TEST_ZERO
Zero Finder Tests
TEST_ZERO,
an Octave 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 function TEST_ZERO evaluates one of several scalar
nonlinear test functions, and has the form:
function [ fx, fp ] = test_zero ( x )
To make use of this function while running MATLAB, you must issue
the command
global IPROB
and then set IPROB to a value between 1 and 18 to select a
particular problem.
The functions, which are accessible by number, are
-
f(x) = sin ( x ) - x / 2.
-
f(x) = 2 * x - exp ( - x ).
-
f(x) = x * exp ( - x ).
-
f(x) = exp ( x ) - 1 / ( 10 * x )^2.
-
f(x) = ( x + 3 ) * ( x - 1 )^2.
-
f(x) = exp ( x ) - 2 - 1 / ( 10 * x )^2 + 2 / ( 100 * x )^3.
-
f(x) = x^3.
-
f(x) = cos ( x ) - x.
-
the Newton Baffler.
-
the Repeller.
-
the Pinhead.
-
Flat Stanley.
-
Lazy Boy.
-
the Camel.
-
a pathological function for Newton's method.
-
Kepler's Equation.
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f(x) = x^3 - 2*x - 5, Wallis's function.
-
f(x) = (x-1)^7, written term by term.
-
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 FORTRAN90 version and
a MATLAB version and
an Octave version and
a Python version.
Related Data and Programs:
fsolve_test,
an Octave code which
calls fsolve() which
seeks the solution x of one or more nonlinear equations f(x)=0.
test_zero_test
Reference:
-
George Donovan, Arnold Miller, Timothy Moreland,
Pathological Functions for Newton's Method,
American Mathematical Monthly, January 1993, pages 53-58.
-
Peter Colwell,
Solving Kepler's Equation Over Three Centuries,
Willmann-Bell, 1993
-
Arnold Krommer, Christoph Ueberhuber,
Numerical Integration on Advanced Systems,
Springer, 1994.
Source Code:
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bisection.m,
carries out the bisection method on any problem.
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brent.m,
carries out Brent's method on any problem.
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muller.m,
carries out Muller's method on any problem.
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newton.m,
carries out Newton's method on any problem.
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p00_fx.m,
evaluates a function for any problem.
-
p00_fx1.m,
first derivative of a function for any problem.
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p00_fx2.m,
second derivative of a function for any problem.
-
p00_prob_num.m,
returns the number of problems.
-
p00_range.m,
returns an interval bounding the root for any problem.
-
p00_root.m,
returns a known root for any problem.
-
p00_root_num.m,
returns the number of known roots for any problem.
-
p00_start.m,
returns a starting point for any problem.
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p00_start_num.m,
returns the number of starting points for any problem.
-
p00_title.m,
returns the title for any problem.
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p01_fx.m,
evaluates the function for problem 1.
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p01_fx1.m,
first derivative of the function for problem 1.
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p01_fx2.m,
second derivative of the function for problem 1.
-
p01_range.m,
returns an interval bounding the root for problem 1.
-
p01_root.m,
returns a known root for problem 1.
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p01_root_num.m,
returns the number of known roots for problem 1.
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p01_start.m,
returns a starting point for problem 1.
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p01_start_num.m,
returns the number of starting points for problem 1.
-
p01_title.m,
returns the title for problem 1.
-
p02_fx.m,
evaluates the function for problem 2.
-
p02_fx1.m,
first derivative of the function for problem 2.
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p02_fx2.m,
second derivative of the function for problem 2.
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p02_range.m,
returns an interval bounding the root for problem 2.
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p02_root.m,
returns a known root for problem 2.
-
p02_root_num.m,
returns the number of known roots for problem 2.
-
p02_start.m,
returns a starting point for problem 2.
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p02_start_num.m,
returns the number of starting points for problem 2.
-
p02_title.m,
returns the title for problem 2.
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p03_fx.m,
evaluates the function for problem 3.
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p03_fx1.m,
first derivative of the function for problem 3.
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p03_fx2.m,
second derivative of the function for problem 3.
-
p03_range.m,
returns an interval bounding the root for problem 3.
-
p03_root.m,
returns a known root for problem 3.
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p03_root_num.m,
returns the number of known roots for problem 3.
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p03_start.m,
returns a starting point for problem 3.
-
p03_start_num.m,
returns the number of starting points for problem 3.
-
p03_title.m,
returns the title for problem 3.
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p04_fx.m,
evaluates the function for problem 4.
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p04_fx1.m,
first derivative of the function for problem 4.
