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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)
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)
4 Acknowledgement I would like to thank my mentor Prof. Dr. Kre simir Veseli c for introducing me to the exciting eld 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 Drma c und Xiaofeng Wang for the possibility to check my ideas in numerous(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)
Two new algorithms for one-sided bidiagonalization are presented. The first is a block version which improves execution time by improving cache utilization from the use of BLAS 2.5 operations and more BLAS 3 operations. The second is adapted to parallel computation. When incorporated into singular value decomposition software, the second algorithm is faster(More)
An error analysis result is given for classical Gram–Schmidt fac-torization of a full rank matrix A into A = QR where Q is left orthogonal (has orthonormal columns) and R is upper triangular. The work presented here shows that the computed R satisfies R T R = A T A + E where E is an appropriately small backward error, but only if the diagonals of R are(More)