# Research Seminars in Mathematics

Please contact Andrii Dmytryshyn if you have any questions regarding this seminar series.

The research seminars are in the subjects of pure, applied, and computational mathematics and are usually held at the afternoons on Fridays. All are welcome to attend!

## 2022

**Speaker:** Andrii Dmytryshyn, Örebro University

**Date:** Friday, September 16 , at 13.15, Zoom meeting

**Title:** Versal deformations of matrices

**Abstract: ** Jordan canonical form for matrices is well known and studied with various purposes but reduction to this form is an unstable operation: both the corresponding canonical form and the reduction transformation depend discontinuously on the entries of an original matrix. This issue complicates the use of the canonical form for numerical purposes. Therefore V.I. Arnold introduced a normal form to which an arbitrary family of matrices A' close to a given matrix A can be reduced by similarity transformation smoothly depending on the entries of A’. He called such a normal form a versal deformation of A.

In this presentation we will discuss versal deformations and their use in investigation of possible changes in canonical forms (eigenstructures), reduction of unstructured perturbations to structured perturbations, and codimension computations.

**Speaker:** Fernando De Terán, Universidad Carlos III de Madrid

**Date:** Friday, May13 , at 13.15, Zoom meeting

**Title:** On the consistency of the matrix equation X^TAX=B when B is either symmetric or skew

**Abstract:** In this talk, we analyze the consistency of the matrix equation X^T AX=B, (1) where A in an mxm complex matrix, X in an mxn complex matrix (unknown), and B (complex nxn) is either symmetric or skew-symmetric (and (·)^T means the transpose). In particular, we first provide a necessary condition for (1) to have a solution X. Then, we will prove that this condition is also sufficient for most matrices A and an arbitrary symmetric (or skew) matrix B. We want to emphasize that the question on the consistency of (1), when B is symmetric (respectively, skew), is equivalent to the following problem: given a bilinear form over C^m (represented by the matrix A), find the maximum dimension of a subspace such that the restriction of the bilinear form to this subspace is a symmetric (resp., skew) non-degenerate bilinear form.

**Speaker: **Johan Hellsvik, KTH

**Date:** Friday, April 1 , at 13.15, Zoom meeting

**Title:** The Dardel HPE Cray EX supercomputer at PDC

**Abstract: **The CPU partition of the HPE Cray EX supercomputer Dardel has now entered regular operation. In this talk I will give an introduction to Dardel, the Cray programming environment, and the experiences learned so far on how to obtain good performance for scientific codes. An overview will be given on some of the scientific application programs that are now running on Dardel and how these how been tuned for AMD EPYC CPUs. In the spring the hardware for the GPU partition will be delivered to PDC. The GPU nodes with AMD Instinct MI250X GPUs will provide the main part of the total performance of Dardel. Examples will be given on some of the software porting activities that are currently ongoing to get codes ready for the GPU partition.

**Speaker: **Alan Edelman, Massachusetts Institute of Technology

**Date:** Friday, March 4, at 13.15, Zoom meeting

**Title:** Generalizing Orthogonal Matrices: On the Structure of the Solutions to the Matrix Equation G*JG=J

**Abstract: **We study the mathematical structure of the solution set to the matrix equation G*JG=J for a given square matrix J. In the language of pure mathematics, this is a Lie group which is the isometry group for a bilinear (or a sesquilinear) form.

We found that on its own, the related (tangent space) equation X*J+JX= 0 is hard to solve. By throwing into the mix the complementary linear equation X*J−JX= 0, we find that rather than increasing the complexity, we reduce the complexity.

We explicitly demonstrate computation of solutions, visualizations, and closure hierarchies that connect to previous work by Dmytryshyn, Futorny, Kågström, Klimenko, and Sergeichuk.

Joint work with Sung Woo Jeong.

**Speaker: **Jens Fjelstad, Örebro University

**Date:** Friday, February 4, at 13.15, Zoom meeting

**Title:** On quantum representations of mapping class groups from a finite group

**Abstract: **Topological quantum field theory (TQFT) is a gadget that produces two kinds of data, topological invariants and finite dimensional representations of mapping class groups of surfaces, so called quantum representations. These data are of interest both for Mathematics and Physics, one recent application being in quantum computing. One construction of TQFT is via a certain Hopf algebra constructed from a finite group, producing the quantum representations relevant for this talk. I will briefly introduce the ingredients such as mapping class groups of surfaces and the class of quantum representations associated to a finite group, and then discuss work (partly in collaboration with Jürgen Fuchs) to determine their structure in terms of finite group data. Recent results are focused on the restriction of a quantum representation to the terms in a filtration of a mapping class group known as the Johnson filtration.

