File: oct-norm.cc

package info (click to toggle)
octave 4.0.3-3
  • links: PTS, VCS
  • area: main
  • in suites: stretch
  • size: 94,200 kB
  • ctags: 52,925
  • sloc: cpp: 316,850; ansic: 43,469; fortran: 23,670; sh: 13,805; yacc: 8,204; objc: 7,939; lex: 3,631; java: 2,127; makefile: 1,746; perl: 1,022; awk: 988
file content (578 lines) | stat: -rw-r--r-- 15,110 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
519
520
521
522
523
524
525
526
527
528
529
530
531
532
533
534
535
536
537
538
539
540
541
542
543
544
545
546
547
548
549
550
551
552
553
554
555
556
557
558
559
560
561
562
563
564
565
566
567
568
569
570
571
572
573
574
575
576
577
578
/*

Copyright (C) 2008-2015 VZLU Prague, a.s.

This file is part of Octave.

Octave 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 3 of the License, or (at your
option) any later version.

Octave 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 Octave; see the file COPYING.  If not, see
<http://www.gnu.org/licenses/>.

*/

// author: Jaroslav Hajek <highegg@gmail.com>

#ifdef HAVE_CONFIG_H
#include <config.h>
#endif

#include <cassert>
#include <cfloat>
#include <math.h>

#include <iostream>
#include <vector>

#include "oct-cmplx.h"
#include "lo-error.h"
#include "lo-ieee.h"
#include "mx-cm-s.h"
#include "mx-s-cm.h"
#include "mx-fcm-fs.h"
#include "mx-fs-fcm.h"
#include "Array.h"
#include "Array-util.h"
#include "CMatrix.h"
#include "dMatrix.h"
#include "fCMatrix.h"
#include "fMatrix.h"
#include "CColVector.h"
#include "dColVector.h"
#include "CRowVector.h"
#include "dRowVector.h"
#include "fCColVector.h"
#include "fColVector.h"
#include "fCRowVector.h"
#include "fRowVector.h"
#include "CSparse.h"
#include "dSparse.h"
#include "dbleSVD.h"
#include "CmplxSVD.h"
#include "floatSVD.h"
#include "fCmplxSVD.h"

// Theory: norm accumulator is an object that has an accum method able
// to handle both real and complex element, and a cast operator
// returning the intermediate norm. Reference: Higham, N. "Estimating
// the Matrix p-Norm." Numer. Math. 62, 539-555, 1992.

// norm accumulator for the p-norm
template <class R>
class norm_accumulator_p
{
  R p,scl,sum;
public:
  norm_accumulator_p () {} // we need this one for Array
  norm_accumulator_p (R pp) : p(pp), scl(0), sum(1) {}

  template<class U>
  void accum (U val)
  {
    octave_quit ();
    R t = std::abs (val);
    if (scl == t) // we need this to handle Infs properly
      sum += 1;
    else if (scl < t)
      {
        sum *= std::pow (scl/t, p);
        sum += 1;
        scl = t;
      }
    else if (t != 0)
      sum += std::pow (t/scl, p);
  }
  operator R () { return scl * std::pow (sum, 1/p); }
};

// norm accumulator for the minus p-pseudonorm
template <class R>
class norm_accumulator_mp
{
  R p,scl,sum;
public:
  norm_accumulator_mp () {} // we need this one for Array
  norm_accumulator_mp (R pp) : p(pp), scl(0), sum(1) {}

  template<class U>
  void accum (U val)
  {
    octave_quit ();
    R t = 1 / std::abs (val);
    if (scl == t)
      sum += 1;
    else if (scl < t)
      {
        sum *= std::pow (scl/t, p);
        sum += 1;
        scl = t;
      }
    else if (t != 0)
      sum += std::pow (t/scl, p);
  }
  operator R () { return scl * std::pow (sum, -1/p); }
};

// norm accumulator for the 2-norm (euclidean)
template <class R>
class norm_accumulator_2
{
  R scl,sum;
  static R pow2 (R x) { return x*x; }
public:
  norm_accumulator_2 () : scl(0), sum(1) {}

  void accum (R val)
  {
    R t = std::abs (val);
    if (scl == t)
      sum += 1;
    else if (scl < t)
      {
        sum *= pow2 (scl/t);
        sum += 1;
        scl = t;
      }
    else if (t != 0)
      sum += pow2 (t/scl);
  }

  void accum (std::complex<R> val)
  {
    accum (val.real ());
    accum (val.imag ());
  }

  operator R () { return scl * std::sqrt (sum); }
};

// norm accumulator for the 1-norm (city metric)
template <class R>
class norm_accumulator_1
{
  R sum;
public:
  norm_accumulator_1 () : sum (0) {}
  template<class U>
  void accum (U val)
  {
    sum += std::abs (val);
  }
  operator R () { return sum; }
};

