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<html>
<head>
<title>
TEST_ZERO - Zero Finder Tests
</title>
</head>
<body bgcolor="#EEEEEE" link="#CC0000" alink="#FF3300" vlink="#000055">
<h1 align = "center">
TEST_ZERO <br> Zero Finder Tests
</h1>
<hr>
<p>
<b>TEST_ZERO</b>
is a FORTRAN90 library which
defines nonlinear functions that may be used to test zero finders.
</p>
<p>
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.
</p>
<p>
<b>TEST_ZERO</b> 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.
</p>
<p>
<b>TEST_ZERO</b> also includes implementations of some simple
zero finders, as a demonstration of how the package might be used.
</p>
<p>
The functions, which are accessible by number, are
<ol>
<li>
f(x) = sin ( x ) - x / 2.
</li>
<li>
f(x) = 2 * x - exp ( - x ).
</li>
<li>
f(x) = x * exp ( - x ).
</li>
<li>
f(x) = exp ( x ) - 1 / ( 10 * x )^2.
</li>
<li>
f(x) = ( x + 3 ) * ( x - 1 )^2.
</li>
<li>
f(x) = exp ( x ) - 2 - 1 / ( 10 * x )^2 + 2 / ( 100 * x )^3.
</li>
<li>
f(x) = x^3.
</li>
<li>
f(x) = cos ( x ) - x.
</li>
<li>
the Newton Baffler.
</li>
<li>
the Repeller.
</li>
<li>
the Pinhead.
</li>
<li>
Flat Stanley.
</li>
<li>
Lazy Boy.
</li>
<li>
the Camel.
</li>
<li>
a pathological function for Newton's method.
</li>
<li>
Kepler's Equation.
</li>
<li>
f(x) = x^3 - 2*x - 5, Wallis's function.
</li>
<li>
f(x) = (x-1)^7, written term by term.
</li>
</ol>
</p>
<h3 align = "center">
Licensing:
</h3>
<p>
The computer code and data files described and made available on this web page
are distributed under
<a href = "../../txt/gnu_lgpl.txt">the GNU LGPL license.</a>
</p>
<h3 align = "center">
Languages:
</h3>
<p>
<b>TEST_ZERO</b> is available in
<a href = "../../c_src/test_zero/test_zero.html">a C version</a> and
<a href = "../../cpp_src/test_zero/test_zero.html">a C++ version</a> and
<a href = "../../f77_src/test_zero/test_zero.html">a FORTRAN77 version</a> and
<a href = "../../f_src/test_zero/test_zero.html">a FORTRAN90 version</a> and
<a href = "../../m_src/test_zero/test_zero.html">a MATLAB version.</a>
</p>
<h3 align = "center">
Related Data and Programs:
</h3>
<p>
<a href = "../../f_src/bisection_integer/bisection_integer.html">
BISECTION_INTEGER</a>,
a FORTRAN90 library which
seeks an integer solution to the equation F(X)=0,
using bisection within a user-supplied change of sign interval [A,B].
</p>
<p>
<a href = "../../f_src/brent/brent.html">
BRENT</a>,
a FORTRAN90 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.
</p>
<p>
<a href = "../../cpp_src/gsl/gsl.html">
GSL</a>,
a C++ library which
includes rootfinding routines.
</p>
<p>
<a href = "../../f_src/nms/nms.html">
NMS</a>,
a FORTRAN90 library which
includes the zero finder FZERO.
</p>
<p>
<a href = "../../f_src/slatec/slatec.html">
SLATEC</a>,
a FORTRAN90 library which
includes the zero finder FZERO.
</p>
<p>
<a href = "../../f_src/zoomin/zoomin.html">
ZOOMIN</a>,
a FORTRAN90 library which
includes various zero finder routines.
</p>
<h3 align = "center">
Reference:
</h3>
<p>
<ol>
<li>
Richard Brent,<br>
Algorithms for Minimization without Derivatives,<br>
Dover, 2002,<br>
ISBN: 0-486-41998-3,<br>
LC: QA402.5.B74.
</li>
<li>
Peter Colwell,<br>
Solving Kepler's Equation Over Three Centuries,<br>
Willmann-Bell, 1993,<br>
ISBN: 0943396409,<br>
LC: QB355.5.C65.
</li>
<li>
George Donovan, Arnold Miller, Timothy Moreland,<br>
Pathological Functions for Newton's Method,<br>
American Mathematical Monthly, January 1993, pages 53-58.
