In this paper 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

Lopes, Ana Paula

Pascoal, António José

Scientific Repository of the Polytechnic Institute of Porto

English

A new approach to numerical algorithms

Repositório Científico do Instituto Politécnico do Porto

14-08-20091A.P. Lopes*, A.J. Pascoal*** ISCAP – IPP – Porto ** Universidade Portucalense – PortoPortugalIstanbul,  4 – 10 August, 2009Ana Paula Lopes 1A New Approach to Numerical Algorithms                               ICMS09Ana Paula Lopes 2A New Approach to Numerical Algorithms                               ICMS0914-08-20092Introduction In this paper we developed a new Lanczos   algorithm on the Grassmann manifold. This work comes in the wake of the article byA. Edelman, T. A. Arias and S. T. Smith,“The geometry of algorithms withorthogonality constraints”Ana Paula Lopes 3A New Approach to Numerical Algorithms                               ICMS09Introduction The Grassmann and Stiefel manifolds are   based on orthogonality constraints The Lanczos method and the conjugate gradients method are closely related One of the main problems of the Lanczos method is the loss of orthogonalityAna Paula Lopes 4A New Approach to Numerical Algorithms                               ICMS0914-08-20093Numerical linear algebra problems The problem of computing eigenvalues, eigenvectors and invariant subspaces is always present in areas as diverse as Engineering, Physics, Computer Sciences and Mathematics. Lately, it has been verified that the iterations of eigenvaluesand eigenvectors problems are best analyzed in some special spaces.  A bridge between the geometry of the abstract spaces and the well known algorithms of numerical linear algebra. Ana Paula Lopes A New Approach to Numerical Algorithms                               ICMS09 5Optimization Problem The optimization problem of the estimative of the invariant subspaces is made explicit with a geometric approach. However a geometrical treatment on the Grassmann manifold appropriate for numerical linear algebra is not present in standard references.  Ana Paula Lopes A New Approach to Numerical Algorithms                               ICMS09 614-08-20094Grassmann  Manifold Gr(p,n) – Grassmann Manifold : p – dimensional subspaces in  Identify matrix algorithms that induce iterations on the Grassmann manifold Translate abstractly defined Grassmannian algorithms into tractable numerical algorithms Use suitable matrix representations that facilitate convergence analysisAna Paula Lopes A New Approach to Numerical Algorithms                               ICMS09 7nIterations on the Grassmann  Manifold One possible way to represent numerically  an element     of Gr(p,n) consists of specifying an n p full column rank matrix Y whose columns span the space     , and we can write  is called the column space of Y The set of all the matrices that have the same column space is a fiber over YAna Paula Lopes A New Approach to Numerical Algorithms                               ICMS09 8( )span Y14-08-20095Iterations on the Grassmann  ManifoldWe have an iteration on the Grassmann Manifold if a fiber is mapped into a fiber The concept of the fiber bundle structure allows us to describe the relationship between subspaces and matrices representationsAna Paula Lopes A New Approach to Numerical Algorithms                               ICMS09 9Lanczos on the Grassmann manifold The Lanczos algorithm is a method for computing some eigenvalues of a large symmetric matrix A and their eigenvectors The idea consists in building a sequence of nested subspaces   span                              and solving the eigenproblem reduced to these subspacesAna Paula Lopes A New Approach to Numerical Algorithms                               ICMS09 102, , ,...x Ax A x14-08-20096Algorithm Grassmann-Lanczos (GL)Let A be an symmetric matrix Consider      an  p-dimensional subspace  of       i.e., The algorithm produce a sequence of subspaces Ana Paula Lopes A New Approach to Numerical Algorithms                               ICMS09 11n n nGr( , )p n+Gr( , ) Gr( , )           p n p n Algorithm Grassmann-Lanczos (GL)1. Pick an orthonormal matrix Y,         , being 2. Create an orthonormal basis Q for the Krylov subspace3. Calculate the matrix Rayleigh quotient                    that  represents the projection of A into4. Calculate X, an orthonormal basis for the      p- dimensional dominated eigenspace of M5. Let      be the span of Ana Paula Lopes A New Approach to Numerical Algorithms                               ICMS09 12n p span( )Y1( ) , ,..., mm Y span Y AY A YTM Q AQ( )m Y  QX14-08-20097Algorithm Grassmann-Lanczos (GL) Doesn’t have a function that maps fibers to fibers The algorithm is defined as a mapping  This algorithm is well - definedAna Paula Lopes A New Approach to Numerical Algorithms                               ICMS09 13Gr( , ) Gr( , )p n p nConclusion and Future Work This paper offers a new approach to the Lanczos algorithm Lanczos algorithm is a very competitive method whose main problem is the loss of orthogonality, but the introduction of Grassmann manifolds seems to solve this problem This method might be easily implemented by blocksAna Paula Lopes A New Approach to Numerical Algorithms                               ICMS09 1414-08-20098Conclusion and Future Work We believe that this method is competitive in sequential computation, and also in parallel computing Until now, as far as we know, people haven’t though of Lanczos as a subspace iteration  We intend to obtain new convergence results and make extensive comparisons with other Grassmannian methodsAna Paula Lopes A New Approach to Numerical Algorithms                               ICMS09 15References1] P. A Absil, R. Sepulchre, P. Van Dooren and R. Mahony, (2004), Cubically convergent iterations for invariant subspace computation, SIAM  J. Matrix Analysis, vol 26, 1,  70-96.[2] W. M. Boothby, (1975), An introduction to differentiable manifolds and Riemannian geometry, Academic Press.[3] J. Baglama, D. Calvetti and L. Reichel, (2003), IRBL: An Implicitly Restarted Block Lanczos Method for Large-Scale Hermitian Eigenproblems, SIAM J. Sci. Comput., 24, nº5, 1650-1677.Ana Paula Lopes A New Approach to Numerical Algorithms                               ICMS09 1614-08-20099References[4] Jane K. Cullum and  Ralph A. Willoughby, (1985), Lanczos Algoritms for Large Symmetric Eigenvalue Computations; Vol.I and Vol. II, Birkhauser.[5] A. Edelman, T. A. Arias and S. T. Smith, (1998), The geometry of algorithms with orthogonality constraints, SIAM J. Matrix Anal. Appl. 20, nº 2,  303-353. [6] B. N. Parlett, (1998), The symmetric eigenvalue problem, Prentice Hall series in Computational Mathematics. Ana Paula Lopes A New Approach to Numerical Algorithms                               ICMS09 17A New Approach to Numerical AlgorithmsAna Paula Lopes A New Approach to Numerical Algorithms                               ICMS09 18www.paulalopes.comaplopes@iscap.ipp.pt - Ana Paula LopesThank youTeşekkür ederimObrigado

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A new approach to numerical algorithms

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