The Scaling and Squaring Method for the Matrix Exponential Revisited

By Nicholas John Higham. SIAM Journal on Matrix Analysis and Applications, 2005

763 citations recorded by the source corpus.

Subfields: Computational Theory and Mathematics, Numerical Analysis

View Paper: https://www.semanticscholar.org/paper/ef714d7b12754359e7f812905954c9f7c75c7842

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Citations in the corpus

Deformable Medical Image Registration: A Survey (2013, methodology): “Given an even number of steps, this was performed efficiently by a scaling and squaring approach [45], [46].”
Interpolation among reduced‐order matrices to obtain parameterized models for design, optimization and probabilistic analysis (2009, methodology): “which is obtained by calculating the matrix exponential (exp) of Ĥ using a Padé approximation with scaling and squaring [44].”
Cheap Orthogonal Constraints in Neural Networks: A Simple Parametrization of the Orthogonal and Unitary Group (2019, methodology): “The best one known to the authors is based on the paper (Al-Mohy & Higham, 2009b).”
FOURIER TRANSFORM METHODS FOR REGIME-SWITCHING JUMP-DIFFUSIONS AND THE PRICING OF FORWARD STARTING OPTIONS (2011, methodology): “For M > 2 the cumbersome task is the calculation of Φ for which efficient numerical techniques are available (see Higham (2009)).”
Differential Geometry and Lie Groups (2020, methodology): “The key idea is that exp(M) is easy to compute if M is close zero, since in this case, one can use a few terms of the exponential series, or better a Padé approximant (see Higham [59]).”
Collected Matrix Derivative Results for Forward and Reverse Mode Algorithmic Differentiation (2008, background): “where m is a power of 2, and p1 and p2 are polynomials such that p2(x)/p1(x) is a Pad´ approximation to exp(x/m) [ 9 ].”
The Physics of Communicability in Complex Networks (2011, methodology): “One of the most efficient and accurate available methods is the one based on the Padé approximation combined with the scaling and squaring method (Higham, 2005; 2008).”
QTT approximation of elliptic solution operators in higher dimensions (2011, methodology): “For the (approximate) computation of the matrix exponential the scaling-and-squaring method [17, 27] can be used, which is the most suitable for the rank-structured format.”
A New Scaling and Squaring Algorithm for the Matrix Exponential (2009, neutral): “The algorithm of [10] can be summarized as follows.”
Implementation of a restarted Krylov subspace method for the evaluation of matrix functions (2008, background): “13 for the exponential function and s ∈ N 0 depends on (cid:9) (cid:4) H k (cid:9) 1 (see [12]).”
Error analysis of finite difference and Markov chain approximations for option pricing (2018, positive): “Alternatively, one can also use the algorithm in Higham (2005) to directly compute the matrix exponential in (2.”
Quadrature rule-based bounds for functions of adjacency matrices (2010, methodology): “Some of these quantities are expressed in terms of adjacency matrices; in particular, we will use here thenotions of Estrada index, subgraph centrality and communicability, which are presented indetail in [5,8–16]; see also the discussion in [20].”
A Log-Euclidean Framework for Statistics on Diffeomorphisms (2006, background): “obtain exp(M) = exp( M 2 )(2) N by squaring recursively N times the result [7].”
Geometric Means in a Novel Vector Space Structure on Symmetric Positive-Definite Matrices (2007, methodology): “These formulae provide a particularly efficient method to calculate the logarithms and exponentials of symmetric matrices, whenever the cost of a diagonalization is less than that of the many matrix multiplications (in the case of the exponential) and inversions (in the case of the logarithm) used i”