We consider matrix-valued functions in the form $\mathcal{F}(\lambda)=\sum_{i=1}^d f_i(\lambda) A_i$, where $A_i \in \mathbb{C}^{n\times n}$ and $f_i: \mathbb{C} \mapsto \mathbb{C}$ entire functions, for $i=1,\ldots,d$. Given a regular matrix-valued function, that is a function whose determinant $\det \left( \mathcal{F}(\lambda) \right)$ is not identically zero, we discuss the problem of computing the singular matrix-valued function closest to it in the Frobenius norm. This problem is known in literature as the computation of the distance to singularity. More precisely, we are interested in approximating the nearest matrix-valued function $\mathcal{F}(\lambda) + \Delta \mathcal{F}(\lambda)$ such that its determinant is identically equal to zero, where $\Delta \mathcal{F}(\lambda)= \sum_{i=1}^{d} f_i(\lambda) \Delta A_i$, with $\Delta A_i \in \mathbb{C}^{n \times n}$, for $i=1,\ldots,d$. The problem of the numerical approximation of the distance to singularity is well-known to be challenging, even for linear cases, where it reduces to the computation of the distance to singularity for matrix pencils. Recently, the problem has gained increasing attention and numerical approaches have been developed both for matrix pencils and in the case of polynomial nonlinearities. Nevertheless, none of the currently available techniques has been applied to the approximation of the distance to singularity for general nonlinearities. The solution of this problem for general nonlinearities becomes important in the context of differential algebraic equations and delay differential algebraic equations. A major difficulty is due to the presence of nonlinearities in the matrix-valued function, which represents a delicate point of the problem, since a general matrix-valued function may have an infinite number of eigenvalues. Observe that this feature of the problem does not arise when dealing with matrix pencils and matrix polynomials, and, to our knowledge, this characteristic may prevent the extension of the currently available methods to nonlinearities different from the polynomial one. We propose a method for the numerical approximation of the distance to singularity for nonlinear matrix-valued functions. In particular, we show that the problem can be rephrased as a nearness problem and the property of singularity of the matrix-valued function is translated into a discrete numerical constraint for a suitable minimization problem. Nevertheless, this resulting problem turns out to be highly non-convex. In order to solve it, we propose an iterative procedure made by two nested optimization subproblems, of whose the inner one introduces a constraint gradient system of matrix differential equations and the outer one consists in the optimization of the norm of the perturbations via a Newton-like method. We dedicate special attention to the numerical treatment of the continuous constraint onto the determinant, since a careful translation of this condition into its discrete version is an essential step for the applicability of our numerical approach. To this purpose, we employ results from approximation theory for analytic functions. In many practical applications, such as in the ones arising from engineering and mechanical modeling, matrix-valued functions are often endowed with additional structures. Indeed, the coefficients frequently encode data coming from the underlying application: for instance, they may represent the stiffness or damping matrix in a PDE setting. In this framework, it is important to employ an approach with the desired feature of addressing different structures, in which case the search of the closest singular function is restricted to the class of functions preserving the structure of the matrices. Nevertheless, the possibility of including additional structural constraints into a nearness problem is not an easy task and it leads to a more challenging version of the problem. Indeed, techniques that are able to compute the unstructured distance to singularity often can not be directly extended to their structured counterparts. One of the advantages of the nested approach we propose consists in the fact that it can be naturally extended to its structured version, with minors changes, and, therefore, it is able to tackle nearness problems with the additional constraint of structured perturbations. We practically demonstrate this feature of our technique, by providing a number of case studies. For example, the method allows us to limit the perturbations to just a few matrices and also to include individual structures, such as the preservation of the sparsity pattern of one or more coefficient matrices, and collective-like properties, like a palindromic structure of the matrix-valued function. Among general nonlinearities, a relevant subset consists in the class of matrix polynomials. For this special set, we propose an alternative approach, which relies on a Riemannian optimization based method. This second proposal consists in a general optimizer, which can be employed in several distinct settings, such as the computation of the closest unstable matrix to a stable one and the approximate gcd between to scalar polynomials. Nevertheless, in order to be coherent with the subject of this work, we limit the discussion to the case of the computation of the distance to singularity for matrix polynomials. In this proposal, one main issue to address consists in the fact that the minimizing function is discontinuous. Taking into account this feature of the problem is particularly challenging. We propose to incorporate a regularization technique in our approach, in order to overcome this issue. The theoretical results about this proposal are provided in the illustrative case of the computation of the distance to singularity for matrices. This choice is motivated by the fact that most of nearness problems in matrix theory can be reduced to this case study. We then show the applicability of our approach to matrix polynomials, both by a theoretical point of view and by numerical evidence.

On the approximation of the closest singular matrix-valued functions / Gnazzo, Miryam. - (2025 Jan 23).