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p04_fx2.m,
second derivative of the function for problem 4.
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p04_range.m,
returns an interval bounding the root for problem 4.
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p04_root.m,
returns a known root for problem 4.
-
p04_root_num.m,
returns the number of known roots for problem 4.
-
p04_start.m,
returns a starting point for problem 4.
-
p04_start_num.m,
returns the number of starting points for problem 4.
-
p04_title.m,
returns the title for problem 4.
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p05_fx.m,
evaluates the function for problem 5.
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p05_fx1.m,
first derivative of the function for problem 5.
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p05_fx2.m,
second derivative of the function for problem 5.
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p05_range.m,
returns an interval bounding the root for problem 5.
-
p05_root.m,
returns a known root for problem 5.
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p05_root_num.m,
returns the number of known roots for problem 5.
-
p05_start.m,
returns a starting point for problem 5.
-
p05_start_num.m,
returns the number of starting points for problem 5.
-
p05_title.m,
returns the title for problem 5.
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p06_fx.m,
evaluates the function for problem 6.
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p06_fx1.m,
first derivative of the function for problem 6.
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p06_fx2.m,
second derivative of the function for problem 6.
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p06_range.m,
returns an interval bounding the root for problem 6.
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p06_root.m,
returns a known root for problem 6.
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p06_root_num.m,
returns the number of known roots for problem 6.
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p06_start.m,
returns a starting point for problem 6.
-
p06_start_num.m,
returns the number of starting points for problem 6.
-
p06_title.m,
returns the title for problem 6.
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p07_fx.m,
evaluates the function for problem 7.
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p07_fx1.m,
first derivative of the function for problem 7.
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p07_fx2.m,
second derivative of the function for problem 7.
-
p07_range.m,
returns an interval bounding the root for problem 7.
-
p07_root.m,
returns a known root for problem 7.
-
p07_root_num.m,
returns the number of known roots for problem 7.
-
p07_start.m,
returns a starting point for problem 7.
-
p07_start_num.m,
returns the number of starting points for problem 7.
-
p07_title.m,
returns the title for problem 7.
-
p08_fx.m,
evaluates the function for problem 8.
-
p08_fx1.m,
first derivative of the function for problem 8.
-
p08_fx2.m,
second derivative of the function for problem 8.
-
p08_range.m,
returns an interval bounding the root for problem 8.
-
p08_root.m,
returns a known root for problem 8.
-
p08_root_num.m,
returns the number of known roots for problem 8.
-
p08_start.m,
returns a starting point for problem 8.
-
p08_start_num.m,
returns the number of starting points for problem 8.
-
p08_title.m,
returns the title for problem 8.
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p09_fx.m,
evaluates the function for problem 9.
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p09_fx1.m,
first derivative of the function for problem 9.
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p09_fx2.m,
second derivative of the function for problem 9.
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p09_range.m,
returns an interval bounding the root for problem 9.
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p09_root.m,
returns a known root for problem 9.
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p09_root_num.m,
returns the number of known roots for problem 9.
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p09_start.m,
returns a starting point for problem 9.
-
p09_start_num.m,
returns the number of starting points for problem 9.
-
p09_title.m,
returns the title for problem 9.
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p10_fx.m,
evaluates the function for problem 10.
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p10_fx1.m,
first derivative of the function for problem 10.
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p10_fx2.m,
second derivative of the function for problem 10.
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p10_range.m,
returns an interval bounding the root for problem 10.
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p10_root.m,
returns a known root for problem 10.
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p10_root_num.m,
returns the number of known roots for problem 10.
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p10_start.m,
returns a starting point for problem 10.
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p10_start_num.m,
returns the number of starting points for problem 10.
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p10_title.m,
returns the title for problem 10.
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p11_fx.m,
evaluates the function for problem 11.
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p11_fx1.m,
first derivative of the function for problem 11.
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p11_fx2.m,
second derivative of the function for problem 11.
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p11_range.m,
returns an interval bounding the root for problem 11.
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p11_root.m,
returns a known root for problem 11.
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p11_root_num.m,
returns the number of known roots for problem 11.
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p11_start.m,
returns a starting point for problem 10.
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p11_start_num.m,
returns the number of starting points for problem 11.
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p11_title.m,
returns the title for problem 11.
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p12_fx.m,
evaluates the function for problem 12.
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p12_fx1.m,
first derivative of the function for problem 12.
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p12_fx2.m,
second derivative of the function for problem 12.
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p12_range.m,
returns an interval bounding the root for problem 12.
