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    Generalized Complex Step Approximation to Estimate the First and Second Order Frechet Derivative of Matrix Functions
    (Univ Nis, Fac Sci Math, 2022) Arslan, Bahar; Al-Mohy, Awad H.
    Applications of Frechet derivative emerge in the sensitivity analysis of matrix functions. Our work extends the generalized complex step approximation using the complex computation f (A + e(i theta)hE) as a tool to matrix case, and combines it with finite difference formula to estimate the Frechet derivative. We provide numerical results for the approximation to the first and the second order Frechet derivative of the matrix exponential and matrix square root.
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    Nonstandard Finite Difference Theta Approaches to the Predator-Prey System
    (Wiley, 2026) Ozdogan, Nihal; Arslan, Bahar
    In this paper, we propose nonstandard finite difference theta schemes for a well-known Lotka-Volterra model without the Allee effect and investigate the dynamical behavior of the discretized system. We prove that the scheme is dynamically consistent with the continuous model, preserving key properties such as the positivity of solutions and the stability of equilibrium points. Numerical experiments are conducted to validate the theoretical results and demonstrate the superiority of the proposed schemes over standard methods in maintaining these critical properties.
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    ODE Solvers for Computing Matrix Functions
    (Mustafa UÇKUN, 2026) Arslan, Bahar
    This study investigates the use of ordinary differential equation solvers to estimate the action of matrix functions on vectors. In addition, an evaluation is conducted to ascertain the degree of computational efficiency that is achieved by executing matrix factorisation prior to the implementation of ODE solvers. Three models are employed to analyse the computation of matrix functions acting on vectors, including fractional matrix powers, the matrix exponential, and matrix cosine functions. The performance of the improved Euler, Taylor, Runge–Kutta and Adams–Bashforth methods are compared within these models.
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    Structure Preserving Algorithm for the Logarithm of Symplectic Matrices
    (Murat TOSUN, 2021) Arslan, Bahar
    The current algorithms use either the full form or the Schur decomposition of the matrix in the inverse scaling and squaring method to compute the matrix logarithm. The inverse scaling and squaring method consists of two main calculations: taking a square root and evaluating the Padé approximants. In this work, we suggest using the structure preserving iteration as an alternative to Denman-Beavers iteration for taking a square root. Numerical experiments show that while using the structure preserving square root iteration in the inverse scaling and squaring method preserves the Hamiltonian structure of matrix logarithm, Denman-Beavers iteration and Schur decomposition cause a structure loss.
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    Structured Condition Number for a Certain Class of Functions of Non-commuting Matrices
    (Springer Basel Ag, 2023) Arslan, Bahar; Cardoso, Joao R.
    We investigate the structured condition number of a class of functions of two non-commuting matrices with some type of structure, in order to assess the sensitivity of structured perturbations in the input matrices. This might have interest in several practical applications, like the computation of the geometric mean of two symmetric positive definite matrices or the exp-log mean of two symplectic matrices. These two particular matrix functions deserve a particular study, as well as the matrix-matrix exponentiation. Algorithms for computing the structured condition number are proposed, but they are in general expensive. To round this issue, we derive lower and upper bounds for estimating the structured condition number. Results regarding the comparison between the structured and unstructured condition numbers for pairs of symmetric or skew-symmetric input matrices are provided. Several numerical experiments involving many structured matrices are carried out to compare the structured condition number with the bounds and also with the unstructured condition number.
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    STRUCTURED LEVEL-2 CONDITION NUMBERS OF MATRIX FUNCTIONS
    (Int Linear Algebra Soc, 2024) Arslan, Bahar; Relton, Samuel D.; Schweitzer, Marcel
    Matrix functions play an increasingly important role in many areas of scientific computing and engineering disciplines. In such real-world applications, algorithms working in floating-point arithmetic are used for computing matrix functions and additionally input data might be unreliable, e.g., due to measurement errors. Therefore, it is crucial to understand the sensitivity of matrix functions to perturbations, which is measured by condition numbers. However, the condition number itself might not be computed exactly as well due to round-off and errors in the input. The sensitivity of the condition number is measured by the so-called level-2 condition number. For the usual (level-1) condition number, it is well known that structured condition numbers (i.e., where only perturbations are taken into account that preserve the structure of the input matrix) might be much smaller than unstructured ones, which, e.g., suggests that structure-preserving algorithms for matrix functions might yield much more accurate results than general-purpose algorithms. In this work, we present a novel upper bound on the structured level-2 condition number, focusing on perturbation matrices within an automorphism group, a Lie or Jordan algebra, or the space of quasi-triangular matrices. In numerical experiments, we then compare the unstructured level-2 condition number with the structured one for some specific matrix functions such as the matrix logarithm, matrix square root, and matrix exponential.
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    The complex step approximation to the higher order Frechet derivatives of a matrix function
    (Springer, 2021) Al-Mohy, Awad H.; Arslan, Bahar
    Thekth Frechet derivative of a matrix functionfis a multilinear operator from a cartesian product ofksubsets of the spaceDOUBLE-STRUCK CAPITAL C-nxn into itself. We show that thekth Frechet derivative of a real-valued matrix functionfat a real matrixAin real direction matrices E-1, E-2, horizontal ellipsis, E-k can be computed using the complex step approximation. We exploit the algorithm of Higham and Relton (SIAM J. Matrix Anal. Appl.35(3):1019-1037,2014) with the complex step approximation and mixed derivative of complex step and central finite difference scheme. Comparing with their approach, our cost analysis and numerical experiment reveal thathalfandseven-eighthsof the computational cost can be saved for the complex step and mixed derivative, respectively. Whenfhas an algorithm that computes its action on a vector, the computational cost drops down significantly as the dimension of the problem andkincrease.
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    THE STRUCTURED CONDITION NUMBER OF A DIFFERENTIABLE MAP BETWEEN MATRIX MANIFOLDS, WITH APPLICATIONS
    (Siam Publications, 2019) Arslan, Bahar; Noferini, Vanni; Tisseur, Francoise
    We study the structured condition number of differentiable maps between smooth matrix manifolds, extending previous results to maps that are only R-differentiable for complex manifolds. We present algorithms to compute the structured condition number. As special cases of smooth manifolds, we analyze automorphism groups, and Lie and Jordan algebras associated with a scalar product. For such manifolds, we derive a lower bound on the structured condition number that is cheaper to compute than the structured condition number. We provide numerical comparisons between the structured and unstructured condition numbers for the principal matrix logarithm and principal matrix square root of matrices in automorphism groups as well as for the map between matrices in automorphism groups and their polar decomposition. We show that our lower bound can be used as a good estimate for the structured condition number when the matrix argument is well conditioned. We show that the structured and unstructured condition numbers can differ by many orders of magnitude, thus motivating the development of algorithms preserving structure.