On the approximation of the closest singular matrix-valued functions

GNAZZO, MIRYAM
2025-01-23

Abstract

We consider matrix-valued functions in the form $\mathcal{F}(\lambda)=\sum_{i=1}^d f_i(\lambda) A_i$, where $A_i \in \mathbb{C}^{n\times n}$ and $f_i: \mathbb{C} \mapsto \mathbb{C}$ entire functions, for $i=1,\ldots,d$. Given a regular matrix-valued function, that is a function whose determinant $\det \left( \mathcal{F}(\lambda) \right)$ is not identically zero, we discuss the problem of computing the singular matrix-valued function closest to it in the Frobenius norm. This problem is known in literature as the computation of the distance to singularity. More precisely, we are interested in approximating the nearest matrix-valued function $\mathcal{F}(\lambda) + \Delta \mathcal{F}(\lambda)$ such that its determinant is identically equal to zero, where $\Delta \mathcal{F}(\lambda)= \sum_{i=1}^{d} f_i(\lambda) \Delta A_i$, with $\Delta A_i \in \mathbb{C}^{n \times n}$, for $i=1,\ldots,d$. The problem of the numerical approximation of the distance to singularity is well-known to be challenging, even for linear cases, where it reduces to the computation of the distance to singularity for matrix pencils. Recently, the problem has gained increasing attention and numerical approaches have been developed both for matrix pencils and in the case of polynomial nonlinearities. Nevertheless, none of the currently available techniques has been applied to the approximation of the distance to singularity for general nonlinearities. The solution of this problem for general nonlinearities becomes important in the context of differential algebraic equations and delay differential algebraic equations. A major difficulty is due to the presence of nonlinearities in the matrix-valued function, which represents a delicate point of the problem, since a general matrix-valued function may have an infinite number of eigenvalues. Observe that this feature of the problem does not arise when dealing with matrix pencils and matrix polynomials, and, to our knowledge, this characteristic may prevent the extension of the currently available methods to nonlinearities different from the polynomial one. We propose a method for the numerical approximation of the distance to singularity for nonlinear matrix-valued functions. In particular, we show that the problem can be rephrased as a nearness problem and the property of singularity of the matrix-valued function is translated into a discrete numerical constraint for a suitable minimization problem. Nevertheless, this resulting problem turns out to be highly non-convex. In order to solve it, we propose an iterative procedure made by two nested optimization subproblems, of whose the inner one introduces a constraint gradient system of matrix differential equations and the outer one consists in the optimization of the norm of the perturbations via a Newton-like method. We dedicate special attention to the numerical treatment of the continuous constraint onto the determinant, since a careful translation of this condition into its discrete version is an essential step for the applicability of our numerical approach. To this purpose, we employ results from approximation theory for analytic functions. In many practical applications, such as in the ones arising from engineering and mechanical modeling, matrix-valued functions are often endowed with additional structures. Indeed, the coefficients frequently encode data coming from the underlying application: for instance, they may represent the stiffness or damping matrix in a PDE setting. In this framework, it is important to employ an approach with the desired feature of addressing different structures, in which case the search of the closest singular function is restricted to the class of functions preserving the structure of the matrices. Nevertheless, the possibility of including additional structural constraints into a nearness problem is not an easy task and it leads to a more challenging version of the problem. Indeed, techniques that are able to compute the unstructured distance to singularity often can not be directly extended to their structured counterparts. One of the advantages of the nested approach we propose consists in the fact that it can be naturally extended to its structured version, with minors changes, and, therefore, it is able to tackle nearness problems with the additional constraint of structured perturbations. We practically demonstrate this feature of our technique, by providing a number of case studies. For example, the method allows us to limit the perturbations to just a few matrices and also to include individual structures, such as the preservation of the sparsity pattern of one or more coefficient matrices, and collective-like properties, like a palindromic structure of the matrix-valued function. Among general nonlinearities, a relevant subset consists in the class of matrix polynomials. For this special set, we propose an alternative approach, which relies on a Riemannian optimization based method. This second proposal consists in a general optimizer, which can be employed in several distinct settings, such as the computation of the closest unstable matrix to a stable one and the approximate gcd between to scalar polynomials. Nevertheless, in order to be coherent with the subject of this work, we limit the discussion to the case of the computation of the distance to singularity for matrix polynomials. In this proposal, one main issue to address consists in the fact that the minimizing function is discontinuous. Taking into account this feature of the problem is particularly challenging. We propose to incorporate a regularization technique in our approach, in order to overcome this issue. The theoretical results about this proposal are provided in the illustrative case of the computation of the distance to singularity for matrices. This choice is motivated by the fact that most of nearness problems in matrix theory can be reduced to this case study. We then show the applicability of our approach to matrix polynomials, both by a theoretical point of view and by numerical evidence.
23-gen-2025
Singular matrix-valued functions; matrix nearness; gradient flow; matrix ODEs; approximation of analytic functions; delay differential equations; nearest singular matrix polynomial; Riemannian optimization; Riemann-Oracle
On the approximation of the closest singular matrix-valued functions / Gnazzo, Miryam. - (2025 Jan 23).
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Utilizza questo identificativo per citare o creare un link a questo documento: https://hdl.handle.net/20.500.12571/34304
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