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<html>
<head>
<title>
TEST_NLS - Nonlinear least squares test problems
</title>
</head>
<body bgcolor="#EEEEEE" link="#CC0000" alink="#FF3300" vlink="#000055">
<h1 align = "center">
TEST_NLS <br> Nonlinear least squares test problems
</h1>
<hr>
<p>
<b>TEST_NLS</b>
is a FORTRAN90 library which
defines test problems for
least squares minimization.
</p>
<p>
The typical problem comprises M functions in N variables,
and it is desired to minimize the sum of squares of the values of
the functions. Each problem is specified by a starting point, a
function and jacobian routines.
</p>
<p>
The problems include:
<ol>
<li>
Linear function, full rank
</li>
<li>
Linear function, rank 1.
</li>
<li>
Linear function, rank 1, zero columns and rows.
</li>
<li>
Rosenbrock function.
</li>
<li>
Helical valley function.
</li>
<li>
Powell singular function.
</li>
<li>
Freudenstein/Roth function.
</li>
<li>
Bard function.
</li>
<li>
Kowalik and Osborne function.
</li>
<li>
Meyer function.
</li>
<li>
Watson function.
</li>
<li>
Box 3-dimensional function.
</li>
<li>
Jennrich and Sampson function.
</li>
<li>
Brown and Dennis function.
</li>
<li>
Chebyquad function.
</li>
<li>
Brown almost-linear function.
</li>
<li>
Osborne function 1.
</li>
<li>
Osborne function 2.
</li>
<li>
Hanson function 1
</li>
<li>
Hanson function 2
</li>
<li>
McKeown function 1
</li>
<li>
McKeown function 2
</li>
<li>
McKeown function 3
</li>
<li>
Devilliers and Glasser function 1
</li>
<li>
Devilliers and Glasser function 2
</li>
<li>
Madsen example
</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">
Related Data and Programs:
</h3>
<p>
<a href = "../../f_src/dqed/dqed.html">
DQED</a>,
a FORTRAN90 library which
seeks the solution of bounded or constrained minimization problems.
</p>
<p>
<a href = "../../f_src/minpack/minpack.html">
MINPACK</a>,
a FORTRAN90 library which
seeks the solution of nonlinear equations, or the least squares minimization
of the residual.
</p>
<h3 align = "center">
Reference:
</h3>
<p>
<ol>
<li>
John Dennis, David Gay, and Phuong Vu, <br>
A new nonlinear equations test problem, <br>
Technical Report 83-16, <br>
Mathematical Sciences Department,<br>
Rice University, (1983 - revised 1985).
</li>
<li>
N. de Villiers and D. Glasser, <br>
A continuation method for nonlinear regression, <br>
SIAM Journal of Numerical Analysis, <br>
Volume 18, pages 1139-1154, 1981.
</li>
<li>
C. Fraley, <br>
Solution of nonlinear least-squares problems, <br>
Technical Report STAN-CS-1165, <br>
Computer Science Department, <br>
Stanford University, 1987.
</li>
<li>
C. Fraley, <br>
Software performance on nonlinear least-squares problems, <br>
Technical Report SOL 88-17, <br>
Systems Optimization Laboratory, <br>
Department of Operations Research, <br>
Stanford University, 1988.
</li>
<li>
JJ McKeown, <br>
Specialized versus general-purpose algorithms for functions that
are sums of squared terms, <br>
Math. Prog., <br>
Volume 9, pages 57-68, 1975.
</li>
<li>
JJ McKeown, <br>
On algorithms for sums of squares problems, <br>
in Towards Global Optimization, <br>
L. C. W. Dixon and G. Szego (editors), <br>
North-Holland, pages 229-257, 1975.
</li>
<li>
Jorge More, Burton Garbow, Kenneth Hillstrom,<br>
Testing unconstrained optimization software,<br>
ACM Transactions on Mathematical Software,<br>
Volume 7, Number 1, March 1981, pages 17-41.
</li>
<li>
Jorge More, Burton Garbow, Kenneth Hillstrom,<br>
Algorithm 566:
FORTRAN Subroutines for Testing unconstrained optimization software,<br>
ACM Transactions on Mathematical Software,<br>
Volume 7, Number 1, March 1981, pages 136-140.