// norm accumulator for the inf-norm (max metric)
template <class R>
class norm_accumulator_inf
{
  R max;
public:
  norm_accumulator_inf () : max (0) {}
  template<class U>
  void accum (U val)
  {
    max = std::max (max, std::abs (val));
  }
  operator R () { return max; }
};

// norm accumulator for the -inf pseudonorm (min abs value)
template <class R>
class norm_accumulator_minf
{
  R min;
public:
  norm_accumulator_minf () : min (octave_Inf) {}
  template<class U>
  void accum (U val)
  {
    min = std::min (min, std::abs (val));
  }
  operator R () { return min; }
};

// norm accumulator for the 0-pseudonorm (hamming distance)
template <class R>
class norm_accumulator_0
{
  unsigned int num;
public:
  norm_accumulator_0 () : num (0) {}
  template<class U>
  void accum (U val)
  {
    if (val != static_cast<U> (0)) ++num;
  }
  operator R () { return num; }
};


// OK, we're armed :) Now let's go for the fun

template <class T, class R, class ACC>
inline void vector_norm (const Array<T>& v, R& res, ACC acc)
{
  for (octave_idx_type i = 0; i < v.numel (); i++)
    acc.accum (v(i));

  res = acc;
}

// dense versions
template <class T, class R, class ACC>
void column_norms (const MArray<T>& m, MArray<R>& res, ACC acc)
{
  res = MArray<R> (dim_vector (1, m.columns ()));
  for (octave_idx_type j = 0; j < m.columns (); j++)
    {
      ACC accj = acc;
      for (octave_idx_type i = 0; i < m.rows (); i++)
        accj.accum (m(i, j));

      res.xelem (j) = accj;
    }
}

template <class T, class R, class ACC>
void row_norms (const MArray<T>& m, MArray<R>& res, ACC acc)
{
  res = MArray<R> (dim_vector (m.rows (), 1));
  std::vector<ACC> acci (m.rows (), acc);
  for (octave_idx_type j = 0; j < m.columns (); j++)
    {
      for (octave_idx_type i = 0; i < m.rows (); i++)
        acci[i].accum (m(i, j));
    }

  for (octave_idx_type i = 0; i < m.rows (); i++)
    res.xelem (i) = acci[i];
}

// sparse versions
template <class T, class R, class ACC>
void column_norms (const MSparse<T>& m, MArray<R>& res, ACC acc)
{
  res = MArray<R> (dim_vector (1, m.columns ()));
  for (octave_idx_type j = 0; j < m.columns (); j++)
    {
      ACC accj = acc;
      for (octave_idx_type k = m.cidx (j); k < m.cidx (j+1); k++)
        accj.accum (m.data (k));

      res.xelem (j) = accj;
    }
}

template <class T, class R, class ACC>
void row_norms (const MSparse<T>& m, MArray<R>& res, ACC acc)
{
  res = MArray<R> (dim_vector (m.rows (), 1));
  std::vector<ACC> acci (m.rows (), acc);
  for (octave_idx_type j = 0; j < m.columns (); j++)
    {
      for (octave_idx_type k = m.cidx (j); k < m.cidx (j+1); k++)
        acci[m.ridx (k)].accum (m.data (k));
    }

  for (octave_idx_type i = 0; i < m.rows (); i++)
    res.xelem (i) = acci[i];
}

// now the dispatchers
#define DEFINE_DISPATCHER(FUNC_NAME, ARG_TYPE, RES_TYPE) \
template <class T, class R> \
RES_TYPE FUNC_NAME (const ARG_TYPE& v, R p) \
{ \
  RES_TYPE res; \
  if (p == 2) \
    FUNC_NAME (v, res, norm_accumulator_2<R> ()); \
  else if (p == 1) \
    FUNC_NAME (v, res, norm_accumulator_1<R> ()); \
  else if (lo_ieee_isinf (p)) \
    { \
      if (p > 0) \
        FUNC_NAME (v, res, norm_accumulator_inf<R> ()); \
      else \
        FUNC_NAME (v, res, norm_accumulator_minf<R> ()); \
    } \
  else if (p == 0) \
    FUNC_NAME (v, res, norm_accumulator_0<R> ()); \
  else if (p > 0) \
    FUNC_NAME (v, res, norm_accumulator_p<R> (p)); \
  else \
    FUNC_NAME (v, res, norm_accumulator_mp<R> (p)); \
  return res; \
}

DEFINE_DISPATCHER (vector_norm, MArray<T>, R)
DEFINE_DISPATCHER (column_norms, MArray<T>, MArray<R>)
DEFINE_DISPATCHER (row_norms, MArray<T>, MArray<R>)
DEFINE_DISPATCHER (column_norms, MSparse<T>, MArray<R>)
DEFINE_DISPATCHER (row_norms, MSparse<T>, MArray<R>)