**Speaker: **David Cohen, Chalmers University of Technology

**Date:** Friday, December 10, at 13.15, Zoom meeting

**Title:** Efficient discretisations of stochastic Hamiltonian and Poisson systems

**Abstract: ** We start by recalling classical results on time discretisations of (deterministic) Hamiltonian and Poisson systems. We will then randomly perturbed such systems, present and analyse various time integrators for an efficient simulations of stochastic Hamiltonian and Poisson systems.

The presentation is based on joint works with C-E. Bréhier, C. Chen, R. D'Ambrosio, K. Debrabant, T. Jahnke, A. Lang, A. Rößler and G. Vilmart.

**Speaker: **Jens Fjelstad, Örebro University

**Date:** Friday, November 12, at 13.15 in T213

**Title:** On quantum representations of mapping class groups from a finite group

**Abstract: **Topological quantum field theory (TQFT) is a gadget that produces two kinds of data, topological invariants and finite dimensional representations of mapping class groups of surfaces, so called quantum representations. These data are of interest both for Mathematics and Physics, one recent application being in quantum computing. One construction of TQFT is via a certain Hopf algebra constructed from a finite group, producing the quantum representations relevant for this talk. I will briefly introduce the ingredients such as mapping class groups of surfaces and the class of quantum representations associated to a finite group, and then discuss work (partly in collaboration with Jürgen Fuchs) to determine their structure in terms of finite group data. Recent results are focused on the restriction of a quantum representation to the terms in a filtration of a mapping class group known as the Johnson filtration.

**Speaker: **Massimiliano Fasi, Durham University, UK

**Date:** Friday, October 15, at 13.15, Zoom meeting

**Title:** CPFloat: A C Library for Emulating Low-Precision Arithmetic

**Abstract: **Low-precision floating-point arithmetic can be simulated via software by executing each arithmetic operation in hardware and rounding the result to the desired number of significant bits. For IEEE-compliant formats, rounding requires only standard mathematical library functions, but handling subnormals, underflow, and overflow demands special attention, and numerical errors can cause mathematically correct formulae to behave incorrectly in finite arithmetic. Moreover, the ensuing algorithms are not necessarily efficient, as the library functions these techniques build upon are typically designed to handle a broad range of cases and may not be optimized for the specific needs of floating-point rounding algorithms. CPFloat is a C library that offers efficient routines for rounding arrays of binary32 and binary64 numbers to lower precision. The software exploits the bit level representation of the underlying formats and performs only low-level bit manipulation and integer arithmetic, without relying on costly library calls. In numerical experiments the new techniques bring a considerable speedup (typically one order of magnitude or more) over existing alternatives in C, C++, and MATLAB. To the best of our knowledge, CPFloat is currently the most efficient and complete library for experimenting with custom low-precision floating-point arithmetic available in any language.

**Speaker: **Hugo U.R. Strand, Örebro University

**Date:** Friday, September 17, at 13.15, Zoom meeting

**Title:** Hands-on high-order orthogonal-polynomial methods for an integrodifferential equation

**Abstract: **High-order orthogonal-polynomial approximation methods and their application is still an evolving field in numerical mathematics [1]. Recent advances has enabled extreme high-order approximations [2] and new algorithmic developments has opened the door for applications on integral equations [3].

In this talk we will present an application of these methods to the integrodifferential "Dyson" equation, that appears in the context of perturbation theory in many-body quantum physics [4].