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p12_root.m,
returns a known root for problem 12.
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p12_root_num.m,
returns the number of known roots for problem 12.
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p12_start.m,
returns a starting point for problem 10.
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p12_start_num.m,
returns the number of starting points for problem 12.
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p12_title.m,
returns the title for problem 12.
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p13_fx.m,
evaluates the function for problem 13.
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p13_fx1.m,
first derivative of the function for problem 13.
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p13_fx2.m,
second derivative of the function for problem 13.
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p13_range.m,
returns an interval bounding the root for problem 13.
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p13_root.m,
returns a known root for problem 13.
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p13_root_num.m,
returns the number of known roots for problem 13.
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p13_start.m,
returns a starting point for problem 10.
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p13_start_num.m,
returns the number of starting points for problem 13.
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p13_title.m,
returns the title for problem 13.
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p14_fx.m,
evaluates the function for problem 14.
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p14_fx1.m,
first derivative of the function for problem 14.
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p14_fx2.m,
second derivative of the function for problem 14.
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p14_range.m,
returns an interval bounding the root for problem 14.
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p14_root.m,
returns a known root for problem 14.
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p14_root_num.m,
returns the number of known roots for problem 14.
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p14_start.m,
returns a starting point for problem 14.
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p14_start_num.m,
returns the number of starting points for problem 14.
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p14_title.m,
returns the title for problem 14.
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p15_fx.m,
evaluates the function for problem 15.
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p15_fx1.m,
first derivative of the function for problem 15.
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p15_fx2.m,
second derivative of the function for problem 15.
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p15_range.m,
returns an interval bounding the root for problem 15.
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p15_root.m,
returns a known root for problem 15.
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p15_root_num.m,
returns the number of known roots for problem 15.
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p15_start.m,
returns a starting point for problem 15.
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p15_start_num.m,
returns the number of starting points for problem 15.
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p15_title.m,
returns the title for problem 15.
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p16_fx.m,
evaluates the function for problem 16.
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p16_fx1.m,
first derivative of the function for problem 16.
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p16_fx2.m,
second derivative of the function for problem 16.
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p16_range.m,
returns an interval bounding the root for problem 16.
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p16_root.m,
returns a known root for problem 16.
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p16_root_num.m,
returns the number of known roots for problem 160.
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p16_start.m,
returns a starting point for problem 16.
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p16_start_num.m,
returns the number of starting points for problem 16.
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p16_title.m,
returns the title for problem 16.
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p17_fx.m,
evaluates the function for problem 17.
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p17_fx1.m,
first derivative of the function for problem 17.
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p17_fx2.m,
second derivative of the function for problem 17.
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p17_range.m,
returns an interval bounding the root for problem 17.
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p17_root.m,
returns a known root for problem 17.
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p17_root_num.m,
returns the number of known roots for problem 17.
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p17_start.m,
returns a starting point for problem 17.
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p17_start_num.m,
returns the number of starting points for problem 17.
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p17_title.m,
returns the title for problem 17.
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p18_fx.m,
evaluates the function for problem 18.
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p18_fx1.m,
first derivative of the function for problem 18.
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p18_fx2.m,
second derivative of the function for problem 18.
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p18_range.m,
returns an interval bounding the root for problem 18.
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p18_root.m,
returns a known root for problem 18.
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p18_root_num.m,
returns the number of known roots for problem 18.
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p18_start.m,
returns a starting point for problem 18.
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p18_start_num.m,
returns the number of starting points for problem 18.
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p18_title.m,
returns the title for problem 18.
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p19_fx.m,
evaluates the function for problem P19.
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p19_fx1.m,
first derivative of the function for problem P19.
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p19_fx2.m,
second derivative of the function for problem P19.
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p19_range.m,
returns an interval bounding the root for problem P19.
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p19_root.m,
returns a known root for problem P19.
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p19_root_num.m,
returns the number of known roots for problem P19.
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p19_start.m,
returns a starting point for problem P19.
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p19_start_num.m,
returns the number of starting points for problem P19.
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p19_title.m,
returns the title for problem P19.
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r8_cube_root.m,
computes the cube root of an R8.
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r8_sign.m,
returns the sign of an R8.
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r8poly2_rroot.m,
returns the real parts of the roots of a quadratic polynomial.
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regula_falsi.m,
carries out the regula falsi method on any problem.
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secant.m,
carries out the secant method on any problem.
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timestamp.m,
prints the YMDHMS date as a timestamp.
Last revised on 10 October 2020.