</li>
<li>
Douglas Salane, <br>
A continuation approach for solving large residual nonlinear
least squares problems, <br>
SIAM Journal of Scientific and Statistical Computing, <br>
Volume 8, pages 655-671, 1987.
</lli>
</ol>
</p>
<h3 align = "center">
Source Code:
</h3>
<p>
<ul>
<li>
<a href = "test_nls.f90">test_nls.f90</a>, the source code.
</li>
<li>
<a href = "test_nls.sh">test_nls.sh</a>,
commands to compile the source code.
</li>
</ul>
</p>
<h3 align = "center">
Examples and Tests:
</h3>
<p>
<ul>
<li>
<a href = "test_nls_prb.f90">test_nls_prb.f90</a>, a sample problem.
</li>
<li>
<a href = "test_nls_prb.sh">test_nls_prb.sh</a>,
commands to compile, link and run the sample problem.
</li>
<li>
<a href = test_nls_prb_output.txt">test_nls_prb_output.txt</a>,
the output file.
</li>
</ul>
</p>
<h3 align = "center">
List of Routines:
</h3>
<p>
<ul>
<li>
<b>P00_F</b> evaluates the M nonlinear functions for any problem.
</li>
<li>
<b>P00_G</b> evaluates the least squares gradient for any problem.
</li>
<li>
<b>P00_J</b> evaluates the jacobian for any problem.
</li>
<li>
<b>P00_MN</b> reports the default sizes M and N for the least squares problems.
</li>
<li>
<b>P00_MPROB</b> returns the number of problems available.
</li>
<li>
<b>P00_SOL</b> returns the solution of the problem, if known.
</li>
<li>
<b>P00_START</b> sets a starting point for any problem.
</li>
<li>
<b>P00_title</b> sets the title of any problem.
</li>
<li>
<b>P01_F</b> evaluates the M nonlinear functions for problem 1.
</li>
<li>
<b>P01_J</b> evaluates the jacobian for problem 1.
</li>
<li>
<b>P01_SOL</b> returns the solution of problem 1.
</li>
<li>
<b>P01_START</b> sets a starting point for problem 1.
</li>
<li>
<b>P01_title</b> specifies the title for problem 1.
</li>
<li>
<b>P02_F</b> evaluates the M nonlinear functions for problem 2.
</li>
<li>
<b>P02_J</b> evaluates the jacobian for problem 2.
</li>
<li>
<b>P02_SOL</b> returns the solution of problem 2.
</li>
<li>
<b>P02_START</b> sets a starting point for problem 2.
</li>
<li>
<b>P02_title</b> specifies the title for problem 2.
</li>
<li>
<b>P03_F</b> evaluates the M nonlinear functions for problem 3.
</li>
<li>
<b>P03_J</b> evaluates the jacobian for problem 3.
</li>
<li>
<b>P03_START</b> sets a starting point for problem 3.
</li>
<li>
<b>P03_SOL</b> returns the solution of problem 3.
</li>
<li>
<b>P03_title</b> specifies the title for problem 3.
</li>
<li>
<b>P04_F</b> evaluates the M nonlinear functions for problem 4.
</li>
<li>
<b>P04_J</b> evaluates the jacobian for problem 4.
</li>
<li>
<b>P04_SOL</b> returns the solution of problem 4.
</li>
<li>
<b>P04_START</b> sets a starting point for problem 4.
</li>
<li>
<b>P04_title</b> specifies the title for problem 4.
</li>
<li>
<b>P05_F</b> evaluates the M nonlinear functions for problem 5.
</li>
<li>
<b>P05_J</b> evaluates the jacobian for problem 5.
</li>
<li>
<b>P05_SOL</b> returns the solution of problem 5.
</li>
<li>
<b>P05_START</b> sets a starting point for problem 5.
</li>
<li>
<b>P05_title</b> specifies the title for problem 5.
</li>
<li>
<b>P06_F</b> evaluates the M nonlinear functions for problem 6.
</li>
<li>
<b>P06_J</b> evaluates the jacobian for problem 6.
</li>
<li>
<b>P06_SOL</b> returns the solution of problem 6.
</li>
<li>
<b>P06_START</b> sets a starting point for problem 6.
</li>
<li>
<b>P06_title</b> specifies the title for problem 6.
</li>
<li>
<b>P07_F</b> evaluates the M nonlinear functions for problem 7.
