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linear_solve.hxx |

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35
36
37 #ifndef VIGRA_LINEAR_SOLVE_HXX
38 #define VIGRA_LINEAR_SOLVE_HXX
39
40 #include <ctype.h>
41 #include <string>
42 #include "mathutil.hxx"
43 #include "matrix.hxx"
44 #include "singular_value_decomposition.hxx"
45
46
47 namespace vigra
48 {
49
50 namespace linalg
51 {
52
53 namespace detail {
54
55 template <class T, class C1>
56 T determinantByLUDecomposition(MultiArrayView<2, T, C1> const & a)
57 {
58 typedef MultiArrayShape<2>::type Shape;
59
60 MultiArrayIndex m = rowCount(a), n = columnCount(a);
61 vigra_precondition(n == m,
62 "determinant(): square matrix required.");
63
64 Matrix<T> LU(a);
65 T det = 1.0;
66
67 for (MultiArrayIndex j = 0; j < n; ++j)
68 {
69 // Apply previous transformations.
70 for (MultiArrayIndex i = 0; i < m; ++i)
71 {
72 MultiArrayIndex end = std::min(i, j);
73 T s = dot(rowVector(LU, Shape(i,0), end), columnVector(LU, Shape(0,j), end));
74 LU(i,j) = LU(i,j) -= s;
75 }
76
77 // Find pivot and exchange if necessary.
78 MultiArrayIndex p = j + argMax(abs(columnVector(LU, Shape(j,j), m)));
79 if (p != j)
80 {
81 rowVector(LU, p).swapData(rowVector(LU, j));
82 det = -det;
83 }
84
85 det *= LU(j,j);
86
87 // Compute multipliers.
88 if (LU(j,j) != 0.0)
89 columnVector(LU, Shape(j+1,j), m) /= LU(j,j);
90 else
91 break; // det is zero
92 }
93 return det;
94 }
95
96 // returns the new value of 'a' (when this Givens rotation is applied to 'a' and 'b')
97 // the new value of 'b' is zero, of course
98 template <class T>
99 T givensCoefficients(T a, T b, T & c, T & s)
100 {
101 if(abs(a) < abs(b))
102 {
103 T t = a/b,
104 r = std::sqrt(1.0 + t*t);
105 s = 1.0 / r;
106 c = t*s;
107 return r*b;
108 }
109 else if(a != 0.0)
110 {
111 T t = b/a,
112 r = std::sqrt(1.0 + t*t);
113 c = 1.0 / r;
114 s = t*c;
115 return r*a;
116 }
117 else // a == b == 0.0
118 {
119 c = 1.0;
120 s = 0.0;
121 return 0.0;
122 }
123 }
124
125 // see Golub, van Loan: Algorithm 5.1.3 (p. 216)
126 template <class T>
127 bool givensRotationMatrix(T a, T b, Matrix<T> & gTranspose)
128 {
129 if(b == 0.0)
130 return false; // no rotation needed
131 givensCoefficients(a, b, gTranspose(0,0), gTranspose(0,1));
132 gTranspose(1,1) = gTranspose(0,0);
133 gTranspose(1,0) = -gTranspose(0,1);
134 return true;
135 }
136
137 // reflections are symmetric matrices and can thus be applied to rows
138 // and columns in the same way => code simplification relative to rotations
139 template <class T>
140 inline bool
141 givensReflectionMatrix(T a, T b, Matrix<T> & g)
142 {
143 if(b == 0.0)
144 return false; // no reflection needed
145 givensCoefficients(a, b, g(0,0), g(0,1));
146 g(1,1) = -g(0,0);
147 g(1,0) = g(0,1);
148 return true;
149 }
150
151 // see Golub, van Loan: Algorithm 5.2.2 (p. 227) and Section 12.5.2 (p. 608)
152 template <class T, class C1, class C2>
153 bool
154 qrGivensStepImpl(MultiArrayIndex i, MultiArrayView<2, T, C1> r, MultiArrayView<2, T, C2> rhs)
155 {
156 typedef typename Matrix<T>::difference_type Shape;
157
158 const MultiArrayIndex m = rowCount(r);
159 const MultiArrayIndex n = columnCount(r);
160 const MultiArrayIndex rhsCount = columnCount(rhs);
161 vigra_precondition(m == rowCount(rhs),
162 "qrGivensStepImpl(): Matrix shape mismatch.");
163
164 Matrix<T> givens(2,2);
165 for(int k=m-1; k>(int)i; --k)
166 {
167 if(!givensReflectionMatrix(r(k-1,i), r(k,i), givens))
168 continue; // r(k,i) was already zero
169
170 r(k-1,i) = givens(0,0)*r(k-1,i) + givens(0,1)*r(k,i);
171 r(k,i) = 0.0;
172
173 r.subarray(Shape(k-1,i+1), Shape(k+1,n)) = givens*r.subarray(Shape(k-1,i+1), Shape(k+1,n));
174 rhs.subarray(Shape(k-1,0), Shape(k+1,rhsCount)) = givens*rhs.subarray(Shape(k-1,0), Shape(k+1,rhsCount));
175 }
176 return r(i,i) != 0.0;
177 }
178
179 // see Golub, van Loan: Section 12.5.2 (p. 608)
180 template <class T, class C1, class C2, class Permutation>
181 void
182 upperTriangularCyclicShiftColumns(MultiArrayIndex i, MultiArrayIndex j,
183 MultiArrayView<2, T, C1> &r, MultiArrayView<2, T, C2> &rhs, Permutation & permutation)
184 {
185 typedef typename Matrix<T>::difference_type Shape;
186
187 const MultiArrayIndex m = rowCount(r);
188 const MultiArrayIndex n = columnCount(r);
189 const MultiArrayIndex rhsCount = columnCount(rhs);
190 vigra_precondition(i < n && j < n,
191 "upperTriangularCyclicShiftColumns(): Shift indices out of range.");
192 vigra_precondition(m == rowCount(rhs),
193 "upperTriangularCyclicShiftColumns(): Matrix shape mismatch.");
194
195 if(j == i)
196 return;
197 if(j < i)
198 std::swap(j,i);
199
200 Matrix<T> t = columnVector(r, i);
201 MultiArrayIndex ti = permutation[i];
202 for(MultiArrayIndex k=i; k<j;++k)
203 {
204 columnVector(r, k) = columnVector(r, k+1);
205 permutation[k] = permutation[k+1];
206 }
207 columnVector(r, j) = t;
208 permutation[j] = ti;
209
210 Matrix<T> givens(2,2);
211 for(MultiArrayIndex k=i; k<j; ++k)
212 {
213 if(!givensReflectionMatrix(r(k,k), r(k+1,k), givens))
214 continue; // r(k+1,k) was already zero
215
216 r(k,k) = givens(0,0)*r(k,k) + givens(0,1)*r(k+1,k);
217 r(k+1,k) = 0.0;
218
219 r.subarray(Shape(k,k+1), Shape(k+2,n)) = givens*r.subarray(Shape(k,k+1), Shape(k+2,n));
220 rhs.subarray(Shape(k,0), Shape(k+2,rhsCount)) = givens*rhs.subarray(Shape(k,0), Shape(k+2,rhsCount));
221 }
222 }
223
224 // see Golub, van Loan: Section 12.5.2 (p. 608)
225 template <class T, class C1, class C2, class Permutation>
226 void
227 upperTriangularSwapColumns(MultiArrayIndex i, MultiArrayIndex j,
228 MultiArrayView<2, T, C1> &r, MultiArrayView<2, T, C2> &rhs, Permutation & permutation)
229 {
230 typedef typename Matrix<T>::difference_type Shape;
231
232 const MultiArrayIndex m = rowCount(r);
233 const MultiArrayIndex n = columnCount(r);
234 const MultiArrayIndex rhsCount = columnCount(rhs);
235 vigra_precondition(i < n && j < n,
