test_zero, a FORTRAN90 code which defines nonlinear functions that may be used to test zero finders.

Zero finders are programs that seek a (scalar) root of a scalar equation F(X) = 0. Some zero finders require that an initial "change-of-sign" interval [A,B] be supplied, with the function having opposite sign at the two endpoints, thus guaranteeing that there is some value C between A and B for which F(C) = 0 (assuming that the function F is continuous). In other cases, a particular zero finder may want information about the first or second derivative of the function. And some zero finders can handle situations where the function has a multiple root, or where the function is a polynomial.

The code supplies a set of nonlinear functions, along with change of sign interval, first and second derivatives, suggested starting points, so that the behavior of any zero finder can be analyzed.

The code also includes implementations of some simple zero finders, as a demonstration of how the package might be used.

The functions, which are accessible by number, are

  1. f(x) = sin ( x ) - x / 2.
  2. f(x) = 2 * x - exp ( - x ).
  3. f(x) = x * exp ( - x ).
  4. f(x) = exp ( x ) - 1 / ( 10 * x )^2.
  5. f(x) = ( x + 3 ) * ( x - 1 )^2.
  6. f(x) = exp ( x ) - 2 - 1 / ( 10 * x )^2 + 2 / ( 100 * x )^3.
  7. f(x) = x^3.
  8. f(x) = cos ( x ) - x.
  9. the Newton Baffler.
  10. the Repeller.
  11. the Pinhead.
  12. Flat Stanley.
  13. Lazy Boy.
  14. the Camel.
  15. a pathological function for Newton's method.
  16. Kepler's Equation.
  17. f(x) = x^3 - 2*x - 5, Wallis's function.
  18. f(x) = (x-1)^7, written term by term.
  19. f(x) = cos(100*x)-4*erf(30*x-10), the jumping cosine.


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


test_zero is available in a C version and a C++ version and a FORTRAN90 version and a MATLAB version and an Octave version and a Python version.

Related Data and Programs:

BISECTION_INTEGER, a FORTRAN90 code which seeks an integer solution to the equation F(X)=0, using bisection within a user-supplied change of sign interval [A,B].

BISECTION_RC, a FORTRAN90 code 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.

BRENT, a FORTRAN90 code which contains Richard Brent's routines for finding the zero, local minimizer, or global minimizer of a scalar function of a scalar argument, without the use of derivative information.

NMS, a FORTRAN90 code which includes the zero finder FZERO.


ZERO_RC, a FORTRAN90 code which seeks solutions of a scalar nonlinear equation f(x) = 0, or a system of nonlinear equations, using reverse communication.

ZOOMIN, a FORTRAN90 code which includes various zero finder routines.


  1. Richard Brent,
    Algorithms for Minimization without Derivatives,
    Dover, 2002,
    ISBN: 0-486-41998-3,
    LC: QA402.5.B74.
  2. Peter Colwell,
    Solving Kepler's Equation Over Three Centuries,
    Willmann-Bell, 1993,
    ISBN: 0943396409,
    LC: QB355.5.C65.
  3. George Donovan, Arnold Miller, Timothy Moreland,
    Pathological Functions for Newton's Method,
    American Mathematical Monthly, January 1993, pages 53-58.
  4. Arnold Krommer, Christoph Ueberhuber,
    Numerical Integration on Advanced Computer Systems,
    Springer, 1994,
    ISBN: 3540584102,
    LC: QA299.3.K76.
  5. Jean Meeus,
    Astronomical Algorithms,
    Second Edition,
    Willman-Bell, 1998,
    ISBN: 0943396611,
    LC: QB51.3.E43M42.

Source Code:

Last revised on 05 September 2020.