Real-time soft shadows in dynamic scenes using spherical harmonic exponentiation
ACM SIGGRAPH 2006 Papers
An error analysis of the modified scaling and squaring method
Computers & Mathematics with Applications
Comparison of methods for evaluating functions of a matrix exponential
Applied Numerical Mathematics
The scaling and modified squaring method for matrix functions related to the exponential
Applied Numerical Mathematics
Algorithm 894: On a block Schur--Parlett algorithm for ϕ-functions based on the sep-inverse estimate
ACM Transactions on Mathematical Software (TOMS)
A Fast and Log-Euclidean Polyaffine Framework for Locally Linear Registration
Journal of Mathematical Imaging and Vision
Journal of Computational and Applied Mathematics
Learning averages over the lie group of unitary matrices
IJCNN'09 Proceedings of the 2009 international joint conference on Neural Networks
Exponentials of skew-symmetric matrices and logarithms of orthogonal matrices
Journal of Computational and Applied Mathematics
An algorithm to compute averages on matrix Lie groups
IEEE Transactions on Signal Processing
The Immersed Structural Potential Method for haemodynamic applications
Journal of Computational Physics
A New Scaling and Squaring Algorithm for the Matrix Exponential
SIAM Journal on Matrix Analysis and Applications
Diffeomorphic registration of images with variable contrast enhancement
Journal of Biomedical Imaging - Special issue on modern mathematics in biomedical imaging
Shift-Invert Arnoldi Approximation to the Toeplitz Matrix Exponential
SIAM Journal on Scientific Computing
Smooth conditional transition paths in dynamical gaussian networks
KI'11 Proceedings of the 34th Annual German conference on Advances in artificial intelligence
Computing the Action of the Matrix Exponential, with an Application to Exponential Integrators
SIAM Journal on Scientific Computing
A log-euclidean framework for statistics on diffeomorphisms
MICCAI'06 Proceedings of the 9th international conference on Medical Image Computing and Computer-Assisted Intervention - Volume Part I
Journal of Computational and Applied Mathematics
ACM Transactions on Mathematical Software (TOMS)
A log-euclidean polyaffine framework for locally rigid or affine registration
WBIR'06 Proceedings of the Third international conference on Biomedical Image Registration
Accurate matrix exponential computation to solve coupled differential models in engineering
Mathematical and Computer Modelling: An International Journal
Complexity reduction of stochastic master equation simulation based on Kronecker product analysis
Proceedings of the ACM Conference on Bioinformatics, Computational Biology and Biomedicine
Comparative performance of exponential, implicit, and explicit integrators for stiff systems of ODEs
Journal of Computational and Applied Mathematics
Lie bodies: a manifold representation of 3D human shape
ECCV'12 Proceedings of the 12th European conference on Computer Vision - Volume Part I
Circuit simulation via matrix exponential method for stiffness handling and parallel processing
Proceedings of the International Conference on Computer-Aided Design
Exponential Taylor methods: Analysis and implementation
Computers & Mathematics with Applications
Reducing the influence of tiny normwise relative errors on performance profiles
ACM Transactions on Mathematical Software (TOMS)
A modal precise integration method for the calculation of footbridge vibration response
Computers and Structures
Stochastic Model Simulation Using Kronecker Product Analysis and Zassenhaus Formula Approximation
IEEE/ACM Transactions on Computational Biology and Bioinformatics (TCBB)
Process-Variation and Temperature Aware SoC Test Scheduling Technique
Journal of Electronic Testing: Theory and Applications
Lie-group interpolation and variational recovery for internal variables
Computational Mechanics
Hi-index | 0.01 |
The scaling and squaring method is the most widely used method for computing the matrix exponential, not least because it is the method implemented in MATLAB's {\tt expm} function. The method scales the matrix by a power of 2 to reduce the norm to order 1, computes a Padé approximant to the matrix exponential, and then repeatedly squares to undo the effect of the scaling. We give a new backward error analysis of the method (in exact arithmetic) that employs sharp bounds for the truncation errors and leads to an implementation of essentially optimal efficiency. We also give new rounding error analysis that shows the computed Padé approximant of the scaled matrix to be highly accurate. For IEEE double precision arithmetic the best choice of degree of Padé approximant turns out to be 13, rather than the 6 or 8 used by previous authors. Our implementation of the scaling and squaring method always requires at least two fewer matrix multiplications than {\tt expm} when the matrix norm exceeds 1, which can amount to a 37% saving in the number of multiplications, and it is typically more accurate, owing to the fewer required squarings. We also investigate a different scaling and squaring algorithm proposed by Najfeld and Havel that employs a Padé approximation to the function $x \coth(x)$. This method is found to be essentially a variation of the standard one with weaker supporting error analysis.