Algebra with Mathematica

As you have seen, matrix algebra can be tedious. Mathematica has all of the matrix operations built into it, so that you can form matrix products and carry out matrix inversion automatically. Mathematica treats matrices as lists of lists, with the elements of each row entered as a list. A list is entered inside curly brackets ( braces ) with the elements separated by commas. A list of lists requires braces around the set of lists with braces and commas. For example, to enter the following 3 by 3 matrix 

Note that the symbol that we chose for the matrix name is in lower case and requires no auxiliary labels. You should start the names of all Mathematica variables with lowercase letters to avoid possible confusion with Mathematica operators and functions. If you want to see the matrix A in standard form, type the statement MatrixForm[a] and press the Enter key or the Shift-Return.  

Mathematica treats vectors as a single list. It does not distinguish between row vectors and column vectors. If you want to enter a vector v = (2, 4, 6), you enter the components inside curly brackets separated by commas as follows v= 2, 4, 6  

After two matrices A and B have been entered, a matrix multiplication is carried 

The inverse of a matrix is obtained with the statement Inverse[a] 

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Although matrix multiplications, row reductions, and calculation of null spaces can be done by hand for small matrices, a computer with programs for linear algebra are needed for large matrices. Mathematica is very convenient for this purpose. More information about the operations of linear algebra can be obtained from textbooks (Strang, 1988), but this section provides a brief introduction to making calculations with Mathematica (Wolfram, 1999). 

Kirkwood and Buff [15] obtained expressions for those quantities in compact matrix forms. For binary mixtures, Kirkwood and Buff provided the results listed in Appendix A. Starting from the matrix form and employing the algebraic software Mathematica [16], analytical expressions for the partial molar volumes, the isothermal compressibility and the derivatives of the chemical potentials for ternary mixtures were obtained by us. They are listed in Appendix B together with the expressions at infinite dilution for the partial molar volumes and isothermal compressibility. 

Computer algebra) Using Mathematica, Maple, or some other computer algebra package, apply the Poincare-Lindstedt method to the problem X + X - ex = 0, with x(0) = a, and x(0) = 0. Find the frequency O) of periodic solutions, up to and including the ( ( ) term. 

For this three-dimensional electrode model, with or without decoupling of the two coupled equations, the approximate solutions can be obtained by using the Mathematica codes of the ADM given in the Appendix.17 The algebraic expressions of dimensionless potential and concentration are in a series form with even orders as... 

Throughout this book, we have seen that when more than one species is involved in a process or when energy balances are required, several balance equations must be derived and solved simultaneously. For steady-state systems the equations are algebraic, but when the systems are transient, simultaneous differential equations must be solved. For the simplest systems, analytical solutions may be obtained by hand, but more commonly numerical solutions are required. Software packages that solve general systems of ordinary differential equations— such as Mathematica , Maple , Matlab , TK-Solver , Polymath , and EZ-Solve —are readily obtained for most computers. Other software packages have been designed specifically to simulate transient chemical processes. Some of these dynamic process simulators run in conjunction with the steady-state flowsheet simulators mentioned in Chapter 10 (e.g.. SPEEDUP, which runs with Aspen Plus, and a dynamic component of HYSYS ) and so have access to physical property databases and thermodynamic correlations. 

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. 

In Chapter 3, we introduced the use of Mathematica to solve a single algebraic equation, using the Solve statement and the NSolve statement. The Solve statement can also be used to solve simultaneous equations. The equations are typed inside curly brackets with commas between them, and the variables are listed inside curly brackets. To solve the equations ax + by = c gx + hy = k we type the input entry... 

There is much more that can be done to manipulate equations and their solutions. For example, there is a set of commands for doing algebra that mimics what we do by hand (Expand, Factor, Simplify, FullSimplify, PowerExpand...). We observed here that we obtained imaginary roots to these equations. If our problems demand only real roots, then we can have Mathematica filter out the imaginaries and return just the real roots (Miscellaneous RealOnly ). But we should not get too far ahead of ourselves. It is better that we learn Mathematica in natural stages that follow our level of need. In other words, we will find and introduce more sophisticated commands, routines, and procedures as we need them, so that their function is understood and retained, rather than trying to cover everything at once. With this in mind let us turn now to some Calculus functions. 

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]. 

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 . 

Four different Fock/Kohn-Sham operators have been applied to obtain the orbitals, which are subsequently localized by the standard Foster-Boys procedure. In addition to the local/semi-local functionals LDA and PBE, the range-separated hybrid RSHLDA [37, 56] with a range-separation parameter of /r = 0.5 a.u. as well as the standard restricted Hartree-Fock (RHF) method were used. The notations LDA[M] and LDA[0] refer to the procedure applied to obtain the matrix elanents either by the matrix algebra [M] or by the operator algebra [O] method. All calculations were done with the aug-cc-pVTZ basis set, using the MOLPRO quantum chemical program package [57]. The matrix elements were obtained by the MATROP facility of MOLPRO [57] the Cg coefficients were calculated by Mathematica. 

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