8 The training of some types of neural networks leads to separable non-linear least squares problems. These problems may be 9 ill-conditioned and require special techniques. A robust algorithm based on the Variable Projections method of Golub and Pereyra 10 is designed for a class of feed-forward neural networks and tested on benchmark examples and real data.

We study the combinatorics of addition using balanced digits, deriving an analog of Holte’s “amazing matrix” for carries in usual addition. The eigenvalues of this matrix for base b balanced addition of n numbers are found to be 1, 1/b, · · · , 1/b, and formulas are given for its left and right eigenvectors. It is shown that the left eigenvectors can be identified with hyperoctahedral Foulkes c...

Reliable estimates for the condition number of a large, sparse, real matrix A are important in many applications. To get an approximation for the condition number κ(A), an approximation for the smallest singular value is needed. Standard Krylov subspaces are usually unsuitable for finding a good approximation to the smallest singular value. Therefore, we study extended Krylov subspaces which tu...

Stephen Walker, M.Sc., Ph.D., Associate Professor, Department of Oral Biology and Pathology, School of Dental Medicine Stony Brook University, State University of New York Stony Brook, N.Y., U.S.A. And Lorne M. Golub, D.M.D., M.Sc., M.D. (honorary), S.U.N.Y. Distinguished Professor Department of Oral Biology and Pathology School of Dental Medicine Stony Brook University, State University of New...

4 Global Testing of a Single Pathway 4 4.1 Golub Data and Cell Cycle Pathway . . . . . . . . . . . . . . . . 4 4.1.1 Testing all Genes . . . . . . . . . . . . . . . . . . . . . . . 4 4.1.2 Testing the Cell Cycle Pathway . . . . . . . . . . . . . . . 6 4.1.3 Adjusting for Covariates . . . . . . . . . . . . . . . . . . . 7 4.2 van’t Veer Data and p53–Signalling Pathway . . . . . . . . . . . 8 4.2...

Research assistance from Valentin Shmidov is gratefully acknowledged. Comments from Don Fullerton, Sasha Golub, Rob Williams, Kerry Smith and Nat Keohane have been appreciated. Research supported in part by the University of California Center for Energy and Environmental Economics (UCE3). The views expressed herein are those of the author and do not necessarily reflect the views of the National...

We discuss an investigation into parallelizing the computation of a singular value decomposition (SVD). We break the process into three steps: bidiagonalization, computation of the singular values, and computation of the singular vectors. We discuss the algorithms, parallelism, implementation, and performance of each of these three steps. The original goal was to accomplish all three tasks usin...

In this paper, we present the block least squares method for solving nonsymmetric linear systems with multiple righthand sides. This method is based on the block bidiagonalization. We first derive two algorithms by using two different convergence criteria. The first one is based on independently minimizing the 2-norm of each column of the residual matrix and the second approach is based on mini...

We introduce a randomized algorithm for overdetermined linear least-squares regression. Given an arbitrary full-rank m x n matrix A with m >/= n, any m x 1 vector b, and any positive real number epsilon, the procedure computes an n x 1 vector x such that x minimizes the Euclidean norm ||Ax - b || to relative precision epsilon. The algorithm typically requires ((log(n)+log(1/epsilon))mn+n(3)) fl...