</li>
<li>
Arnold Krommer, Christoph Ueberhuber,<br>
Numerical Integration on Advanced Computer Systems,<br>
Springer, 1994,<br>
ISBN: 3540584102,<br>
LC: QA299.3.K76.
</li>
<li>
Jean Meeus,<br>
Astronomical Algorithms,<br>
Second Edition,<br>
Willman-Bell, 1998,<br>
ISBN: 0943396611,<br>
LC: QB51.3.E43M42.
</li>
</ol>
</p>
<h3 align = "center">
Source Code:
</h3>
<p>
<ul>
<li>
<a href = "test_zero.f90">test_zero.f90</a>, the source code;
</li>
<li>
<a href = "test_zero.sh">test_zero.sh</a>,
commands to compile the source code;
</li>
</ul>
</p>
<h3 align = "center">
Examples and Tests:
</h3>
<p>
<ul>
<li>
<a href = "test_zero_prb.f90">test_zero_prb.f90</a>, the calling
program;
</li>
<li>
<a href = "test_zero_prb.sh">test_zero_prb.sh</a>,
commands to compile, link and run the calling program;
</li>
<li>
<a href = "test_zero_prb_output.txt">test_zero_prb_output.txt</a>,
the output file.
</li>
</ul>
</p>
<p>
PNG images of the graphs of some of the functions were made using MATLAB:
<ul>
<li>
<a href = "p01_fx.png">p01_fx.png</a>,
an image of P01_FX(X) over [-4,+4].
</li>
<li>
<a href = "p02_fx.png">p02_fx.png</a>,
an image of P02_FX(X) over [-0.5, +3.0].
</li>
<li>
<a href = "p03_fx.png">p03_fx.png</a>,
an image of P03_FX(X) over [-0.1,+4].
</li>
<li>
<a href = "p04_fx.png">p04_fx.png</a>,
an image of P04_FX(X) over [-4,+2].
</li>
<li>
<a href = "p05_fx.png">p05_fx.png</a>,
an image of P05_FX(X) over [-4,+2].
</li>
<li>
<a href = "p06_fx.png">p06_fx.png</a>,
an image of P06_FX(X) over [-4,+2].
</li>
<li>
<a href = "p07_fx.png">p07_fx.png</a>,
an image of P07_FX(X) over [-1,+1].
</li>
<li>
<a href = "p08_fx.png">p08_fx.png</a>,
an image of P08_FX(X) over [-4,+4].
</li>
<li>
<a href = "p09_fx.png">p09_fx.png</a>,
an image of P09_FX(X) over [5,7].
</li>
<li>
<a href = "p10_fx.png">p10_fx.png</a>,
an image of P10_FX(X) over [-2,+2].
</li>
<li>
<a href = "p11_fx.png">p11_fx.png</a>,
an image of P11_FX(X) over [+1,+10].
</li>
<li>
<a href = "p12_fx.png">p12_fx.png</a>,
an image of P12_FX(X) over [-0.5,+0.5].
</li>
<li>
<a href = "p13_fx.png">p13_fx.png</a>,
an image of P13_FX(X) over [0,100].
</li>
<li>
<a href = "p14_fx.png">p14_fx.png</a>,
an image of P14_FX(X) over [-0.5,+2.0].
</li>
<li>
<a href = "p15_fx.png">p15_fx.png</a>,
an image of P15_FX(X) over [-4,+4].
</li>
<li>
<a href = "p16_fx.png">p16_fx.png</a>,
an image of P16_FX(X) over [0,50].
</li>
<li>
<a href = "p17_fx.png">p17_fx.png</a>,
an image of P17_FX(X) over [-2,+4].
</li>
<li>
<a href = "p18_fx.png">p18_fx.png</a>,
an image of P18_FX(X) over [0.988,1.012].
</li>
</ul>
</p>
<h3 align = "center">
List of Routines:
</h3>
<p>
<ul>
<li>
<b>BISECTION</b> carries out the bisection method to seek a root of F(X) = 0.
</li>
<li>
<b>BRENT</b> implements the Brent bisection-based zero finder.
</li>
<li>
<b>MULLER</b> carries out Muller's method for seeking a real root of a nonlinear function.
</li>
<li>
<b>NEWTON</b> carries out Newton's method to seek a root of F(X) = 0.
</li>
<li>
<b>P00_FX</b> evaluates a function specified by problem number.
</li>
<li>
<b>P00_FX1</b> evaluates the first derivative of a function specified by problem number.
</li>
<li>
<b>P00_FX2</b> evaluates the second derivative of a function specified by problem number.
