A Differential Equation Approach to the Singular Value Decomposition of Bidiagonal Matrices
نویسندگان
چکیده
We consider the problem of approximating the singular value decomposition of a bidiagonal matrix by a one-parameter family of differentiable matrix flows. It is shown that this approach can be fully expressed as an autonomous, homogeneous, and cubic dynamical system. Asymptotic behavior is justified by the established theory of the Toda lattice.
منابع مشابه
More Accurate Bidiagonal Reduction for Computing the Singular Value Decomposition
Bidiagonal reduction is the preliminary stage for the fastest stable algorithms for computing the singular value decomposition. However, the best error bounds on bidiagonal reduction methods are of the form A + A = UBV T ; kAk 2 " M f(n)kAk 2 where B is bidiagonal, U and V are orthogonal, " M is machine precision, and f(n) is a modestly growing function of the dimensions of A. A Givens-based bi...
متن کاملA QR-method for computing the singular values via semiseparable matrices
A QR–method for computing the singular values via semiseparable matrices. Abstract The standard procedure to compute the singular value decomposition of a dense matrix, first reduces it into a bidiagonal one by means of orthogonal transformations. Once the bidiagonal matrix has been computed, the QR–method is applied to reduce the latter matrix into a diagonal one. In this paper we propose a ne...
متن کاملA Dimensionless Parameter Approach based on Singular Value Decomposition and Evolutionary Algorithm for Prediction of Carbamazepine Particles Size
The particle size control of drug is one of the most important factors affecting the efficiency of the nano-drug production in confined liquid impinging jets. In the present research, for this investigation the confined liquid impinging jet was used to produce nanoparticles of Carbamazepine. The effects of several parameters such as concentration, solution and anti-solvent flow rate and solvent...
متن کاملA Toolbox for Computing the Singular Value Decomposition on Distributed Memory Computers a Toolbox for Computing the Singular Value Decomposition on Distributed Memory Computers
We present a parallel software implementation for computing the singular value decomposition (SVD) of general, banded or bidiagonal matrices. First, the matrix is reduced to bidiagonal form. This reduction can be rearranged in a way that allows heavy use of matrix-matrix operations. Then the singular values are computed in an iterative process. Finally the singular vectors are computed independ...
متن کاملThe Bidiagonal Singular Value Decomposition and Hamiltonian Mechanics
We consider computing the singular value decomposition of a bidiagonal matrix B. This problem arises in the singular value decomposition of a general matrix, and in the eigenproblem for a symmetric positive de nite tridiagonal matrix. We show that if the entries of B are known with high relative accuracy, the singular values and singular vectors of B will be determined to much higher accuracy t...
متن کامل