File: Fitting-large-linear-systems-example.html

package info (click to toggle)
gsl-ref-html 2.3-1
  • links: PTS
  • area: non-free
  • in suites: bullseye, buster, sid
  • size: 6,876 kB
  • ctags: 4,574
  • sloc: makefile: 35
file content (290 lines) | stat: -rw-r--r-- 10,813 bytes parent folder | download
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
34
35
36
37
38
39
40
41
42
43
44
45
46
47
48
49
50
51
52
53
54
55
56
57
58
59
60
61
62
63
64
65
66
67
68
69
70
71
72
73
74
75
76
77
78
79
80
81
82
83
84
85
86
87
88
89
90
91
92
93
94
95
96
97
98
99
100
101
102
103
104
105
106
107
108
109
110
111
112
113
114
115
116
117
118
119
120
121
122
123
124
125
126
127
128
129
130
131
132
133
134
135
136
137
138
139
140
141
142
143
144
145
146
147
148
149
150
151
152
153
154
155
156
157
158
159
160
161
162
163
164
165
166
167
168
169
170
171
172
173
174
175
176
177
178
179
180
181
182
183
184
185
186
187
188
189
190
191
192
193
194
195
196
197
198
199
200
201
202
203
204
205
206
207
208
209
210
211
212
213
214
215
216
217
218
219
220
221
222
223
224
225
226
227
228
229
230
231
232
233
234
235
236
237
238
239
240
241
242
243
244
245
246
247
248
249
250
251
252
253
254
255
256
257
258
259
260
261
262
263
264
265
266
267
268
269
270
271
272
273
274
275
276
277
278
279
280
281
282
283
284
285
286
287
288
289
290
<!DOCTYPE html PUBLIC "-//W3C//DTD HTML 4.01 Transitional//EN" "http://www.w3.org/TR/html4/loose.dtd">
<html>
<!-- Copyright (C) 1996, 1997, 1998, 1999, 2000, 2001, 2002, 2003, 2004, 2005, 2006, 2007, 2008, 2009, 2010, 2011, 2012, 2013, 2014, 2015, 2016 The GSL Team.

Permission is granted to copy, distribute and/or modify this document
under the terms of the GNU Free Documentation License, Version 1.3 or
any later version published by the Free Software Foundation; with the
Invariant Sections being "GNU General Public License" and "Free Software
Needs Free Documentation", the Front-Cover text being "A GNU Manual",
and with the Back-Cover Text being (a) (see below). A copy of the
license is included in the section entitled "GNU Free Documentation
License".

(a) The Back-Cover Text is: "You have the freedom to copy and modify this
GNU Manual." -->
<!-- Created by GNU Texinfo 5.1, http://www.gnu.org/software/texinfo/ -->
<head>
<title>GNU Scientific Library &ndash; Reference Manual: Fitting large linear systems example</title>

<meta name="description" content="GNU Scientific Library &ndash; Reference Manual: Fitting large linear systems example">
<meta name="keywords" content="GNU Scientific Library &ndash; Reference Manual: Fitting large linear systems example">
<meta name="resource-type" content="document">
<meta name="distribution" content="global">
<meta name="Generator" content="makeinfo">
<meta http-equiv="Content-Type" content="text/html; charset=utf-8">
<link href="index.html#Top" rel="start" title="Top">
<link href="Function-Index.html#Function-Index" rel="index" title="Function Index">
<link href="Fitting-Examples.html#Fitting-Examples" rel="up" title="Fitting Examples">
<link href="Fitting-References-and-Further-Reading.html#Fitting-References-and-Further-Reading" rel="next" title="Fitting References and Further Reading">
<link href="Fitting-robust-linear-regression-example.html#Fitting-robust-linear-regression-example" rel="previous" title="Fitting robust linear regression example">
<style type="text/css">
<!--
a.summary-letter {text-decoration: none}
blockquote.smallquotation {font-size: smaller}
div.display {margin-left: 3.2em}
div.example {margin-left: 3.2em}
div.indentedblock {margin-left: 3.2em}
div.lisp {margin-left: 3.2em}
div.smalldisplay {margin-left: 3.2em}
div.smallexample {margin-left: 3.2em}
div.smallindentedblock {margin-left: 3.2em; font-size: smaller}
div.smalllisp {margin-left: 3.2em}
kbd {font-style:oblique}
pre.display {font-family: inherit}
pre.format {font-family: inherit}
pre.menu-comment {font-family: serif}
pre.menu-preformatted {font-family: serif}
pre.smalldisplay {font-family: inherit; font-size: smaller}
pre.smallexample {font-size: smaller}
pre.smallformat {font-family: inherit; font-size: smaller}
pre.smalllisp {font-size: smaller}
span.nocodebreak {white-space:nowrap}
span.nolinebreak {white-space:nowrap}
span.roman {font-family:serif; font-weight:normal}
span.sansserif {font-family:sans-serif; font-weight:normal}
ul.no-bullet {list-style: none}
-->
</style>


</head>

<body lang="en" bgcolor="#FFFFFF" text="#000000" link="#0000FF" vlink="#800080" alink="#FF0000">
<a name="Fitting-large-linear-systems-example"></a>
<div class="header">
<p>
Previous: <a href="Fitting-robust-linear-regression-example.html#Fitting-robust-linear-regression-example" accesskey="p" rel="previous">Fitting robust linear regression example</a>, Up: <a href="Fitting-Examples.html#Fitting-Examples" accesskey="u" rel="up">Fitting Examples</a> &nbsp; [<a href="Function-Index.html#Function-Index" title="Index" rel="index">Index</a>]</p>
</div>
<hr>
<a name="Large-Dense-Linear-Regression-Example"></a>
<h4 class="subsection">38.8.6 Large Dense Linear Regression Example</h4>

