File: lsquare.cpp

package info (click to toggle)
pluto-find-orb 0.0~git20180227-2
  • links: PTS, VCS
  • area: main
  • in suites: experimental
  • size: 2,668 kB
  • sloc: cpp: 30,743; makefile: 263
file content (518 lines) | stat: -rw-r--r-- 16,003 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
291
292
293
294
295
296
297
298
299
300
301
302
303
304
305
306
307
308
309
310
311
312
313
314
315
316
317
318
319
320
321
322
323
324
325
326
327
328
329
330
331
332
333
334
335
336
337
338
339
340
341
342
343
344
345
346
347
348
349
350
351
352
353
354
355
356
357
358
359
360
361
362
363
364
365
366
367
368
369
370
371
372
373
374
375
376
377
378
379
380
381
382
383
384
385
386
387
388
389
390
391
392
393
394
395
396
397
398
399
400
401
402
403
404
405
406
407
408
409
410
411
412
413
414
415
416
417
418
419
420
421
422
423
424
425
426
427
428
429
430
431
432
433
434
435
436
437
438
439
440
441
442
443
444
445
446
447
448
449
450
451
452
453
454
455
456
457
458
459
460
461
462
463
464
465
466
467
468
469
470
471
472
473
474
475
476
477
478
479
480
481
482
483
484
485
486
487
488
489
490
491
492
493
494
495
496
497
498
499
500
501
502
503
504
505
506
507
508
509
510
511
512
513
514
515
516
517
518
/* lsquare.cpp: least-squares computations

Copyright (C) 2010, Project Pluto

This program is free software; you can redistribute it and/or
modify it under the terms of the GNU General Public License
as published by the Free Software Foundation; either version 2
of the License, or (at your option) any later version.

This program is distributed in the hope that it will be useful,
but WITHOUT ANY WARRANTY; without even the implied warranty of
MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE.  See the
GNU General Public License for more details.

You should have received a copy of the GNU General Public License
along with this program; if not, write to the Free Software
Foundation, Inc., 51 Franklin Street, Fifth Floor, Boston, MA
02110-1301, USA.    */

/* 2013 October:  Revised to use long doubles for matrix inversion,
and using a better choice for partial pivoting (see comments
below above the pivot_value( ) function).  I did this thinking
it would reduce roundoff errors in matrix inversion,  resulting
in more stable results for short-arc solutions.  In practice,  it
doesn't really seem to help or hinder,  but it's still a good
idea for nearly-singular matrices.  */

// #define LSQUARE_ERROR_DEBUGGING

#include <stdlib.h>
#include <string.h>
#include <math.h>
#include <assert.h>
#include "lsquare.h"

#ifdef DEBUG_MEM
#include "checkmem.h"
#endif

/*    Matrix inversion offers plenty of opportunity to lose precision.  So
      I decided to use GNU C's __float128 type where possible (seems to be
      versions 4.6 and up) and long doubles otherwise.  Plain old long
      doubles usually offer 64 bits of precision;  __float128 offers
      112 bits of precision.  Either is better than "ordinary" doubles
      with a mere 52 bits of precision.   */

#if ((__GNUC__ * 100) + __GNUC_MINOR__) >= 406
#define ldouble    __float128
#else
#define ldouble    long double
#endif

#define LSQUARE struct lsquare

LSQUARE
   {
   int n_params, n_obs;
   ldouble *wtw, *uw;
   };

void *lsquare_init( const int n_params)
{
   LSQUARE *rval = (LSQUARE *)calloc( 1, sizeof( LSQUARE));

   if( !rval)
      return( (void *)rval);
   rval->n_params = n_params;
   rval->n_obs = 0;
   rval->uw = (ldouble *)calloc( (n_params + 1) * n_params, sizeof( ldouble));
   rval->wtw = rval->uw + n_params;
   return(( void *)rval);
}

#ifdef LSQUARE_ERROR_DEBUGGING
static inline ldouble lfabs( ldouble ival)
{
   return( ival > 0. ? ival : -ival);
}

