We propose a novel approach to solve systems of multivariate polynomial equations, using the column space of the Macaulay matrix that is constructed from the coefficients of these polynomials. A multidimensional reali...
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We propose a novel approach to solve systems of multivariate polynomial equations, using the column space of the Macaulay matrix that is constructed from the coefficients of these polynomials. A multidimensional realization problem in the null space of the Macaulay matrix results in an eigenvalue problem, the eigenvalues and eigenvectors of which yield the common roots of the system. Since this null space based algorithm uses well-established numerical linear algebra tools, like the singular value and eigenvalue decomposition, it finds the solutions within machine precision. In this paper, on the other hand, we determine a complementary approach to solve systems of multivariate polynomial equations, which considers the column space of the Macaulay matrix instead of its null space. This approach works directly on the data in the Macaulay matrix, which is sparse and structured. We provide a numerical example to illustrate our new approach and to compare it with the existing null space based root-finding algorithm.
Abstract We consider a dynamic programming approach for solving optimal control problems of sampled continuous-time systems. Robustness to bounded noise and model uncertainties is provided by formulating the problem i...
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Abstract We consider a dynamic programming approach for solving optimal control problems of sampled continuous-time systems. Robustness to bounded noise and model uncertainties is provided by formulating the problem in the framework of differential games. We use a semi-Lagrangian scheme for the computation of the value function and the state feedback control law for constrained infinite horizon optimal control problems. The approach is illustrated with the optimal control of linear systems and nonlinear systems subject to bounded disturbances.
In this paper we discuss the one dimensional heat equation and the wave equation subject to nonlocal conditions. We use the method of Laplace transforms. Finally, we obtain the solution by using a numerical technique ...
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In this paper we discuss the one dimensional heat equation and the wave equation subject to nonlocal conditions. We use the method of Laplace transforms. Finally, we obtain the solution by using a numerical technique for inverting the Laplace transforms.
Controlled measurements of the sound field from a point source above a curved surface are described. The measurements were made in the frequency range between 0.3 and 10 kHz, in the case of a rigid boundary and a surf...
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Controlled measurements of the sound field from a point source above a curved surface are described. The measurements were made in the frequency range between 0.3 and 10 kHz, in the case of a rigid boundary and a surface of finite impedance. Receiver positions include all of the area within, and above, the shadow zone and for various source heights. Particular attention is given to the region across the shadow boundary. The measurements are compared to diffraction theory expressed in terms of a residue series, or creeping wave solution. The calculation is extended by removing restrictive approximations and by carrying the computation to higher‐order terms. A numerical algorithm allows the extension to the general case of a finite impedance. Above the shadow boundary, the sound field is calculated using geometrical theory that accounts for reflections from a curved surface. Deep within the shadow, theory and measurements agree to, typically, 0.5 dB. The same agreement is obtained between measurements and the geometrical theory well above the shadow boundary. In the vicinity of the shadow boundary, both theories agree to within 0.5 dB but differ from the measured results by 2 to 5 dB. Finally, the theory is compared to measurements obtained outdoors above a grass covered curved ground with no refraction and above flat ground with refraction.
The Bezout equation over the rings of proper stable rational functions and matrices is studied in this paper. First, a relationship between the rational Bezout equation and a combined serial/parallel interconnection o...
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The Bezout equation over the rings of proper stable rational functions and matrices is studied in this paper. First, a relationship between the rational Bezout equation and a combined serial/parallel interconnection of linear systems is established. The controllability and observability properties that this scheme has to fulfill in order the Bezout equation to be satisfied yield a numerical procedure for finding a particular solution of the concerned equation. This routine is usable for problems of small-to-medium size as demonstrated by numerical experiments.
Output feedback pole placement problem is not solvable analytically for the plants having either number of inputs or number of outputs, more than two. There is even lack of a good numerical method which solves any gen...
