Simultaneous algebraic equations

Consider the balance in the shallow Atlantic reservoir. The flux of phosphorus is 

The equations for the remaining two reservoirs, Antarctic and deep Atlantic, are 

The complete system consists of five linear algebraic equations in the five unknown concentrations. The order of the terms in these equations can be rearranged so that the first term in each equation is a number (which may be 0) times sat, the second is a number times sind, the third is a number times dind, the fourth is a number times ant, the fifth is a number times dat, and the constant terms, which correspond in this system to the river-borne source, appear on the right-hand sides of the equations. After this rearrangement, the equations become 

For convenience in notation and computation this system can be represented by an array of coefficients, which I call sleq  

The solution begins with a method called Gaussian elimination that converts the elements on the diagonal to 1 and the elements below the 

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To describe the basic concept of the Gaussian elimination method we consider the following system of simultaneous algebraic equations... 

To estimate the computational time required in a Gaussian elimination procedure we need to evaluate the number of arithmetic operations during the forward reduction and back substitution processes. Obviously multiplication and division take much longer time than addition and subtraction and hence the total time required for the latter operations, especially in large systems of equations, is relatively small and can be ignored. Let us consider a system of simultaneous algebraic equations, the representative calculation for forward reduction at stage is expressed as... 

Then, the overall problem is determinate and reduces to the solution of the following set of simultaneous algebraic equations for A, B, and C ... 

The application of the reverse Euler method of solution to a system of coupled differential equations yields a system of coupled algebraic equations that can be solved by the method of Gaussian elimination and back substitution. In this chapter I demonstrated the solution of simultaneous algebraic equations by means of this method and showed how the solution of algebraic equations can be used to solve the related differential equations. In the process, I presented subroutine GAUSS, the computational engine of all of the programs discussed in the chapters that follow. 

The Determinant Method. For a mechanism involving several enzyme-containing species, derivation of the rate equation can be done by solving the simultaneous algebraic equations by the determinant method. Consider the mechanism described by Eq. (1) with the addition of an EP intermediate. 

Note how much simpler this problem is to solve in the CSTR than in the PFTR (Figure 4—6), where we had to solve simultaneous differential equations. The CSTR involves only simultaneous algebraic equations so we just need to find roots of polynomials.]... 

In this text all numerical problems involve integration of simultaneous ordinary differential equations or solution of simultaneous algebraic equations. You should have no trouble finding ways to solve algebraic equations with a calculator, a spreadsheet, a personal computer, etc. 

The method of lines is called an explicit method because the new value T(r, z + Az) is given as an explicit function of the old values T(r, z),T(r — Ar, z),. See, for example, Equation (8.57). This explicit scheme is obtained by using a first-order, forward difference approximation for the axial derivative. See, for example, Equation (8.16). Other approximations for dTjdz are given in Appendix 8.2. These usually give rise to implicit methods where T(r,z Az) is not found directly but is given as one member of a set of simultaneous algebraic equations. The simplest implicit scheme is known as backward differencing and is based on a first-order, backward difference approximation for dT/dz. Instead of Equation (8.57), we obtain... 

The solution, based on the variational method, leads to a set of simultaneous algebraic equations, like what was found in Eq. (1-26) for diatomic molecules ... 

Many real-world applications of chemistry and biochemistry involve fairly complex sets of reactions occurring in sequence and/or in parallel. Each of these individual reactions is governed by its own equilibrium constant. How do we describe the overall progress of the entire coupled set of reactions We write all the involved equilibrium expressions and treat them as a set of simultaneous algebraic equations, because the concentrations of various chemical species appear in several expressions in the set. Examination of relative values of equilibrium constants shows that some reactions dominate the overall coupled set of reactions, and this chemical insight enables mathematical simplifications in the simultaneous equations. We study coupled equilibria in considerable detail in Chapter 15 on acid-base equilibrium. Here, we provide a brief introduction to this topic in the context of an important biochemical reaction. 

Mathematica can carry out both symbolic and numerical solutions of equations, including single algebraic equations, simultaneous algebraic equations, and differential equations, which we discuss later. Mathematica contains the rules needed for the symbolic solution of polynomial equations up to the fourth degree, and can... 

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