[1] S. Olver, R.M. Slevisksy, A. Townsend, Acta Numerica pp. 573-699 (2020)

[2] I. Boagert, SIAM J. Sci. Comput., v36 no. 3 pp. A1008-A1026 (2014)

[3] N. Hale, A. Townsend, SIAM J. Sci. Comput., v36 no. 3 pp. A1207-A1220 (2014)

[4] X. Dong, D. Zgid, E. Gull, H.U.R. Strand, J. Chem. Phys. 152, 134107 (2020)

**Speaker: **Attila Szilva, Uppsala University

**Date:** Friday, May 7, at 13.15, Zoom meeting

**Title:** From elections to science of cities: an introduction to complexity

**Abstract: **Animals from rats to the blues whales are built up from cells arranged in networks. The topology of the underlying networks explains the so-called Kleiber’ scaling law, which states that an animal's metabolic rate scales to the ¾ power of the animal's mass (a cat having a mass 100 times that of a mouse will consume only about 32 times the energy the mouse uses). This scaling is sublinear because the power is less than 1. In the presentation, it will be shown that the infrastructure of cities (the length of electric cables or the number of gas stations) also follows universal sublinear scaling law while in socio-economic dimensions (GDP per capita, innovation, crime) cities are superlinear. They are as a result of the individual interactions proven by a large set of mobile phone data. The concept of scaling and universality is originated in statistical physics where a large complex system emerges from simple interactions, and its behavior is almost totally independent of its microscopic structure. Another examples are the democratic elections for which physics inspired models will be also discussed in the presentation.

The talk is based mostly on the books of Geoffrey West: Scale: The Universal Laws of Growth, Innovation, Sustainability, and the Pace of Life in Organisms, Cities, Economies, and Companies and Stefan Thurner, Rudolf Hanel and Peter Klimek: Introduction to the theory of complex systems and the paper Social influence with recurrent mobility and multiple options by Jerome Michaud and Attila Szilva published in Physical Review E 97 (6), 062313.

**Speaker: **Jakob Palmkvist, Örebro University

**Date:** Friday, April 9, at 13.15, Zoom meeting

**Title:** Infinite-dimensional Lie superalgebras

**Abstract:** I will review how the classification of simple finite-dimensional Lie algebras leads to the construction of contragredient Lie algebras, which in general are infinite-dimensional, and how these can be further generalised to contragredient Lie superalgebras. I will then explain how a modification of the construction gives rise to a new class of non-contragredient Lie superalgebras, called tensor hierarchy algebras, whose finite-dimensional members are the simple Lie superalgebras of Cartan type. Tensor hierarchy algebras have proven useful in describing gauge structures in physical models related to string theory. I will describe some of the remarkable features they exhibit.

**Speaker: **Maria ElGhaoui, Sorbonne University, Paris (France) and Saint Joseph University, Beirut (Lebanon)

**Date:** Friday, March 5, at 13.15, Zoom meeting

**Title:** A Trefftz Method With Reconstruction of The Normal Derivative Applied to Elliptic Equations

**Abstract:** There are many classical numerical methods for solving boundary value problems on general domains. The Trefftz method is an approximation method for solving linear boundary value problems arising in applied mathematics and engineering sciences. This method consists to approximate the exact solution through a linear combination of trial functions satisfying exactly the governing differential equation. One of the advantages of this method is that the number of trial functions per cell is O(m), asymptotically much less than the quadratic estimate O(m^2) for finite element and discontinuous Galerkin approximations.

For a Laplace model equation, we present a high order Trefftz method with quadrature formula for calculation of normal derivative at interfaces. We introduce a discrete variational formulation and study the existence and uniqueness of the discrete solution. A priori error estimate is then established and finally, several numerical experiments are shown

**Speaker: **Mårten Gulliksson, Örebro University

**Date:** Friday, February 5, at 13.15, Zoom meeting

**Title:** My Least Squares Problems

**Abstract:** During a period of more than 30 years I have now and then worked with different kinds of least squares problems. These include simple linear least squares in different settings but also much more abstract ill-posed nonlinear problems in Hilbert spaces. Some of the problems were applied such as, e.g., surface fitting, neural networks, and estimating elastic properties in paper. Others were more theoretical in nature involving convergence aspects of methods, perturbation theory, and sparsity. The talk will end with some open least squares problems.

**Speaker: **Andrii Dmytryshyn, Örebro University

**Date:** Friday, November 11, at 13.15 in T211

**Title:** Recovering a perturbation of a matrix polynomial from a perturbation of its linearization

**Abstract:** A number of theoretical and computational problems for matrix polynomials are solved by passing to linearizations. Therefore, a perturbation theory results for the linearizations need to be related back to matrix polynomials. We present an algorithm that finds which perturbation of matrix coefficients of a matrix polynomial corresponds to a given perturbation of the entire linearization pencil. We consider general matrix polynomials as well as polynomials of odd grade whose coefficients are (skew) symmetric. Moreover, we find transformation matrices that, via strict equivalence or, respectively, congruence, transform a perturbation of the linearization to the linearization of a perturbed polynomial.