// The approximate subproblem in Higham's method. Find lambda and mu such that
// norm ([lambda, mu], p) == 1 and norm (y*lambda + col*mu, p) is maximized.
// Real version. As in Higham's paper.
template <class ColVectorT, class R>
static void
higham_subp (const ColVectorT& y, const ColVectorT& col,
             octave_idx_type nsamp, R p, R& lambda, R& mu)
{
  R nrm = 0;
  for (octave_idx_type i = 0; i < nsamp; i++)
    {
      octave_quit ();
      R fi = i * static_cast<R> (M_PI) / nsamp;
      R lambda1 = cos (fi);
      R mu1 = sin (fi);
      R lmnr = std::pow (std::pow (std::abs (lambda1), p) +
                         std::pow (std::abs (mu1), p), 1/p);
      lambda1 /= lmnr; mu1 /= lmnr;
      R nrm1 = vector_norm (lambda1 * y + mu1 * col, p);
      if (nrm1 > nrm)
        {
          lambda = lambda1;
          mu = mu1;
          nrm = nrm1;
        }
    }
}

// Complex version. Higham's paper does not deal with complex case, so we use a
// simple extension. First, guess the magnitudes as in real version, then try
// to rotate lambda to improve further.
template <class ColVectorT, class R>
static void
higham_subp (const ColVectorT& y, const ColVectorT& col,
             octave_idx_type nsamp, R p,
             std::complex<R>& lambda, std::complex<R>& mu)
{
  typedef std::complex<R> CR;
  R nrm = 0;
  lambda = 1.0;
  CR lamcu = lambda / std::abs (lambda);
  // Probe magnitudes
  for (octave_idx_type i = 0; i < nsamp; i++)
    {
      octave_quit ();
      R fi = i * static_cast<R> (M_PI) / nsamp;
      R lambda1 = cos (fi);
      R mu1 = sin (fi);
      R lmnr = std::pow (std::pow (std::abs (lambda1), p) +
                         std::pow (std::abs (mu1), p), 1/p);
      lambda1 /= lmnr; mu1 /= lmnr;
      R nrm1 = vector_norm (lambda1 * lamcu * y + mu1 * col, p);
      if (nrm1 > nrm)
        {
          lambda = lambda1 * lamcu;
          mu = mu1;
          nrm = nrm1;
        }
    }
  R lama = std::abs (lambda);
  // Probe orientation
  for (octave_idx_type i = 0; i < nsamp; i++)
    {
      octave_quit ();
      R fi = i * static_cast<R> (M_PI) / nsamp;
      lamcu = CR (cos (fi), sin (fi));
      R nrm1 = vector_norm (lama * lamcu * y + mu * col, p);
      if (nrm1 > nrm)
        {
          lambda = lama * lamcu;
          nrm = nrm1;
        }
    }
}

// the p-dual element (should work for both real and complex)
template <class T, class R>
inline T elem_dual_p (T x, R p)
{
  return signum (x) * std::pow (std::abs (x), p-1);
}

// the VectorT is used for vectors, but actually it has to be
// a Matrix type to allow all the operations. For instance SparseMatrix
// does not support multiplication with column/row vectors.
// the dual vector
template <class VectorT, class R>
VectorT dual_p (const VectorT& x, R p, R q)
{
  VectorT res (x.dims ());
  for (octave_idx_type i = 0; i < x.numel (); i++)
    res.xelem (i) = elem_dual_p (x(i), p);
  return res / vector_norm (res, q);
}

// Higham's hybrid method
template <class MatrixT, class VectorT, class R>
R higham (const MatrixT& m, R p, R tol, int maxiter,
          VectorT& x)
{
  x.resize (m.columns (), 1);
  // the OSE part
  VectorT y(m.rows (), 1, 0), z(m.rows (), 1);
  typedef typename VectorT::element_type RR;
  RR lambda = 0;
  RR mu = 1;
  for (octave_idx_type k = 0; k < m.columns (); k++)
    {
      octave_quit ();
      VectorT col (m.column (k));
      if (k > 0)
        higham_subp (y, col, 4*k, p, lambda, mu);
      for (octave_idx_type i = 0; i < k; i++)
        x(i) *= lambda;
      x(k) = mu;
      y = lambda * y + mu * col;
    }

  // the PM part
  x = x / vector_norm (x, p);
  R q = p/(p-1);

  R gamma = 0, gamma1;
  int iter = 0;
  while (iter < maxiter)
    {
      octave_quit ();
      y = m*x;
      gamma1 = gamma;
      gamma = vector_norm (y, p);
      z = dual_p (y, p, q);
      z = z.hermitian ();
      z = z * m;

      if (iter > 0 && (vector_norm (z, q) <= gamma
                       || (gamma - gamma1) <= tol*gamma))
        break;

      z = z.hermitian ();
      x = dual_p (z, q, p);
      iter ++;
    }

  return gamma;
}

// derive column vector and SVD types

static const char *p_less1_gripe = "xnorm: p must be at least 1";

