The Solution of Simultaneous Algebraic Equations

If there are two variables in an equation, such as F x, y) — 0, then the equation can be solved for y as a function of v or v as a function of y, but in order to solve for constant values of both variables, a second equation, such as G(x, 3 ) = 0, is required, and the two equations must be solved simultaneously. If there are n variables, n independent and consistent equations are required. In this chapter, we discuss various methods for finding the roots to sets of simultaneous equations. 

To solve for n variables, n equations are required, and these equations must be independent and consistent. 

Simultaneous linear inhomogeneous equations can be solved with various techniques, including elimination, use of Cramer s formula, and by matrix inversion. 

Linear homogeneous simultaneous equations have a nontrivial solution only when a certain dependence condition is met. 

After studying this chapter, you should be able to  

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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 Solution of Simultaneous Algebraic Equations with More than Two Unknowns... 

Under the first assumption, each electron moves as an independent particle and is described by a one-electron orbital similar to those of the hydrogen atom. The wave function for the atom then becomes a product of these one-electron orbitals, which we denote P (r,). For example, the wave function for lithium (Li) has the form i/ atom = Pa ri) Pp r2) Py r3). This product form is called the orbital approximation for atoms. The second and third assumptions in effect convert the exact Schrodinger equation for the atom into a set of simultaneous equations for the unknown effective field and the unknown one-electron orbitals. These equations must be solved by iteration until a self-consistent solution is obtained. (In spirit, this approach is identical to the solution of complicated algebraic equations by the method of iteration described in Appendix C.) Like any other method for solving the Schrodinger equation, Hartree s method produces two principal results energy levels and orbitals. 

Stagewise calculations require the simultaneous solution of material and energy balances with equilibrium relationships. It was demonstrated in Example 1.1 that the design of a simple extraction system reduces to the solution of linear algebraic equations if (1) no energy balances are needed and (2) the equilibrium relationship is linear. 

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 determination of the steam density, pj, therefore requires the simultaneous solution of two algebraic equations. This represents an IMPLICIT algebraic loop and cannot be solved within a simulation program without the incorporation of a trial and error convergence procedure. 

The formal, algebraic, method. The presence of recycle implies that some of the mass balance equations will have to be solved simultaneously. The equations are set up with the recycle flows as unknowns and solved using standard methods for the solution of simultaneous equations. 

One of the most common problems in digital simulation is the solution of simultaneous nonlinear algebraic equations. If these equations contain transcendental functions, analytical solutions are impossible. Therefore, an iterative trial-and-error procedure of some sort must be devised. If there is only one unknown, a value for the solution is guessed. It is plugged into the equation or equations to see if it satisfies them. If not, a new guess is made and the whole process is repeated until the iteration eonverges (we hope) to the right value. 

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 ... 

Thus, the required %v/v propylene glycol is (27.1/ 46.5) X 100 = 58.3. Alternatively, the %v/v of the new cosolvent can be solved using an algebraic method involving the solution of simultaneous equations however, aligation is a simpler method when more than one cosolvent is to be included in the formulation. When a vehicle is to be formulated for the first time, it is necessary to experimentally determine the concentration of some cosolvent necessary to maintain the required concentration of drug in solution. This value can then be used to calculate the ADR and the final vehicle calculated as illustrated previously. 

Gimbun, j., Nagy, Z. K. Rielly, C. D. 2009 Simultaneous quadrature method of moments for the solution of population balance equations, using a differential algebraic equation framework. Industrial U Engineering Chemistry Research 48, 7798-7812. 

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