</li>
<li>
<b>P00_INTERVAL</b> returns an interval bounding the root for any problem.
</li>
<li>
<b>P00_NPROB</b> returns the number of problems available.
</li>
<li>
<b>P00_ROOT</b> returns a root for any problem.
</li>
<li>
<b>P00_START</b> returns one or more starting points for any problem.
</li>
<li>
<b>P00_TITLE</b> returns the title for a given problem.
</li>
<li>
<b>P01_FX</b> evaluates sin ( x ) - x / 2.
</li>
<li>
<b>P01_FX1</b> evaluates the derivative of the function for problem 1.
</li>
<li>
<b>P01_FX2</b> evaluates the second derivative of the function for problem 1.
</li>
<li>
<b>P01_INTERVAL</b> returns an interval bounding the root for problem 1.
</li>
<li>
<b>P01_ROOT</b> returns a root for problem 1.
</li>
<li>
<b>P01_START</b> returns one or more starting points for problem 1.
</li>
<li>
<b>P01_TITLE</b> returns the title of problem 1.
</li>
<li>
<b>P02_FX</b> evaluates 2 * x - exp ( - x ).
</li>
<li>
<b>P02_FX1</b> evaluates the derivative of the function for problem 2.
</li>
<li>
<b>P02_FX2</b> evaluates the second derivative of the function for problem 2.
</li>
<li>
<b>P02_INTERVAL</b> returns an interval bounding the root for problem 2.
</li>
<li>
<b>P02_ROOT</b> returns a root for problem 2.
</li>
<li>
<b>P02_START</b> returns one or more starting points for problem 2.
</li>
<li>
<b>P02_TITLE</b> returns the title of problem 2.
</li>
<li>
<b>P03_FX</b> evaluates x * exp ( - x ).
</li>
<li>
<b>P03_FX1</b> evaluates the derivative of the function for problem 3.
</li>
<li>
<b>P03_FX2</b> evaluates the second derivative of the function for problem 3.
</li>
<li>
<b>P03_INTERVAL</b> returns an interval bounding the root for problem 3.
</li>
<li>
<b>P03_ROOT</b> returns a root for problem 3.
</li>
<li>
<b>P03_START</b> returns one or more starting points for problem 3.
</li>
<li>
<b>P03_TITLE</b> returns the title of problem 3.
</li>
<li>
<b>P04_FX</b> evaluates exp ( x ) - 1 / ( 10 * x )^2.
</li>
<li>
<b>P04_FX1</b> evaluates the derivative of the function for problem 4.
</li>
<li>
<b>P04_FX2</b> evaluates the second derivative of the function for problem 4.
</li>
<li>
<b>P04_INTERVAL</b> returns an interval bounding the root for problem 4.
</li>
<li>
<b>P04_ROOT</b> returns a root for problem 4.
</li>
<li>
<b>P04_START</b> returns one or more starting points for problem 4.
</li>
<li>
<b>P04_TITLE</b> returns the title of problem 4.
</li>
<li>
<b>P05_FX</b> evaluates ( x + 3 ) * ( x - 1 )^2.
</li>
<li>
<b>P05_FX1</b> evaluates the derivative of the function for problem 5.
</li>
<li>
<b>P05_FX2</b> evaluates the second derivative of the function for problem 5.
</li>
<li>
<b>P05_INTERVAL</b> returns an interval bounding the root for problem 5.
</li>
<li>
<b>P05_ROOT</b> returns a root for problem 5.
</li>
<li>
<b>P05_START</b> returns one or more starting points for problem 5.
</li>
<li>
<b>P05_TITLE</b> returns the title of problem 5.
</li>
<li>
<b>P06_FX</b> evaluates exp ( x ) - 2 - 1 / ( 10 * x )^2 + 2 / ( 100 * x )^3.
</li>
<li>
<b>P06_FX1</b> evaluates the derivative of the function for problem 6.
</li>
<li>
<b>P06_FX2</b> evaluates the second derivative of the function for problem 6.
</li>
<li>
<b>P06_INTERVAL</b> returns an interval bounding the root for problem 6.
</li>
<li>
<b>P06_ROOT</b> returns a root for problem 6.
</li>
<li>
<b>P06_START</b> returns one or more starting points for problem 6.
</li>
<li>
<b>P06_TITLE</b> returns the title of problem 6.
</li>
<li>
<b>P07_FX</b> evaluates x^3.
</li>
<li>
<b>P07_FX1</b> evaluates the derivative of the function for problem 7.
</li>
<li>
<b>P07_FX2</b> evaluates the second derivative of the function for problem 7.
