nl2sol/nl2sol.html

<html>

  <head>
    <title>
      NL2SOL - An Adaptive Nonlinear Least-Squares Algorithm
    </title>
  </head>

  <body bgcolor="#EEEEEE" link="#CC0000" alink="#FF3300" vlink="#000055">

    <h1 align = "center">
      NL2SOL <br> An Adaptive Nonlinear Least-Squares Algorithm
    </h1>

    <hr>

    <p>
      <b>NL2SOL</b>
      is a FORTRAN90 library which
      implements an adaptive nonlinear least-squares algorithm,
      by John Dennis, David Gay, Roy Welsch.
    </p>

    <p>
      <b>NL2SOL</b> assumes that there are <b>NVAR</b> variables <b>X</b>,
      and <b>MEQN</b> nonlinear "residual functions" <b>F(X)</b>, and that
      the task is to find a solution <b>X</b> which minimizes the Euclidean
      norm of <b>||F(X)||</b>.
    </p>

    <p>
      <b>NL2SOL</b> is ACM TOMS algorithm 573.
      The original, true, correct version of ACM TOMS 573 is available
      in the TOMS subdirectory of
      <a href = "http://www.netlib.org/">the NETLIB web site</a>.
    </p>

    <h3 align = "center">
      Languages:
    </h3>

    <p>
      <b>NL2SOL</b> is available in
      <a href = "../../f77_src/nl2sol/nl2sol.html">a FORTRAN77 version</a> and
      <a href = "../../f_src/nl2sol/nl2sol.html">a FORTRAN90 version</a>.
    </p>

    <h3 align = "center">
      Related Data and Programs:
    </h3>

    <p>
      <a href = "../../f_src/bvls/bvls.html">
      BVLS</a>,
      a FORTRAN90 library which
      applies least squares methods to solve a linear system for which
      lower and upper constraints may have been placed on every variable.
    </p>

    <p>
      <a href = "../../f_src/dqed/dqed.html">
      DQED</a>,
      a FORTRAN90 library which
      solves constrained least squares problems.
    </p>

    <p>
      <a href = "../../f_src/minpack/minpack.html">
      MINPACK</a>,
      a FORTRAN90 library which
      carries out the least squares minimization of the residual
      of a set of linear or nonlinear equations.
    </p>

    <p>
      <a href = "../../f_src/praxis/praxis.html">
      PRAXIS</a>,
      a FORTRAN90 library which
      minimizes a scalar
      function of several variables.
    </p>

    <p>
      <a href = "../../f_src/test_opt/test_opt.html">
      TEST_OPT</a>,
      a FORTRAN90 library which 
      defines test problems
      requiring the minimization of a scalar function of several variables.
    </p>

    <p>
      <a href = "../../f_src/toms611/toms611.html">
      TOMS611</a>,
      a FORTRAN90 library which
      can be used to seek the minimizer of a scalar functional
      of multiple variables.
    </p>

    <h3 align = "center">
      Author:
    </h3>

    <p>
      John Dennis, David Gay, Roy Welsch
    </p>

    <h3 align = "center">
      Reference:
    </h3>

    <p>
      <ol>
        <li>
          K M Brown,<br>
          A Quadratically Convergent Newton-like Method Based upon
          Gaussian Elimination,<br>
          SIAM Journal on Numerical Analysis,<br>
          Volume 6, pages 560-569, 1969.
        </li>
        <li>
          John Dennis, David Gay, Roy Welsch,<br>
          Algorithm 573:<br>
          An Adaptive Nonlinear Least-Squares Algorithm, <br>
          ACM Transactions on Mathematical Software,<br>
          Volume 7, Number 3, 1981, pages 367-383.
        </li>
        <li>
          David Gay,<br>
          Computing Optimal Locally Constrained Steps,<br>
          SIAM Journal on Scientific and Statistical Computing, <br>
          Volume 2, Number 2, pages 186-197, 1981.
        </li>
        <li>
          Philip Gill, Walter Murray,<br>
          Algorithms for the Solution of the
          Non-linear Least-squares Problem, <br>
          SIAM Journal on Numerical Analysis,<br>
          Volume 15, Number 5, pages 977-991, 1978.<br>
        </li>
        <li>
          Charles Lawson, Richard Hanson, <br>
          Solving Least Squares Problems, <br>
          Prentice-Hall.
        </li>
      </ol>
    </p>

    <h3 align = "center">
      Source Code:
    </h3>

    <p>
      <ul>
        <li>
          <a href = "nl2sol.f90">nl2sol.f90</a>, the source code.
        </li>
        <li>
          <a href = "nl2sol.sh">nl2sol.sh</a>,
          commands to compile the source code.
        </li>
      </ul>
    </p>