**Speaker: **Massimiliano Fasi, Örebro University

**Date:** Friday, October 9, at 13.15 in T207

**Title:** Generating extreme-scale matrices with specified singular values or condition number

**Abstract:** The randsvd matrix is a widely used test matrix constructed as the product A = USV, where U and V are random orthogonal or unitary matrices from the Haar distribution and S is a diagonal matrix of singular values. Such matrices are random but have a specified singular value distribution. Forming an m-by-n randsvd matrix requires a number of floating-point operations cubic in both m and n, which is prohibitively expensive at extreme scale. Moreover the randsvd construction requires a significant amount of communication, making it unsuitable for distributed memory environments. By dropping the requirement that U and V be Haar distributed and that both be random, we derive new algorithms for forming A that have cost linear in the number of matrix elements and require a low amount of communication and synchronisation. We specialise these algorithms to generating matrices with specified 2-norm condition number. Numerical experiments show that the algorithms have excellent efficiency and scalability.

**Speaker: **Malte Litsgård, Uppsala University

**Date:** Friday, September 11, at 13.15 in T207

**Title:** Degenerate Kolmogorov-type equations with rough coefficients - Potential theory and boundary regularity

**Abstract:** Today Kolmogorov-type operators with low-regularity coefficients have applications in many different areas: analysis, physics, and finance, to name a few. In particular the study of local regularity of weak solutions to PDEs associated to such operators is relevant in for example kinetic theory, where these equations appear in relation to the study of conditional regularity of the Boltzmann and Landau equations. In this talk I will give some background for these operators and present some results concerning potential theory and boundary regularity in Lipschitz-type domains.

**Speaker: **Magnus Ögren, Örebro universitet

**Date:** Friday, March 6, at 13.15 in T1210

**Title:** A numerical damped oscillator approach to constrained Schrödinger equations

**Abstract:** This article explains and illustrates the use of a set of coupled dynamical equations, second order in a fictitious time, which converges to solutions of stationary Schr\"{o}dinger equations with additional constraints.

We include three qualitative different numerical examples: the radial Schr\"{o}dinger equation for the hydrogen atom; the two-dimensional harmonic oscillator with degenerate excited states; and finally a non-linear Schr\"{o}dinger equation for rotating states.

The presented method is intuitive, with analogies in classical mechanics for damped oscillators, and easy to implement, either in own coding, or with software for dynamical systems.

Hence, we find it suitable to introduce it in a continuation course in quantum mechanics or generally in applied mathematics courses which contain computational parts.

**Speaker: **Mikael Hansson, Örebro universitet

**Date:** Friday, February 7, at 13.15 in T1210

**Title:** An introduction to Coxeter groups, the word problem, and twisted involutions

**Abstract:** Given a group generated by an alphabet S, the word problem is the problem of deciding whether two words with letters from S represent the same group element. For the important class of Coxeter groups, there is a solution based on the so-called word property, which will be described. I shall motivate and define Coxeter groups, so no knowledge of these groups is assumed. Then, twisted involutions will be introduced. We shall see that in this case, too, there is a word property which solves the word problem.

This talk is based on joint work with Axel Hultman, Linköping University.

**Speaker: **Per Enflo

**Date:** Friday, January 10, at 13.15 in T1210

**Title:** Några aktuella problemområden inom funktionalanalysen och dess tillämpningar

**Abstract: **1/ För 200 år sedan visade Fourier att en godtycklig periodisk funktion kan delas upp och skrivas som en summa (linjär kombination) av sinus- och cosinusfunktioner, som alltså utgör ett system av genererande funktioner. Men sinus- och cosinusfunktioner kan inte alltid användas och idag bedrivs mycket forskning att hitta lämpliga system av genererande funktioner (baser) i många olika sammanhang, såväl inom ”ren” matematik som inom dess tillämpningar.

2/ Vid studiet av nxn-matriser är det ofta viktigt att finna matrisernas egenvektorer och deras egenvärden. Men för generaliseringen av matriser till en oändligt-dimensionell situation (de kallas då linjära operatorer) så finns inte alltid egenvektorer. Men kanske finns det ”invarianta underrum”

(en svagare men ändå viktig egenskap). Det nu drygt 80 år gamla problemet, huruvida operatorer på Hilbertrummet alltid har invarianta underrum, är nog det mest berömda olösta problemet inom funktionalanalysen.

