NAG Library Chapter Introduction
c05 – Roots of One or More Transcendental Equations
1 Scope of the Chapter
This chapter is concerned with the calculation of zeros of continuous functions of one or more variables. The majority of problems considered are for realvalued functions of real variables, in which case complex equations must be expressed in terms of the equivalent larger system of real equations.
2 Background to the Problems
The chapter divides naturally into two parts.
2.1 A Single Equation
The first deals with the real zeros of a real function of a single variable $f\left(x\right)$.
There are three functions with simple calling sequences. The first assumes that you can determine an initial interval
$\left[a,b\right]$ within which the desired zero lies, (that is, where
$f\left(a\right)\times f\left(b\right)<0$), 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 signchange of the function; none of the available functions are capable of making any finer distinction.) However, as with the other functions described below, a nonsimple zero might be determined and it is left to you to check for this. The algorithm used is due to
Brent (1973).
The two other functions are both designed for the case where you are unable to specify an interval containing the simple zero. One starts from an initial point and performs a search for an interval containing a simple zero. If such an interval is computed then the method described above is used next to determine the zero accurately. The other method uses a ‘continuation’ method based on a secant iteration. A sequence of subproblems is solved; the first of these is trivial and the last is the actual problem of finding a zero of $f\left(x\right)$. The intermediate problems employ the solutions of earlier problems to provide initial guesses for the secant iterations used to calculate their solutions.
Three other functions are also supplied. They employ reverse communication and use the same core algorithms as the functions described above.
Finally, two functions are provided to return values of Lambert's
$W$ function (sometimes known as the ‘product log’ or ‘Omega’ function), which is the inverse function of
that is, if Lambert's
$W$ function
$W\left(x\right)=a$ for
$x,a\in C$, then
$a$ is a zero of the function
$F\left(w\right)=w{e}^{w}x$. One function uses the iterative method described in
Barry et al. (1995) to return values from the real branches of
$W$ (restricting
$x,a\in R$). The second function enforces no such restriction, and uses the approach described in
Corless et al. (1996).
2.2 Systems of Equations
The functions in the second part of this chapter are designed to solve a set of nonlinear equations in
$n$ unknowns
where
$\mathrm{T}$ 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 ${J}_{ij}\left(x\right)=\left(\frac{\partial {f}_{i}}{\partial {x}_{j}}\right)$ evaluated at the point $x$, exists, though it may not be possible to calculate it directly.
The functions ${f}_{i}$ 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.
3 Recommendations on Choice and Use of Available Functions
3.1 Zeros of Functions of One Variable
The functions can be divided into two classes. There are three functions (
nag_interval_zero_cont_func (c05avc),
nag_zero_cont_func_cntin_rcomm (c05axc) and
nag_zero_cont_func_brent_rcomm (c05azc)) all written in reverse communication form and three (
nag_zero_cont_func_brent_binsrch (c05auc),
nag_zero_cont_func_cntin (c05awc) and
nag_zero_cont_func_brent (c05ayc)) written in direct communication form. The direct communication functions are designed for inexperienced users and, in particular, for solving problems where the function
$f\left(x\right)$ whose zero is to be calculated, can be coded as a usersupplied (sub)program. These functions find the zero by using the same core algorithms as the reverse communication functions. Experienced users are recommended to use the reverse communication functions directly as they permit you more control of the calculation. Indeed, if the zerofinding process is embedded in a much larger program then the reverse communication functions should always be used.
The recommendation as to which function should be used depends mainly on whether you can supply an interval
$\left[a,b\right]$ containing the zero; that is, where
$f\left(a\right)\times f\left(b\right)<0$. If the interval can be supplied, then
nag_zero_cont_func_brent (c05ayc) (or, in reverse communication,
nag_zero_cont_func_brent_rcomm (c05azc)) should be used, in general. This recommendation should be qualified in the case when the only interval which can be supplied is very long relative to your error requirements
and you can also supply a good approximation to the zero. In this case
nag_zero_cont_func_cntin (c05awc) (or, in reverse communication,
nag_zero_cont_func_cntin_rcomm (c05axc))
may prove more efficient (though these latter functions will not provide the error bound available from
nag_zero_cont_func_brent_rcomm (c05azc)).
If an interval containing the zero cannot be supplied then you must choose between
nag_zero_cont_func_brent_binsrch (c05auc) (or, in reverse communication,
nag_interval_zero_cont_func (c05avc) followed by
nag_zero_cont_func_brent_rcomm (c05azc)) and
nag_zero_cont_func_cntin (c05awc) (or, in reverse communication,
nag_zero_cont_func_cntin_rcomm (c05axc)).
