Jesse L. Barlow

Computing accurate eigensystems of scaled diagonally dominant matrices

When computing eigenvalues of symmetric matrices and singular values of general matrices in finite precision arithmetic we in general only expect to compute them with an error bound proportional to the product of machine precision and the norm of the matrix. In particular, we do not expect to compute tiny eigenvalues and singular values to high relative accuracy. There are some important classes of matrices where we can do much better, including bidiagonal matrices, scaled diagonally dominant matrices, and scaled diagonally dominant definite pencils. These classes include many graded matrices, and all symmetric positive definite matrices which can be consistently ordered (and thus all symmetric positive definite tridiagonal matrices). In particular, the singular values and eigenvalues are determined to high relative precision independent of their magnitudes, and there are algorithms to compute them this accurately. The eigenvectors are also determined more accurately than for general matrices, and may be computed more accurately as well. This work extends results of Kahan and Demmel for bidiagnoal and tridiagonal matrices. 17 refs.

Efficient Minimization Methods of Mixed l 2- l 1 and l 1- l 1 Norms for Image Restoration

Image restoration problems are often solved by finding the minimizer of a suitable objective function. Usually this function consists of a data-fitting term and a regularization term. For the least squares solution, both the data-fitting and the regularization terms are in the

Semi-supervised nonlinear dimensionality reduction

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On roundoff error distributions in floating point and logarithmic arithmetic

2 norm. In this paper, we consider the least absolute deviation (LAD) solution and the least mixed norm (LMN) solution. For the LAD solution, both the data-fitting and the regularization terms are in the

Iterative Methods for Equality-Constrained Least Squares Problems

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Error Analysis and Implementation Aspects of Deferred Correction for Equality Constrained Least Squares Problems

1 norm. For the LMN solution, the regularization term is in the

Scaled givens rotations for the solution of linear least squares problems on systolic arrays

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An algorithm and a stability theory for downdating the ULV decomposition

1 norm but the data-fitting term is in the

Optimal perturbation bounds for the Hermitian eigenvalue problem

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The Direct Solution of Weighted and Equality Constrained Least-Squares Problems

2 norm. Since images often have nonnegative intensity values, the proposed algorithms provide the option of taking into account the nonnegativity constraint. The LMN and LAD solutions are formulated as the solution to a linear or quadratic programming problem which is solved by interior point methods. At each iteration of the interior point method, a structured linear system must be solved. The preconditioned conjugate gradient method with factorized sparse inverse preconditioners is employed to solve such structured inner systems. Experimental results are used to demonstrate the effectiveness of our approach. We also show the quality of the restored images, using the minimization of mixed