**Speaker: **Danny Thonig, Örebro University.

**Date:** Wednesday, December 4, at 15.00 in T217

**Title:** An introduction to ab-initio magnetization dynamics

**Abstract: **The time-integrated amount of stored information is doubled roughly every eighteen months, and since the majority of the world’s information is stored in magnetic media, the possibility to write and retrieve information in a magnetic material at ever greater speed, bigger data transmission rates and with lower energy consumption, has obvious benefits for our society. Hence the seemingly simple switching of a magnetic unit, a magnetic bit, is a crucial process which defines how efficiently information can be stored and retrieved from a magnetic memory. From an application point of view, it is apparent that it is advantageous to be able to switch the magnetization of a bit as fast as possible while minimizing energy losses. It implies also to understand the microscopic origin of the magnetization dynamics in magnetic materials.

Within my talk I will give a general introduction about what magnetism is, from where it originates, and how it can be treated in modeling. Why magnetism is materials specific and - although known as rather static - magnetism is dynamic will be motived after the introduction part. Here, the equation of motion will be discussed more in detail and applied to certain questions that occurred from experimental observations. All this is implemented in the software package "Cahmd" (https://cahmd.gitlab.io/cahmdweb/), which was developed by the PI. I will finish my presentation with an outlook on future challenges.

**Speaker:** Zhaojun Bai, University of California, Davis.

**Date:** Tuesday, November 26, at 15.00 in T215

**Title:** Rayleigh quotient optimizations and eigenvalue problems

**Abstract:** Many computational science and data analysis techniques lead to optimizing Rayleigh quotient (RQ) and RQ type objective functions, such as computing excitation states (energies) of electronic structures, robust classification to handle uncertainty and constrained data clustering to incorporate a prior information. We will discuss origins of recently emerging RQ optimization problem, variational principles, and reformulations to algebraic linear and nonlinear eigenvalue problems. We will show how to exploit underlying properties of eigenvalue problems for designing eigensolvers, and illustrate the efficacy of these solvers in applications.

**Speaker: **Per Bäck, Mälardalen University

**Date:** Friday, November 8, at 13.15 in T1210

**Title:** Theory of non-commutative and non-associative polynomial rings

**Abstract:** In this talk, I will give an introduction to the theory of non-commutative polynomial rings known as Ore extensions, and then show how this can be generalized to the non-associative setting. In particular, I will discuss hom-associative Ore extensions and how they provide a framework for deforming otherwise rigid algebras appearing in e.g. quantum physics, such as the Weyl algebras. I will also show how these deformations induce deformations of the corresponding commutator Lie algebras into so-called hom-Lie algebras.

In the end, we will see how the classical Hilbert’s basis theorem for commutative polynomial rings can be extended to a non-commutative and non-associative version, and applications thereof including quaternionic and octonionic rings.

**Speaker: **Anders Tengstrand

**Date: **Wednesday, October 30, at 13.15 in T217

**Title: **Det oändligt stora och det oändligt lilla

**Abstract: **Det oändligt stora och det oändligt lilla har i alla tider utmanat matematiker. Med hjälp av dessa begrepp har man skapat effektiva verktyg för att lösa problem inom matematiken och dess tillämpningar. Samtidigt har matematiker som använt sig av oändligt små och oändligt stora tal utsatts för kritik för otydlighet och brist på konkretion. Jag ska i min föreläsning belysa denna problematik med exempel från antiken fram till 1900-talet.

**Speaker:** Johan Andersson (Örebro Universitet)

**Date:** Friday, October 11, at 13.15-14.30 in T2102

**Title: **Universality of zeta and L-functions

**Abstract:** Voronin proved in the seventies that any zero-free analytic function f(s) on a disc |s-3/4|<r<1/4 which is continuous up to its boundary may be uniformly approximated as closely as desired by imaginary shifts of the Riemann zeta-function. We say that the Riemann zeta-function is universal. In the last 40 years this property has been proved for a wide range of zeta and L-functions, such as automorphic L-functions and the Selberg zeta-function. I will talk about some of my recent work in the field. In particular I have recently proved (arXiv:1809.03444) that the Euler-Zagier multiple zeta-function is universal in several complex variables, thus giving the first example of a Dirichlet series that is universal in more than one complex variable. I will also talk about recent work in progress where I give the first universality theorem for the Hurwitz zeta-function with an algebraic irrational parameter.