    <h3 align = "center">
      Examples and Tests:
    </h3>

    <p>
      <b>NL2SOL_PRB1</b> is a simple test.
      <ul>
        <li>
          <a href = "nl2sol_prb1.f90">nl2sol_prb1.f90</a>,
          a sample problem.
        </li>
        <li>
          <a href = "nl2sol_prb1.sh">nl2sol_prb1.sh</a>,
          commands to compile and run the problem.
        </li>
        <li>
          <a href = "nl2sol_prb1_output.txt">nl2sol_prb1_output.txt</a>,
          the output file.
        </li>
      </ul>
    </p>

    <p>
      <b>NL2SOL_PRB2</b> is an extensive set of tests.
      <ul>
        <li>
          <a href = "nl2sol_prb2.f90">nl2sol_prb2.f90</a>,
          a sample problem.
        </li>
        <li>
          <a href = "nl2sol_prb2.sh">nl2sol_prb2.sh</a>,
          commands to compile and run the problem.
        </li>
        <li>
          <a href = "nl2sol_prb2_output.txt">nl2sol_prb2_output.txt</a>,
          the output file.
        </li>
      </ul>
    </p>

    <h3 align = "center">
      List of Routines:
    </h3>

    <p>
      <ul>
        <li>
          <b>NL2SOL</b> minimizes nonlinear sum of squares using analytic jacobian.
        </li>
        <li>
          <b>NL2SNO</b> is like NL2SOL, but uses a finite difference jacobian.
        </li>
        <li>
          <b>NL2ITR</b> carries out nl2sol (nonlinear least-squares) iterations.
        </li>
        <li>
          <b>ASSESS</b> assesses a candidate step.
        </li>
        <li>
          <b>COVCLC</b> computes the covariance matrix for NL2ITR.
        </li>
        <li>
          <b>DFAULT</b> supplies default values to IV and V.
        </li>
        <li>
          <b>DOTPRD</b> returns the inner product of the p-vectors x and y.
        </li>
        <li>
          <b>DUPDAT</b> updates the scale vector D for NL2ITR.
        </li>
        <li>
          <b>GQTSTP</b> computes the Goldfeld-Quandt-Trotter step by More-Hebden technique.
        </li>
        <li>
          <b>ITSMRY</b> prints an iteration summary.
        </li>
        <li>
          <b>LINVRT</b> computes the inverse of a lower triangular matrix.
        </li>
        <li>
          <b>LITVMU</b> solves L' * X = Y, where L is a lower triangular matrix.
        </li>
        <li>
          <b>LIVMUL</b> solves L * X = Y, where L is a lower triangular matrix.
        </li>
        <li>
          <b>LMSTEP</b> computes the Levenberg-Marquardt step using More-Hebden technique.
        </li>
        <li>
          <b>LSQRT</b> computes the Cholesky factor of a lower triangular matrix.
        </li>
        <li>
          <b>LSVMIN</b> estimates the smallest singular value of a lower triangular matrix.
        </li>
        <li>
          <b>LTSQAR</b> sets A to the lower triangle of L' * L.
        </li>
        <li>
          <b>PARCHK</b> checks the NL2SOL parameters.
        </li>
        <li>
          <b>QAPPLY</b> applies orthogonal transformation to the residual R.
        </li>
        <li>
          <b>QRFACT</b> computes the QR decomposition of a matrix.
        </li>
        <li>
          <b>RELDST</b> computes the  relative difference between two real values.
        </li>
        <li>
          <b>RPTMUL</b> computes R * P' * X or P * R' * R * P' * X or P * R' * X.
        </li>
        <li>
          <b>SLUPDT</b> updates a symmetric matrix A so that A * STEP = Y.
        </li>
        <li>
          <b>SLVMUL</b> sets Y = S * X, where S is a P by P symmetric matrix.
        </li>
        <li>
          <b>STOPX</b> is called to stop execution.
        </li>
        <li>
          <b>TIMESTAMP</b> prints the current YMDHMS date as a time stamp.
        </li>
        <li>
          <b>VAXPY</b> sets W = A*X + Y where A is a scalar.
        </li>
        <li>
          <b>VCOPY</b> copies a vector.
        </li>
        <li>
          <b>VSCOPY</b> sets a vector to a scalar value.
        </li>
        <li>
          <b>V2NORM</b> computes the L2 norm of a vector.
        </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 01 February 2008.
    </i>

    <!-- John Burkardt -->


  </body>

  <!-- Initial HTML skeleton created by HTMLINDEX. -->

</html>