236 "upperTriangularSwapColumns(): Swap indices out of range.");
237 vigra_precondition(m == rowCount(rhs),
238 "upperTriangularSwapColumns(): Matrix shape mismatch.");
239
240 if(j == i)
241 return;
242 if(j < i)
243 std::swap(j,i);
244
245 columnVector(r, i).swapData(columnVector(r, j));
246 std::swap(permutation[i], permutation[j]);
247
248 Matrix<T> givens(2,2);
249 for(int k=m-1; k>(int)i; --k)
250 {
251 if(!givensReflectionMatrix(r(k-1,i), r(k,i), givens))
252 continue; // r(k,i) was already zero
253
254 r(k-1,i) = givens(0,0)*r(k-1,i) + givens(0,1)*r(k,i);
255 r(k,i) = 0.0;
256
257 r.subarray(Shape(k-1,i+1), Shape(k+1,n)) = givens*r.subarray(Shape(k-1,i+1), Shape(k+1,n));
258 rhs.subarray(Shape(k-1,0), Shape(k+1,rhsCount)) = givens*rhs.subarray(Shape(k-1,0), Shape(k+1,rhsCount));
259 }
260 MultiArrayIndex end = std::min(j, m-1);
261 for(MultiArrayIndex k=i+1; k<end; ++k)
262 {
263 if(!givensReflectionMatrix(r(k,k), r(k+1,k), givens))
264 continue; // r(k+1,k) was already zero
265
266 r(k,k) = givens(0,0)*r(k,k) + givens(0,1)*r(k+1,k);
267 r(k+1,k) = 0.0;
268
269 r.subarray(Shape(k,k+1), Shape(k+2,n)) = givens*r.subarray(Shape(k,k+1), Shape(k+2,n));
270 rhs.subarray(Shape(k,0), Shape(k+2,rhsCount)) = givens*rhs.subarray(Shape(k,0), Shape(k+2,rhsCount));
271 }
272 }
273
274 // see Lawson & Hanson: Algorithm H1 (p. 57)
275 template <class T, class C1, class C2, class U>
276 bool householderVector(MultiArrayView<2, T, C1> const & v, MultiArrayView<2, T, C2> & u, U & vnorm)
277 {
278 vnorm = (v(0,0) > 0.0)
279 ? -norm(v)
280 : norm(v);
281 U f = std::sqrt(vnorm*(vnorm - v(0,0)));
282
283 if(f == NumericTraits<U>::zero())
284 {
285 u.init(NumericTraits<T>::zero());
286 return false;
287 }
288 else
289 {
290 u(0,0) = (v(0,0) - vnorm) / f;
291 for(MultiArrayIndex k=1; k<rowCount(u); ++k)
292 u(k,0) = v(k,0) / f;
293 return true;
294 }
295 }
296
297 // see Lawson & Hanson: Algorithm H1 (p. 57)
298 template <class T, class C1, class C2, class C3>
299 bool
300 qrHouseholderStepImpl(MultiArrayIndex i, MultiArrayView<2, T, C1> & r,
301 MultiArrayView<2, T, C2> & rhs, MultiArrayView<2, T, C3> & householderMatrix)
302 {
303 typedef typename Matrix<T>::difference_type Shape;
304
305 const MultiArrayIndex m = rowCount(r);
306 const MultiArrayIndex n = columnCount(r);
307 const MultiArrayIndex rhsCount = columnCount(rhs);
308
309 vigra_precondition(i < n && i < m,
310 "qrHouseholderStepImpl(): Index i out of range.");
311
312 Matrix<T> u(m-i,1);
313 T vnorm;
314 bool nontrivial = householderVector(columnVector(r, Shape(i,i), m), u, vnorm);
315
316 r(i,i) = vnorm;
317 columnVector(r, Shape(i+1,i), m).init(NumericTraits<T>::zero());
318
319 if(columnCount(householderMatrix) == n)
320 columnVector(householderMatrix, Shape(i,i), m) = u;
321
322 if(nontrivial)
323 {
324 for(MultiArrayIndex k=i+1; k<n; ++k)
325 columnVector(r, Shape(i,k), m) -= dot(columnVector(r, Shape(i,k), m), u) * u;
326 for(MultiArrayIndex k=0; k<rhsCount; ++k)
327 columnVector(rhs, Shape(i,k), m) -= dot(columnVector(rhs, Shape(i,k), m), u) * u;
328 }
329 return r(i,i) != 0.0;
330 }
331
332 template <class T, class C1, class C2>
333 bool
334 qrColumnHouseholderStep(MultiArrayIndex i, MultiArrayView<2, T, C1> &r, MultiArrayView<2, T, C2> &rhs)
335 {
336 Matrix<T> dontStoreHouseholderVectors; // intentionally empty
337 return qrHouseholderStepImpl(i, r, rhs, dontStoreHouseholderVectors);
338 }
339
340 template <class T, class C1, class C2>
341 bool
342 qrRowHouseholderStep(MultiArrayIndex i, MultiArrayView<2, T, C1> &r, MultiArrayView<2, T, C2> & householderMatrix)
343 {
344 Matrix<T> dontTransformRHS; // intentionally empty
345 MultiArrayView<2, T, StridedArrayTag> rt = transpose(r),
346 ht = transpose(householderMatrix);
347 return qrHouseholderStepImpl(i, rt, dontTransformRHS, ht);
348 }
349
350 // O(n) algorithm due to Bischof: Incremental Condition Estimation, 1990
351 template <class T, class C1, class C2, class SNType>
352 void
353 incrementalMaxSingularValueApproximation(MultiArrayView<2, T, C1> const & newColumn,
354 MultiArrayView<2, T, C2> & z, SNType & v)
355 {
356 typedef typename Matrix<T>::difference_type Shape;
357 MultiArrayIndex n = rowCount(newColumn) - 1;
358
359 SNType vneu = squaredNorm(newColumn);
360 T yv = dot(columnVector(newColumn, Shape(0,0),n), columnVector(z, Shape(0,0),n));
361 // use atan2 as it is robust against overflow/underflow
362 T t = 0.5*std::atan2(T(2.0*yv), T(sq(v)-vneu)),
363 s = std::sin(t),
364 c = std::cos(t);
365 v = std::sqrt(sq(c*v) + sq(s)*vneu + 2.0*s*c*yv);
366 columnVector(z, Shape(0,0),n) = c*columnVector(z, Shape(0,0),n) + s*columnVector(newColumn, Shape(0,0),n);
367 z(n,0) = s*newColumn(n,0);
368 }
369
370 // O(n) algorithm due to Bischof: Incremental Condition Estimation, 1990
371 template <class T, class C1, class C2, class SNType>
372 void
373 incrementalMinSingularValueApproximation(MultiArrayView<2, T, C1> const & newColumn,
374 MultiArrayView<2, T, C2> & z, SNType & v, double tolerance)
375 {
376 typedef typename Matrix<T>::difference_type Shape;
377
378 if(v <= tolerance)
379 {
380 v = 0.0;
381 return;
382 }
383
384 MultiArrayIndex n = rowCount(newColumn) - 1;
385
386 T gamma = newColumn(n,0);
387 if(gamma == 0.0)
388 {
389 v = 0.0;
390 return;
391 }
392
393 T yv = dot(columnVector(newColumn, Shape(0,0),n), columnVector(z, Shape(0,0),n));
394 // use atan2 as it is robust against overflow/underflow
395 T t = 0.5*std::atan2(T(-2.0*yv), T(squaredNorm(gamma / v) + squaredNorm(yv) - 1.0)),
396 s = std::sin(t),
397 c = std::cos(t);
398 columnVector(z, Shape(0,0),n) *= c;
399 z(n,0) = (s - c*yv) / gamma;
400 v *= norm(gamma) / hypot(c*gamma, v*(s - c*yv));
401 }
402
403 // QR algorithm with optional column pivoting
404 template <class T, class C1, class C2, class C3>
405 unsigned int
406 qrTransformToTriangularImpl(MultiArrayView<2, T, C1> & r, MultiArrayView<2, T, C2> & rhs, MultiArrayView<2, T, C3> & householder,