</li>
<li>
<b>P07_INTERVAL</b> returns an interval bounding the root for problem 7.
</li>
<li>
<b>P07_ROOT</b> returns a root for problem 7.
</li>
<li>
<b>P07_START</b> returns one or more starting points for problem 7.
</li>
<li>
<b>P07_TITLE</b> returns the title of problem 7.
</li>
<li>
<b>P08_FX</b> evaluates cos ( x ) - x.
</li>
<li>
<b>P08_FX1</b> evaluates the derivative of the function for problem 8.
</li>
<li>
<b>P08_FX2</b> evaluates the second derivative of the function for problem 8.
</li>
<li>
<b>P08_INTERVAL</b> returns an interval bounding the root for problem 8.
</li>
<li>
<b>P08_ROOT</b> returns a root for problem 8.
</li>
<li>
<b>P08_START</b> returns one or more starting points for problem 8.
</li>
<li>
<b>P08_TITLE</b> returns the title of problem 8.
</li>
<li>
<b>P09_FX</b> evaluates the Newton Baffler.
</li>
<li>
<b>P09_FX1</b> evaluates the derivative of the function for problem 9.
</li>
<li>
<b>P09_FX2</b> evaluates the second derivative of the function for problem 9.
</li>
<li>
<b>P09_INTERVAL</b> returns an interval bounding the root for problem 9.
</li>
<li>
<b>P09_ROOT</b> returns a root for problem 9.
</li>
<li>
<b>P09_START</b> returns one or more starting points for problem 9.
</li>
<li>
<b>P09_TITLE</b> returns the title of problem 9.
</li>
<li>
<b>P10_FX</b> evaluates the Repeller.
</li>
<li>
<b>P10_FX1</b> evaluates the derivative of the function for problem 10.
</li>
<li>
<b>P10_FX2</b> evaluates the second derivative of the function for problem 10.
</li>
<li>
<b>P10_INTERVAL</b> returns an interval bounding the root for problem 10.
</li>
<li>
<b>P10_ROOT</b> returns a root for problem 10.
</li>
<li>
<b>P10_START</b> returns one or more starting points for problem 10.
</li>
<li>
<b>P10_TITLE</b> returns the title of problem 10.
</li>
<li>
<b>P11_FX</b> evaluates the Pinhead.
</li>
<li>
<b>P11_FX1</b> evaluates the derivative of the function for problem 11.
</li>
<li>
<b>P11_FX2</b> evaluates the second derivative of the function for problem 11.
</li>
<li>
<b>P11_INTERVAL</b> returns an interval bounding the root for problem 11.
</li>
<li>
<b>P11_ROOT</b> returns a root for problem 11.
</li>
<li>
<b>P11_START</b> returns one or more starting points for problem 11.
</li>
<li>
<b>P11_TITLE</b> returns the title of problem 11.
</li>
<li>
<b>P12_FX</b> evaluates Flat Stanley.
</li>
<li>
<b>P12_FX1</b> evaluates the derivative of the function for problem 12.
</li>
<li>
<b>P12_FX2</b> evaluates the second derivative of the function for problem 12.
</li>
<li>
<b>P12_INTERVAL</b> returns an interval bounding the root for problem 12.
</li>
<li>
<b>P12_ROOT</b> returns a root for problem 12.
</li>
<li>
<b>P12_START</b> returns one or more starting points for problem 12.
</li>
<li>
<b>P12_TITLE</b> returns the title of problem 12.
</li>
<li>
<b>P13_FX</b> evaluates Lazy Boy.
</li>
<li>
<b>P13_FX1</b> evaluates the derivative of the function for problem 13.
</li>
<li>
<b>P13_FX2</b> evaluates the second derivative of the function for problem 13.
</li>
<li>
<b>P13_INTERVAL</b> returns an interval bounding the root for problem 13.
</li>
<li>
<b>P13_ROOT</b> returns a root for problem 13.
</li>
<li>
<b>P13_START</b> returns one or more starting points for problem 13.
</li>
<li>
<b>P13_TITLE</b> returns the title of problem 13.
</li>
<li>
<b>P14_FX</b> evaluates the Camel.
</li>
<li>
<b>P14_FX1</b> evaluates the derivative of the function for problem 14.
</li>
<li>
<b>P14_FX2</b> evaluates the second derivative of the function for problem 14.
</li>
<li>
<b>P14_INTERVAL</b> returns an interval bounding the root for problem 14.
</li>
<li>
<b>P14_ROOT</b> returns a root for problem 14.
