Parallel Algorithm for Solving Some Spectral Problems of Linear Algebra
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TLDR
The algorithm is intended to compute the projection matrices P and I - P onto the deflating subspaces of matrix pencils corresponding to the eigenvalues inside and outside the unit circle to solve the Riccati equation.About:
This article is published in Linear Algebra and its Applications.The article was published on 1993-07-01 and is currently open access. It has received 71 citations till now. The article focuses on the topics: Eigenvalues and eigenvectors & Matrix (mathematics).read more
Citations
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The matrix sign function
Charles Kenney,Alan J. Laub +1 more
TL;DR: A survey of the matrix sign function is presented including some historical background, definitions and properties, approximation theory and computational methods, and condition theory and estimation procedures.
Journal ArticleDOI
Fast linear algebra is stable
TL;DR: It is shown that essentially all standard linear algebra operations, including LU decompositions, QR decomposition, linear equation solving, matrix inversion, solving least squares problems, (generalized) eigenvalue problems and the singular value decomposition can also be done stably (in a normwise sense) in O(nω+η) operations.
Journal ArticleDOI
Fast linear algebra is stable
TL;DR: In this article, it was shown that a large class of fast recursive matrix multiplication algorithms are stable in a norm-wise sense, including LU decomposition, QR decomposition and linear equation solving, matrix inversion, solving least squares problems, eigenvalue problems and singular value decomposition.
Journal ArticleDOI
An inverse free parallel spectral divide and conquer algorithm for nonsymmetric eigenproblems
Zhaojun Bai,James Demmel,Ming Gu +2 more
TL;DR: An inverse-free, highly parallel, spectral divide and conquer algorithm that can compute either an invariant subspace of a nonsymmetric matrix, or a pair of left and right deflating subspaces of a regular matrix pencil.
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Stable and Efficient Spectral Divide and Conquer Algorithms for the Symmetric Eigenvalue Decomposition and the SVD
TL;DR: New spectral divide and conquer algorithms for the symmetric eigenvalue problem and the singular value decomposition that are backward stable, achieve lower bounds on communication costs recently derived by Ballard, Demmel, Holtz, and Schwartz, and have operation counts within a small constant factor of those for the standard algorithms.
References
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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
Linear Optimal Control Systems
Huibert Kwakernaak,Raphael Sivan +1 more
TL;DR: In this article, the authors provide an excellent introduction to feedback control system design, including a theoretical approach that captures the essential issues and can be applied to a wide range of practical problems.
Journal ArticleDOI
Nineteen Dubious Ways to Compute the Exponential of a Matrix
Cleve B. Moler,Charles Van Loan +1 more
TL;DR: In this article, the exponential of a matrix could be computed in many ways, including approximation theory, differential equations, the matrix eigenvalues, and the matrix characteristic polynomial.
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An Algorithm for Generalized Matrix Eigenvalue Problems.
Cleve B. Moler,G. W. Stewart +1 more
TL;DR: A new method, called the QZ algorithm, is presented for the solution of the matrix eigenvalue problem $Ax = \lambda Bx$ with general square matrices A and B with particular attention to the degeneracies which result when B is singular.
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Computing integrals involving the matrix exponential
TL;DR: A new algorithm for computing integrals involving the matrix exponential is given, which employs diagonal Pade approximation with scaling and squaring and is compared with existing techniques.