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<a name="Nonlinear-Least_002dSquares-Large-Example"></a>
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<p>
Previous: <a href="Nonlinear-Least_002dSquares-Comparison-Example.html#Nonlinear-Least_002dSquares-Comparison-Example" accesskey="p" rel="previous">Nonlinear Least-Squares Comparison Example</a>, Up: <a href="Nonlinear-Least_002dSquares-Examples.html#Nonlinear-Least_002dSquares-Examples" accesskey="u" rel="up">Nonlinear Least-Squares Examples</a> &nbsp; [<a href="Function-Index.html#Function-Index" title="Index" rel="index">Index</a>]</p>
</div>
<hr>
<a name="Large-Nonlinear-Least-Squares-Example"></a>
<h4 class="subsection">39.12.4 Large Nonlinear Least Squares Example</h4>

<p>The following program illustrates the large nonlinear least
squares solvers on a system with significant sparse structure
in the Jacobian. The cost function is given by
</p>
<div class="example">
<pre class="example">\Phi(x) &amp;= 1/2 \sum_{i=1}^{p+1} f_i^2
f_i &amp;= \sqrt{\alpha} (x_i - 1), 1 \le i \le p
f_{p+1} &amp;= ||x||^2 - 1/4
</pre></div>

<p>with <em>\alpha = 10^{-5}</em>. The residual <em>f_{p+1}</em> imposes a constraint on the <em>p</em>
parameters <em>x</em>, to ensure that <em>||x||^2 \approx {1 \over 4}</em>.
The <em>(p+1)</em>-by-<em>p</em> Jacobian for this system is given by
</p>
<div class="example">
<pre class="example">J(x) = [ \sqrt{alpha} I_p; 2 x^T ]
</pre></div>

<p>and the normal equations matrix is given by
</p>
<div class="example">
<pre class="example">J^T J = [ \alpha I_p + 4 x x^T ]
</pre></div>

<p>Finally, the second directional derivative of <em>f</em> for the
geodesic acceleration method is given by
</p>
<div class="example">
<pre class="example">fvv = [ 0; 2 ||v||^2 ]
</pre></div>

<p>Since the upper <em>p</em>-by-<em>p</em> block of <em>J</em> is diagonal,
this sparse structure should be exploited in the nonlinear solver.
For comparison, the following program solves the system for <em>p = 2000</em>
using the dense direct Cholesky solver based on the normal equations matrix
<em>J^T J</em>, as well as the iterative Steihaug-Toint solver, based on
sparse matrix-vector products <em>J u</em> and <em>J^T u</em>. The
program output is shown below.
</p>
<div class="smallformat">
<pre class="verbatim">Method                    NITER NFEV NJUEV NJTJEV NAEV Init Cost  Final cost cond(J) Final |x|^2 Time (s)  
levenberg-marquardt       25    31   26    26     0    7.1218e+18 1.9555e-02 447.50  2.5044e-01  46.28
levenberg-marquardt+accel 22    23   45    23     22   7.1218e+18 1.9555e-02 447.64  2.5044e-01  33.92
dogleg                    37    87   36    36     0    7.1218e+18 1.9555e-02 447.59  2.5044e-01  56.05
double-dogleg             35    88   34    34     0    7.1218e+18 1.9555e-02 447.62  2.5044e-01  52.65
2D-subspace               37    88   36    36     0    7.1218e+18 1.9555e-02 447.71  2.5044e-01  59.75
steihaug-toint            35    88   345   0      0    7.1218e+18 1.9555e-02 inf     2.5044e-01  0.09
</pre></div>

<p>The first five rows use methods based on factoring the dense <em>J^T J</em> matrix
while the last row uses the iterative Steihaug-Toint method. While the number
of Jacobian matrix-vector products (NJUEV) is less for the dense methods, the added time
to construct and factor the <em>J^T J</em> matrix (NJTJEV) results in a much larger runtime than the
iterative method (see last column).
</p>
<p>The program is given below.
</p>
<div class="example">
<pre class="verbatim">#include &lt;stdlib.h&gt;
#include &lt;stdio.h&gt;
#include &lt;sys/time.h&gt;
#include &lt;gsl/gsl_vector.h&gt;
#include &lt;gsl/gsl_matrix.h&gt;
#include &lt;gsl/gsl_blas.h&gt;
#include &lt;gsl/gsl_multilarge_nlinear.h&gt;
#include &lt;gsl/gsl_spblas.h&gt;
#include &lt;gsl/gsl_spmatrix.h&gt;

