Mathematical Methods for Physics and Engineering - Matematica.NET

NUMERICAL METHODSf(x)141210 f(x) =x 5 − 2x 2 − 3864200.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6x1.8−2−4Figure 27.10 ≤ x ≤ 1.9.A graph of the function f(x) =x 5 − 2x 2 − 3forx in the rangeExamples of the types of equation mentioned are the quartic equation,and the transcendental equation,ax 4 + bx + c =0,x − 3 tanh x =0.The latter type is characterised by the fact that it contains, in effect, a polynomialof infinite order on the LHS.We will discuss four methods that, in various circumstances, can be used toobtain the real roots of equations of the above types. In all cases we will take asthe specific equation to be solved the fifth-order polynomial equationf(x) ≡ x 5 − 2x 2 − 3=0. (27.1)The reasons for using the same equation each time were discussed in the introductionto this chapter.For future reference, and so that the reader may follow some of the calculationsleading to the evaluation of the real root of (27.1), a graph of f(x) in the range0 ≤ x ≤ 1.9 is shown in figure 27.1.Equation (27.1) is one for which no solution can be found in closed form, thatis in the form x = a, wherea does not explicitly contain x. The general scheme tobe employed will be an iterative one in which successive approximations to a realroot of (27.1) will be obtained, each approximation, it is to be hoped, being betterthan the preceding one; certainly, we require that the approximations convergeand that they have as their limit the sought-for root. Let us denote the required986