A Taylor expansion of the square root matrix function

Moral, Pierre Del; Niclas, Angèle

doi:10.1016/j.jmaa.2018.05.005

Cited by 33 publications

(21 citation statements)

References 23 publications

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“…Using hypothesis (H2.a) and the main result of Ref. [16] to expand √˜ + L around √˜ , we find that the expressions of the forward and backward propagation operators T (±) are simplified to:…”

Section: Paraxial Schemementioning

confidence: 93%

Physics-based multistep beam propagation in inhomogeneous birefringent media

Poy¹,

Žumer²

2020

Opt. Express

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We present a unified theoretical framework for paraxial and wide-angle beam propagation methods in inhomogeneous birefringent media based on a minimal set of physical assumptions. The advantage of our schemes is that they are based on differential operators with a clear physical interpretation and easy numerical implementation based on sparse matrices. We demonstrate the validity of our schemes on three simple two-dimensional birefringent systems and introduce an example of application on complex three-dimensional systems by showing that topological solitons in frustrated cholesteric liquid-crystals can be used as light waveguides.

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Section: Paraxial Schemementioning

confidence: 93%

Physics-based multistep beam propagation in inhomogeneous birefringent media

Poy¹,

Žumer²

2020

Opt. Express

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“…. , E k ∈ C m,n , where L [1] f (X) ≡ L f (X) denotes the first Fréchet derivative obtained from (8) [18,Sec. 2].…”

Section: Fréchet Derivativesmentioning

confidence: 99%

“…for X [8], [20]. We now try to present a similar direct approach for computing the Fréchet derivatives of the sign and polar function using (4) and the common differentiation rules.…”

Section: Direct Approachesmentioning

confidence: 99%

On using the complex step method for the approximation of Fréchet derivatives of matrix functions in automorphism groups

Werner¹

2021

Preprint

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We show, that the Complex Step approximation Im(f (A + ihE))/h to the Fréchet derivative of matrix functions f : R m,n → R m,n is applicable to the matrix sign, square root and polar mapping using iterative schemes. While this property was already discovered for the matrix sign using Newtons method, we extend the research to the family of Padé iterations, that allows us to introduce iterative schemes for finding function and derivative values while approximately preserving automorphism group structure.

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“…Proof We shall use the following result as a lemma (see Del Moral and Niclas for a proof):

{\frac{d}{d t}|}_{t = 0} {(A + t X)}^{1 / 2} = \int_{0}^{\infty} \exp (- t A^{1 / 2}) X \exp (- t A^{1 / 2}) d t .

By the commutativity of matrix trace operation with differentiation and integration, we have

\begin{matrix} {(|, \frac{normald}{normald t})}_{t = 0} tr false[{false(A + t X false)}^{1 false/ 2} false] & = \int_{0}^{\infty} tr false[\exp false(- t A^{1 false/ 2} false) X \exp false(- t A^{1 false/ 2} false) false] normald t, \\ = \int_{0}^{\infty} tr false[\exp false(- 2 t A^{1 false/ 2} false) X false] normald t, \\ = tr \int_{0}^{\infty} \exp false(- 2 t A^{1 false/ 2} false) X normald t, \\ = tr ([], \int_{0}^{\infty} \exp false(- 2 t A^{1 false/ 2} false) normald t X) . \end{matrix}

Now it suffices to show that …”

Section: Numerical Computation For Locating the Wasserstein Barycentermentioning

confidence: 99%

A Riemannian submersion‐based approach to the Wasserstein barycenter of positive definite matrices

Sun

2020

Math Meth Appl Sci

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In this paper, we introduce a novel geometrization on the space of positive definite matrices, derived from the Riemannian submersion from the general linear group to the space of positive definite matrices, resulting in easier computation of its geometric structure. The related metric is found to be the same as a particular Wasserstein metric. Based on this metric, the Wasserstein barycenter problem is studied. To solve this problem, some schemes of the numerical computation based on gradient descent algorithms are proposed and compared. As an application, we test the k‐means clustering of positive definite matrices with different choices of matrix mean.

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A Taylor expansion of the square root matrix function

Cited by 33 publications

References 23 publications

Physics-based multistep beam propagation in inhomogeneous birefringent media

Physics-based multistep beam propagation in inhomogeneous birefringent media

On using the complex step method for the approximation of Fréchet derivatives of matrix functions in automorphism groups

A Riemannian submersion‐based approach to the Wasserstein barycenter of positive definite matrices

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