#include <stdio.h>

static FILE *debug_file;

static void dump_matrix( const ldouble *matrix, const int xsize, const int ysize)
{
   int i;
   ldouble largest_element = 0., sum = 0.;

   for( i = 0; i < xsize * ysize; i++)
      {
      ldouble curr = lfabs( *matrix);

      if( largest_element < curr)
         largest_element = curr;
      sum += *matrix * *matrix;
      fprintf( debug_file, "%11.2e%s", (double)*matrix++, !((i + 1) % xsize) ? "\n" : "");
      }
   sum /= (ldouble)( ysize * xsize);
   fprintf( debug_file, "Largest element: %11.2e; RMS: %11.2e\n",
               (double)largest_element, sqrt( (double)sum));
}
#endif

double levenberg_marquardt_lambda = 0.;      /* damping factor */

int lsquare_add_observation( void *lsquare, const double residual,
                                  const double weight, const double *obs)
{
   LSQUARE *lsq = (LSQUARE *)lsquare;
   int i, j;
   const int n_params = lsq->n_params;

   for( i = 0; i < n_params; i++)
      {
      const ldouble w2_obs_i = (ldouble)( weight * weight * obs[i]);

      lsq->uw[i] += (ldouble)residual * w2_obs_i;
      for( j = 0; j < n_params; j++)
         lsq->wtw[i + j * n_params] += w2_obs_i * (ldouble)obs[j];
      lsq->wtw[i + i * n_params] +=
                  w2_obs_i * (ldouble)( obs[i] * levenberg_marquardt_lambda);
      }
   lsq->n_obs++;
   return( lsq->n_obs);
}

ldouble lsquare_determinant;

   /* A simple Gauss-Jordan matrix inverter,  with partial pivoting.  It
      first extends the size x size square matrix into a size-high by
      (2*size) wide one,  with the expanded space on the right side filled
      with an identity matrix.  It then zeroes out the lower left triangle
      of the original size x size matrix.  Some row-swapping is done in
      this procedure (that's the "partial pivoting" aspect;  see _Numerical
      Recipes_,  chap. 2.1 for details.)   */


/* Commonly,  in partial pivoting schemes,  you just use whichever
line has the greatest absolute value.  So,  when starting with a matrix
such as

87   32  -12
99 1e+30 -4e+29
.7   .3   -.2

   you'd pivot on the second line.  The problem is that,  while
99 is indeed the largest of the three values in the first column,
you'll get some horrendous loss of precision because of the remaining
values in the second line.

   This "standard" scheme would correspond to the pivot_value function
below returning fabs( *line).  Instead,  it returns *line^2 divided
by the sum of the squares of all values in the row (i.e.,  the square
of the vector norm for that row).  This is somewhat similar to the
'scaled pivoting' scheme described at

https://en.wikipedia.org/wiki/Pivot_element
*/

static ldouble pivot_value( const ldouble *line, const unsigned line_size)
{
   ldouble sum_squares = line[0] * line[0], square0 = sum_squares;
   unsigned i;

   for( i = 1; i < line_size; i++)
      sum_squares += line[i] * line[i];
   return( sum_squares ? square0 / sum_squares : 0.);
}

#define swap_ldoubles( a, b)   { const ldouble temp = a;  a = b;  b = temp; }

static ldouble *calc_inverse( const ldouble *src, const int size)
{
   ldouble *rval;
   ldouble *temp = (ldouble *)calloc( 2 * size * size, sizeof( ldouble));
   ldouble *tptr = temp;
   ldouble *tptr1, *tptr2;
   int i, j, k;
   const int dsize = 2 * size;

   lsquare_determinant = 1.;
   if( !temp)
      return( NULL);
               /* copy 'src' to 'temp',  doubling its width and putting */
               /* an identity matrix in the right half: */
   for( i = 0; i < size; i++)
      {
      for( j = 0; j < size; j++)
         *tptr++ = (ldouble)*src++;
      for( j = 0; j < size; j++)
         *tptr++ = ((i == j) ? 1. : 0.);
      }
/* src -= size * size;    Restores src to its original value.  But we */
/*         aren't actually using it anyway,  so we'll comment it out. */

   tptr1 = temp;
   for( i = 0; i < size; i++, tptr1 += dsize)
      {
      int pivot = -1;
      ldouble best_val = 0.;

      tptr = tptr1;
      for( j = i; j < size; j++, tptr += dsize)
         {
         const ldouble zval = pivot_value( tptr + i, size - i);

         if( zval > best_val)
            {
            best_val = zval;
            pivot = j;
            }
         }

      if( pivot == -1)     /* un-invertable matrix:  return NULL */
         {
         free( temp);
         return( NULL);
         }

      if( pivot != i)                  /* swap rows */
         {
         tptr2 = temp + dsize * pivot;
         for( j = i; j < dsize; j++)
            swap_ldoubles( tptr1[j], tptr2[j]);
         }

      for( j = i + 1; j < size; j++)
         {
         ldouble tval;

         tptr2 = temp + dsize * j;
         tval = tptr2[i] / tptr1[i];
         for( k = i; k < dsize; k++)
            tptr2[k] -= tptr1[k] * tval;
         }
      }
                  /* the lower left triangle is now cleared;  time to */
                  /* zero out the upper right triangle: */

