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C05 (Roots)
Roots of One or More Transcendental Equations

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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 real-valued 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(x) .
There are three functions with simple calling sequences. The first assumes that you can determine an initial interval [a,b] within which the desired zero lies, (that is, where f(a)×f(b)<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, as with the other functions described below, a non-simple 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(x) . 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
f(w)=wew  for  w;  
that is, if Lambert's W function W(x)=a for x,a, then a is a zero of the function F(w)=wew-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). 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
fi(x) = 0 ,   i= 1,2,,n ,   x= (x1,x2,,xn) T , (1)
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.

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 (c05avc, c05axc and c05azc) all written in reverse communication form and three (c05auc, c05awc and c05ayc) written in direct communication form (see Section 7 in How to Use the NAG Library for a description of the difference between these two conventions). The direct communication functions are designed for inexperienced users and, in particular, for solving problems where the function f(x) whose zero is to be calculated, can be coded as a user-supplied (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 zero-finding 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 [a,b] containing the zero; that is, where f(a)×f(b)<0 . If the interval can be supplied, then c05ayc (or, in reverse communication, 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 c05awc (or, in reverse communication, c05axc) may prove more efficient (though these latter functions will not provide the error bound available from c05azc).
If an interval containing the zero cannot be supplied then you must choose between c05auc (or, in reverse communication, c05avc followed by c05azc) and c05awc (or, in reverse communication, c05axc). c05auc first determines an interval containing the zero, and then proceeds as in 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 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
fi(x1,x2,,xn)=0,  i=1,2,,n (2)
can be regarded as a special case of the problem of finding a minimum of a sum of squares
s (x) = i = 1 m [ f i ( x 1 , x 2 ,, x n )] 2 ,   (mn) . (3)
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 six functions (c05mbc, c05qbc, c05qcc, c05qsc, c05rbc and c05rcc) all written in direct communication form and three (c05mdc, c05qdc and c05rdc) written in reverse communication form. The direct communication functions are designed for inexperienced users and, in particular, these functions require the fi (and possibly their derivatives) to be calculated in user-supplied 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 zero-finding 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 fi xj . 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.
c05rbc, c05rcc and c05rdc require you to provide the derivatives, whilst c05mbc, c05mdc, c05qbc, c05qcc, c05qdc and c05qsc do not. c05qbc, c05qsc and c05rbc are easy-to-use functions; greater flexibility may be obtained using c05qcc and c05rcc (or, in reverse communication, c05qdc and c05rdc), but these have longer argument lists. c05qbc, c05qcc, c05qdc and c05qsc approximate the derivatives internally using finite differences. c05mbc and c05mdc do not use derivatives at all, and may be useful when the cost of evaluating f(x) is high. If f(x) can be evaluated cheaply, then functions which use the Jacobian or its approximation may converge more quickly.
c05qsc is an easy-to-use 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.
c05zdc is provided for use in conjunction with c05rbc, c05rcc and c05rdc to check the user-supplied derivatives for consistency with the functions themselves. You are strongly advised to make use of this function whenever c05rbc, c05rcc or c05rdc is used.
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 fi should be of comparable size.
The accuracy is usually determined by the accuracy arguments of the functions, but the following points may be useful.
  1. (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.
  2. (ii)Some idea of the accuracies of the xi may be obtained by monitoring the progress of the function to see how many figures remain unchanged during the last few iterations.
  3. (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(x) = (f1(x),f2(x),,fn(x)) T .
    Note that the QR decomposition of J is available from c05qcc and c05rcc (or, in reverse communication, c05qdc and c05rdc) so that
    and QTf is also provided by these functions.
  4. (iv)If the functions fi(x) are changed by small amounts ε i , for i=1,2,,n, then the corresponding change in the solution x is given approximately by σ, where σ is the solution of the set of linear equations
    J(x)σ = -ε .  
    Thus one can estimate the sensitivity of x to any uncertainties in the specification of fi(x) , for i=1,2,,n. As noted above, the sophisticated functions c05qcc and c05rcc (or, in reverse communication, c05qdc and c05rdc) provide the QR decomposition of J.

3.3 Values of Lambert's W function

If you require purely-real values of W, these will be evaluated marginally more efficiently by c05bac than c05bbc owing to the differing iterative procedures used by each function.

4 Decision Trees

Tree 1: Functions of One Variable

Is reverse communication required?   Is there available an interval [a,b] containing a simple zero, and no others?   c05azc
  no   no
Is a good approximation to the zero available?   c05axc
c05avc followed by c05azc
Do you wish to compute the values of Lambert's W function?   do you require values from only the real branches?   c05bac
  no   no
Is there available an interval [a,b] containing a simple zero, and no others?   c05ayc
Is a good approximation to the zero available?   c05awc

Tree 2: Functions of several variables

Do you wish to avoid the use of the Jacobian matrix or its approximation?   c05mbc and c05mdc
Is the Jacobian matrix sparse?   c05qsc
Is reverse communication required?   Is the Jacobian matrix available?   c05rdc and c05zdc
  no   no
Is the Jacobian matrix available?   Is flexibility required?   c05rcc and c05zdc
  no   no
c05rbc and c05zdc
Is flexibility required?   c05qcc

5 Functionality Index

Lambert's W function,  
complex values   c05bbc
real values   c05bac
Zeros of functions of one variable,  
direct communication,  
binary search followed by Brent algorithm   c05auc
Brent algorithm   c05ayc
continuation method   c05awc
reverse communication,  
binary search   c05avc
Brent algorithm   c05azc
continuation method   c05axc
Zeros of functions of several variables,  
checking function,  
checks user-supplied Jacobian   c05zdc
direct communication,  
Anderson acceleration   c05mbc
derivatives required   c05rbc
no derivatives required   c05qbc
no derivatives required, sparse   c05qsc
derivatives required   c05rcc
no derivatives required   c05qcc
reverse communication,  
Anderson acceleration   c05mdc
derivatives required   c05rdc
no derivatives required   c05qdc

6 Auxiliary Functions Associated with Library Function Arguments


7 Withdrawn or Deprecated Functions

The following lists all those functions that have been withdrawn since Mark 24 of the Library or are in the Library, but deprecated.
Function Status Replacement Function(s)
c05adc Withdrawn at Mark 24 c05ayc
c05agc Withdrawn at Mark 25 c05auc
c05nbc Withdrawn at Mark 24 c05qbc
c05pbc Withdrawn at Mark 24 c05rbc
c05sdc Withdrawn at Mark 25 c05ayc
c05tbc Withdrawn at Mark 24 c05qbc
c05ubc Withdrawn at Mark 25 c05rbc
c05zbc Withdrawn at Mark 24 c05zdc
c05zcc Withdrawn at Mark 24 c05zdc

8 References

Anderson D G (1965) Iterative Procedures for Nonlinear Integral Equations J. Assoc. Comput. Mach. 12 547–560
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 least squares problem Report NAC 71 National Physical Laboratory
Moré J J, Garbow B S and Hillstrom K E (1980) User guide for MINPACK-1 Technical Report ANL-80-74 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