407 ArrayVector<MultiArrayIndex> & permutation, double epsilon)
408 {
409 typedef typename Matrix<T>::difference_type Shape;
410 typedef typename NormTraits<MultiArrayView<2, T, C1> >::NormType NormType;
411 typedef typename NormTraits<MultiArrayView<2, T, C1> >::SquaredNormType SNType;
412
413 const MultiArrayIndex m = rowCount(r);
414 const MultiArrayIndex n = columnCount(r);
415 const MultiArrayIndex maxRank = std::min(m, n);
416
417 vigra_precondition(m >= n,
418 "qrTransformToTriangularImpl(): Coefficient matrix with at least as many rows as columns required.");
419
420 const MultiArrayIndex rhsCount = columnCount(rhs);
421 bool transformRHS = rhsCount > 0;
422 vigra_precondition(!transformRHS || m == rowCount(rhs),
423 "qrTransformToTriangularImpl(): RHS matrix shape mismatch.");
424
425 bool storeHouseholderSteps = columnCount(householder) > 0;
426 vigra_precondition(!storeHouseholderSteps || r.shape() == householder.shape(),
427 "qrTransformToTriangularImpl(): Householder matrix shape mismatch.");
428
429 bool pivoting = permutation.size() > 0;
430 vigra_precondition(!pivoting || n == (MultiArrayIndex)permutation.size(),
431 "qrTransformToTriangularImpl(): Permutation array size mismatch.");
432
433 if(n == 0)
434 return 0; // trivial solution
435
436 Matrix<SNType> columnSquaredNorms;
437 if(pivoting)
438 {
439 columnSquaredNorms.reshape(Shape(1,n));
440 for(MultiArrayIndex k=0; k<n; ++k)
441 columnSquaredNorms[k] = squaredNorm(columnVector(r, k));
442
443 int pivot = argMax(columnSquaredNorms);
444 if(pivot != 0)
445 {
446 columnVector(r, 0).swapData(columnVector(r, pivot));
447 std::swap(columnSquaredNorms[0], columnSquaredNorms[pivot]);
448 std::swap(permutation[0], permutation[pivot]);
449 }
450 }
451
452 qrHouseholderStepImpl(0, r, rhs, householder);
453
454 MultiArrayIndex rank = 1;
455 NormType maxApproxSingularValue = norm(r(0,0)),
456 minApproxSingularValue = maxApproxSingularValue;
457
458 double tolerance = (epsilon == 0.0)
459 ? m*maxApproxSingularValue*NumericTraits<T>::epsilon()
460 : epsilon;
461
462 bool simpleSingularValueApproximation = (n < 4);
463 Matrix<T> zmax, zmin;
464 if(minApproxSingularValue <= tolerance)
465 {
466 rank = 0;
467 pivoting = false;
468 simpleSingularValueApproximation = true;
469 }
470 if(!simpleSingularValueApproximation)
471 {
472 zmax.reshape(Shape(m,1));
473 zmin.reshape(Shape(m,1));
474 zmax(0,0) = r(0,0);
475 zmin(0,0) = 1.0 / r(0,0);
476 }
477
478 for(MultiArrayIndex k=1; k<maxRank; ++k)
479 {
480 if(pivoting)
481 {
482 for(MultiArrayIndex l=k; l<n; ++l)
483 columnSquaredNorms[l] -= squaredNorm(r(k, l));
484 int pivot = k + argMax(rowVector(columnSquaredNorms, Shape(0,k), n));
485 if(pivot != (int)k)
486 {
487 columnVector(r, k).swapData(columnVector(r, pivot));
488 std::swap(columnSquaredNorms[k], columnSquaredNorms[pivot]);
489 std::swap(permutation[k], permutation[pivot]);
490 }
491 }
492
493 qrHouseholderStepImpl(k, r, rhs, householder);
494
495 if(simpleSingularValueApproximation)
496 {
497 NormType nv = norm(r(k,k));
498 maxApproxSingularValue = std::max(nv, maxApproxSingularValue);
499 minApproxSingularValue = std::min(nv, minApproxSingularValue);
500 }
501 else
502 {
503 incrementalMaxSingularValueApproximation(columnVector(r, Shape(0,k),k+1), zmax, maxApproxSingularValue);
504 incrementalMinSingularValueApproximation(columnVector(r, Shape(0,k),k+1), zmin, minApproxSingularValue, tolerance);
505 }
506
507 #if 0
508 Matrix<T> u(k+1,k+1), s(k+1, 1), v(k+1,k+1);
509 singularValueDecomposition(r.subarray(Shape(0,0), Shape(k+1,k+1)), u, s, v);
510 std::cerr << "estimate, svd " << k << ": " << minApproxSingularValue << " " << s(k,0) << "\n";
511 #endif
512
513 if(epsilon == 0.0)
514 tolerance = m*maxApproxSingularValue*NumericTraits<T>::epsilon();
515
516 if(minApproxSingularValue > tolerance)
517 ++rank;
518 else
519 pivoting = false; // matrix doesn't have full rank, triangulize the rest without pivoting
520 }
521 return (unsigned int)rank;
522 }
523
524 template <class T, class C1, class C2>
525 unsigned int
526 qrTransformToUpperTriangular(MultiArrayView<2, T, C1> & r, MultiArrayView<2, T, C2> & rhs,
527 ArrayVector<MultiArrayIndex> & permutation, double epsilon = 0.0)
528 {
529 Matrix<T> dontStoreHouseholderVectors; // intentionally empty
530 return qrTransformToTriangularImpl(r, rhs, dontStoreHouseholderVectors, permutation, epsilon);
531 }
532
533 // QR algorithm with optional row pivoting
534 template <class T, class C1, class C2, class C3>
535 unsigned int
536 qrTransformToLowerTriangular(MultiArrayView<2, T, C1> & r, MultiArrayView<2, T, C2> & rhs, MultiArrayView<2, T, C3> & householderMatrix,
537 double epsilon = 0.0)
538 {
539 ArrayVector<MultiArrayIndex> permutation((unsigned int)rowCount(rhs));
540 for(MultiArrayIndex k=0; k<(MultiArrayIndex)permutation.size(); ++k)
541 permutation[k] = k;
542 Matrix<T> dontTransformRHS; // intentionally empty
543 MultiArrayView<2, T, StridedArrayTag> rt = transpose(r),
544 ht = transpose(householderMatrix);
545 unsigned int rank = qrTransformToTriangularImpl(rt, dontTransformRHS, ht, permutation, epsilon);
546
547 // apply row permutation to RHS
548 Matrix<T> tempRHS(rhs);
549 for(MultiArrayIndex k=0; k<(MultiArrayIndex)permutation.size(); ++k)
550 rowVector(rhs, k) = rowVector(tempRHS, permutation[k]);
551 return rank;
552 }
553
554 // QR algorithm without column pivoting
555 template <class T, class C1, class C2>
556 inline bool
557 qrTransformToUpperTriangular(MultiArrayView<2, T, C1> & r, MultiArrayView<2, T, C2> & rhs,
558 double epsilon = 0.0)
559 {
560 ArrayVector<MultiArrayIndex> noPivoting; // intentionally empty
561
562 return (qrTransformToUpperTriangular(r, rhs, noPivoting, epsilon) ==
563 (unsigned int)columnCount(r));
564 }
565
566 // QR algorithm without row pivoting
567 template <class T, class C1, class C2>
568 inline bool
