchre, P. Van Dooren and R. Mahony, Cubically convergent iterations for invariant subspace computation, SIAM J. Matrix Analysis, vol 26, 1, 70-96, 2004.

[2] W. M. Boothby, An introduction to differentiable manifolds and Riemannian geometry, Academic Press, 1975.

[3] J. Baglama, D. Calvetti and L. Reichel, IRBL: An Implicitly Restarted Block

Lanczos Method for Large-Scale Hermitian Eigenproblems, SIAM J. Sci. Comput.,

24, 5, 1650-1677, 2003.

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[4] Jane K. Cullum and Ralph A. Willoughby, Lanczos Algoritms for Large Symmetric Eigenvalue Computations; Vol.I Theory, Birkhauser, 1985.

[5] Jane K. Cullum and Ralph A. Willoughby, Lanczos Algoritms for Large Symmetric Eigenvalue Computations; Vol.II Programs, Birkhauser, 1985.

[6] J.W. Demmel, Three Methods for finding estimates of invariant subspaces Computing. 38, 43-57, 1987.

[7] A. Edelman, T. A. Arias and S. T. Smith, The geometry of algorithms with orthogonality constraints, SIAM J. Matrix Anal. Appl. 20, 2, 303-353, 1998.

[8] A.P. Lopes, O método de Lanczos na Variedade de Grassmann, PhD theses, Universidade Portucalense, 2006.

[9] B. N. Parlett, The symmetric eigenvalue problem, Prentice Hall series in Computational

Mathematics, 1998.

Referências

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