   for( i = size - 1; i >= 0; i--)
      {
      tptr1 = temp + i * dsize;
      for( j = size; j < dsize; j++)
         {
         lsquare_determinant /= tptr1[i];
         tptr1[j] /= tptr1[i];
         }
      tptr2 = temp;
      for( k = 0; k < i; k++, tptr2 += dsize)
         for( j = size; j < dsize; j++)
            tptr2[j] -= tptr2[i] * tptr1[j];
      }

   rval = (ldouble *)calloc( size * size, sizeof( ldouble));
   if( rval)            /* copy the right-hand half of 'temp',  which */
      {                 /* now has the inverse we wanted              */
      tptr1 = rval;
      for( i = 0; i < size; i++)
         for( j = 0; j < size; j++)
            *tptr1++ = temp[(i * 2 + 1) * size + j];
      }
   free( temp);
   return( rval);
}

#ifdef CHOLESKY_INVERSION

   /* Cholesky decomposition,  as described in _Numerical Recipes_ 2.9.
      Cholesky decomposition appears to be widely recommended for inverting
      covariance matrices,  and I may end up going that route.  I started
      out with Gauss-Jackson,  though,  and it appears to work Just Fine.
      (Admittedly,  with tweaks to improve pivot selection and the
      calc_improved_matrix() trick given below to "polish" an initial
      inversion.)  I may take the time,  at some point,  to try to get
      Cholesky inversion to work... though I don't think it'll actually
      get me anything at this point.  At least for the nonce,  I'm leaving
      it ifdeffed out. */

static int cholesky_decomposition( ldouble *a, const int size, ldouble *diag)
{
   int i, j;

   for( i = 0; i < size; i++)
      for( j = i; j < size; j++)
         {
         ldouble sum = a[i * size + j];
         int k;

         for( k = i - 1; k >= 0; k--)
            sum -= a[i * size + k] * a[j * size + k];
         if( i == j)
            {
            if( sum < 0.)       /* not actually positive definite after all */
               return( -1);
            else
               diag[i] = sqrt( sum);
            }
         else
            a[j * size + i] = sum / diag[i];
         }
   return( 0);
}

   /* Inverting a symmetric positive-definite matrix,  again as described */
   /* in _Numerical Recipes_,  with slight modifications.  In their       */
   /* version,  the lower triangle is computed;  I added a few lines to   */
   /* get the upper.  NOTE: does not work yet.                            */

static int cholesky_inversion( ldouble *a, const int size)
{
   ldouble diag[30];
   int i, j, k, rval = cholesky_decomposition( a, size, diag);

   assert( !rval);
   if( !rval)
      {
      for( i = 0; i < size; i++)
         {
         a[i * (size + 1)] = 1. / diag[i];
         for( j = i + 1; j < size; j++)
            {
            ldouble sum = 0;

            for( k = i; k < j; k++)
               sum -= a[j * size + k] * a[k * size + i];
            a[j * size + i] = sum / diag[j];
            }
         }
      for( i = 1; i < size; i++)          /* copy lower diag to upper */
         for( j = 0; j < i; j++)
            a[j * size + i] = a[i * size + j];
      }
   return( rval);
}

static ldouble *invert_symmetric_positive_definite_matrix( const ldouble *a,
                           const int size)
{
   ldouble *rval = (ldouble *)calloc( size * size, sizeof( ldouble));
   int cholesky_rval;

   memcpy( rval, a, size * size * sizeof( ldouble));
   cholesky_rval = cholesky_inversion( rval, size);
   if( cholesky_rval)
      {
      free( rval);
      rval = NULL;
      }
   return( rval);
}
#endif

static void mult_matrices( ldouble *prod, const ldouble *a, const int awidth,
                  const int aheight, const ldouble *b, const int bwidth)
{
   int i, j;

   for( j = 0; j < aheight; j++)
      for( i = 0; i < bwidth; i++, prod++)
         {
         int k;
         const ldouble *aptr = a + j * awidth, *bptr = b + i;

         *prod = 0.;
         for( k = awidth; k; k--, bptr += bwidth)
            *prod += *aptr++ * (*bptr);
         }
}

/* calc_inverse_improved() computes a matrix inverse using the simpler
   'calc_inverse()' function,  a plain ol' Gauss-Jordan inverter (see above).
   It then uses a rather simple trick from _Numerical Recipes_,  chap. 2.5,
   "Iterative Improvement of a Solution to Linear Equations",  to "polish"
   the result.  The idea is this.  Suppose you've inverted A to create a matrix
   B,  with (inevitably) some error in it,  so that instead of AB=I,  you have
   AB=I+E,  where E is an "error matrix" with lots of small values.  In that
   case,  A(B - BE) = AB - (AB)E = I+E - (I+E)E = I+E - E - E^2 = I - E^2.
   In other words,  B-BE is a better approximation to the inverse of A.

