test_zero

test_zero, a Python code 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 information on this web page is 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:

bisection_rc, a Python 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 (RC).

fsolve_test, a Python code which calls fsolve() which seeks the solution x of one or more nonlinear equations f(x)=0.

zero_itp, a Python code which finds a zero of a scalar function of a scalar variable, starting from a change of sign interval, using the Interpolate/Truncate/Project (ITP) method, which has faster convergence than the bisection method.

zero_laguerre, a Python code which uses Laguerre's method to find the zero of a function. The method needs first and second derivative information. The method almost always works when the function is a polynomial.

zero_muller, a Python code which seeks a root of a nonlinear equation using the Muller method, with complex arithmetic.

zero_rc, a Python code 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:

• test_zero.py, tests the code.
• test_zero.sh, runs all the tests.
• test_zero.txt, the output file.