</li>
<li>
<b>P14_START</b> returns one or more starting points for problem 14.
</li>
<li>
<b>P14_TITLE</b> returns the title of problem 14.
</li>
<li>
<b>P15_FX</b> evaluates a pathological function for Newton's method.
</li>
<li>
<b>P15_FX1</b> evaluates the derivative of the function for problem 15.
</li>
<li>
<b>P15_FX2</b> evaluates the second derivative of the function for problem 15.
</li>
<li>
<b>P15_INTERVAL</b> returns an interval bounding the root for problem 15.
</li>
<li>
<b>P15_ROOT</b> returns a root for problem 15.
</li>
<li>
<b>P15_START</b> returns one or more starting points for problem 15.
</li>
<li>
<b>P15_TITLE</b> returns the title of problem 15.
</li>
<li>
<b>P16_FX</b> evaluates Kepler's Equation.
</li>
<li>
<b>P16_FX1</b> evaluates the derivative of the function for problem 16.
</li>
<li>
<b>P16_FX2</b> evaluates the second derivative of the function for problem 16.
</li>
<li>
<b>P16_INTERVAL</b> returns an interval bounding the root for problem 16.
</li>
<li>
<b>P16_P</b> sets, gets, or prints the parameter values for problem 16.
</li>
<li>
<b>P16_P_GET</b> gets the parameter values for problem 16.
</li>
<li>
<b>P16_P_PRINT</b> prints the parameter values for problem 16.
</li>
<li>
<b>P16_P_SET</b> sets the parameter values for problem 16.
</li>
<li>
<b>P16_ROOT</b> returns a root for problem 16.
</li>
<li>
<b>P16_START</b> returns one or more starting points for problem 16.
</li>
<li>
<b>P16_TITLE</b> returns the title of problem 16.
</li>
<li>
<b>P17_FX</b> evaluates the bogus function (returns random numbers).
</li>
<li>
<b>P17_FX1</b> evaluates the derivative of the function for problem 17.
</li>
<li>
<b>P17_FX2</b> evaluates the second derivative of the function for problem 17.
</li>
<li>
<b>P17_INTERVAL</b> returns an interval bounding the root for problem 17.
</li>
<li>
<b>P17_ROOT</b> returns a root for problem 17.
</li>
<li>
<b>P17_START</b> returns one or more starting points for problem 17.
</li>
<li>
<b>P17_TITLE</b> returns the title of problem 17.
</li>
<li>
<b>P18_FX</b> evaluates Wallis's function, f(x) = x^3 - 2*x - 5.
</li>
<li>
<b>P18_FX1</b> evaluates the derivative of the function for problem 18.
</li>
<li>
<b>P18_FX2</b> evaluates the second derivative of the function for problem 18.
</li>
<li>
<b>P18_INTERVAL</b> returns an interval bounding the root for problem 18.
</li>
<li>
<b>P18_ROOT</b> returns a root for problem 18.
</li>
<li>
<b>P18_START</b> returns one or more starting points for problem 18.
</li>
<li>
<b>P18_TITLE</b> returns the title of problem 18.
</li>
<li>
<b>P19_FX</b> evaluates the function with a very thin pole.
</li>
<li>
<b>P19_FX1</b> evaluates the derivative of the function for problem 19.
</li>
<li>
<b>P19_FX2</b> evaluates the second derivative of the function for problem 19.
</li>
<li>
<b>P19_INTERVAL</b> returns an interval bounding the root for problem 19.
</li>
<li>
<b>P19_ROOT</b> returns a root for problem 19.
</li>
<li>
<b>P19_START</b> returns one or more starting points for problem 19.
</li>
<li>
<b>P19_TITLE</b> returns the title of problem 19.
</li>
<li>
<b>R8_CUBE_ROOT</b> returns the cube root of a real number.
</li>
<li>
<b>R8_SWAP</b> switches two R8's.
</li>
<li>
<b>R8POLY2_RROOT</b> returns the real parts of the roots of a quadratic polynomial.
</li>
<li>
<b>REGULA_FALSI</b> carries out the Regula Falsi method to seek a root of F(X) = 0.
</li>
<li>
<b>SECANT</b> carries out the secant method to seek a root of F(X) = 0.
</li>
<li>
<b>TIMESTAMP</b> prints the current YMDHMS date as a time stamp.
</li>
</ul>
</p>
<p>
You can go up one level to <a href = "../f_src.html">
the FORTRAN90 source codes</a>.
</p>
<hr>
<i>
Last revised on 12 April 2008.
</i>
<!-- John Burkardt -->
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