/* parameters for functions */
struct model_params
{
  double alpha;
  gsl_spmatrix *J;
};

/* penalty function */
int
penalty_f (const gsl_vector * x, void *params, gsl_vector * f)
{
  struct model_params *par = (struct model_params *) params;
  const double sqrt_alpha = sqrt(par-&gt;alpha);
  const size_t p = x-&gt;size;
  size_t i;
  double sum = 0.0;

  for (i = 0; i &lt; p; ++i)
    {
      double xi = gsl_vector_get(x, i);

      gsl_vector_set(f, i, sqrt_alpha*(xi - 1.0));

      sum += xi * xi;
    }

  gsl_vector_set(f, p, sum - 0.25);

  return GSL_SUCCESS;
}

int
penalty_df (CBLAS_TRANSPOSE_t TransJ, const gsl_vector * x,
            const gsl_vector * u, void * params, gsl_vector * v,
            gsl_matrix * JTJ)
{
  struct model_params *par = (struct model_params *) params;
  const size_t p = x-&gt;size;
  size_t j;

  /* store 2*x in last row of J */
  for (j = 0; j &lt; p; ++j)
    {
      double xj = gsl_vector_get(x, j);
      gsl_spmatrix_set(par-&gt;J, p, j, 2.0 * xj);
    }

  /* compute v = op(J) u */
  if (v)
    gsl_spblas_dgemv(TransJ, 1.0, par-&gt;J, u, 0.0, v);

  if (JTJ)
    {
      gsl_vector_view diag = gsl_matrix_diagonal(JTJ);

      /* compute J^T J = [ alpha*I_p + 4 x x^T ] */
      gsl_matrix_set_zero(JTJ);

      /* store 4 x x^T in lower half of JTJ */
      gsl_blas_dsyr(CblasLower, 4.0, x, JTJ);

      /* add alpha to diag(JTJ) */
      gsl_vector_add_constant(&amp;diag.vector, par-&gt;alpha);
    }

  return GSL_SUCCESS;
}

int
penalty_fvv (const gsl_vector * x, const gsl_vector * v,
             void *params, gsl_vector * fvv)
{
  const size_t p = x-&gt;size;
  double normv = gsl_blas_dnrm2(v);

  gsl_vector_set_zero(fvv);
  gsl_vector_set(fvv, p, 2.0 * normv * normv);

  (void)params; /* avoid unused parameter warning */

  return GSL_SUCCESS;
}

void
solve_system(const gsl_vector *x0, gsl_multilarge_nlinear_fdf *fdf,
             gsl_multilarge_nlinear_parameters *params)
{
  const gsl_multilarge_nlinear_type *T = gsl_multilarge_nlinear_trust;
  const size_t max_iter = 200;
  const double xtol = 1.0e-8;
  const double gtol = 1.0e-8;
  const double ftol = 1.0e-8;
  const size_t n = fdf-&gt;n;
  const size_t p = fdf-&gt;p;
  gsl_multilarge_nlinear_workspace *work =
    gsl_multilarge_nlinear_alloc(T, params, n, p);
  gsl_vector * f = gsl_multilarge_nlinear_residual(work);
  gsl_vector * x = gsl_multilarge_nlinear_position(work);
  int info;
  double chisq0, chisq, rcond, xsq;
  struct timeval tv0, tv1;

  gettimeofday(&amp;tv0, NULL);

  /* initialize solver */
  gsl_multilarge_nlinear_init(x0, fdf, work);

  /* store initial cost */
  gsl_blas_ddot(f, f, &amp;chisq0);

  /* iterate until convergence */
  gsl_multilarge_nlinear_driver(max_iter, xtol, gtol, ftol,
                                NULL, NULL, &amp;info, work);

  gettimeofday(&amp;tv1, NULL);

  /* store final cost */
  gsl_blas_ddot(f, f, &amp;chisq);

  /* compute final ||x||^2 */
  gsl_blas_ddot(x, x, &amp;xsq);

  /* store cond(J(x)) */
  gsl_multilarge_nlinear_rcond(&amp;rcond, work);