nag_zero_cont_func_brent_binsrch (c05auc) first determines an interval containing the zero, and then proceeds as in
nag_zero_cont_func_brent (c05ayc); it is particularly recommended when you do not have a good initial approximation to the zero. If a good initial approximation to the zero is available then
nag_zero_cont_func_cntin (c05awc) is to be preferred. Since neither of these latter functions has guaranteed convergence to the zero, you are recommended to experiment with both in case of difficulty.
3.2 Solution of Sets of Nonlinear Equations
The solution of a set of nonlinear equations
can be regarded as a special case of the problem of finding a minimum of a sum of squares
So the functions in
Chapter e04 are relevant as well as the special nonlinear equations functions.
The functions for solving a set of nonlinear equations can also be divided into classes. There are five functions (
nag_zero_nonlin_eqns_easy (c05qbc),
nag_zero_nonlin_eqns_expert (c05qcc),
nag_zero_sparse_nonlin_eqns_easy (c05qsc),
nag_zero_nonlin_eqns_deriv_easy (c05rbc) and
nag_zero_nonlin_eqns_deriv_expert (c05rcc)) all written in direct communication form and two (
nag_zero_nonlin_eqns_rcomm (c05qdc) and
nag_zero_nonlin_eqns_deriv_rcomm (c05rdc)) written in reverse communication form. The direct communication functions are designed for inexperienced users and, in particular, these functions require the
${f}_{i}$ (and possibly their derivatives) to be calculated in usersupplied functions. These should be set up carefully so the Library functions can work as efficiently as possible. Experienced users are recommended to use the reverse communication functions as they permit you more control of the calculation. Indeed, if the zerofinding process is embedded in a much larger program then the reverse communication functions should always be used.
The main decision you have to make is whether to supply the derivatives $\frac{\partial {f}_{i}}{\partial {x}_{j}}$. It is advisable to do so if possible, since the results obtained by algorithms which use derivatives are generally more reliable than those obtained by algorithms which do not use derivatives.
nag_zero_sparse_nonlin_eqns_easy (c05qsc) is an easytouse function specially adapted for sparse problems, that is, problems where each function depends on a small subset of the
$n$ variables so that the Jacobian matrix has many zeros. It employs sparse linear algebra methods and consequently is expected to take significantly less time to complete than the other functions, especially if
$n$ is large.
Firstly, the calculation of the functions and their derivatives should be ordered so that cancellation errors are avoided. This is particularly important in a function that uses these quantities to build up estimates of higher derivatives.
Secondly, scaling of the variables has a considerable effect on the efficiency of a function. The problem should be designed so that the elements of $x$ are of similar magnitude. The same comment applies to the functions, i.e., all the ${f}_{i}$ should be of comparable size.
The accuracy is usually determined by the accuracy arguments of the functions, but the following points may be useful.
(i) 
Greater accuracy in the solution may be requested by choosing smaller input values for the accuracy arguments. However, if unreasonable accuracy is demanded, rounding errors may become important and cause a failure. 
(ii) 
Some idea of the accuracies of the ${x}_{i}$ may be obtained by monitoring the progress of the function to see how many figures remain unchanged during the last few iterations. 
(iii) 
An approximation to the error in the solution $x$ is given by $e$ where $e$ is the solution to the set of linear equations
where $f\left(x\right)={\left({f}_{1}\left(x\right),{f}_{2}\left(x\right),\dots ,{f}_{n}\left(x\right)\right)}^{\mathrm{T}}$.
Note that the $QR$ decomposition of $J$ is available from nag_zero_nonlin_eqns_expert (c05qcc) and nag_zero_nonlin_eqns_deriv_expert (c05rcc) (or, in reverse communication, nag_zero_nonlin_eqns_rcomm (c05qdc) and nag_zero_nonlin_eqns_deriv_rcomm (c05rdc)) so that
and ${Q}^{\mathrm{T}}f$ is also provided by these functions. 
(iv) 
If the functions
${f}_{i}\left(x\right)$ are changed by small amounts ${\epsilon}_{i}$, for $\mathit{i}=1,2,\dots ,n$, then the corresponding change in the solution $x$ is given approximately by $\sigma $, where $\sigma $ is the solution of the set of linear equations
Thus one can estimate the sensitivity of $x$ to any uncertainties in the specification of
${f}_{i}\left(x\right)$, for $\mathit{i}=1,2,\dots ,n$. As noted above, the sophisticated functions nag_zero_nonlin_eqns_expert (c05qcc) and nag_zero_nonlin_eqns_deriv_expert (c05rcc) (or, in reverse communication, nag_zero_nonlin_eqns_rcomm (c05qdc) and nag_zero_nonlin_eqns_deriv_rcomm (c05rdc)) provide the $QR$ decomposition of $J$. 
3.3 Values of Lambert's $W$ function
If you require purelyreal values of
$W$, these will be evaluated marginally more efficiently by
nag_lambertW (c05bac) than
nag_lambertW_complex (c05bbc) owing to the differing iterative procedures used by each function.
4 Decision Trees
Tree 1: Functions of One Variable
Is reverse communication required? 
_ yes 
Is there available an interval $\left[a,b\right]$ containing a simple zero, and no others? 
_ yes 
c05azc 
 

