Programming techniques

Here we describe numerical techniques for the MD method that have been mentioned in the literature and others that we have found useful in our own work. Unfortunately the applicability and desirability of many techniques depend upon the computer hardware in question and different techniques are required to take advantage of new types of hardware. 

Hie simplicity of the trajectory calculation for hard-core particles permits an extremely simple (albeit inefficient) MD program. We begin by describing such a program so as to establish a point of departure for discussing more efficient programs. 

In the discussion that follows we use the configuration vector r (t) (in which the N particles remain in the primary cell) and the velocity vector v (r). In actual fact it is usually more convenient to use velocities v relative to the center of mass. The distinction is of no consequence here. In computing (r), however, it must be remembered that there are applications in which the checkerboard coordinates are required, e.g., the calculation of the mean square displacement A (r), Eq. (55). In such cases, provision must be made to obtain R (f)- One could equally well compute R (t), without appreciable change in the numerical methods. 

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Grossmann, I. E., Mixed-Integer Non-Linear Programming Techniques for the Synthesis of Engineering Systems, Res. Eng. Design, 1 205, 1990. 

Freeman, J. A., Skapura, D. M. Neural Networks Algorithms, Applications and Programming Techniques, Computation and Neural systems Series. Addison Wesley Publishing Company, 1991... 

Dynamic programming Technique widely used in sequence alignment... 

A. M. Morshedi, C. R. Cutier, and T. A. Skrovanek, "Optimal Solution of Dynamic Matrix Control with Linear Programming Techniques,"... 

If the source fingerprints, for each of n sources are known and the number of sources is less than or equal to the number of measured species (n < m), an estimate for the solution to the system of equations (3) can be obtained. If m > n, then the set of equations is overdetermined, and least-squares or linear programming techniques are used to solve for L. This is the basis of the chemical mass balance (CMB) method (20,21). If each source emits a particular species unique to it, then a very simple tracer technique can be used (5). Examples of commonly used tracers are lead and bromine from mobile sources, nickel from fuel oil, and sodium from sea salt. The condition that each source have a unique tracer species is not often met in practice. 

One important class of nonlinear programming techniques is called quadratic programming (QP), where the objective function is quadratic and the constraints are hnear. While the solution is iterative, it can be obtained qmckly as in linear programming. This is the basis for the newest type of constrained multivariable control algorithms called model predic tive control. The dominant method used in the refining industiy utilizes the solution of a QP and is called dynamic matrix con-... 

Mathematical modeling, using digital computers, aids in performing a systems-type analysis for either the entire system or parts of it. By means of integer or linear-programming techniques, optimum systems can be identified. The dynamic performance of these can then be determined by simulation techniques. 

Stanley, C., and El-Halw, M. M. (1995). Synthesis of mass exchange networks using linear programming techniques. In Waste Minimization nirough Process Design (A. P. Rossiter, ed.), pp. 209-224. McGraw Hill, New York. 

The treatment in this chapter has been theoretical. For a brief, dear, and very practical description of computational details for a number of standard problems, [10] is unsurpassed, and [12] can be recommended for programming techniques for automatic computers. For information on ordinary differential equations, the reader should consult [2], and for partial differential equations, [1]. For general methods of reduction to algebraic form as well as methods of solution, see [5], [7], and [8]. 

Another illustration of operations research is the use of linear programming techniques (Section 5.14) to obtain optimal mixtures of gasoline ingredients that will produce a result suitable for different climatic conditions and subject to demand constraints on a long-range basis. 

Applications of nested summation symbols to Quantum Chemistry formalism and programming techniques R. Carbo and E. Besalu... 

J.A. Freeman and D.M. Skapura, Neural Networks, Algorithms, Applications and Programming Techniques. Addison-Wesley, Reading, MA, 1991. 

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