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Root Finding and Nonlinear Sets of Equations part 1

We now consider that most basic of tasks, solving equations numerically. While most equations are born with both a right-hand side and a left-hand side, one traditionally moves all terms to the left, leaving f(x) = 0 (9.0.1) whose solution or solutions are desired. When there is only one | Chapter 9. Root Finding and Nonlinear Sets of Equations 9.0 Introduction We now consider that most basic of tasks solving equations numerically. While most equations are born with both a right-hand side and a left-hand side one traditionally moves all terms to the left leaving f x 0 9.0.1 whose solutionor solutionsare desired. When there is only one independent variable the problem is one-dimensional namely to find the root or roots of a function. With more than one independent variable more than one equation can be satisfied simultaneously. You likely once learned the implicit function theorem which in this context gives us the hope of satisfying N equations in N unknowns simultaneously. Note that we have only hope not certainty. A nonlinear set of equations may have no real solutions at all. Contrariwise it may have more than one solution. The implicit function theorem tells us that generically the solutions will be distinct pointlike and separated from each other. If however life is so unkind as to present you with a nongeneric i.e. degenerate case then you can get a continuous family of solutions. In vector notation we want to find one or more N-dimensional solution vectors x such that f x 0 9.0.2 where f is the N-dimensional vector-valued function whose components are the individual equations to be satisfied simultaneously. Don t be fooled by the apparent notational similarity of equations 9.0.2 and 9.0.1 . Simultaneous solution of equations in N dimensions is much more difficult than finding roots in the one-dimensional case. The principal difference between one and many dimensions is that in one dimension it is possible to bracket or trap a root between bracketing values and then hunt it down like a rabbit. In multidimensions you can never be sure that the root is there at all until you have found it. Except in linear problems root finding invariably proceeds by iteration and this is equally true in one or in many dimensions. Starting from some approximate .