  /* print summary */
  fprintf(stderr, &quot;%-25s %-5zu %-4zu %-5zu %-6zu %-4zu %-10.4e %-10.4e %-7.2f %-11.4e %.2f\n&quot;,
          gsl_multilarge_nlinear_trs_name(work),
          gsl_multilarge_nlinear_niter(work),
          fdf-&gt;nevalf,
          fdf-&gt;nevaldfu,
          fdf-&gt;nevaldf2,
          fdf-&gt;nevalfvv,
          chisq0,
          chisq,
          1.0 / rcond,
          xsq,
          (tv1.tv_sec - tv0.tv_sec) + 1.0e-6 * (tv1.tv_usec - tv0.tv_usec));

  gsl_multilarge_nlinear_free(work);
}

int
main (void)
{
  const size_t p = 2000;
  const size_t n = p + 1;
  gsl_vector *f = gsl_vector_alloc(n);
  gsl_vector *x = gsl_vector_alloc(p);

  /* allocate sparse Jacobian matrix with 2*p non-zero elements in triplet format */
  gsl_spmatrix *J = gsl_spmatrix_alloc_nzmax(n, p, 2 * p, GSL_SPMATRIX_TRIPLET);

  gsl_multilarge_nlinear_fdf fdf;
  gsl_multilarge_nlinear_parameters fdf_params =
    gsl_multilarge_nlinear_default_parameters();
  struct model_params params;
  size_t i;

  params.alpha = 1.0e-5;
  params.J = J;

  /* define function to be minimized */
  fdf.f = penalty_f;
  fdf.df = penalty_df;
  fdf.fvv = penalty_fvv;
  fdf.n = n;
  fdf.p = p;
  fdf.params = &amp;params;

  for (i = 0; i &lt; p; ++i)
    {
      /* starting point */
      gsl_vector_set(x, i, i + 1.0);

      /* store sqrt(alpha)*I_p in upper p-by-p block of J */
      gsl_spmatrix_set(J, i, i, sqrt(params.alpha));
    }

  fprintf(stderr, &quot;%-25s %-4s %-4s %-5s %-6s %-4s %-10s %-10s %-7s %-11s %-10s\n&quot;,
          &quot;Method&quot;, &quot;NITER&quot;, &quot;NFEV&quot;, &quot;NJUEV&quot;, &quot;NJTJEV&quot;, &quot;NAEV&quot;, &quot;Init Cost&quot;,
          &quot;Final cost&quot;, &quot;cond(J)&quot;, &quot;Final |x|^2&quot;, &quot;Time (s)&quot;);
  
  fdf_params.scale = gsl_multilarge_nlinear_scale_levenberg;

  fdf_params.trs = gsl_multilarge_nlinear_trs_lm;
  solve_system(x, &amp;fdf, &amp;fdf_params);

  fdf_params.trs = gsl_multilarge_nlinear_trs_lmaccel;
  solve_system(x, &amp;fdf, &amp;fdf_params);

  fdf_params.trs = gsl_multilarge_nlinear_trs_dogleg;
  solve_system(x, &amp;fdf, &amp;fdf_params);

  fdf_params.trs = gsl_multilarge_nlinear_trs_ddogleg;
  solve_system(x, &amp;fdf, &amp;fdf_params);

  fdf_params.trs = gsl_multilarge_nlinear_trs_subspace2D;
  solve_system(x, &amp;fdf, &amp;fdf_params);

  fdf_params.trs = gsl_multilarge_nlinear_trs_cgst;
  solve_system(x, &amp;fdf, &amp;fdf_params);

  gsl_vector_free(f);
  gsl_vector_free(x);
  gsl_spmatrix_free(J);

  return 0;
}
</pre></div>

<hr>
<div class="header">
<p>
Previous: <a href="Nonlinear-Least_002dSquares-Comparison-Example.html#Nonlinear-Least_002dSquares-Comparison-Example" accesskey="p" rel="previous">Nonlinear Least-Squares Comparison Example</a>, Up: <a href="Nonlinear-Least_002dSquares-Examples.html#Nonlinear-Least_002dSquares-Examples" accesskey="u" rel="up">Nonlinear Least-Squares Examples</a> &nbsp; [<a href="Function-Index.html#Function-Index" title="Index" rel="index">Index</a>]</p>
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