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We develop normwise backward errors and condition numbers for the polynomial eigenvalue problem. The standard way of dealing with this problem is to reformulate it as a generalized eigenvalue problem (GEP). For the special case of the quadratic ei-genvalue problem (QEP), we show that solving the QEP by applying the QZ algorithm to a corresponding GEP can be(More)
n fi When com puting eigenvalu es of sym m etric m atrices an d singular valu es of general m atrices i nite precision arithm etic we in general only expect to com pute them with an error bound pro-n portional to the product of m ach ine precision an d the norm of the m atrix. In particular, we do ot expect to com pute tiny eigenvalu es an d singular valu(More)
Let LDL t be the triangular factorization of an unreduced symmetric tridiagonal matrix T − τ I. Small relative changes in the nontrivial entries of L and D may be represented by diagonal scaling matrices 1 and 2 ; LDL t −→ 2 L 1 D 1 L t 2. The effect of 2 on the eigenvalues λ i − τ is benign. In this paper we study the inner perturbations induced by 1.(More)
  • Hagen K Veseli´c, Zweitgutachter, Berkeley J Demmel, Jesse Barlow
  • 1992
Acknowledgement I would like to thank my mentor Prof. Dr. Krešimir Veseli´c for introducing me to the exciting field of relative error analysis, for devoting to me a lot of his time, and for sharing with me so many of his ideas. I also thank my colleagues Eberhard Pietzsch, Zlatko Drmač und Xiaofeng Wang for the possibility to check my ideas in numerous(More)
The problem of nonlinear dimensionality reduction is considered. We focus on problems where prior information is available, namely, semi-supervised dimensionality reduction. It is shown that basic nonlinear dimensionality reduction algorithms, such as Locally Linear Embedding (LLE), Isometric feature mapping (ISOMAP), and Local Tangent Space Alignment(More)
A new algorithm of Demmel et al. for computing the singular value decomposition (SVD) to high relative accuracy begins by computing a rank-revealing decomposition (RRD). Dem-mel et al. analyse the use of Gaussian elimination with complete pivoting (GECP) for computing the RRD. We investigate the use of QR factorization with complete pivoting (that is,(More)
This paper studies the solution of the linear least squares problem for a large and sparse m by n matrix A with m n by QR factorization of A and transformation of the right-hand side vector b to Q T b. A multifrontal-based method for computing Q T b using Householder factorization is presented. A theoretical operation count for the K by K unbordered grid(More)