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38.12 Fitting Examples for large linear systems

The following program demostrates the large linear least squares solvers. This example is adapted from Trefethen and Bau, and fits the function f(t) = \exp{(\sin{4(t - 10)}}) on the interval [10,11] with a degree 14 polynomial. The program generates n = 50000 equally spaced points t_i on this interval, calculates the function value and adds random noise to determine the observation value y_i. The entries of the least squares matrix are X_{ij} = t_i^j, representing a polynomial fit. The matrix is highly ill-conditioned, and we accumulate the matrix into the least squares system in 5 blocks, each with 10000 rows. This way the full matrix X is never stored in memory. We solve the system with both the normal equations and TSQR methods. The results are shown in the plot below. In the left plot, we see the unregularized normal equations solution has large error due to the ill-conditioning of the matrix, while TSQR provides a good fit. In the right panel, we add regularization with \lambda = 1 which allows the normal equations method to converge to a good solution.

#include <gsl/gsl_math.h>
#include <gsl/gsl_vector.h>
#include <gsl/gsl_matrix.h>
#include <gsl/gsl_rng.h>
#include <gsl/gsl_randist.h>
#include <gsl/gsl_multifit.h>
#include <gsl/gsl_multilarge.h>
#include <gsl/gsl_blas.h>

/* function to be fitted */
double
func(const double t)
{
  return exp(sin(4.0 * (t - 10.0)));
}

/* construct a row of the least squares matrix */
int
build_row(const double t, gsl_vector *row)
{
  const size_t p = row->size;
  double Xj = 1.0;
  size_t j;

  for (j = 0; j < p; ++j)
    {
      gsl_vector_set(row, j, Xj);
      Xj *= t;
    }

  return 0;
}

int
solve_system(const gsl_multilarge_linear_type * T,
             const double lambda, const size_t n, const size_t p,
             gsl_vector * c)
{
  const size_t nblock = 5;         /* number of blocks to accumulate */
  const size_t nrows = n / nblock; /* number of rows per block */
  gsl_multilarge_linear_workspace * w =
    gsl_multilarge_linear_alloc(T, p);
  gsl_matrix *X = gsl_matrix_alloc(nrows, p);
  gsl_vector *y = gsl_vector_alloc(nrows);
  gsl_rng *r = gsl_rng_alloc(gsl_rng_default);
  size_t rowidx = 0;
  double rnorm, snorm, rcond;
  double t = 10.0;
  double dt = 1.0 / (n - 1.0);

  while (rowidx < n)
    {
      size_t nleft = n - rowidx;         /* number of rows left to accumulate */
      size_t nr = GSL_MIN(nrows, nleft); /* number of rows in this block */
      gsl_matrix_view Xv = gsl_matrix_submatrix(X, 0, 0, nr, p);
      gsl_vector_view yv = gsl_vector_subvector(y, 0, nr);
      size_t i;

      /* build (X,y) block with 'nr' rows */
      for (i = 0; i < nr; ++i)
        {
          gsl_vector_view row = gsl_matrix_row(&Xv.matrix, i);
          double yi = func(t);
          double ei = gsl_ran_gaussian (r, 0.3 * yi); /* noise */

          /* construct this row of LS matrix */
          build_row(t, &row.vector);

          /* set right hand side value with added noise */
          gsl_vector_set(&yv.vector, i, yi + ei);

          t += dt;
        }

      /* accumulate (X,y) block into LS system */
      gsl_multilarge_linear_accumulate(&Xv.matrix, &yv.vector, w);

      rowidx += nr;
    }

  /* solve large LS system and store solution in c */
  gsl_multilarge_linear_solve(lambda, c, &rnorm, &snorm, w);

  /* compute reciprocal condition number */
  gsl_multilarge_linear_rcond(&rcond, w);

  fprintf(stderr, "=== Method %s ===\n", gsl_multilarge_linear_name(w));
  if (rcond != 0.0)
    fprintf(stderr, "matrix condition number = %e\n", 1.0 / rcond);
  fprintf(stderr, "residual norm  = %e\n", rnorm);
  fprintf(stderr, "solution norm  = %e\n", snorm);

  gsl_matrix_free(X);
  gsl_vector_free(y);
  gsl_multilarge_linear_free(w);
  gsl_rng_free(r);

  return 0;
}

int
main(int argc, char *argv[])
{
  const size_t n = 50000;   /* number of observations */
  const size_t p = 15;      /* polynomial order + 1 */
  double lambda = 0.0;      /* regularization parameter */
  gsl_vector *c_tsqr = gsl_vector_alloc(p);
  gsl_vector *c_normal = gsl_vector_alloc(p);

  if (argc > 1)
    lambda = atof(argv[1]);

  /* solve system with TSQR method */
  solve_system(gsl_multilarge_linear_tsqr, lambda, n, p, c_tsqr);

  /* solve system with Normal equations method */
  solve_system(gsl_multilarge_linear_normal, lambda, n, p, c_normal);

  /* output solutions */
  {
    gsl_vector *v = gsl_vector_alloc(p);
    double t;

    for (t = 10.0; t <= 11.0; t += 0.01)
      {
        double f_exact = func(t);
        double f_tsqr, f_normal;

        build_row(t, v);
        gsl_blas_ddot(v, c_tsqr, &f_tsqr);
        gsl_blas_ddot(v, c_normal, &f_normal);

        printf("%f %e %e %e\n", t, f_exact, f_tsqr, f_normal);
      }

    gsl_vector_free(v);
  }

  gsl_vector_free(c_tsqr);
  gsl_vector_free(c_normal);

  return 0;
}

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