Listar por autoría "49f49a14-5052-44bd-8570-56b1f8e52f90"
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A general formula for the Magnus expansion in terms of iterated integrals of right-nested commutators
Arnal, A.; Casas, Fernando; Chiralt, Cristina IOP (2018-03)We present a general expression for any term of the Magnus series as an iterated integral of a linear combination of independent right-nested commutators with given coefficients. The relation with the Malvenuto–Reutenauer ... -
A Lie-Deprit perturbation algorithm for linear differential equations with periodic coefficients
Casas, Fernando; Chiralt, Cristina American Institute of Mathematical Sciences (AIMS) (2014-03)A perturbative procedure based on the Lie-Deprit algorithm of classical mechanics is proposed to compute analytic approximations to the fundamental matrix of linear di erential equations with periodic coe cients. These ... -
A New Optimality Property of Strang’s Splitting
Casas, Fernando; Sanz-Serna, JM; Shaw, Luke Society for Industrial and Applied Mathematics (2023)For systems of the form ̇q = M−1 p, ̇p = −Aq + f(q), common in many applications, we analyze splitting integrators based on the (linear/nonlinear) split systems ̇q = M−1 p, ̇p = −Aq and ̇q = 0, ̇p = f(q). We show ... -
A note on the Baker–Campbell–Hausdorff series in terms of right-nested commutators
Arnal, A.; Casas, Fernando; Chiralt, Cristina Springer (2021-01-27)We get compact expressions for the Baker–Campbell–Hausdorff series Z = log(eX eY ) in terms of right-nested commutators. The reduction in the number of terms originates from two facts: (i) we use as a starting point an ... -
A note on trigonometric identities involving non-commuting matrices
Arnal, A.; Casas, Fernando; Chiralt, Cristina Springer Milan (2017)An algorithm is presented for generating successive approximations to trigonometric functions of sums of non-commuting matrices. The resulting expressions involve nested commutators of the respective matrices. The procedure ... -
A pedagogical approach to the Magnus expansion
Blanes, Sergio; Casas, Fernando; Oteo, J. A.; Ros, J. Institute of Physics (2010-07)Time-dependent perturbation theory as a tool to compute approximate solutions of the Schrödinger equation does not preserve unitarity. Here we present, in a simple way, how the Magnus expansion (also known as exponential ... -
A Unifying Framework for Perturbative Exponential Factorizations
Arnal, A.; Casas, Fernando; Chiralt, Cristina; Oteo, José-Angel MDPI (2021-03-17)We propose a framework where Fer and Wilcox expansions for the solution of differential equations are derived from two particular choices for the initial transformation that seeds the product expansion. In this scheme, ... -
An efficient algorithm based on splitting for the time integration of the Schrödinger equation
Blanes, Sergio; Casas, Fernando; Murua, Ander Elsevier (2015-12)We present a practical algorithm based on symplectic splitting methods intended for the numerical integration in time of the Schrödinger equation when the Hamiltonian operator is either time-independent or changes slowly ... -
An efficient algorithm for computing the Baker–Campbell–Hausdorff series and some of its applications
Casas, Fernando; Murua, Ander American Institute of Physics (2009-03)We provide a new algorithm for generating the Baker–Campbell–Hausdorff (BCH) seriesZ = log(eXeY) in an arbitrary generalized Hall basis of the free Lie algebra L(X,Y)generated by X and Y. It is based on the close relationship ... -
An efficient algorithm to compute the exponential of skew-Hermitian matrices for the time integration of the Schrödinger equation
Bader, Philipp; Blanes, Sergio; Casas, Fernando; Seydaoğlu, Muaz Elsevier B.V. (2022-04-30)We present a practical algorithm to approximate the exponential of skew-Hermitian matrices up to round-off error based on an efficient computation of Chebyshev polynomials of matrices and the corresponding error analysis. ... -
Applying splitting methods with complex coefficients to the numerical integration of unitary problems
Blanes, Sergio; Casas, Fernando; Escorihuela-Tomàs, Alejandro American Institute of Mathematical Sciences (AIMS) (2022)We explore the applicability of splitting methods involving complex coefficients to solve numerically the time-dependent Schr¨odinger equation. We prove that a particular class of integrators are conjugate to unitary ... -
Composition Methods for Dynamical Systems Separable into Three Parts
Casas, Fernando; Escorihuela-Tomàs, Alejandro MDPI (2020)New families of fourth-order composition methods for the numerical integration of initial value problems defined by ordinary differential equations are proposed. They are designed when the problem can be separated into ... -
Compositions of pseudo-symmetric integrators with complex coefficients for the numerical integration of differential equations
Casas, Fernando; Chartier, Philippe; Escorihuela-Tomàs, Alejandro; Zhang, Yong Elsevier (2020-05-29)In this paper, we are concerned with the construction and analysis of a new class of methods obtained as double jump compositions with complex coefficients and projection on the real axis. It is shown in particular that ... -
Computing the Matrix Exponential with an Optimized Taylor Polynomial Approximation
Bader, Philipp; Blanes, Sergio; Casas, Fernando MDPI (2019)A new way to compute the Taylor polynomial of a matrix exponential is presented which reduces the number of matrix multiplications in comparison with the de-facto standard Paterson-Stockmeyer method for polynomial ... -
Computing the matrix sine and cosine simultaneously with a reduced number of products
Seydaoğlu, Muaz; Bader, Philipp; Blanes, Sergio; Casas, Fernando Elsevier (2021-05)A new procedure is presented for computing the matrix cosine and sine simultaneously by means of Taylor polynomial approximations. These are factorized so as to reduce the number of matrix products involved. Two versions ... -
Continuous changes of variables and the Magnus expansion
Casas, Fernando; Chartier, Philippe; Murua, Ander IOP (2019-09-23)In this paper, we are concerned with a formulation of Magnus and Floquet-Magnus expansions for general nonlinear differential equations. To this aim, we introduce suitable continuous variable transformations generated by ... -
Convergence analysis of high-order commutator-free quasi-Magnus exponential integrators for nonautonomous linear evolution equations of parabolic type
Blanes, Sergio; Casas, Fernando; Thalhammer, Mechthild Oxford University Press (2018-04)The main objective of this work is to provide a stability and error analysis of high-order commutator-free quasi-Magnus (CFQM) exponential integrators. These time integration methods for nonautonomous linear evolution ... -
Convergence analysis of high-order commutator-free quasi-Magnus exponential integrators for nonautonomous linear Schrödinger equations
Blanes, Sergio; Casas, Fernando; González, Cesáreo; Thalhammer, Mechthild Oxford University Press (2020-03-02)This work is devoted to the derivation of a convergence result for high-order commutator-free quasi-Magnus (CFQM) exponential integrators applied to nonautonomous linear Schrödinger equations; a detailed stability and local ... -
Efficient computation of the Zassenhaus formula
Casas, Fernando; Murua, Ander; Nadinic, Mladen Elsevier (2012-11)A new recursive procedure to compute the Zassenhaus formula up to high order is presented, providing each exponent in the factorization directly as a linear combination of independent commutators and thus containing the ... -
Efficient numerical integration of NNth-order non-autonomous linear differential equations
Bader, Philipp; Blanes, Sergio; Casas, Fernando; Ponsoda, Enrique Elsevier (2016)We consider the numerical integration of high-order linear non-homogeneous differential equations, written as first order homogeneous linear equations, and using exponential methods. Integrators like Magnus expansions or ...