test_zero

test_zero, a Fortran77 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.

TEST_ZERO 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)

Licensing:

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

Languages:

test_zero is available in a C version and a C++ version and a Fortran77 version and a Fortran90 version and a MATLAB version and an Octave version and a Python version.

Related Data and Programs:

test_zero_test

bisection_rc, a Fortran77 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 Fortran77 library 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 Fortran77 library which includes the zero finder FZERO.

slatec, a Fortran90 library which includes the zero finder FZERO.

zoomin, a Fortran90 library which includes various zero finder routines.

Reference:

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:

• test_zero.f, the source code;
• test_zero.sh, commands to compile the source code;