</li>
<li>
<b>P07_J</b> evaluates the jacobian for problem 7.
</li>
<li>
<b>P07_SOL</b> returns the solution of problem 7.
</li>
<li>
<b>P07_START</b> sets a starting point for problem 7.
</li>
<li>
<b>P07_title</b> specifies the title for problem 7.
</li>
<li>
<b>P08_F</b> evaluates the M nonlinear functions for problem 8.
</li>
<li>
<b>P08_J</b> evaluates the jacobian for problem 8.
</li>
<li>
<b>P08_SOL</b> returns the solution of problem 8.
</li>
<li>
<b>P08_START</b> sets a starting point for problem 8.
</li>
<li>
<b>P08_title</b> specifies the title for problem 8.
</li>
<li>
<b>P09_F</b> evaluates the M nonlinear functions for problem 9.
</li>
<li>
<b>P09_J</b> evaluates the jacobian for problem 9.
</li>
<li>
<b>P09_SOL</b> returns the solution of problem 9.
</li>
<li>
<b>P09_START</b> sets a starting point for problem 9.
</li>
<li>
<b>P09_title</b> specifies the title for problem 9.
</li>
<li>
<b>P10_F</b> evaluates the M nonlinear functions for problem 10.
</li>
<li>
<b>P09_J</b> evaluates the jacobian for problem 9.
</li>
<li>
<b>P10_SOL</b> returns the solution of problem 10.
</li>
<li>
<b>P10_START</b> sets a starting point for problem 10.
</li>
<li>
<b>P10_title</b> specifies the title for problem 10.
</li>
<li>
<b>P11_F</b> evaluates the M nonlinear functions for problem 11.
</li>
<li>
<b>P11_J</b> evaluates the jacobian for problem 11.
</li>
<li>
<b>P11_SOL</b> returns the solution of problem 11.
</li>
<li>
<b>P11_START</b> sets a starting point for problem 11.
</li>
<li>
<b>P11_title</b> specifies the title for problem 11.
</li>
<li>
<b>P12_F</b> evaluates the M nonlinear functions for problem 12.
</li>
<li>
<b>P12_J</b> evaluates the jacobian for problem 12.
</li>
<li>
<b>P12_SOL</b> returns the solution of problem 12.
</li>
<li>
<b>P12_START</b> sets a starting point for problem 12.
</li>
<li>
<b>P12_title</b> specifies the title for problem 12.
</li>
<li>
<b>P13_F</b> evaluates the M nonlinear functions for problem 13.
</li>
<li>
<b>P13_J</b> evaluates the jacobian for problem 13.
</li>
<li>
<b>P13_SOL</b> returns the solution of problem 13.
</li>
<li>
<b>P13_START</b> sets a starting point for problem 13.
</li>
<li>
<b>P13_title</b> specifies the title for problem 13.
</li>
<li>
<b>P14_F</b> evaluates the M nonlinear functions for problem 14.
</li>
<li>
<b>P14_J</b> evaluates the jacobian for problem 14.
</li>
<li>
<b>P14_SOL</b> returns the solution of problem 14.
</li>
<li>
<b>P14_START</b> sets a starting point for problem 14.
</li>
<li>
<b>P14_title</b> specifies the title for problem 14.
</li>
<li>
<b>P15_F</b> evaluates the M nonlinear functions for problem 15.
</li>
<li>
<b>P15_J</b> evaluates the jacobian for problem 15.
</li>
<li>
<b>P15_SOL</b> returns the solution of problem 15.
</li>
<li>
<b>P15_START</b> sets a starting point for problem 15.
</li>
<li>
<b>P15_title</b> specifies the title for problem 15.
</li>
<li>
<b>P16_F</b> evaluates the M nonlinear functions for problem 16.
</li>
<li>
<b>P16_J</b> evaluates the jacobian for problem 16.
</li>
<li>
<b>P16_SOL</b> returns the solution of problem 16.
</li>
<li>
<b>P16_START</b> sets a starting point for problem 16.
</li>
<li>
<b>P16_title</b> specifies the title for problem 16.
</li>
<li>
<b>P17_F</b> evaluates the M nonlinear functions for problem 17.
</li>
<li>
<b>P17_J</b> evaluates the jacobian for problem 17.