569 qrTransformToLowerTriangular(MultiArrayView<2, T, C1> & r, MultiArrayView<2, T, C2> & householder,
570 double epsilon = 0.0)
571 {
572 Matrix<T> noPivoting; // intentionally empty
573
574 return (qrTransformToLowerTriangular(r, noPivoting, householder, epsilon) ==
575 (unsigned int)rowCount(r));
576 }
577
578 // restore ordering of result vector elements after QR solution with column pivoting
579 template <class T, class C1, class C2, class Permutation>
580 void inverseRowPermutation(MultiArrayView<2, T, C1> &permuted, MultiArrayView<2, T, C2> &res,
581 Permutation const & permutation)
582 {
583 for(MultiArrayIndex k=0; k<columnCount(permuted); ++k)
584 for(MultiArrayIndex l=0; l<rowCount(permuted); ++l)
585 res(permutation[l], k) = permuted(l,k);
586 }
587
588 template <class T, class C1, class C2>
589 void applyHouseholderColumnReflections(MultiArrayView<2, T, C1> const &householder, MultiArrayView<2, T, C2> &res)
590 {
591 typedef typename Matrix<T>::difference_type Shape;
592 MultiArrayIndex n = rowCount(householder);
593 MultiArrayIndex m = columnCount(householder);
594 MultiArrayIndex rhsCount = columnCount(res);
595
596 for(int k = m-1; k >= 0; --k)
597 {
598 MultiArrayView<2, T, C1> u = columnVector(householder, Shape(k,k), n);
599 for(MultiArrayIndex l=0; l<rhsCount; ++l)
600 columnVector(res, Shape(k,l), n) -= dot(columnVector(res, Shape(k,l), n), u) * u;
601 }
602 }
603
604 } // namespace detail
605
606 template <class T, class C1, class C2, class C3>
607 unsigned int
608 linearSolveQRReplace(MultiArrayView<2, T, C1> &A, MultiArrayView<2, T, C2> &b,
609 MultiArrayView<2, T, C3> & res,
610 double epsilon = 0.0)
611 {
612 typedef typename Matrix<T>::difference_type Shape;
613
614 MultiArrayIndex n = columnCount(A);
615 MultiArrayIndex m = rowCount(A);
616 MultiArrayIndex rhsCount = columnCount(res);
617 MultiArrayIndex rank = std::min(m,n);
618 Shape ul(MultiArrayIndex(0), MultiArrayIndex(0));
619
620
621 vigra_precondition(rhsCount == columnCount(b),
622 "linearSolveQR(): RHS and solution must have the same number of columns.");
623 vigra_precondition(m == rowCount(b),
624 "linearSolveQR(): Coefficient matrix and RHS must have the same number of rows.");
625 vigra_precondition(n == rowCount(res),
626 "linearSolveQR(): Mismatch between column count of coefficient matrix and row count of solution.");
627 vigra_precondition(epsilon >= 0.0,
628 "linearSolveQR(): 'epsilon' must be non-negative.");
629
630 if(m < n)
631 {
632 // minimum norm solution of underdetermined system
633 Matrix<T> householderMatrix(n, m);
634 MultiArrayView<2, T, StridedArrayTag> ht = transpose(householderMatrix);
635 rank = (MultiArrayIndex)detail::qrTransformToLowerTriangular(A, b, ht, epsilon);
636 res.subarray(Shape(rank,0), Shape(n, rhsCount)).init(NumericTraits<T>::zero());
637 if(rank < m)
638 {
639 // system is also rank-deficient => compute minimum norm least squares solution
640 MultiArrayView<2, T, C1> Asub = A.subarray(ul, Shape(m,rank));
641 detail::qrTransformToUpperTriangular(Asub, b, epsilon);
642 linearSolveUpperTriangular(A.subarray(ul, Shape(rank,rank)),
643 b.subarray(ul, Shape(rank,rhsCount)),
644 res.subarray(ul, Shape(rank, rhsCount)));
645 }
646 else
647 {
648 // system has full rank => compute minimum norm solution
649 linearSolveLowerTriangular(A.subarray(ul, Shape(rank,rank)),
650 b.subarray(ul, Shape(rank, rhsCount)),
651 res.subarray(ul, Shape(rank, rhsCount)));
652 }
653 detail::applyHouseholderColumnReflections(householderMatrix.subarray(ul, Shape(n, rank)), res);
654 }
655 else
656 {
657 // solution of well-determined or overdetermined system
658 ArrayVector<MultiArrayIndex> permutation((unsigned int)n);
659 for(MultiArrayIndex k=0; k<n; ++k)
660 permutation[k] = k;
661
662 rank = (MultiArrayIndex)detail::qrTransformToUpperTriangular(A, b, permutation, epsilon);
663
664 Matrix<T> permutedSolution(n, rhsCount);
665 if(rank < n)
666 {
667 // system is rank-deficient => compute minimum norm solution
668 Matrix<T> householderMatrix(n, rank);
669 MultiArrayView<2, T, StridedArrayTag> ht = transpose(householderMatrix);
670 MultiArrayView<2, T, C1> Asub = A.subarray(ul, Shape(rank,n));
671 detail::qrTransformToLowerTriangular(Asub, ht, epsilon);
672 linearSolveLowerTriangular(A.subarray(ul, Shape(rank,rank)),
673 b.subarray(ul, Shape(rank, rhsCount)),
674 permutedSolution.subarray(ul, Shape(rank, rhsCount)));
675 detail::applyHouseholderColumnReflections(householderMatrix, permutedSolution);
676 }
677 else
678 {
679 // system has full rank => compute exact or least squares solution
680 linearSolveUpperTriangular(A.subarray(ul, Shape(rank,rank)),
681 b.subarray(ul, Shape(rank,rhsCount)),
682 permutedSolution);
683 }
684 detail::inverseRowPermutation(permutedSolution, res, permutation);
685 }
686 return (unsigned int)rank;
687 }
688
689 template <class T, class C1, class C2, class C3>
690 unsigned int linearSolveQR(MultiArrayView<2, T, C1> const & A, MultiArrayView<2, T, C2> const & b,
691 MultiArrayView<2, T, C3> & res)
692 {
693 Matrix<T> r(A), rhs(b);
694 return linearSolveQRReplace(r, rhs, res);
695 }
696
697 /** \defgroup MatrixAlgebra Advanced Matrix Algebra
698
699 \brief Solution of linear systems, eigen systems, linear least squares etc.
700
701 \ingroup LinearAlgebraModule
702 */
703 //@{
704 /** Create the inverse or pseudo-inverse of matrix \a v.
705
706 If the matrix \a v is square, \a res must have the same shape and will contain the
707 inverse of \a v. If \a v is rectangular, \a res must have the transposed shape
708 of \a v. The inverse is then computed in the least-squares
709 sense, i.e. \a res will be the pseudo-inverse (Moore-Penrose inverse).
710 The function returns <tt>true</tt> upon success, and <tt>false</tt> if \a v
711 is not invertible (has not full rank). The inverse is computed by means of QR
712 decomposition. This function can be applied in-place.
