Nonlinear Equation Solver, Reverse Communication

ROOTS_RC is a C++ library which seeks a solution of a system of nonlinear equations, using reverse communication (RC), by Gaston Gonnet.

One of the standard problems in numerical analysis is to determine an approximate solution to a scalar nonlinear equation of the form f(x)=0.

Reliable and efficient procedures for this task, sometimes called "zero finders" or "root solvers" are available in many libraries. These procedures typically require the user to write a sub-procedure to evaluate the function for any argument x; the form and list of arguments for this sub-procedure are strictly prescribed by the library procedure. The user must somehow make this sub-procedure available to the solver, either by using a fixed name for the sub-procedure, or by passing in the actual name as an argument.

In many cases, it can be inconvenient to use such a library routine, because it is inserted between the main user program and the user function. This means that data defining the function, which might be chosen in the main program, must then somehow be communicated to the user function, perhaps by declaring some kind of global memory, or by sneaking the data through in an extra argument provided by the library procedure, or by the use of auxilliary data files.

Moreover, the user essentially loses control of the process until the library procedure returns. It might, however, be the case that the user could detect important error conditions, or gather useful information, if the intermediate x values generated by the library procedure were visible.

If we denote this typical method of interaction between a user and a library an instance of "forward communication", then there is an alternative approach, known as "reverse communication", which allows the user much more freedom in designing the function evaluation, and in observing and intervening in the iteration that is seeking the solution.

An idealized version of the use of a reverse communication zero finder might look like this:

        x = initial approximation.
        while ( not satisfied )
          fx = f(x)
          x = root ( x, fx )
Here, "not satisfied" might simply be a test of the magnitude of f(x). But note two things:

Reverse communication zero finders can be very useful in situations where the function to be evaluated is actually the outcome of a complicated process. For instance, if we are seeking an eigenvalue x that makes the determinant of some matrix zero, then our function evaluation may require us to form a large matrix and to factor it in order to evaluate the determinant. This may be cumbersome to do if we must perform all these operations in a sub-procedure.

Similarly, we might be solving a boundary value problem using the shooting method, and f(x) might be the deviation at the final time between the computed and desired boundary values. In that case, a subprocedure formulation would require us to set up and solve a boundary value problem repeatedly in an isolated piece of code.


The computer code and data files described and made available on this web page are distributed under the GNU LGPL license.


ROOTS_RC is available in a C version and a C++ version and a FORTRAN90 version and a MATLAB version and a Python version.

Related Data and Programs:

BACKTRACK_BINARY_RC, a C++ library which carries out a backtrack search for a set of binary decisions, using reverse communication (RC).

BISECTION_RC, a C++ library which seeks a solution to the equation F(X)=0 using bisection within a user-supplied change of sign interval [A,B]. The procedure is written using reverse communication (RC).

BRENT, a C++ library which contains routines for finding zeroes or minima of a scalar function of a scalar variable, without the use of derivative information, including a reverse communication (RC) option, by Richard Brent.

CG_RC, a C++ library which implements the conjugate gradient method for solving a positive definite sparse linear system A*x=b, using reverse communication (RC).

LOCAL_MIN_RC, a C++ library which finds a local minimum of a scalar function of a scalar variable, without the use of derivative information, using reverse communication (RC), by Richard Brent.

ROOT_RC, a C++ library which seeks a solution of a scalar nonlinear equation f(x) = 0, using reverse communication (RC), by Gaston Gonnet.

SORT_RC, a C++ library which can sort a list of any kind of objects, using reverse communication (RC).

TEST_ZERO, a C++ library which implements test problems for the solution of a single nonlinear equation in one variable.

ZERO_RC, a C++ library which seeks a solution of a scalar nonlinear equation f(x) = 0, using reverse communication (RC), by Richard Brent.


  1. Gaston Gonnet,
    On the Structure of Zero Finders,
    BIT Numerical Mathematics,
    Volume 17, Number 2, June 1977, pages 170-183.
  2. Werner Rheinboldt,
    Algorithms for finding zeros of a function,
    UMAP Journal,
    Volume 2, Number 1, 1981, pages 43-72.

Source Code:

Examples and Tests:

List of Routines:

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Last revised on 04 December 2016.