Matrix exponentials, SU(N) group elements, and real polynomial roots

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.

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On the structure of polynomial roots

The Mathematical Gazette ◽

10.1017/mag.2021.55 ◽

2021 ◽

Vol 105 (563) ◽

pp. 253-262

Author(s):

R. W. D. Nickalls

Keyword(s):

Underlying Structure ◽

Polynomial Roots ◽

Polynomial Degree

This Article explores how root multiplicity and polynomial degree influence the structure of the roots of a univariant polynomial. After setting up the notation, we draw upon a result derived in [1], and show that all polynomial roots have a common underlying structure comprising just five parameters. Finally we present some examples involving the lower polynomials.

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Bounds on polynomial roots using intercyclic companion matrices

Linear Algebra and its Applications ◽

10.1016/j.laa.2017.11.002 ◽

2018 ◽

Vol 539 ◽

pp. 94-116

Author(s):

Kevin N. Vander Meulen ◽

Trevor Vanderwoerd

Keyword(s):

Polynomial Roots ◽

Companion Matrices

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Hyperbolic Polynomials and Convex Analysis

Canadian Journal of Mathematics ◽

10.4153/cjm-2001-020-6 ◽

2001 ◽

Vol 53 (3) ◽

pp. 470-488 ◽

Cited By ~ 28

Author(s):

Heinz H. Bauschke ◽

Osman Güler ◽

Adrian S. Lewis ◽

Hristo S. Sendov

Keyword(s):

Convex Function ◽

Interior Point ◽

Convex Analysis ◽

Symmetric Functions ◽

Interior Point Methods ◽

Optimization Problems ◽

Real Polynomial ◽

Hyperbolic Polynomials ◽

Real Roots ◽

Univariate Polynomial

AbstractA homogeneous real polynomial p is hyperbolic with respect to a given vector d if the univariate polynomial t ⟼ p(x − td) has all real roots for all vectors x. Motivated by partial differential equations, Gårding proved in 1951 that the largest such root is a convex function of x, and showed various ways of constructing new hyperbolic polynomials. We present a powerful new such construction, and use it to generalize Gårding’s result to arbitrary symmetric functions of the roots. Many classical and recent inequalities follow easily. We develop various convex-analytic tools for such symmetric functions, of interest in interior-point methods for optimization problems over related cones.

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Polynomial remainders and plane automorphisms

Bulletin of the Australian Mathematical Society ◽

10.1017/s0004972700037424 ◽

2003 ◽

Vol 68 (1) ◽

pp. 73-79

Author(s):

Takis Sakkalis

Keyword(s):

Jacobian Conjecture ◽

Real Polynomial ◽

Polynomial Automorphisms ◽

Polynomial Map

This note relates polynomial remainders with polynomial automorphisms of the plane. It also formulates a conjecture, equivalent to the famous Jacobian Conjecture. The latter provides an algorithm for checking when a polynomial map is an automorphism. In addition, a criterion is presented for a real polynomial map to be bijective.

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