713
714 <b>\#include</b> <vigra/linear_solve.hxx> or<br>
715 <b>\#include</b> <vigra/linear_algebra.hxx><br>
716 Namespaces: vigra and vigra::linalg
717 */
718 template <class T, class C1, class C2>
719 bool inverse(const MultiArrayView<2, T, C1> &v, MultiArrayView<2, T, C2> &res)
720 {
721 typedef typename MultiArrayShape<2>::type Shape;
722
723 const MultiArrayIndex n = columnCount(v);
724 const MultiArrayIndex m = rowCount(v);
725 vigra_precondition(n == rowCount(res) && m == columnCount(res),
726 "inverse(): shape of output matrix must be the transpose of the input matrix' shape.");
727
728 if(m < n)
729 {
730 MultiArrayView<2, T, StridedArrayTag> vt = transpose(v);
731 Matrix<T> r(vt.shape()), q(n, n);
732 if(!qrDecomposition(vt, q, r))
733 return false; // a didn't have full rank
734 linearSolveUpperTriangular(r.subarray(Shape(0,0), Shape(m,m)),
735 transpose(q).subarray(Shape(0,0), Shape(m,n)),
736 transpose(res));
737 }
738 else
739 {
740 Matrix<T> r(v.shape()), q(m, m);
741 if(!qrDecomposition(v, q, r))
742 return false; // a didn't have full rank
743 linearSolveUpperTriangular(r.subarray(Shape(0,0), Shape(n,n)),
744 transpose(q).subarray(Shape(0,0), Shape(n,m)),
745 res);
746 }
747 return true;
748 }
749
750 /** Create the inverse or pseudo-inverse of matrix \a v.
751
752 The result is returned as a temporary matrix. If the matrix \a v is square,
753 the result will have the same shape and contains the inverse of \a v.
754 If \a v is rectangular, the result will have the transposed shape of \a v.
755 The inverse is then computed in the least-squares
756 sense, i.e. \a res will be the pseudo-inverse (Moore-Penrose inverse).
757 The inverse is computed by means of QR decomposition. If \a v
758 is not invertible, <tt>vigra::PreconditionViolation</tt> exception is thrown.
759 Usage:
760
761 \code
762 vigra::Matrix<double> v(n, n);
763 v = ...;
764
765 vigra::Matrix<double> m = inverse(v);
766 \endcode
767
768 <b>\#include</b> <vigra/linear_solve.hxx> or<br>
769 <b>\#include</b> <vigra/linear_algebra.hxx><br>
770 Namespaces: vigra and vigra::linalg
771 */
772 template <class T, class C>
773 TemporaryMatrix<T> inverse(const MultiArrayView<2, T, C> &v)
774 {
775 TemporaryMatrix<T> ret(columnCount(v), rowCount(v)); // transpose shape
776 vigra_precondition(inverse(v, ret),
777 "inverse(): matrix is not invertible.");
778 return ret;
779 }
780
781 template <class T>
782 TemporaryMatrix<T> inverse(const TemporaryMatrix<T> &v)
783 {
784 if(columnCount(v) == rowCount(v))
785 {
786 vigra_precondition(inverse(v, const_cast<TemporaryMatrix<T> &>(v)),
787 "inverse(): matrix is not invertible.");
788 return v;
789 }
790 else
791 {
792 TemporaryMatrix<T> ret(columnCount(v), rowCount(v)); // transpose shape
793 vigra_precondition(inverse(v, ret),
794 "inverse(): matrix is not invertible.");
795 return ret;
796 }
797 }
798
799 /** Compute the determinant of a square matrix.
800
801 \a method must be one of the following:
802 <DL>
803 <DT>"Cholesky"<DD> Compute the solution by means of Cholesky decomposition. This
804 method is faster than "LU", but requires the matrix \a a
805 to be symmetric positive definite. If this is
806 not the case, a <tt>ContractViolation</tt> exception is thrown.
807
808 <DT>"LU"<DD> (default) Compute the solution by means of LU decomposition.
809 </DL>
810
811 <b>\#include</b> <vigra/linear_solve.hxx> or<br>
812 <b>\#include</b> <vigra/linear_algebra.hxx><br>
813 Namespaces: vigra and vigra::linalg
814 */
815 template <class T, class C1>
816 T determinant(MultiArrayView<2, T, C1> const & a, std::string method = "LU")
817 {
818 MultiArrayIndex n = columnCount(a);
819 vigra_precondition(rowCount(a) == n,
820 "determinant(): Square matrix required.");
821
822 method = tolower(method);
823
824 if(n == 1)
825 return a(0,0);
826 if(n == 2)
827 return a(0,0)*a(1,1) - a(0,1)*a(1,0);
828 if(method == "lu")
829 {
830 return detail::determinantByLUDecomposition(a);
831 }
832 else if(method == "cholesky")
833 {
834 Matrix<T> L(a.shape());
835 vigra_precondition(choleskyDecomposition(a, L),
836 "determinant(): Cholesky method requires symmetric positive definite matrix.");
837 T det = L(0,0);
838 for(MultiArrayIndex k=1; k<n; ++k)
839 det *= L(k,k);
840 return sq(det);
841 }
842 else
843 {
844 vigra_precondition(false, "determinant(): Unknown solution method.");
845 }
846 return T();
847 }
848
849 /** Compute the logarithm of the determinant of a symmetric positive definite matrix.
850
851 This is useful to avoid multiplication of very large numbers in big matrices.
852 It is implemented by means of Cholesky decomposition.
853
854 <b>\#include</b> <vigra/linear_solve.hxx> or<br>
855 <b>\#include</b> <vigra/linear_algebra.hxx><br>
856 Namespaces: vigra and vigra::linalg
857 */
858 template <class T, class C1>
859 T logDeterminant(MultiArrayView<2, T, C1> const & a)
860 {
861 MultiArrayIndex n = columnCount(a);
862 vigra_precondition(rowCount(a) == n,
863 "logDeterminant(): Square matrix required.");
864 if(n == 1)
865 {
866 vigra_precondition(a(0,0) > 0.0,
867 "logDeterminant(): Matrix not positive definite.");
868 return std::log(a(0,0));
869 }
870 if(n == 2)
871 {
872 T det = a(0,0)*a(1,1) - a(0,1)*a(1,0);
873 vigra_precondition(det > 0.0,
874 "logDeterminant(): Matrix not positive definite.");
875 return std::log(det);
876 }
877 else
878 {
879 Matrix<T> L(a.shape());
880 vigra_precondition(choleskyDecomposition(a, L),
881 "logDeterminant(): Matrix not positive definite.");
882 T logdet = std::log(L(0,0));
883 for(MultiArrayIndex k=1; k<n; ++k)
884 logdet += std::log(L(k,k)); // L(k,k) is guaranteed to be positive
885 return 2.0*logdet;
886 }
887 }
888
889 /** Cholesky decomposition.
890
891 \a A must be a symmetric positive definite matrix, and \a L will be a lower
892 triangular matrix, such that (up to round-off errors):
893
894 \code
895 A == L * transpose(L);
896 \endcode
897
898 This implementation cannot be applied in-place, i.e. <tt>&L == &A</tt> is an error.
899 If \a A is not symmetric, a <tt>ContractViolation</tt> exception is thrown. If it
900 is not positive definite, the function returns <tt>false</tt>.
