Solving Sets of Equations: Linear and Nonlinear

The previous chapter has discussed the solution of a single linear or nonlinear equation to find the roots of the equation or values of a variable for which the equation is zero. This chapter extends that discussion to sets of equations in several variables. Only cases where the number of variables is equal to the number of equations will be considered with equations of the form:

(4.1)

$$\begin{array}{*{20}c} {f_1 \left( {x_1,x_2, \ldots \ldots,x_n } \right) = 0} \\ {f_2 \left( {x_1,x_2, \ldots \ldots,x_n } \right) = 0} \\ \vdots \\ {f_n \left( {x_1,x_2, \ldots \ldots,x_n } \right) = 0} \\ \end{array}$$

In general it will be assumed that this is a nonlinear set of equations for which a solution set of x

1

, x

2

,....., x

n

values is desired that satisfy the equations. In the general case there may be (a) no solution values, (b) one set of values or (c) many sets of solution values. The fact that one is trying to find a set of solution values implies that he/she believes that the set of equations has at least one set of solution values. In most physical problems, one has some general idea as to the range of solution values for the variables which can be used as initial guesses at the solution values.