Solving nonlinear equations

Of such schemes, two of the most robust and powerful are Newton s method for solving an equation with one unknown variable, and Newton-Raphson iteration, which treats systems of equations in more than one unknown. I will briefly describe these methods here before I approach the solution of chemical problems. Further details can be found in a number of texts on numerical analysis, such as Carnahan et al. (1969). 

In Newton s method, we seek a value of x that satisfies 

The method s goal is to make the residual vanish by successively improving our guess. To find an improved value x(1), we take the tangent line to the residual function at point x(o) and project it to the zero line (Fig. 5.1). We repeat the projection from x(l) to give x(2), and so on. The process continues until we reach a value x(ri on the ( )-th iteration that satisfies our equation to within a small tolerance. 

The method can be expressed mathematically by noting that the slope of the residual function plotted against x is df/dx. Geometrically, the slope is rise over run, so 

The multidimensional counterpart to Newton s method is Newton-Raphson iteration. A mathematics professor once complained to me, with apparent sincerity, that he could visualize surfaces in no more than twelve dimensions. My perspective on hyperspace is less incisive, as perhaps is the reader s, so we will consider first a system of two nonlinear equations f-a and g=b with unknowns x and y. 

We pose the problem for the remaining equations by specifying the total mole numbers Mw, Mi, and of the basis entries. Our task in this case is to solve the equations for the values of nw, mt, and - The solution is more difficult now because the unknown values appear raised to their reaction coefficients and multiplied by each other in the mass action Equation 4.7. In the next two sections we discuss how such nonlinear equations can be solved numerically. 

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USE MULLER.S METK)0 TO SOLVE NONLINEAR EQUATIONS FOP THE TRUE VAPOR-PHASE COMPOSITION. 

Difference Methods for Solving Nonlinear Equations of Mathematical Physics... 

Similarly to the most robust methods of solving nonlinear equations, we start with bracketing. Assume that the interval [xy, X ] contains a single minimum point r, i.e., the function f is decreasing up to r and increasing afterwards. Then the function is said to be unimodal on the interval [xy, Xy], This property is exploited in cut-off methods, purported to reduce the length of the interval which will, however, include the minimum point in all iterations. 

Methods for solving nonlinear equations such as these are discussed in Chapter 5. 

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