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The Bidiagonal Singular Value Decomposition and Hamiltonian Mechanics
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In this article, the singular value decomposition of a bidiagonal matrix B is considered, and it is shown 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 than the standard perturbation theory suggests.Abstract:
Computing the singular value decomposition of a bidiagonal matrix B is considered. This problem arises in the singular value decomposition of a general matrix, and in the eigenproblem for a symmetric positive-definite tridiagonal matrix. It is shown 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 than the standard perturbation theory suggests. It is also shown that the algorithm in [Demmel and Kahan, SIAM I. Sci. Statist. Comput., 11 (1990), pp. 873–912] computes the singular vectors as well as the singular values to this accuracy. A Hamiltonian interpretation of the algorithm is also given, and differential equation methods are used to prove many of the basic facts. The Hamiltonian approach suggests a way to use flows to predict the accumulation of error in other eigenvalue algorithms as well.read more
Citations
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Journal ArticleDOI
On the early history of the singular value decomposition
TL;DR: This paper surveys the contributions of five mathematicians who were responsible for establishing the existence of the singular value decomposition and developing its theory.
Journal ArticleDOI
Computing the Singular Value Decomposition with High Relative Accuracy
TL;DR: This paper analyzes when it is possible to compute the singular values and singular vectors of a matrix with high relative accuracy, which means that each computed singular value is guaranteed to have some correct digits, even if the singularvalues have widely varying magnitudes.
Computing accurate eigensystems of scaled diagonally dominant matrices: LAPACK working note No. 7
J. Barlow,J. Demmel +1 more
TL;DR: In this article, the singular values and eigenvalues of symmetric positive definite tridiagonal matrices are determined to high relative precision independent of their magnitudes, and there are algorithms to compute them this accurately.
Book
Computing Accurate Eigensystems of Scaled Diagonally Dominant Matrices
Jesse L. Barlow,James Demmel +1 more
TL;DR: This work extends results of Kahan and Demmel for bidiagnoal and tridiagonal matrices and finds that the singular values and eigenvalues are determined to high relative precision independent of their magnitudes, and there are algorithms to compute them this accurately.
Journal ArticleDOI
Relative Perturbation Theory: I. Eigenvalue and Singular Value Variations
Ren-Cang Li,Ren-Cang Li +1 more
TL;DR: In this article, it was shown that small eigenvalues (singular values) are determined to high relative accuracy by the data much more accurately than the classical perturbation theory would indicate.
References
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Book
Mathematical Methods of Classical Mechanics
TL;DR: In this paper, Newtonian mechanics: experimental facts investigation of the equations of motion, variational principles Lagrangian mechanics on manifolds oscillations rigid bodies, differential forms symplectic manifolds canonical formalism introduction to pertubation theory.
Book
The algebraic eigenvalue problem
TL;DR: Theoretical background Perturbation theory Error analysis Solution of linear algebraic equations Hermitian matrices Reduction of a general matrix to condensed form Eigenvalues of matrices of condensed forms The LR and QR algorithms Iterative methods Bibliography.
Book
Theory of Ordinary Differential Equations
TL;DR: The prerequisite for the study of this book is a knowledge of matrices and the essentials of functions of a complex variable as discussed by the authors, which is a useful text in the application of differential equations as well as for the pure mathematician.
Journal ArticleDOI
The Symmetric Eigenvalue Problem.
TL;DR: Parlett as discussed by the authors presents mathematical knowledge that is needed in order to understand the art of computing eigenvalues of real symmetric matrices, either all of them or only a few.
Book
The Symmetric Eigenvalue Problem
TL;DR: Parlett as discussed by the authors presents mathematical knowledge that is needed in order to understand the art of computing eigenvalues of real symmetric matrices, either all of them or only a few.