Mathematical Methods for Physics and Engineering - Matematica.NET

27.1 ALGEBRAIC AND TRANSCENDENTAL EQUATIONSerrors introduced as a result of approximations made in setting up the numericalprocedures (truncation errors). For this scale of application, books specificallydevoted to numerical analysis, data analysis and computer programming shouldbe consulted.So far as is possible, the method of presentation here is that of indicatingand discussing in a qualitative way the main steps in the procedure, and thenof following this with an elementary worked example. The examples have beenrestricted in complexity to a level at which they can be carried out with a pocketcalculator. Naturally it will not be possible for the student to check all thenumerical values presented, unless he or she has a programmable calculator orcomputer readily available, and even then it might be tedious to do so. However,it is advisable to check the initial step and at least one step in the middle ofeach repetitive calculation given in the text, so that how the symbolic equationsare used with actual numbers is understood. Clearly the intermediate step shouldbe chosen to be at a point in the calculation at which the changes are stillsufficiently large that they can be detected by whatever calculating device isused.Where alternative methods for solving the same type of problem are discussed,for example in finding the roots of a polynomial equation, we have usuallytaken the same example to illustrate each method. This could give the mistakenimpression that the methods are very restricted in applicability, but it is felt bythe authors that using the same examples repeatedly has sufficient advantages, interms of illustrating the relative characteristics of competing methods, to justifydoing so. Once the principles are clear, little is to be gained by using newexamples each time, and, in fact, having some prior knowledge of the ‘correctanswer’ should allow the reader to judge the efficiency and dangers of particularmethods as the successive steps are followed through.One other point remains to be mentioned. Here, in contrast with every otherchapter of this book, the value of a large selection of exercises is not clear cut.The reader with sufficient computing resources to tackle them can easily devisealgebraic or differential equations to be solved, or functions to be integrated(which perhaps have arisen in other contexts). Further, the solutions of theseproblems will be self-checking, for the most part. Consequently, although anumber of exercises are included, no attempt has been made to test the full rangeof ideas treated in this chapter.27.1 Algebraic and transcendental equationsThe problem of finding the real roots of an equation of the form f(x) =0,wheref(x) is an algebraic or transcendental function of x, is one that can sometimesbe treated numerically, even if explicit solutions in closed form are not feasible.985