no  

 

Is a good approximation to the zero available? 
_ yes 
c05axc 
 

no  

 

c05avc followed by c05azc 
no  

Do you wish to compute the values of Lambert's $W$ function? 
_ yes 
do you require values from only the real branches? 
_ yes 
c05bac 
 

no  

 

c05bbc 
no  

Is there available an interval $\left[a,b\right]$ containing a simple zero, and no others? 
_ yes 
c05ayc 
no  

Is a good approximation to the zero available? 
_ yes 
c05awc 
no  

c05auc 
Tree 2: Functions of several variables
Is the Jacobian matrix sparse? 
_ yes 
c05qsc 
no  

Is reverse communication required? 
_ yes 
Is the Jacobian matrix available? 
_ yes 
c05rdc and c05zdc 
 

no  

 

c05qdc 
no  

Is the Jacobian matrix available? 
_ yes 
Is flexibility required? 
_ yes 
c05rcc and c05zdc 
 

no  

 

c05rbc and c05zdc 
no  

Is flexibility required? 
_ yes 
c05qcc 
no  

c05qbc 
5 Functionality Index
Zeros of functions of one variable:   
Zeros of functions of several variables:   
6 Functions Withdrawn or Scheduled for Withdrawal
7 References
Barry D J, Culligan–Hensley P J, and Barry S J (1995) Real values of the $W$function ACM Trans. Math. Software 21(2) 161–171
Brent R P (1973) Algorithms for Minimization Without Derivatives Prentice–Hall
Corless R M, Gonnet G H, Hare D E G, Jeffrey D J and Knuth D
E (1996) On the Lambert $W$ function Advances in Comp. Math. 3 329–359
Gill P E and Murray W (1976) Algorithms for the solution of the nonlinear leastsquares problem Report NAC 71 National Physical Laboratory
Moré J J, Garbow B S and Hillstrom K E (1980) User guide for MINPACK1 Technical Report ANL8074 Argonne National Laboratory
Ortega J M and Rheinboldt W C (1970) Iterative Solution of Nonlinear Equations in Several Variables Academic Press
Rabinowitz P (1970) Numerical Methods for Nonlinear Algebraic Equations Gordon and Breach