Matrix Exponentials

SIAM Review ◽  
1996 ◽  
Vol 38 (2) ◽  
pp. 313-314
Author(s):  
Robert E. Terrell
Keyword(s):  
Matrix Exponentials

scholarly journals Markov chain approximation of one-dimensional sticky diffusions

2021 ◽  
Vol 53 (2) ◽  
pp. 335-369
Author(s):  
Christian Meier ◽  
Lingfei Li ◽  
Gongqiu Zhang
Keyword(s):  
Markov Chain ◽  
Approximate Solutions ◽  
Price Model ◽  
One Dimensional ◽  
Markov Chain Approximation ◽  
Alternative Approaches ◽  
Second Order Convergence ◽  
Chain Approximation ◽  
Matrix Exponentials ◽  
Interior Points

AbstractWe develop a continuous-time Markov chain (CTMC) approximation of one-dimensional diffusions with sticky boundary or interior points. Approximate solutions to the action of the Feynman–Kac operator associated with a sticky diffusion and first passage probabilities are obtained using matrix exponentials. We show how to compute matrix exponentials efficiently and prove that a carefully designed scheme achieves second-order convergence. We also propose a scheme based on CTMC approximation for the simulation of sticky diffusions, for which the Euler scheme may completely fail. The efficiency of our method and its advantages over alternative approaches are illustrated in the context of bond pricing in a sticky short-rate model for a low-interest environment and option pricing under a geometric Brownian motion price model with a sticky interior point.

scholarly journals Phylosymmetric Algebras: Mathematical Properties of a New Tool in Phylogenetics

2020 ◽  
Vol 82 (12) ◽  
Author(s):  
Michael Hendriksen ◽  
Julia A. Shore
Keyword(s):  
Rooted Tree ◽  
Matrix Multiplication ◽  
Computation Time ◽  
Rate Of Change ◽  
Recent Common Ancestor ◽  
Transition Matrices ◽  
Rate Matrix ◽  
Most Recent Common Ancestor ◽  
Mathematical Properties ◽  
Matrix Exponentials

AbstractIn phylogenetics, it is of interest for rate matrix sets to satisfy closure under matrix multiplication as this makes finding the set of corresponding transition matrices possible without having to compute matrix exponentials. It is also advantageous to have a small number of free parameters as this, in applications, will result in a reduction in computation time. We explore a method of building a rate matrix set from a rooted tree structure by assigning rates to internal tree nodes and states to the leaves, then defining the rate of change between two states as the rate assigned to the most recent common ancestor of those two states. We investigate the properties of these matrix sets from both a linear algebra and a graph theory perspective and show that any rate matrix set generated this way is closed under matrix multiplication. The consequences of setting two rates assigned to internal tree nodes to be equal are then considered. This methodology could be used to develop parameterised models of amino acid substitution which have a small number of parameters but convey biological meaning.

scholarly journals Synthesis on the Acceleration Energies in the Advanced Mechanics of the Multibody Systems

Symmetry ◽  
2019 ◽  
Vol 11 (9) ◽  
pp. 1077 ◽  
Author(s):  
Negrean ◽  
Crișan
Keyword(s):  
Multibody Systems ◽  
Driving Forces ◽  
Higher Order ◽  
Analytical Mechanics ◽  
Polynomial Functions ◽  
Control Functions ◽  
Generalized Variables ◽  
The Matrix ◽  
Matrix Exponentials ◽  
Fifth Order

The present paper’s objective is to highlight some new developments of the main author in the field of advanced dynamics of systems and higher order dynamic equations. These equations have been developed on the basis of the matrix exponentials which prove to have undeniable advantages in the matrix study of any complex mechanical system. The present paper proposes some new approaches, based on differential principles from analytical mechanics, by using some important dynamics notions, regarding the acceleration energies of the first, second and third order. This study extended the equations of the higher order, which provide the possibility of applying the initial motion conditions in the positions, velocities and accelerations of the first and second order. In order to determine the time variation laws for the generalized variables, the driving forces and acceleration energies of the higher order are applied by the time polynomial functions of the fifth order. According to inverse kinematics also named control kinematics of the robots, the applications of polynomial functions lead to the kinematic control functions of mechanical motions, especially the transitory motions. They influence the dynamic behavior of multibody systems, in which robot structures are included.

Industry Tip: Quick and Easy Matrix Exponentials

2020 ◽  
Vol 35 (5) ◽  
pp. 49-52
Author(s):  
James Corwell ◽  
William Dale Blair
Keyword(s):  
Matrix Exponentials
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