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Output feedback pole placement problem is not solvable analytically for the plants having either number of inputs or number of outputs, more than two. There is even lack of a good numerical method which solves any general pole assignment problem. Due to the multi-linear nature of the pole placement problem there is a possibility to utilize multi-linear structure and arrive at a better numerical solution. This paper shows that it is possible to compute analytically the Jacobian and Hessian matrix in an easy manner and utilize them in an iterative numerical method to solve the pole-placement problem. Newton-Raphson method is used with analytical solution of Jacobian matrix to give an iterative solution of pole placement equations with better percentage success rate than the other methods quoted in literature.
A popular version of optimal control problems is discussed and a nonlinear two point boundary value problem embedded in the optimal control problem is then converted into an initial-value problem according to the boun...
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A popular version of optimal control problems is discussed and a nonlinear two point boundary value problem embedded in the optimal control problem is then converted into an initial-value problem according to the boundary value sensitivity differential equations. An integration method for matrix differential equations is also presented. In this novel way, the exact solutions are obtained for the linear optimal control problems. Two illustrative numerical examples are given to show the power and efficiency.
In this paper we introduce a common problem in electronic measurements and electrical engineering: finding the first root from the left of an equation in the presence of some initial conditions. We present examples of...
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In this paper we introduce a common problem in electronic measurements and electrical engineering: finding the first root from the left of an equation in the presence of some initial conditions. We present examples of electrotechnical devices (analog signal filtering), where it is necessary to solve it. Two new methods for solving this problem, based on global optimization ideas, are introduced. The first uses the exact a priori given global Lipschitz constant for the first derivative. The second method adaptively estimates local Lipschitz constants during the search. Both algorithms either find the first root from the left or determine the global minimizers (in the case when the objective function has no roots). Sufficient conditions for convergence of the new methods to the desired solution are established in both cases. The results of numerical experiments for real problems and a set of test functions are also presented.
作者：
COULOUVRAT, FY229)
Université Pierre et Marie Curie (Paris 6) Tour 66 4 place Jussieu 75252 Paris Cedex 05 France
The propagation of large nonlinear bounded sound beams in inviscid fluids is studied by means of the method of renormalization. Starting from a nonuiform quasilinear expansion of a solution of the Khokhlov-Zabolotskay...
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The propagation of large nonlinear bounded sound beams in inviscid fluids is studied by means of the method of renormalization. Starting from a nonuiform quasilinear expansion of a solution of the Khokhlov-Zabolotskaya equation for a Gaussian source, a straining of the retarded time is introduced, which leads to a uniform approximation. The main point is the choice of a nonlinear phase shift, which yields a single smooth continuous representation for the wave, both before and beyond the shock formation point. A computation of the harmonics shows the asymmetrical distortion of the wave profile due to the coupling between nonlinearity and diffraction. A comparison with the results of a finite-difference numerical algorithm turns out favorable. The method is then extended to general plane sources.
Rattlebacks are semi-ellipsoidal tops that have a preferred direction of spin (i.e., a spin-bias). If spun in one direction, the rattleback will exhibit seemingly stable rotary motion. If spun in the other direction, ...
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Rattlebacks are semi-ellipsoidal tops that have a preferred direction of spin (i.e., a spin-bias). If spun in one direction, the rattleback will exhibit seemingly stable rotary motion. If spun in the other direction, the rattleback will being to wobble and subsequently reverse its spin direction. This behavior is often counter-intuitive for physics and engineering students when they first encounter a rattleback, because it appears to oppose the laws of conservation of momenta, thus this simple toy can be a motivator for further study. This paper develops an accurate dynamic model of a rattleback, in a manner accessible to undergraduate physics and engineering students, using concepts from introductory dynamics, calculus, and numerical methods classes. Starting with a simpler, 2D planar rocking semi-ellipse example, we discuss all necessary steps in detail, including computing the mass moment of inertia tensor, choice of reference frame, conservation of momenta equations, application of kinematic constraints, and accounting for slip via a Coulomb friction model. Basic numerical techniques like numerical derivatives and time-stepping algorithms are employed to predict the temporal response of the system. We also present a simple and intuitive explanation for the mechanism causing the spin-bias of the rattleback. It requires no equations and only a basic understanding of particle dynamics, and thus can be used to explain the intriguing rattleback behavior to students at any level of expertise.
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