NAG C Library Chapter Introduction

c05 — Roots of One or More Transcendental Equations

There is one function with simple calling sequences. It first assumes that you can determine an initial interval a,b within which the desired zero lies, that is fa×fb<0, and outside which all other zeros lie. The function then systematically subdivides the interval to produce a final interval containing the zero. This final interval has a length bounded by your specified error requirements; the end of the interval where the function has smallest magnitude is returned as the zero. This function is guaranteed to converge to a simple zero of the function. (Here we define a simple zero as a zero corresponding to a sign-change of the function; none of the available functions are capable of making any finer distinction.) However, a non-simple zero might be determined and it is left to you to check for this. The algorithm used is due to Bus and Dekker.

where T stands for transpose.

It is assumed that the functions are continuous and differentiable so that the matrix of first partial derivatives of the functions, the Jacobian matrix J ij x= ∂fi ∂xj   evaluated at the point x, exists, though it may not be possible to calculate it directly.

The functions fi must be independent, otherwise there will be an infinity of solutions and the methods will fail. However, even when the functions are independent the solutions may not be unique. Since the methods are iterative, an initial guess at the solution has to be supplied, and the solution located will usually be the one closest to this initial guess.

can be regarded as a special case of the problem of finding a minimum of a sum of squares