Computer algebra system

My purpose in this paper is to provide a broad introduction to the capabilities of MACSYMA It is my hope that this information will create new users of Computer Algebra systems by showing what one might expect to gain by using them and what one will lose by not using them ... 

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. 

Notice that KACSYKA has obtained the roots analytically and that numeric approximations have not been made. This demonstrates a fundamental difference between a Computer Algebra system and an ordinary numeric equation solver, namely the ability to obtain a solution without approximations. 1 could have given KACSYKA a "numeric" cubic equation in X by specifying numeric values for A and B. KACSYKA then would have solved the equation and given the numeric roots approximately or exactly depending upon the specified command. 

Note that standard procedures, such as Groebner Bases, that are built into the modern computer algebra systems (e.g. Maple) cannot handle our systems efficiently (see monograph by Bykov et al., 1998). 

All computations used the GAP computer algebra system [GAP02] and the package PlanGraph ([Dut02]) by the second author the programs are available from [Du07]. 

Mag07] The MAGMA Group, Magma Computer Algebra System, Sydney University, 2007. 

The use of computer algebra system is essential to manipulate the permissible functions. 

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. ... 

The total surface area related to the semimajor axis is a function of the two aspect ratios (3 and y. The special functions F(ty, k) and E(ty, k) are incomplete elliptical integrals of the first and second kind, respectively. They depend on the amplitude angle <[> and the modulus k. These special functions can be computed quickly and accurately by means of computer algebra systems such as Mathematica [153]. Their properties are given in Abramowitz and Stegun [1], The relationship between the square root of the total surface area and the semimajor axis is [150] ... 

Transient conduction internal and external to various bodies subjected to the boundary conditions of the (1) first kind (Dirichlet), (2) second kind (Neumann), and (3) third kind (Robin) are presented in this section. Analytical solutions are presented in the form of series or integrals. Since these analytical solutions can be computed quickly and accurately using computer algebra systems, the solutions are not presented in graphic form. 

For Fo < Foc, additional terms in the series solutions must be included. It is therefore necessary to use numerical methods to compute the higher-order eigenvalues 8 that lie in the intervals nn < 8 < (n + 1/2)ji for the plate and (n -1 )rt < 8 < nn for the cylinder and the sphere. Computer algebra systems are very effective in computing the eigenvalues. 

Spreading (constriction) resistance is an important thermal parameter that depends on several factors such as (1) geometry (singly or doubly connected areas, shape, aspect ratio), (2) domain (half-space, flux tube), (3) boundary condition (Dirchlet, Neumann, Robin), and (4) time (steady-state, transient). The results are presented in the form of infinite series and integrals that can be computed quickly and accurately by means of computer algebra systems. Accurate correlation equations are also provided. 

The dimensionless constriction is based on the substrate thermal conductivity k2. The preceding general solution is valid for conductive layers where kllk2 > 1 as well as for resistive layers where kjk2 < 1. The infinite integral can be evaluated numerically by means of computer algebra systems, which provide accurate results. 

Gap Conductance Correlation Equations. Although the gap integral can be computed accurately and easily by means of computer algebra systems, Negus and Yovanovich [73] proposed the following correlation equations for the gap integral ... 

The method, considered in this paper, permits to solve a uniqueness problem of an IP and it also transforms the initial model to a form convenient for tlae estimation of functional combinations. The algorittim has a simple realization in computer algebra systems (such as, e.g. REDUCE [83). 

To solve these problems the resultant method [10] or the Groebner basis method [91 can be used. It appears that the second method is more convenient as far as it has numerical realization in several computer algebraic systems. 

For Problems 4.30-4.38, use either a program similar to that in Table 4.1, a spreadsheet, or a computer-algebra system such as Mathcad. 

Computer algebra systems such as Mathcad and some electronic calculators have built-in commands to easily find eigenvalues and eigenvectors. 

A4 Use a computer-algebra system to find the eigenvalues and normalized eigenvectors of the square matrix B of order six with elements by = + k )/ j + k). 

The Routh-Hurwitz conditions are well known and can be used to determine, in principle, the stability properties of the steady state of any n-variable system. This advantage is, however, balanced by the fact that in practice their use is very cumbersome, even for n as small as 3 or 4. The evaluation, by hand, of all the coefficients Cl of the characteristic polynomial and the Hurwitz determinants A constitutes a rather arduous task. It is for this reason that in the past this tool of linear stability analysis could hardly be found in the literature of nonlinear dynamics. The situation changed with the advent of computer-algebra systems or symbolic computation software. Software such as Mathematica (Wolfram Research, Inc., Champaign, IL) or Maple (Waterloo Maple Inc., Waterloo, Ontario) makes it easy to obtain exact, analytical expressions for the coefficients C/ of the characteristic polynomial (1.12) and the Hurwitz determinants A . 

A parameter sensitivity of a transfer function out of the multiple possible ones of a linear MIMO model is obtained in symbolic form by multiplication of appropriate matrix entries. This can be performed by computer algebra systems. 

---