// Static constant to control the maximum number of iterations.  100 seems to
// be a good value.  Eventually, we can provide a means to change this
// constant from Octave.
static int max_norm_iter = 100;

// version with SVD for dense matrices
template <class MatrixT, class VectorT, class SVDT, class R>
R matrix_norm (const MatrixT& m, R p, VectorT, SVDT)
{
  R res = 0;
  if (p == 2)
    {
      SVDT svd (m, SVD::sigma_only);
      res = svd.singular_values () (0,0);
    }
  else if (p == 1)
    res = xcolnorms (m, 1).max ();
  else if (lo_ieee_isinf (p))
    res = xrownorms (m, 1).max ();
  else if (p > 1)
    {
      VectorT x;
      const R sqrteps = std::sqrt (std::numeric_limits<R>::epsilon ());
      res = higham (m, p, sqrteps, max_norm_iter, x);
    }
  else
    (*current_liboctave_error_handler) (p_less1_gripe);

  return res;
}

// SVD-free version for sparse matrices
template <class MatrixT, class VectorT, class R>
R matrix_norm (const MatrixT& m, R p, VectorT)
{
  R res = 0;
  if (p == 1)
    res = xcolnorms (m, 1).max ();
  else if (lo_ieee_isinf (p))
    res = xrownorms (m, 1).max ();
  else if (p > 1)
    {
      VectorT x;
      const R sqrteps = std::sqrt (std::numeric_limits<R>::epsilon ());
      res = higham (m, p, sqrteps, max_norm_iter, x);
    }
  else
    (*current_liboctave_error_handler) (p_less1_gripe);

  return res;
}

// and finally, here's what we've promised in the header file

#define DEFINE_XNORM_FUNCS(PREFIX, RTYPE) \
  OCTAVE_API RTYPE xnorm (const PREFIX##ColumnVector& x, RTYPE p) \
  { return vector_norm (x, p); } \
  OCTAVE_API RTYPE xnorm (const PREFIX##RowVector& x, RTYPE p) \
  { return vector_norm (x, p); } \
  OCTAVE_API RTYPE xnorm (const PREFIX##Matrix& x, RTYPE p) \
  { return matrix_norm (x, p, PREFIX##Matrix (), PREFIX##SVD ()); } \
  OCTAVE_API RTYPE xfrobnorm (const PREFIX##Matrix& x) \
  { return vector_norm (x, static_cast<RTYPE> (2)); }

DEFINE_XNORM_FUNCS(, double)
DEFINE_XNORM_FUNCS(Complex, double)
DEFINE_XNORM_FUNCS(Float, float)
DEFINE_XNORM_FUNCS(FloatComplex, float)

// this is needed to avoid copying the sparse matrix for xfrobnorm
template <class T, class R>
inline void array_norm_2 (const T* v, octave_idx_type n, R& res)
{
  norm_accumulator_2<R> acc;
  for (octave_idx_type i = 0; i < n; i++)
    acc.accum (v[i]);

  res = acc;
}

#define DEFINE_XNORM_SPARSE_FUNCS(PREFIX, RTYPE) \
  OCTAVE_API RTYPE xnorm (const Sparse##PREFIX##Matrix& x, RTYPE p) \
  { return matrix_norm (x, p, PREFIX##Matrix ()); } \
  OCTAVE_API RTYPE xfrobnorm (const Sparse##PREFIX##Matrix& x) \
  { \
    RTYPE res; \
    array_norm_2 (x.data (), x.nnz (), res); \
    return res; \
  }

DEFINE_XNORM_SPARSE_FUNCS(, double)
DEFINE_XNORM_SPARSE_FUNCS(Complex, double)

#define DEFINE_COLROW_NORM_FUNCS(PREFIX, RPREFIX, RTYPE) \
  extern OCTAVE_API RPREFIX##RowVector xcolnorms (const PREFIX##Matrix& m, RTYPE p) \
  { return column_norms (m, p); } \
  extern OCTAVE_API RPREFIX##ColumnVector xrownorms (const PREFIX##Matrix& m, RTYPE p) \
  { return row_norms (m, p); } \

DEFINE_COLROW_NORM_FUNCS(, , double)
DEFINE_COLROW_NORM_FUNCS(Complex, , double)
DEFINE_COLROW_NORM_FUNCS(Float, Float, float)
DEFINE_COLROW_NORM_FUNCS(FloatComplex, Float, float)

DEFINE_COLROW_NORM_FUNCS(Sparse, , double)
DEFINE_COLROW_NORM_FUNCS(SparseComplex, , double)