Journal Article10.1137/0714039
Numerical Computation of the Matrix Exponential with Accuracy Estimate
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TL;DR: An algorithm for computing the exponential of an arbitrary $n \times n$ matrix is presented and Diagonal Pade table approximations are used in conjunction with several techniques for reducing the norm of the matrix.
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Abstract: This paper presents and analyzes an algorithm for computing the exponential of an arbitrary $n \times n$ matrix. Diagonal Pade table approximations are used in conjunction with several techniques for reducing the norm of the matrix. An important feature of the algorithm is that an estimate for the minimum number of digits accurate in the norm of the computed exponential matrix is returned to the user. In obtaining this estimate, several interesting results concerning rounding errors and Pade approximations are presented.
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Citations
•Posted Content
Near-linear convergence of the Random Osborne algorithm for Matrix Balancing.
TL;DR: In this article, it was shown that a simple random variant of Osborne's algorithm converges in near-linear time in the input sparsity, where the graph with adjacency matrix $K$ is moderately connected.
4
Statistical roundoff error analysis of a padé algorithm for computing the matrix exponential
R. C. Ward
- 01 Jan 1977
TL;DR: A statistical roundoff error analysis of an algorithm to compute the matrix exponential based on diagonal Pade approximations with appropriate scaling and squaring produces an a posteriori estimate for the expectation and variance of the final error.
4
•Dissertation
The propagation of errors in the numerical solution of Markov models
Brenan Joseph McCarragher
- 01 Jan 1989
TL;DR: Equations that bound the roundoff and the integration errors incurred in the numerical solution of the matrix exponential as applied to Markov models are developed and a suggested method for the solution of this constrained minimization problem is presented.
3
Relative error analysis of matrix exponential approximations for numerical integration
TL;DR: Unlike the absolute error, the relative error always grows linearly in time; in the long-time, the contributions to the Relative error relevant to non-rightmost eigenvalues of A disappear.
3
J -lossless factorisations for robust H ∞ -control in delta-domain
TL;DR: In this article, a method for discrete-time H∞-control based on J-lossless factorisations of chain scattering representations of a plant is presented, and a relative condition number is used as a measure of numerical conditioning of the δ-domain algebraic Riccati equation.
3
References
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.
A novel method of evaluating transient response
M.L. Liou
- 01 Jan 1966
TL;DR: In this article, a method of evaluating transient responses of linear time-invariant systems using the state space approach is described, where the Laplace transform of the response function as a ratio of two polynomials in the complex frequency of proper form is formulated.
161
Avoiding the Jordan Canonical Form in the Discussion of Linear Systems with Constant Coefficients
TL;DR: In this paper, the Jordan Canonical Form in the discussion of linear systems with constant coefficients has been avoided in the context of linear system with constant coefficients, and the authors propose an alternative approach to avoid it.
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