<p>The following program demostrates the large dense linear least squares
solvers. This example is adapted from Trefethen and Bau,
and fits the function <em>f(t) = \exp{(\sin^3{(10t)}})</em> on
the interval <em>[0,1]</em> with a degree 15 polynomial. The
program generates <em>n = 50000</em> equally spaced points
<em>t_i</em> on this interval, calculates the function value
and adds random noise to determine the observation value
<em>y_i</em>. The entries of the least squares matrix are
<em>X_{ij} = t_i^j</em>, representing a polynomial fit. The
matrix is highly ill-conditioned, with a condition number
of about <em>1.4 \cdot 10^{11}</em>. The program accumulates the
matrix into the least squares system in 5 blocks, each with
10000 rows. This way the full matrix <em>X</em> 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 top left plot, we see the unregularized
normal equations solution has larger error than TSQR due to
the ill-conditioning of the matrix. In the bottom left plot,
we show the L-curve, which exhibits multiple corners.
In the top right panel, we plot a regularized solution using
<em>\lambda = 10^{-6}</em>. The TSQR and normal solutions now agree,
however they are unable to provide a good fit due to the damping.
This indicates that for some ill-conditioned
problems, regularizing the normal equations does not improve the
solution. This is further illustrated in the bottom right panel,
where we plot the L-curve calculated from the normal equations.
The curve agrees with the TSQR curve for larger damping parameters,
but for small <em>\lambda</em>, the normal equations approach cannot
provide accurate solution vectors leading to numerical
inaccuracies in the left portion of the curve.
</p>

<div class="example">
<pre class="verbatim">#include &lt;gsl/gsl_math.h&gt;
#include &lt;gsl/gsl_vector.h&gt;
#include &lt;gsl/gsl_matrix.h&gt;
#include &lt;gsl/gsl_rng.h&gt;
#include &lt;gsl/gsl_randist.h&gt;
#include &lt;gsl/gsl_multifit.h&gt;
#include &lt;gsl/gsl_multilarge.h&gt;
#include &lt;gsl/gsl_blas.h&gt;

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

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

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

  return 0;
}

int
solve_system(const int print_data, 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);
  const size_t nlcurve = 200;
  gsl_vector *reg_param = gsl_vector_alloc(nlcurve);
  gsl_vector *rho = gsl_vector_alloc(nlcurve);
  gsl_vector *eta = gsl_vector_alloc(nlcurve);
  size_t rowidx = 0;
  double rnorm, snorm, rcond;
  double t = 0.0;
  double dt = 1.0 / (n - 1.0);

  while (rowidx &lt; 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 &lt; nr; ++i)
        {
          gsl_vector_view row = gsl_matrix_row(&amp;Xv.matrix, i);
          double fi = func(t);
          double ei = gsl_ran_gaussian (r, 0.1 * fi); /* noise */
          double yi = fi + ei;

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

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

          if (print_data &amp;&amp; (i % 100 == 0))
            printf(&quot;%f %f\n&quot;, t, yi);

          t += dt;
        }

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

      rowidx += nr;
    }

  if (print_data)
    printf(&quot;\n\n&quot;);

  /* compute L-curve */
  gsl_multilarge_linear_lcurve(reg_param, rho, eta, w);

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

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

  fprintf(stderr, &quot;=== Method %s ===\n&quot;, gsl_multilarge_linear_name(w));
  fprintf(stderr, &quot;condition number = %e\n&quot;, 1.0 / rcond);
  fprintf(stderr, &quot;residual norm    = %e\n&quot;, rnorm);
  fprintf(stderr, &quot;solution norm    = %e\n&quot;, snorm);

  /* output L-curve */
  {
    size_t i;
    for (i = 0; i &lt; nlcurve; ++i)
      {
        printf(&quot;%.12e %.12e %.12e\n&quot;,
               gsl_vector_get(reg_param, i),
               gsl_vector_get(rho, i),
               gsl_vector_get(eta, i));
      }
    printf(&quot;\n\n&quot;);
  }

  gsl_matrix_free(X);
  gsl_vector_free(y);
  gsl_multilarge_linear_free(w);
  gsl_rng_free(r);
  gsl_vector_free(reg_param);
  gsl_vector_free(rho);
  gsl_vector_free(eta);

  return 0;
}

int
main(int argc, char *argv[])
{
  const size_t n = 50000;   /* number of observations */
  const size_t p = 16;      /* 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 &gt; 1)
    lambda = atof(argv[1]);

  /* solve system with TSQR method */
  solve_system(1, gsl_multilarge_linear_tsqr, lambda, n, p, c_tsqr);

  /* solve system with Normal equations method */
  solve_system(0, gsl_multilarge_linear_normal, lambda, n, p, c_normal);

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

    for (t = 0.0; t &lt;= 1.0; t += 0.01)
      {
        double f_exact = func(t);
        double f_tsqr, f_normal;

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

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

    gsl_vector_free(v);
  }

  gsl_vector_free(c_tsqr);
  gsl_vector_free(c_normal);

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

<hr>
<div class="header">
<p>
Previous: <a href="Fitting-robust-linear-regression-example.html#Fitting-robust-linear-regression-example" accesskey="p" rel="previous">Fitting robust linear regression example</a>, Up: <a href="Fitting-Examples.html#Fitting-Examples" accesskey="u" rel="up">Fitting Examples</a> &nbsp; [<a href="Function-Index.html#Function-Index" title="Index" rel="index">Index</a>]</p>
</div>



</body>
</html>