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...

An iterative method LSMR is presented for solving linear systems Ax = b and leastsquares problems min ‖Ax−b‖2, with A being sparse or a fast linear operator. LSMR is based on the Golub-Kahan bidiagonalization process. It is analytically equivalent to the MINRES method applied to the normal equation ATAx = ATb, so that the quantities ‖Ark‖ are monotonically decreasing (where rk = b−Axk is the re...

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...

Abstract Randomized methods can be competitive for the solution of problems with a large matrix low rank. They also have been applied successfully to large-scale linear discrete ill-posed by Tikhonov regularization (Xiang and Zou in Inverse Probl 29:085008, 2013). This entails computation an approximation partial singular value decomposition A that is numerical The present paper compares random...

The joint bidiagonalization (JBD) method has been used to compute some extreme generalized singular values and vectors of a large regular matrix pair . We make numerical analysis the underlying JBD process establish relationships between it two mathematically equivalent Lanczos bidiagonalizations in finite precision. Based on results analysis, we investigate convergence approximate show that, u...