**Speaker: **Mac Panahbehagh (Örebro Universitet)

**Date:** Friday, September 13, at 13.15-14.30 in T1210

**Title: **Simulations of a porous particle settling in density stratified ambient

**Abstract: **We study numerically the settling of a porous sphere in a density-stratified ambient fluid. Simulations are validated against prior laboratory experiments and compared to two mathematical models. Two main effects cause the particle to slow down as it enters a density gradient: lighter fluid within the particle and entrainment of the density-stratified ambient fluid. The numerical simulations accurately capture the particle retention time. We quantify the delay in settling due to ambient fluid entrainment and lighter internal fluid becoming denser through diffusion as a function of different parameters. A simple fitting formula is presented to describe the settling time delay as a function of each of those non-dimensional parameters.

**Speaker: **Simon Streib (Uppsala Universitet)

**Date:** Wednesday, August 21, at 13.15-14.30 in T1210

**Title: **The Barnett/Einstein-de Haas effects and magnetoelastic coupling in magnetic insulators

**Abstract: **The Barnett effect is the magnetization induced by mechanical rotation of an uncharged body while the Einstein-de Haas effect is its reciprocal: rotation induced by magnetization. I give an historical overview of these effects, including a discussion of the recent observation of the nuclear Barnett effect [1], and present a simple derivation of the Barnett effect based on the conservation of energy and angular momentum. Finally, I discuss the physical origin of the magnetoelastic coupling in magnetic insulators and its effect on the lifetime of magnetic excitations [2].

[1] M. Arabgol and T. Sleator, Phys. Rev. Lett. 122, 177202 (2019).

[2] S. Streib, N. Vidal-Silva, K. Shen, and G. E. W. Bauer, Phys. Rev. B 99, 184442 (2019).

**Speaker:** Froilán M. Dopico (Universidad Carlos III de Madrid)

**Date:** Wednesday, May 29, at 15:15 in T1210

**Title: **Local linearizations of rational matrices with application to nonlinear eigenvalue problems

**Abstract: **The numerical solution of nonlinear eigenvalue problems (NLEP) has attracted considerable attention since 2004, mainly as a consequence of the influential reference "Nonlinear eigenvalue problems: a challenge for modern eigenvalue methods" (GAMM Mitt. Ges. Angew. Math. Mech., 2004) by V. Mehrmann and H. Voss. A variety of methods have been developed for these problems and the most successful ones can be found in the recent survey "The nonlinear eigenvalue problem" (Acta Numer., 2017) by S. G\"{u}ttel and F. Tissuer. Both for dense-medium-size and for large-scale problems the preferred methods consist of three steps:

(1) to approximate the NLEP by a rational eigenvalue problem (REP) in a certain region;

(2) to construct a linear eigenvalue problem (LEP) that has the same eigenvalues of the REP in the region of interest;

(3) to compute via the QZ method for dense problems or via structured rational Krylov methods for large-scale problems the eigenvalues of the LEP.

The purpose of this talk is to develop a mathematical local theory of linearizations of REPs that allows us, among other things, to establish rigorously the properties of the LEPs that have been used for solving NLEPs in a number of recent references.

**Speaker:** Massimiliano Fasi (University of Manchester)

**Date:** Tuesday, May 21, at 15:15 in T1210

**Title:** Substitution algorithms for rational matrix equations

**Abstract:** Functions of matrices defined as solutions to matrix equations play an important role in many applications. As rational approximation is a customary tool in algorithms for evaluating this class of functions, one may wonder whether it be possible to solve numerically equations of the form $r(X)=A$, where $A$ and $X$ are square matrices and $r$ is a rational function.

As it turns out, accurate and efficient techniques for this problem can be derived by inverting, through a substitution strategy, computational schemes for evaluating rational matrix functions. The resulting methods all exploit the Schur decomposition of the input matrix to reduce the problem to upper triangular form, and for triangular matrices they yield the same computational cost as the evaluation schemes from which they are obtained. This suggests that solving rational matrix equations is not more difficult than evaluating rational functions at a matrix argument.

These methods can be used in a natural way as building blocks in algorithms for computing functions of matrices defined via matrix equation of the type $f(X) = A$, where $f$ is a primary matrix function.