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Nonrecursive solution for the discrete algebraic Riccati equation and X + A*X -1 A=L

Maria Adam, Nicholas Assimakis (2015)

Open Mathematics

In this paper, we present two new algebraic algorithms for the solution of the discrete algebraic Riccati equation. The first algorithm requires the nonsingularity of the transition matrix and is based on the solution of a standard eigenvalue problem for a new symplectic matrix; the proposed algorithm computes the extreme solutions of the discrete algebraic Riccati equation. The second algorithm solves the Riccati equation without the assumption of the nonsingularity of the transition matrix; the...

Nonsingularity and P -matrices.

Jiří Rohn (1990)

Aplikace matematiky

New proofs of two previously published theorems relating nonsingularity of interval matrices to P -matrices are given.

Nonsingularity, positive definiteness, and positive invertibility under fixed-point data rounding

Jiří Rohn (2007)

Applications of Mathematics

For a real square matrix A and an integer d ≥ 0 , let A ( d ) denote the matrix formed from A by rounding off all its coefficients to d decimal places. The main problem handled in this paper is the following: assuming that A ( d ) has some property, under what additional condition(s) can we be sure that the original matrix A possesses the same property? Three properties are investigated: nonsingularity, positive definiteness, and positive invertibility. In all three cases it is shown that there exists a real number...

Nonsmooth equations approach to a constrained minimax problem

Yan Gao, Xuewen Li (2005)

Applications of Mathematics

An equivalent model of nonsmooth equations for a constrained minimax problem is derived by using a KKT optimality condition. The Newton method is applied to solving this system of nonsmooth equations. To perform the Newton method, the computation of an element of the b -differential for the corresponding function is developed.

Nonstandard Finite Difference Schemes with Application to Finance: Option Pricing

Milev, Mariyan, Tagliani, Aldo (2010)

Serdica Mathematical Journal

2000 Mathematics Subject Classification: 65M06, 65M12.The paper is devoted to pricing options characterized by discontinuities in the initial conditions of the respective Black-Scholes partial differential equation. Finite difference schemes are examined to highlight how discontinuities can generate numerical drawbacks such as spurious oscillations. We analyze the drawbacks of the Crank-Nicolson scheme that is most frequently used numerical method in Finance because of its second order accuracy....

Non-uniqueness of almost unidirectional inviscid compressible flow

Pavel Šolín, Karel Segeth (2004)

Applications of Mathematics

Our aim is to find roots of the non-unique behavior of gases which can be observed in certain axisymmetric nozzle geometries under special flow regimes. For this purpose, we use several versions of the compressible Euler equations. We show that the main reason for the non-uniqueness is hidden in the energy decomposition into its internal and kinetic parts, and their complementary behavior. It turns out that, at least for inviscid compressible flows, a bifurcation can occur only at flow regimes with...

Nonuniqueness of implicit lattice Nagumo equation

Petr Stehlík, Jonáš Volek (2019)

Applications of Mathematics

We consider the implicit discretization of Nagumo equation on finite lattices and show that its variational formulation corresponds in various parameter settings to convex, mountain-pass or saddle-point geometries. Consequently, we are able to derive conditions under which the implicit discretization yields multiple solutions. Interestingly, for certain parameters we show nonuniqueness for arbitrarily small discretization steps. Finally, we provide a simple example showing that the nonuniqueness...

Normal bivariate Birkhoff interpolation schemes and Pell equation

Marius Crainic, Nicolae Crainic (2009)

Commentationes Mathematicae Universitatis Carolinae

Finding the normal Birkhoff interpolation schemes where the interpolation space and the set of derivatives both have a given regular “shape” often amounts to number-theoretic equations. In this paper we discuss the relevance of the Pell equation to the normality of bivariate schemes for different types of “shapes”. In particular, when looking at triangular shapes, we will see that the conjecture in Lorentz R.A., Multivariate Birkhoff Interpolation, Lecture Notes in Mathematics, 1516, Springer, Berlin-Heidelberg,...

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