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Über dieses Buch

Newtonian Nonlinear Dynamics for Complex Linear and Optimization Problems explores how Newton's equation for the motion of one particle in classical mechanics combined with finite difference methods allows creation of a mechanical scenario to solve basic problems in linear algebra and programming. The authors present a novel, unified numerical and mechanical approach and an important analysis method of optimization.



Chapter 1. Elements of Newtonian Mechanics

Within the framework of Newtonian Mechanics, the dynamics of the motion of a single particle is related to the following properties: (1)

the existence of either conservation or variation laws for some quantities according to certain relations

, (2)

the existence of symmetries

and (3)

the stability character of the equilibrium points of the system

. Only in a few cases the analytical solution of the equations of motion is known and, for this reason, numerical simulations are usually necessary.

Luis Vázquez, Salvador Jiménez

Chapter 2. Solution of Systems of Linear Equations

Solving systems of linear equations (or

linear systems

or, also,

simultaneous equations

) is a common situation in many scientific and technological problems. Many methods, either analytical or numerical, have been developed to solve them. A general method most used in Linear Algebra is the Gaussian Elimination, or variations of this. Sometimes they are referred to as “direct methods”. Basically, it is an algorithm that transforms the system into an equivalent one but with a triangular matrix, thus allowing a simpler resolution. In many cases, though, whenever the matrix of the system has a specific structure or is sparse and the like, other methods can be more effective.

Luis Vázquez, Salvador Jiménez

Chapter 3. Solution of Systems of Linear Equations: Numerical Simulations

To check the usefulness of this method, we shall compare it with the simplest and well known iterative methods: Jacobi, Gauss-Seidel, and Steepest Descent [11, 14]. We shall do this through some examples but, first, let us recall how this other methods work.

Luis Vázquez, Salvador Jiménez

Chapter 4. Eigenvalue Problems

Eigenvalue problems are at the base of many scientific and technological issues. They appear at the root of stability problems, differential equations, either ordinary or partial, Mechanics of continuous media, etc.

Luis Vázquez, Salvador Jiménez

Chapter 5. Eigenvalue Problems: Numerical Simulations

In this chapter a numerical scheme is proposed to simulate the evolution of the solution


of dynamical system (4.1) towards an eigenvector of a given matrix, and some examples and applications are presented. The method has a linear convergence rate and we have implemented two potentially second order methods to be combined with the first one to accelerate the convergence.

Luis Vázquez, Salvador Jiménez

Chapter 6. Linear Programming

We may consider the pinball machine as a mechanical device to visualize the minimization of the potential energy of a ball moving in a bounded inclined plane. The potential energy, in this case, is a linear function of the space coordinates. On the other hand, the boundaries of the inclined plane region, where the ball is moving, are represented by a set of inequalities which define the convex region where the motion is possible. The inequalities are linear if the boundary is made of linear segments, while they are nonlinear if the boundary is a piecewise combination of other kinds of curves.

Luis Vázquez, Salvador Jiménez

Chapter 7. Quadratic Programming

In Chap. 6, we associated the minimization of a linear functional with linear constraints to the motion of a Newtonian particle in a constant gravitational field in a bounded region with the frontier made of straight segments. Now, we can extend this mechanical picture to visualize the minimization of a quadratic functional with constraints which can be either linear or nonlinear. Mechanically, the solution is associated to the motion of a Newtonian particle in a quadratic potential with damping and with the associated geometrical constraints. In many cases the analytical estimations are available and we do not need to resort to the numerical simulations. Also, this is the picture in the case of the minimization of a nonlinear functional

Luis Vázquez, Salvador Jiménez


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