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Concept

Exponentielle d'une matrice

Résumé

En mathématiques, et plus particulièrement en analyse, l'exponentielle d'une matrice est une fonction généralisant la fonction exponentielle aux matrices et aux endomorphismes par le calcul fonctionnel. Elle fait en particulier le pont entre un groupe de Lie et son algèbre de Lie. Définition Pour n = 1, on retrouve la définition de l'exponentielle complexe. Propriétés Sauf indication contraire, X, Y désignent des matrices n × n complexes (à coefficients complexes). Propriétés générales

L'exponentielle de la matrice nulle est la matrice identité : \mathrm e^0=I ;
Le déterminant de l'exponentielle d'une matrice est égal à l'exponentielle de sa trace : \det\left(\mathrm e^X\right)=\mathrm e^{\operatorname{tr}(X)};
si Y est une matrice inversible, alors \mathrm e^{YXY^{-1}}=Y,\mathrm e^XY^{-1} ;
l'exponentielle de matrice vérifie la limite : \mathrm e^X=\lim_{

Source officielle

https://fr.wikipedia.org/wiki/Exponentielle_d'une_matrice

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Concepts associés (51)

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Cours associés (24)

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Séances de cours associées (37)

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Chargement

Aggressively Truncated Taylor Series Method For Accurate Computation Of Exponentials Of Essentially Nonnegative Matrices

Meiyue Shao

Small relative perturbations to the entries of an essentially nonnegative matrix introduce small relative errors to entries of its exponential. It is thus desirable to compute the exponential with high componentwise relative accuracy. Taylor series approximation coupled with scaling and squaring is used to compute the exponential of an essentially nonnegative matrix. An a priori componentwise relative error bound of truncation is established, from which one can choose the degree of Taylor series expansion and the scale factor so that the exponential is computed with desired componentwise relative accuracy. To reduce the computational cost, the degree of the Taylor series expansion is chosen small, while the scale factor is chosen sufficiently large to achieve the desired accuracy. The rounding errors in the squaring stage are not serious as squaring is forward stable for nonnegative matrices. We also establish a posteriori componentwise error bounds and derive a novel interval algorithm for the matrix exponential of an essentially nonnegative matrix. Rounding error analysis and numerical experiments demonstrate the efficiency and accuracy of the proposed methods.

Siam Publications2014

Chargement

On the finite section method for computing exponentials of doubly-infinite skew-Hermitian matrices

Meiyue Shao

Computing the exponential of large-scale skew-Hermitian matrices or parts thereof is frequently required in applications. In this work, we consider the task of extracting finite diagonal blocks from a doubly-infinite skew-Hermitian matrix. These matrices usually have unbounded entries which impede the application of many classical techniques from approximation theory. We analyze the decay property of matrix exponentials for several classes of banded skew-Hermitian matrices. Then finite section methods based on the decay property are presented. We use several examples to demonstrate the effectiveness of these methods. (C) 2014 Elsevier Inc. All rights reserved.

Elsevier Science Inc2014

Chargement

The focus of this thesis is on developing efficient algorithms for two important problems arising in model reduction, estimation of the smallest eigenvalue for a parameter-dependent Hermitian matrix and solving large-scale linear matrix equations, by extracting and exploiting underlying low-rank properties. Availability of reliable and efficient algorithms for estimating the smallest eigenvalue of a parameter-dependent Hermitian matrix

A(\mu)

for many parameter values

\mu

is important in a variety of applications. Most notably, it plays a crucial role in \textit{a posteriori} estimation of reduced basis methods for parametrized partial differential equations. We propose a novel subspace approach, which builds upon the current state-of-the-art approach, the Successive Constraint Method (SCM), and improves it by additionally incorporating the sampled smallest eigenvectors and implicitly exploiting their smoothness properties. Like SCM, our approach also provides rigorous lower and upper bounds for the smallest eigenvalues on the parameter domain

D

. We present theoretical and experimental evidence to demonstrate that our approach represents a significant improvement over SCM in the sense that the bounds are often much tighter, at a negligible additional cost. We have successfully applied the approach to computation of the coercivity and the inf-sup constants, as well as computation of

\varepsilon

-pseudospectra. Solving an

m\times n

linear matrix equation

A_1 X B_1^T + \cdots + A_K X B_K^T = C

as an

m n \times m n

linear system, typically limits the feasible values of

m,n

to a few hundreds at most. We propose a new approach, which exploits the fact that the solution

X

can often be well approximated by a low-rank matrix, and computes it by combining greedy low-rank techniques with Galerkin projection as well as preconditioned gradients. This can be implemented in a way where only linear systems of size

m \times m

and

n \times n

need to be solved. Moreover, these linear systems inherit the sparsity of the coefficient matrices, which allows to address linear matrix equations as large as

m = n = O(10^5)

. Numerical experiments demonstrate that the proposed methods perform well for generalized Lyapunov equations, as well as for the standard Lyapunov equations. Finally, we combine the ideas used for addressing matrix equations and parameter-dependent eigenvalue problems, and propose a low-rank reduced basis approach for solving parameter-dependent Lyapunov equations.

EPFL2016

Chargement

A(\mu)

for many parameter values

\mu

D

\varepsilon

-pseudospectra. Solving an

m\times n

linear matrix equation

A_1 X B_1^T + \cdots + A_K X B_K^T = C

as an

m n \times m n

linear system, typically limits the feasible values of

m,n

to a few hundreds at most. We propose a new approach, which exploits the fact that the solution

X

m \times m

and

n \times n

need to be solved. Moreover, these linear systems inherit the sparsity of the coefficient matrices, which allows to address linear matrix equations as large as

m = n = O(10^5)

EPFL2016