TEST_ZERO
Zero Finder Tests

TEST_ZERO is a Python library which defines a number of nonlinear functions.

The nonlinear functions are intended for use in demonstrating or testing zero finder algorithms, that is, programs that seek a root of a scalar equation F(X)=0.

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 GNU LGPL license.

Languages:

TEST_ZERO 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:

BISECTION_RC, a Python 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).

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

Reference:

1. George Donovan, Arnold Miller, Timothy Moreland,
   Pathological Functions for Newton's Method,
   American Mathematical Monthly, January 1993, pages 53-58.
2. Peter Colwell,
   Solving Kepler's Equation Over Three Centuries,
   Willmann-Bell, 1993
3. Arnold Krommer, Christoph Ueberhuber,
   Numerical Integration on Advanced Systems,
   Springer, 1994.

PNG images of the graphs of some of the functions were made using MATLAB:

• p01_fx.png, an image of P01_FX(X) over [-4,+4].
• p02_fx.png, an image of P02_FX(X) over [-0.5, +3.0].
• p03_fx.png, an image of P03_FX(X) over [-0.1,+4].
• p04_fx.png, an image of P04_FX(X) over [-4,+2].
• p05_fx.png, an image of P05_FX(X) over [-4,+2].
• p06_fx.png, an image of P06_FX(X) over [-4,+2].
• p07_fx.png, an image of P07_FX(X) over [-1,+1].
• p08_fx.png, an image of P08_FX(X) over [-4,+4].
• p09_fx.png, an image of P09_FX(X) over [5,7].
• p10_fx.png, an image of P10_FX(X) over [-2,+2].
• p11_fx.png, an image of P11_FX(X) over [+1,+10].
• p12_fx.png, an image of P12_FX(X) over [-0.5,+0.5].
• p13_fx.png, an image of P13_FX(X) over [0,100].
• p14_fx.png, an image of P14_FX(X) over [-0.5,+2.0].
• p15_fx.png, an image of P15_FX(X) over [-4,+4].
• p16_fx.png, an image of P16_FX(X) over [0,50].
• p17_fx.png, an image of P17_FX(X) over [-2,+4].
• p18_fx.png, an image of P18_FX(X) over [0.988,1.012].
• p19_fx.png, an image of P19_FX(X) over [0.0,1.0].

Source Code:

• p00.py, returns information for any problem, given its index.
• p01.py, returns information for problem 01.
• p02.py, returns information for problem 02.
• p03.py, returns information for problem 03.
• p04.py, returns information for problem 04.
• p05.py, returns information for problem 05.
• p06.py, returns information for problem 06.
• p07.py, returns information for problem 07.
• p08.py, returns information for problem 08.
• p09.py, returns information for problem 09.
• p10.py, returns information for problem 10.
• p11.py, returns information for problem 11.
• p12.py, returns information for problem 12.
• p13.py, returns information for problem 13.
• p14.py, returns information for problem 14.
• p15.py, returns information for problem 15.
• p16.py, returns information for problem 16.
• p17.py, returns information for problem 17.
1
• p18.py, returns information for problem 18.
• p19.py, returns information for problem 19.

Methods and utility functions:

• bisection.py, carries out the bisection method on any problem.
• brent.py, carries out Brent's method on any problem.
• muller.py, carries out Muller's method on any problem.
• newton.py, carries out Newton's method on any problem.
• r8_cube_root.py, computes the cube root of an R8.
• r8_sign.py, returns the sign of an R8.
• r8poly2_rroot.py, returns the real parts of the roots of a quadratic polynomial.
• regula_falsi.py, carries out the regula falsi method on any problem.
• secant.py, carries out the secant method on any problem.
• timestamp.py, prints the YMDHMS date as a timestamp.
• test_zero.py, tests the library.
• test_zero.sh, runs all the tests.
• test_zero.txt, the output file.

You can go up one level to the Python source codes.