901
902 <b>\#include</b> <vigra/linear_solve.hxx> or<br>
903 <b>\#include</b> <vigra/linear_algebra.hxx><br>
904 Namespaces: vigra and vigra::linalg
905 */
906 template <class T, class C1, class C2>
907 bool choleskyDecomposition(MultiArrayView<2, T, C1> const & A,
908 MultiArrayView<2, T, C2> &L)
909 {
910 MultiArrayIndex n = columnCount(A);
911 vigra_precondition(rowCount(A) == n,
912 "choleskyDecomposition(): Input matrix must be square.");
913 vigra_precondition(n == columnCount(L) && n == rowCount(L),
914 "choleskyDecomposition(): Output matrix must have same shape as input matrix.");
915 vigra_precondition(isSymmetric(A),
916 "choleskyDecomposition(): Input matrix must be symmetric.");
917
918 for (MultiArrayIndex j = 0; j < n; ++j)
919 {
920 T d(0.0);
921 for (MultiArrayIndex k = 0; k < j; ++k)
922 {
923 T s(0.0);
924 for (MultiArrayIndex i = 0; i < k; ++i)
925 {
926 s += L(k, i)*L(j, i);
927 }
928 L(j, k) = s = (A(j, k) - s)/L(k, k);
929 d = d + s*s;
930 }
931 d = A(j, j) - d;
932 if(d <= 0.0)
933 return false; // A is not positive definite
934 L(j, j) = std::sqrt(d);
935 for (MultiArrayIndex k = j+1; k < n; ++k)
936 {
937 L(j, k) = 0.0;
938 }
939 }
940 return true;
941 }
942
943 /** QR decomposition.
944
945 \a a contains the original matrix, results are returned in \a q and \a r, where
946 \a q is a orthogonal matrix, and \a r is an upper triangular matrix, such that
947 (up to round-off errors):
948
949 \code
950 a == q * r;
951 \endcode
952
953 If \a a doesn't have full rank, the function returns <tt>false</tt>.
954 The decomposition is computed by householder transformations. It can be applied in-place,
955 i.e. <tt>&a == &q</tt> or <tt>&a == &r</tt> are allowed.
956
957 <b>\#include</b> <vigra/linear_solve.hxx> or<br>
958 <b>\#include</b> <vigra/linear_algebra.hxx><br>
959 Namespaces: vigra and vigra::linalg
960 */
961 template <class T, class C1, class C2, class C3>
962 bool qrDecomposition(MultiArrayView<2, T, C1> const & a,
963 MultiArrayView<2, T, C2> &q, MultiArrayView<2, T, C3> &r,
964 double epsilon = 0.0)
965 {
966 const MultiArrayIndex m = rowCount(a);
967 const MultiArrayIndex n = columnCount(a);
968 vigra_precondition(n == columnCount(r) && m == rowCount(r) &&
969 m == columnCount(q) && m == rowCount(q),
970 "qrDecomposition(): Matrix shape mismatch.");
971
972 q = identityMatrix<T>(m);
973 MultiArrayView<2,T, StridedArrayTag> tq = transpose(q);
974 r = a;
975 ArrayVector<MultiArrayIndex> noPivoting; // intentionally empty
976 return ((MultiArrayIndex)detail::qrTransformToUpperTriangular(r, tq, noPivoting, epsilon) == std::min(m,n));
977 }
978
979 /** Deprecated, use \ref linearSolveUpperTriangular().
980 */
981 template <class T, class C1, class C2, class C3>
982 inline
983 bool reverseElimination(const MultiArrayView<2, T, C1> &r, const MultiArrayView<2, T, C2> &b,
984 MultiArrayView<2, T, C3> x)
985 {
986 return linearSolveUpperTriangular(r, b, x);
987 }
988
989 /** Solve a linear system with upper-triangular coefficient matrix.
990
991 The square matrix \a r must be an upper-triangular coefficient matrix as can,
992 for example, be obtained by means of QR decomposition. If \a r doesn't have full rank
993 the function fails and returns <tt>false</tt>, otherwise it returns <tt>true</tt>. The
994 lower triangular part of matrix \a r will not be touched, so it doesn't need to contain zeros.
995
996 The column vectors of matrix \a b are the right-hand sides of the equation (several equations
997 with the same coefficients can thus be solved in one go). The result is returned
998 int \a x, whose columns contain the solutions for the corresponding
999 columns of \a b. This implementation can be applied in-place, i.e. <tt>&b == &x</tt> is allowed.
1000 The following size requirements apply:
1001
1002 \code
1003 rowCount(r) == columnCount(r);
1004 rowCount(r) == rowCount(b);
1005 columnCount(r) == rowCount(x);
1006 columnCount(b) == columnCount(x);
1007 \endcode
1008
1009 <b>\#include</b> <vigra/linear_solve.hxx> or<br>
1010 <b>\#include</b> <vigra/linear_algebra.hxx><br>
1011 Namespaces: vigra and vigra::linalg
1012 */
1013 template <class T, class C1, class C2, class C3>
1014 bool linearSolveUpperTriangular(const MultiArrayView<2, T, C1> &r, const MultiArrayView<2, T, C2> &b,
1015 MultiArrayView<2, T, C3> x)
1016 {
1017 MultiArrayIndex m = rowCount(r);
1018 MultiArrayIndex rhsCount = columnCount(b);
1019 vigra_precondition(m == columnCount(r),
1020 "linearSolveUpperTriangular(): square coefficient matrix required.");
1021 vigra_precondition(m == rowCount(b) && m == rowCount(x) && rhsCount == columnCount(x),
1022 "linearSolveUpperTriangular(): matrix shape mismatch.");
1023
1024 for(MultiArrayIndex k = 0; k < rhsCount; ++k)
1025 {
1026 for(int i=m-1; i>=0; --i)
1027 {
1028 if(r(i,i) == NumericTraits<T>::zero())
1029 return false; // r doesn't have full rank
1030 T sum = b(i, k);
1031 for(MultiArrayIndex j=i+1; j<m; ++j)
1032 sum -= r(i, j) * x(j, k);
1033 x(i, k) = sum / r(i, i);
1034 }
1035 }
1036 return true;
1037 }
1038
1039 /** Solve a linear system with lower-triangular coefficient matrix.
1040
1041 The square matrix \a l must be a lower-triangular coefficient matrix. If \a l
1042 doesn't have full rank the function fails and returns <tt>false</tt>,
1043 otherwise it returns <tt>true</tt>. The upper triangular part of matrix \a l will not be touched,
1044 so it doesn't need to contain zeros.
1045
1046 The column vectors of matrix \a b are the right-hand sides of the equation (several equations
1047 with the same coefficients can thus be solved in one go). The result is returned
1048 in \a x, whose columns contain the solutions for the corresponding
1049 columns of \a b. This implementation can be applied in-place, i.e. <tt>&b == &x</tt> is allowed.
1050 The following size requirements apply:
1051
1052 \code
1053 rowCount(l) == columnCount(l);
1054 rowCount(l) == rowCount(b);
1055 columnCount(l) == rowCount(x);
1056 columnCount(b) == columnCount(x);
1057 \endcode
1058
1059 <b>\#include</b> <vigra/linear_solve.hxx> or<br>
1060 <b>\#include</b> <vigra/linear_algebra.hxx><br>
1061 Namespaces: vigra and vigra::linalg
1062 */
1063 template <class T, class C1, class C2, class C3>
1064 bool linearSolveLowerTriangular(const MultiArrayView<2, T, C1> &l, const MultiArrayView<2, T, C2> &b,
1065 MultiArrayView<2, T, C3> x)
1066 {
1067 MultiArrayIndex m = columnCount(l);
1068 MultiArrayIndex n = columnCount(b);
1069 vigra_precondition(m == rowCount(l),
1070 "linearSolveLowerTriangular(): square coefficient matrix required.");
1071 vigra_precondition(m == rowCount(b) && m == rowCount(x) && n == columnCount(x),
1072 "linearSolveLowerTriangular(): matrix shape mismatch.");
1073
1074 for(MultiArrayIndex k = 0; k < n; ++k)
1075 {
1076 for(MultiArrayIndex i=0; i<m; ++i)
1077 {
1078 if(l(i,i) == NumericTraits<T>::zero())
1079 return false; // l doesn't have full rank
1080 T sum = b(i, k);
1081 for(MultiArrayIndex j=0; j<i; ++j)
1082 sum -= l(i, j) * x(j, k);
1083 x(i, k) = sum / l(i, i);
1084 }
1085 }
1086 return true;
1087 }
1088
1089
1090 /** Solve a linear system when the Cholesky decomposition of the left hand side is given.