      I added this step when I had some concerns that my least squares
   solutions weren't what they ought to be.  The problem lay elsewhere.
   But this _should_ ensure that matrix inversion is more accurate than it
   otherwise would be,  at almost no computational cost.  */

static ldouble *calc_inverse_improved( const ldouble *src, const int size)
{
#ifdef LSQUARE_ERROR_DEBUGGING
   debug_file = fopen( "lsquare.dat", "ab");
#endif

   ldouble *inverse = calc_inverse( src, size);

#ifdef LSQUARE_ERROR_DEBUGGING
   fprintf( debug_file, "Inverse:\n");
   dump_matrix( inverse, size, size);
#endif
   if( inverse)
      {
      ldouble *err_mat = (ldouble *)calloc( 2 * size * size, sizeof( ldouble));
      ldouble *b_times_delta = err_mat + size * size;
      int i;

      mult_matrices( err_mat, src, size, size, inverse, size);
      for( i = 0; i < size; i++)
         err_mat[i * (size + 1)] -= 1.;
      mult_matrices( b_times_delta, inverse, size, size, err_mat, size);
      for( i = 0; i < size * size; i++)
         inverse[i] -= b_times_delta[i];
#ifdef LSQUARE_ERROR_DEBUGGING
      fprintf( debug_file, "%d-square matrix delta (= AB - I):\n", size);
      dump_matrix( err_mat, size, size);
      fprintf( debug_file, "Error matrix after adjustment:\n");
      mult_matrices( err_mat, src, size, size, inverse, size);
      for( i = 0; i < size; i++)
         err_mat[i * (size + 1)] -= 1.;
      dump_matrix( err_mat, size, size);
      fclose( debug_file);
#endif
      free( err_mat);
      }
   return( inverse);
}

int lsquare_solve( const void *lsquare, double *result)
{
   const LSQUARE *lsq = (const LSQUARE *)lsquare;
   int i, j, n_params = lsq->n_params;
   ldouble *inverse;

   if( n_params > lsq->n_obs)       /* not enough observations yet */
      return( -1);

// inverse = invert_symmetric_positive_definite_matrix( lsq->wtw, n_params);
   inverse = calc_inverse_improved( lsq->wtw, n_params);
   if( !inverse)
      return( -2);            /* couldn't invert matrix */

   for( i = 0; i < n_params; i++)
      {
      ldouble temp_result = 0;

      for( j = 0; j < n_params; j++)
         temp_result += inverse[i + j * n_params] * lsq->uw[j];
      result[i] = (double)temp_result;
      }

   free( inverse);
   return( 0);
}

static double *convert_ldouble_matrix_to_double( const ldouble *matrix,
                              const int size)
{
   double *rval = (double *)calloc( size * size, sizeof( double));
   int i;

   for( i = 0; i < size * size; i++)
      rval[i] = (double)matrix[i];
   return( rval);
}

double *lsquare_covariance_matrix( const void *lsquare)
{
   const LSQUARE *lsq = (const LSQUARE *)lsquare;
   ldouble *lrval = NULL;

   if( lsq->n_params <= lsq->n_obs)       /* got enough observations */
      lrval = calc_inverse_improved( lsq->wtw, lsq->n_params);
//    lrval = invert_symmetric_positive_definite_matrix( lsq->wtw, lsq->n_params);
   if( lrval)
      {
      double *rval = convert_ldouble_matrix_to_double( lrval, lsq->n_params);

      free( lrval);
      return( rval);
      }
   else
      return( NULL);
}

double *lsquare_wtw_matrix( const void *lsquare)
{
   const LSQUARE *lsq = (const LSQUARE *)lsquare;

   return( convert_ldouble_matrix_to_double( lsq->wtw, lsq->n_params));
}

void lsquare_free( void *lsquare)
{
   const LSQUARE *lsq = (const LSQUARE *)lsquare;

   free( lsq->uw);
   free( lsquare);
}

#ifdef TEST_CODE
#include <stdio.h>

void main( int argc, char **argv)
{
   FILE *ifile = fopen( "imatrix", "rb");
   int size, i, j;
   double *matrix, *inv;

   fscanf( ifile, "%d", &size);
   matrix = (double *)calloc( size * size, sizeof( double));
   for( i = 0; i < size * size; i++)
      fscanf( ifile, "%lf", matrix + i);
   inv = calc_inverse_improved( matrix, size);
   for( i = 0; i < size; i++)
      {
      printf( "\n");
      for( j = 0; j < size; j++)
         printf( "%10.5lf", *inv++);
      }
   free( inv);
}
#endif