Symbolic, algebraic computation

While analytic linearization is not recommended as appropriate in every case, it is a useful additional tool for critical situations. Its use on the difficult problems likely to be met in practice is being made easier now as a result of the availability of standard computer packages for symbolic, algebraic computation. 

Oonpiter algebra. Symbolic and Algebraic Computation, Edited by B.Buchberger et al, Sprlnger-Verlag, Wien, New York, 1982. 

Wilkes also remarked that Boys tried to explain to him a program that he had written to do symbolic algebra, and Wilkes believed that it was perhaps the first time that a computer had been used for algebraic manipulation. But nothing was ever published on it so there is no way of knowing precisely what was involved. 

Prelie M.J., Proceedings of the 1981 ACM Symposium of Symbolic and Algebraic Computation, Snowbird Utah, 30,1981. 

With the publication of A Symbolic Analysis of Relay and Svntching Circuits (1940) and A Mathematical Theory of Communication (1948), American mathematician Claude Elwood Shannon introduced a new area for the application of Boolean algebra. He showed that the basic properties of series and parallel combinations of electric devices such as relays could be adequately represented by this symbolic algebra. Since then. Boolean algebra has played a significant role in computer science and technology. 

Computer science and algebra The symbolic system of mathematical logic called Boolean algebra represents relationships between entities either ideas or objects. George Boole of England formulated the basic rules of the system in 1847. The Boolean algebra eventually became a cornerstone of computer science. 

The user can explore extremely complex problems that cannot be solved in any other manner. This capability is often thought of as the major use of Computer Algebra systems. However, one should not lose sight of the fact that MACSYMA is often used as an advanced calculator to perform everyday symbolic and numeric problems. It also complements conventional tools such as reference tables or numeric processors. 

The second section, on computer algebra, details chemical applications whose emphasis is on the mathematical nature of chemistry. As chemical theories become increasingly complex, the mathematical equations have become more difficult to apply. Symbolic processing simplifies the construction of mathematical descriptions of chemical phenomena and helps chemists apply numerical techniques to simulate chemical systems. Not only does computer algebra help with complex equations, but the techniques can also help students learn how to manipulate mathematical structures. 

Sedoglavic, A., A probabilistic algorithm to test algebraic observability in polynomial time, J. Symbolic Computation 2002, 33 735-755. 

BASIC, or Beginners Algebraic Symbolic Instruction Code, was developed by Kemeny63 as a "baby FORTRAN" for simple computers (e.g., minicomputers). BASIC does not wait for the whole user-written program to be finished, but compiled each typed line as soon as typed. It was ideally suited for a simple learning environment. Microsoft VISUAL BASIC is a GUI-interfaced version. Microsoft QUICK BASIC 4.5 is much better than FORTRAN embodiments in accessing instruments for real-time data acquisition and control. 

The 3 -j symbols have explicit functional forms dependent on their arguments. We give the simplest of these formulae in General Appendix C at the end of this book. It should also be appreciated that expressions for particular symbols can be derived algebraically by computer software [12] in less time than it takes to look them up in a table. 

In order to understand formal relationships the basic coupling of two angular momenta is expressed in terms of the Clebsch—Gordan coefficients. In setting up the algebra for computation it is more convenient to use the 3-j symbols. Calculations of the Wigner symbols and the Clebsch—Gordan coefficients are found in subroutine libraries. See, for example, Soper (1989). 

In ref. 142 the authors are studied the Numerov-type ODE solvers for the numerical solution of second-order initial value problems. They present a powerful and efficient symbolic code in MATHEMATICA for the derivation of their order conditions and principal truncation error terms. They also present the relative tree theory for such order conditions along with the elements of combinatorial mathematics, partitions of integer numbers and computer algebra which are the basis of the implementation of the S5unbolic code. We must that one of the authors is an expert on this specific field. 

Warning This calculation involves massive amounts of algebra, but if you do it correctly, you ll be rewarded by seeing many wonderful cancellations. Teach yourself Mathematica, Maple, or some other symbolic manipulation language, and do the problem on the computer.)... 

Analytical solutions and correlation equations are presented rather than graphic results. The availability of many computer algebra systems such as Macsyma, MathCad, Maple, MAT-LAB, and Mathematica, as well as spreadsheets such as Excel and Quattro Pro that provide symbolic, numerical, and plotting capabilities, makes the analytical solutions amenable to quick, accurate computations. All equations and correlations reported in this chapter have been verified in Maple worksheets and Mathematica notebooks. These worksheets and notebooks will be available on my home page on the Internet. Some spreadsheet solutions will also be developed and made available on the Internet. ... 

In order to solve a set of simultaneous equations, there must be the same number of equations as there are independent variables. Quite an analogous thing occurs with the simultaneous equations in ordinary algebra. The methods used for the solution of these equations are analogous to those employed for similar equations in algebra. The operations here involved are chiefly processes of elimination and substitution, supplemented by differentiation or integration at various stages of the computation. The use of the symbol of operation D often shortens the work. 

In a similar fashion one can proceed to the second order. However, the algebra becomes rather cumbersome and we performed the symbolic computations with the program Mathematica [19]. 

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