1091
1092 The square matrix \a L must be a lower-triangular matrix resulting from Cholesky
1093 decomposition of some positive definite coefficient matrix.
1094
1095 The column vectors of matrix \a b are the right-hand sides of the equation (several equations
1096 with the same matrix \a L can thus be solved in one go). The result is returned
1097 in \a x, whose columns contain the solutions for the corresponding
1098 columns of \a b. This implementation can be applied in-place, i.e. <tt>&b == &x</tt> is allowed.
1099 The following size requirements apply:
1100
1101 \code
1102 rowCount(L) == columnCount(L);
1103 rowCount(L) == rowCount(b);
1104 columnCount(L) == rowCount(x);
1105 columnCount(b) == columnCount(x);
1106 \endcode
1107
1108 <b>\#include</b> <vigra/linear_solve.hxx> or<br>
1109 <b>\#include</b> <vigra/linear_algebra.hxx><br>
1110 Namespaces: vigra and vigra::linalg
1111 */
1112 template <class T, class C1, class C2, class C3>
1113 inline
1114 void choleskySolve(MultiArrayView<2, T, C1> const & L, MultiArrayView<2, T, C2> const & b, MultiArrayView<2, T, C3> & x)
1115 {
1116 /* Solve L * y = b */
1117 linearSolveLowerTriangular(L, b, x);
1118 /* Solve L^T * x = y */
1119 linearSolveUpperTriangular(transpose(L), x, x);
1120 }
1121
1122 /** Solve a linear system.
1123
1124 <b> Declarations:</b>
1125
1126 \code
1127 // use MultiArrayViews for input and output
1128 template <class T, class C1, class C2, class C3>
1129 bool linearSolve(MultiArrayView<2, T, C1> const & A,
1130 MultiArrayView<2, T, C2> const & b,
1131 MultiArrayView<2, T, C3> res,
1132 std::string method = "QR");
1133
1134 // use TinyVector for RHS and result
1135 template <class T, class C1, int N>
1136 bool linearSolve(MultiArrayView<2, T, C1> const & A,
1137 TinyVector<T, N> const & b,
1138 TinyVector<T, N> & res,
1139 std::string method = "QR");
1140 \endcode
1141
1142 \a A is the coefficient matrix, and the column vectors
1143 in \a b are the right-hand sides of the equation (so, several equations
1144 with the same coefficients can be solved in one go). The result is returned
1145 in \a res, whose columns contain the solutions for the corresponding
1146 columns of \a b. The number of columns of \a A must equal the number of rows of
1147 both \a b and \a res, and the number of columns of \a b and \a res must match.
1148 If right-hand-side and result are specified as TinyVector, the number of columns
1149 of \a A must equal N.
1150
1151 \a method must be one of the following:
1152 <DL>
1153 <DT>"Cholesky"<DD> Compute the solution by means of Cholesky decomposition. The
1154 coefficient matrix \a A must by symmetric positive definite. If
1155 this is not the case, the function returns <tt>false</tt>.
1156
1157 <DT>"QR"<DD> (default) Compute the solution by means of QR decomposition. The
1158 coefficient matrix \a A can be square or rectangular. In the latter case,
1159 it must have more rows than columns, and the solution will be computed in the
1160 least squares sense. If \a A doesn't have full rank, the function
1161 returns <tt>false</tt>.
1162
1163 <DT>"SVD"<DD> Compute the solution by means of singular value decomposition. The
1164 coefficient matrix \a A can be square or rectangular. In the latter case,
1165 it must have more rows than columns, and the solution will be computed in the
1166 least squares sense. If \a A doesn't have full rank, the function
1167 returns <tt>false</tt>.
1168
1169 <DT>"NE"<DD> Compute the solution by means of the normal equations, i.e. by applying Cholesky
1170 decomposition to the equivalent problem <tt>A'*A*x = A'*b</tt>. This only makes sense
1171 when the equation is to be solved in the least squares sense, i.e. when \a A is a
1172 rectangular matrix with more rows than columns. If \a A doesn't have full column rank,
1173 the function returns <tt>false</tt>.
1174 </DL>
1175
1176 This function can be applied in-place, i.e. <tt>&b == &res</tt> or <tt>&A == &res</tt> are allowed
1177 (provided they have the required shapes).
1178
1179 The following size requirements apply:
1180
1181 \code
1182 rowCount(r) == rowCount(b);
1183 columnCount(r) == rowCount(x);
1184 columnCount(b) == columnCount(x);
1185 \endcode
1186
1187 <b>\#include</b> <vigra/linear_solve.hxx> or<br>
1188 <b>\#include</b> <vigra/linear_algebra.hxx><br>
1189 Namespaces: vigra and vigra::linalg
1190 */
1191 doxygen_overloaded_function(template <...> bool linearSolve)
1192
1193 template <class T, class C1, class C2, class C3>
1194 bool linearSolve(MultiArrayView<2, T, C1> const & A,
1195 MultiArrayView<2, T, C2> const & b,
1196 MultiArrayView<2, T, C3> res,
1197 std::string method = "QR")
1198 {
1199 const MultiArrayIndex n = columnCount(A);
1200 const MultiArrayIndex m = rowCount(A);
1201
1202 vigra_precondition(n <= m,
1203 "linearSolve(): Coefficient matrix A must have at least as many rows as columns.");
1204 vigra_precondition(n == rowCount(res) &&
1205 m == rowCount(b) && columnCount(b) == columnCount(res),
1206 "linearSolve(): matrix shape mismatch.");
1207
1208 method = tolower(method);
1209 if(method == "cholesky")
1210 {
1211 vigra_precondition(columnCount(A) == rowCount(A),
1212 "linearSolve(): Cholesky method requires square coefficient matrix.");
1213 Matrix<T> L(A.shape());
1214 if(!choleskyDecomposition(A, L))
1215 return false; // false if A wasn't symmetric positive definite
1216 choleskySolve(L, b, res);
1217 }
1218 else if(method == "qr")
1219 {
1220 return (MultiArrayIndex)linearSolveQR(A, b, res) == n;
1221 }
1222 else if(method == "ne")
1223 {
1224 return linearSolve(transpose(A)*A, transpose(A)*b, res, "Cholesky");
1225 }
1226 else if(method == "svd")
1227 {
1228 MultiArrayIndex rhsCount = columnCount(b);
1229 Matrix<T> u(A.shape()), s(n, 1), v(n, n);
1230
1231 MultiArrayIndex rank = (MultiArrayIndex)singularValueDecomposition(A, u, s, v);
1232
1233 Matrix<T> t = transpose(u)*b;
1234 for(MultiArrayIndex l=0; l<rhsCount; ++l)
1235 {
1236 for(MultiArrayIndex k=0; k<rank; ++k)
1237 t(k,l) /= s(k,0);
1238 for(MultiArrayIndex k=rank; k<n; ++k)
1239 t(k,l) = NumericTraits<T>::zero();
1240 }
1241 res = v*t;
1242
1243 return (rank == n);
1244 }
1245 else
1246 {
1247 vigra_precondition(false, "linearSolve(): Unknown solution method.");
1248 }
1249 return true;
1250 }
1251
1252 template <class T, class C1, int N>
1253 bool linearSolve(MultiArrayView<2, T, C1> const & A,
1254 TinyVector<T, N> const & b,
1255 TinyVector<T, N> & res,
1256 std::string method = "QR")
1257 {
1258 Shape2 shape(N, 1);
1259 return linearSolve(A, MultiArrayView<2, T>(shape, b.data()), MultiArrayView<2, T>(shape, res.data()), method);
1260 }
1261
1262 //@}
1263
1264 } // namespace linalg
1265
1266 using linalg::inverse;
1267 using linalg::determinant;
1268 using linalg::logDeterminant;
1269 using linalg::linearSolve;
1270 using linalg::choleskySolve;
1271 using linalg::choleskyDecomposition;
1272 using linalg::qrDecomposition;
1273 using linalg::linearSolveUpperTriangular;
1274 using linalg::linearSolveLowerTriangular;
1275
1276 } // namespace vigra
1277
1278
1279 #endif // VIGRA_LINEAR_SOLVE_HXX
vigra::linalg::qrDecomposition
bool qrDecomposition(MultiArrayView< 2, T, C1 > const &a, MultiArrayView< 2, T, C2 > &q, MultiArrayView< 2, T, C3 > &r, double epsilon=0.0)
Definition: linear_solve.hxx:962
vigra::atan2
FixedPoint16< 2, OverflowHandling > atan2(FixedPoint16< IntBits, OverflowHandling > y, FixedPoint16< IntBits, OverflowHandling > x)
Arctangent. Accuracy better than 1/3 degree (9 significant bits).
