Simultaneous linear equations, solving

Equation (2.45) represents the weighted residual statement of the original differential equation. Theoretically, this equation provides a system of m simultaneous linear equations, with coefficients Q , i = 1,... m, as unknowns, that can be solved to obtain the unknown coefficients in Equation (2.41). Therefore, the required approximation (i.e. the discrete solution) of the field variable becomes detemfined. 

Because of the work involved in solving large systems of simultaneous linear equations it is desirable that only a small number of us be computed. Thus the gaussian integration formulas are useful because of the economy they offer. See references on numerical solutions of integral equations. 

The known data are substituted and the three linear equations solved simultaneously. 

Note the symmetry of the summation terms in xtj and that numbering of x s in the summations corresponds to matrix indices (rows, columns). This set of p equations in p unknowns can be solved on a computer using one of the many readily available routines for solving simultaneous linear equations. 

Solving Simultaneous Linear Equations with Excel Matrix Operations ... 

Figure 19-5 Solving simultaneous linear equations with Excel.

Balancing chemical reactions is an application of solving multiple simultaneous linear equations. Consider, for example, the complete combustion of one mole of methane to produce carbon dioxide and water ... 

This is just a pair of simultaneous linear equations. To solve for E we set the determinant (product of diagonal terms minus product of antidiagonal terms) to zero ... 

Proceeding similarly with the remaining relations one obtains a set of linearized equilibrium conditions which together with the normalization and consistency conditions yield 11 simultaneous linear equations in the APj mK These may be solved by machine computation to obtain AP/P°. The resulting increments are added to their corresponding zero-order quantities to yield a set of first-order results which replace the Pj m o). The process of solving the linearized... 

Equations (12), constituting a set of n simultaneous linear equations in n unknowns, are known as the normal equations. They can be solved to yield a set of equations in the form... 

A determinant is simply a square matrix. There is a procedure for the numerical evaluation of a determinant, so that anN xN matrix can be reduced to a single numerical value. The value of the determinant has properties that make it useful in certain tests and equations. (See, for example, "Solving Sets of Simultaneous Linear Equations" in Chapter 10.)... 

A set of simultaneous linear equations can also be solved by using matrices, as shown in Chapter 9. The solution matrix is obtained by multiplying the matrix of constants by the inverse of the matrix of coefficients. Applying this simple solution to the spectrophotometric data used above, the inverted matrix is obtained by selecting a 3R x 3C array of cells, entering the array formula... 

SimultEqns.xls illustrates ways to solve sets of simultaneous linear equations by using matrices. 

Numerical methods include those based on finite difference calculus. They are ideally suited for tabulated experimental data such as one finds in thermodynamic tables. They also include methods of solving simultaneous linear equations, curve fitting, numerical solution of ordinary and partial differential equations and matrix operations. In this appendix, numerical interpolation, integration, and differentiation are considered. Information about the other topics is available in monographs by Hornbeck [2] and Lanczos [3]. 

Recognize how determinants can be used to solve simultaneous linear equations... 

We begin our discussion of linear systems by introducing the determinant as a tool for solving sets of simultaneous linear equations in which the indices of the unknown variables are all unity. Consider the pair of equations ... 

If we now measure the rate constant at two different temperatures, T and T2, we obtain a pair of simultaneous linear equations which we can solve for the two unknowns, and In A ... 

The application of matrix algebra for solving sets of simultaneous linear equations - homogeneous and inhomogeneous equations. 

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