Nineteen Dubious Ways to Compute the Exponential of a Matrix

Moler, Cleve B.; Loan, Charles Van

doi:10.1137/1020098

Cited by 1,251 publications

(537 citation statements)

References 84 publications

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“…There are several ways to calculate the matrix exponential functions. In ARTS the Pade-approximation is implemented according to Moler and Loan [1979].…”

Section: Discussionmentioning

confidence: 99%

A polarized discrete ordinate scattering model for simulations of limb and nadir long‐wave measurements in 1‐D/3‐D spherical atmospheres

et al. 2004

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[1] This article describes one of the scattering algorithms of the three-dimensional polarized radiative transfer model ARTS (Atmospheric Radiative Transfer Simulator) which has been implemented to study for example the influence of cirrus clouds on microwave limb sounding. The model uses the DOIT (Discrete Ordinate Iterative) method to solve the vector radiative transfer equation. The implementation of a discrete ordinate method is challenging due to the spherical geometry of the model atmosphere which is required for the simulation of limb radiances. The involved numerical issues, which are grid optimization and interpolation methods, are discussed in this paper. Scattering simulations are presented for limb-and down-looking geometries, for one-dimensional and three-dimensional spherical atmospheres. They show the impact of cloud particle size, shape, and orientation on the brightness temperatures and on the polarization of microwave radiation in the atmosphere. The cloud effect is much larger for limb radiances than for nadir radiances. Particle size is a very important parameter in all simulations. The polarization signal is negligible for simulations with completely randomly oriented particles, whereas for horizontally aligned particles with random azimuthal orientation the polarization signal is significant. Moreover, the effect of particle shape is only relevant for oriented cloud particles. The simulations show that it is essential to use a threedimensional scattering model for inhomogeneous cloud layers.

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“…There are several ways to calculate the matrix exponential functions. In ARTS the Pade-approximation is implemented according to Moler and Loan [1979].…”

Section: Discussionmentioning

confidence: 99%

A polarized discrete ordinate scattering model for simulations of limb and nadir long‐wave measurements in 1‐D/3‐D spherical atmospheres

et al. 2004

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“…Computing this gradient will be the subject of Section 7. For now we assume the existence of a procedure SmoothGrad which takes as input a vector u and returns ∇f α (u) and X u as defined in equations (21) and (22). Once the gradient is explicitly known, a Frank-Wolfe-type gradient descent method takes O( −2 ) iterations to compute -optimalū.…”

Section: Notation and Definitionsmentioning

confidence: 99%

Approximation Algorithms for Semidefinite Packing Problems with Applications to Maxcut and Graph Coloring

Iyengar

Phillips

Stein

2005

Integer Programming and Combinatorial Optimization

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We describe the semidefinite analog of the vector packing problem, and show that the semidefinite programming relaxations for Maxcut [10] and graph coloring [16] are in this class of problems. We extend a method of Bienstock and Iyengar [4] which was based on ideas from Nesterov [24] to design an algorithm for computing -approximate solutions for this class of semidefinite programs. Our algorithm is in the spirit of Klein and Lu [17], and decreases the dependence of the run-time on from −2 to −1 . For sparse graphs, our method is faster than the best specialized interior point methods. A significant feature of our method is that it treats both the Maxcut and the graph coloring problem in a unified manner.

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“…STABILIZATION OF NCSS Similar as in [18], we use the (real) Jordan form of the system matrix A := QJQ −1 [21], with A defined in (2), for the stability analysis and controller synthesis. The general notation of the NCS model is then given by:…”

Section: Ncs Modelmentioning

confidence: 99%