Definition: fixedpoint.hxx:1654
vigra::linalg::columnVector
MultiArrayView< 2, T, C > columnVector(MultiArrayView< 2, T, C > const &m, MultiArrayIndex d)
Definition: matrix.hxx:725
vigra::linalg::rowCount
MultiArrayIndex rowCount(const MultiArrayView< 2, T, C > &x)
Definition: matrix.hxx:669
vigra::linalg::transpose
void transpose(const MultiArrayView< 2, T, C1 > &v, MultiArrayView< 2, T, C2 > &r)
Definition: matrix.hxx:963
vigra::linalg::sin
linalg::TemporaryMatrix< T > sin(MultiArrayView< 2, T, C > const &v)
vigra::linalg::linearSolveLowerTriangular
bool linearSolveLowerTriangular(const MultiArrayView< 2, T, C1 > &l, const MultiArrayView< 2, T, C2 > &b, MultiArrayView< 2, T, C3 > x)
Definition: linear_solve.hxx:1064
vigra::MultiArrayView::shape
const difference_type & shape() const
Definition: multi_array.hxx:1551
vigra::linalg::Matrix< T >
vigra::linalg::reverseElimination
bool reverseElimination(const MultiArrayView< 2, T, C1 > &r, const MultiArrayView< 2, T, C2 > &b, MultiArrayView< 2, T, C3 > x)
Definition: linear_solve.hxx:983
vigra::linalg::isSymmetric
bool isSymmetric(const MultiArrayView< 2, T, C > &v)
Definition: matrix.hxx:777
vigra::linalg::choleskySolve
void choleskySolve(MultiArrayView< 2, T, C1 > const &L, MultiArrayView< 2, T, C2 > const &b, MultiArrayView< 2, T, C3 > &x)
Definition: linear_solve.hxx:1114
vigra::MultiArrayIndex
std::ptrdiff_t MultiArrayIndex
Definition: multi_shape.hxx:55
vigra
Definition: accessor.hxx:43
vigra::hypot
FixedPoint16< IntBits, OverflowHandling > hypot(FixedPoint16< IntBits, OverflowHandling > v1, FixedPoint16< IntBits, OverflowHandling > v2)
Length of hypotenuse.
Definition: fixedpoint.hxx:1640
vigra::ArrayVector< MultiArrayIndex >
vigra::sum
NumericTraits< V >::Promote sum(TinyVectorBase< V, SIZE, D1, D2 > const &l)
sum of the vector&#39;s elements
Definition: tinyvector.hxx:1871
vigra::linalg::sq
linalg::TemporaryMatrix< T > sq(MultiArrayView< 2, T, C > const &v)
vigra::linalg::norm
Matrix< T, ALLLOC >::NormType norm(const Matrix< T, ALLLOC > &a)
vigra::linalg::squaredNorm
Matrix< T, ALLLOC >::SquaredNormType squaredNorm(const Matrix< T, ALLLOC > &a)
vigra::MultiArrayShape::type
TinyVector< MultiArrayIndex, N > type
Definition: multi_shape.hxx:237
vigra::TinyVector
Class for fixed size vectors.This class contains an array of size SIZE of the specified VALUETYPE...
Definition: accessor.hxx:940
vigra::linalg::argMax
int argMax(MultiArrayView< 2, T, C > const &a)
Find the index of the maximum element in a matrix.
Definition: matrix.hxx:1984
vigra::linalg::dot
NormTraits< T >::SquaredNormType dot(const MultiArrayView< 2, T, C1 > &x, const MultiArrayView< 2, T, C2 > &y)
Definition: matrix.hxx:1340
vigra::gamma
double gamma(double x)
The gamma function.
Definition: mathutil.hxx:1549
vigra::linalg::log
linalg::TemporaryMatrix< T > log(MultiArrayView< 2, T, C > const &v)
vigra::linalg::rowVector
MultiArrayView< 2, T, C > rowVector(MultiArrayView< 2, T, C > const &m, MultiArrayIndex d)
Definition: matrix.hxx:695
vigra::linalg::columnCount
MultiArrayIndex columnCount(const MultiArrayView< 2, T, C > &x)
Definition: matrix.hxx:682
vigra::linalg::choleskyDecomposition
bool choleskyDecomposition(MultiArrayView< 2, T, C1 > const &A, MultiArrayView< 2, T, C2 > &L)
Definition: linear_solve.hxx:907
vigra::linalg::logDeterminant
T logDeterminant(MultiArrayView< 2, T, C1 > const &a)
Definition: linear_solve.hxx:859
vigra::MultiArrayView< 2, T, C1 >
vigra::linalg::linearSolveUpperTriangular
bool linearSolveUpperTriangular(const MultiArrayView< 2, T, C1 > &r, const MultiArrayView< 2, T, C2 > &b, MultiArrayView< 2, T, C3 > x)
Definition: linear_solve.hxx:1014
vigra::linalg::singularValueDecomposition
unsigned int singularValueDecomposition(MultiArrayView< 2, T, C1 > const &A, MultiArrayView< 2, T, C2 > &U, MultiArrayView< 2, T, C3 > &S, MultiArrayView< 2, T, C4 > &V)
Definition: singular_value_decomposition.hxx:75
vigra::linalg::inverse
bool inverse(const MultiArrayView< 2, T, C1 > &v, MultiArrayView< 2, T, C2 > &res)
Definition: linear_solve.hxx:719
vigra::linalg::abs
linalg::TemporaryMatrix< T > abs(MultiArrayView< 2, T, C > const &v)
vigra::linalg::cos
linalg::TemporaryMatrix< T > cos(MultiArrayView< 2, T, C > const &v)
vigra::linalg::linearSolve
bool linearSolve(...)
vigra::sqrt
SquareRootTraits< FixedPoint< IntBits, FracBits > >::SquareRootResult sqrt(FixedPoint< IntBits, FracBits > v)
square root.
Definition: fixedpoint.hxx:616
vigra::linalg::determinant
T determinant(MultiArrayView< 2, T, C1 > const &a, std::string method="LU")
Definition: linear_solve.hxx:816

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