</li>
<li>
<b>P17_SOL</b> returns the solution of problem 17.
</li>
<li>
<b>P17_START</b> sets a starting point for problem 17.
</li>
<li>
<b>P17_title</b> specifies the title for problem 17.
</li>
<li>
<b>P18_F</b> evaluates the M nonlinear functions for problem 18.
</li>
<li>
<b>P18_J</b> evaluates the jacobian for problem 18.
</li>
<li>
<b>P18_SOL</b> returns the solution of problem 18.
</li>
<li>
<b>P18_START</b> sets a starting point for problem 18.
</li>
<li>
<b>P18_title</b> specifies the title for problem 18.
</li>
<li>
<b>P19_F</b> evaluates the M nonlinear functions for problem 19.
</li>
<li>
<b>P19_J</b> evaluates the jacobian for problem 19.
</li>
<li>
<b>P19_SOL</b> returns the solution of problem 19.
</li>
<li>
<b>P19_START</b> sets a starting point for problem 19.
</li>
<li>
<b>P19_title</b> specifies the title for problem 19.
</li>
<li>
<b>P20_F</b> evaluates the M nonlinear functions for problem 20.
</li>
<li>
<b>P20_J</b> evaluates the jacobian for problem 20.
</li>
<li>
<b>P20_SOL</b> returns the solution of problem 20.
</li>
<li>
<b>P20_START</b> sets a starting point for problem 20.
</li>
<li>
<b>P20_title</b> specifies the title for problem 20.
</li>
<li>
<b>P21_F</b> evaluates the M nonlinear functions for problem 21.
</li>
<li>
<b>P21_J</b> evaluates the jacobian for problem 21.
</li>
<li>
<b>P21_SOL</b> returns the solution of problem 21.
</li>
<li>
<b>P21_START</b> sets a starting point for problem 21.
</li>
<li>
<b>P21_title</b> specifies the title for problem 21.
</li>
<li>
<b>P22_F</b> evaluates the M nonlinear functions for problem 22.
</li>
<li>
<b>P22_J</b> evaluates the jacobian for problem 22.
</li>
<li>
<b>P22_SOL</b> returns the solution of problem 22.
</li>
<li>
<b>P22_START</b> sets a starting point for problem 22.
</li>
<li>
<b>P22_title</b> specifies the title for problem 22.
</li>
<li>
<b>P23_F</b> evaluates the M nonlinear functions for problem 23.
</li>
<li>
<b>P23_J</b> evaluates the jacobian for problem 23.
</li>
<li>
<b>P23_SOL</b> returns the solution of problem 23.
</li>
<li>
<b>P23_START</b> sets a starting point for problem 23.
</li>
<li>
<b>P23_title</b> specifies the title for problem 23.
</li>
<li>
<b>P24_F</b> evaluates the M nonlinear functions for problem 24.
</li>
<li>
<b>P24_J</b> evaluates the jacobian for problem 24.
</li>
<li>
<b>P24_SOL</b> returns the solution of problem 24.
</li>
<li>
<b>P24_START</b> sets a starting point for problem 24.
</li>
<li>
<b>P24_title</b> specifies the title for problem 24.
</li>
<li>
<b>P25_F</b> evaluates the M nonlinear functions for problem 25.
</li>
<li>
<b>P25_J</b> evaluates the jacobian for problem 25.
</li>
<li>
<b>P25_SOL</b> returns the solution of problem 25.
</li>
<li>
<b>P25_START</b> sets a starting point for problem 25.
</li>
<li>
<b>P25_title</b> specifies the title for problem 25.
</li>
<li>
<b>P26_F</b> evaluates the M nonlinear functions for problem 26.
</li>
<li>
<b>P26_J</b> evaluates the jacobian for problem 26.
</li>
<li>
<b>P26_SOL</b> returns the solution of problem 26.
</li>
<li>
<b>P26_START</b> sets a starting point for problem 26.
</li>
<li>
<b>P26_title</b> specifies the title for problem 26.
</li>
<li>
<b>R_PI</b> returns the value of pi.
</li>
<li>
<b>RMAT_PRINT</b> prints a real matrix.
</li>
<li>
<b>RVEC_PRINT</b> prints a real vector.
</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 30 August 2005.
</i>
<!-- John Burkardt -->
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