1// This file is part of Eigen, a lightweight C++ template library
2// for linear algebra.
3//
4// Copyright (C) 2008-2010 Benoit Jacob <jacob.benoit.1@gmail.com>
5// Copyright (C) 2014 Gael Guennebaud <gael.guennebaud@inria.fr>
6//
7// This Source Code Form is subject to the terms of the Mozilla
8// Public License v. 2.0. If a copy of the MPL was not distributed
9// with this file, You can obtain one at http://mozilla.org/MPL/2.0/.
10
11#ifndef EIGEN_INVERSE_IMPL_H
12#define EIGEN_INVERSE_IMPL_H
13
14namespace Eigen {
15
16namespace internal {
17
18/**********************************
19*** General case implementation ***
20**********************************/
21
22template<typename MatrixType, typename ResultType, int Size = MatrixType::RowsAtCompileTime>
23struct compute_inverse
24{
25 EIGEN_DEVICE_FUNC
26 static inline void run(const MatrixType& matrix, ResultType& result)
27 {
28 result = matrix.partialPivLu().inverse();
29 }
30};
31
32template<typename MatrixType, typename ResultType, int Size = MatrixType::RowsAtCompileTime>
33struct compute_inverse_and_det_with_check { /* nothing! general case not supported. */ };
34
35/****************************
36*** Size 1 implementation ***
37****************************/
38
39template<typename MatrixType, typename ResultType>
40struct compute_inverse<MatrixType, ResultType, 1>
41{
42 EIGEN_DEVICE_FUNC
43 static inline void run(const MatrixType& matrix, ResultType& result)
44 {
45 typedef typename MatrixType::Scalar Scalar;
46 internal::evaluator<MatrixType> matrixEval(matrix);
47 result.coeffRef(0,0) = Scalar(1) / matrixEval.coeff(0,0);
48 }
49};
50
51template<typename MatrixType, typename ResultType>
52struct compute_inverse_and_det_with_check<MatrixType, ResultType, 1>
53{
54 EIGEN_DEVICE_FUNC
55 static inline void run(
56 const MatrixType& matrix,
57 const typename MatrixType::RealScalar& absDeterminantThreshold,
58 ResultType& result,
59 typename ResultType::Scalar& determinant,
60 bool& invertible
61 )
62 {
63 using std::abs;
64 determinant = matrix.coeff(0,0);
65 invertible = abs(determinant) > absDeterminantThreshold;
66 if(invertible) result.coeffRef(0,0) = typename ResultType::Scalar(1) / determinant;
67 }
68};
69
70/****************************
71*** Size 2 implementation ***
72****************************/
73
74template<typename MatrixType, typename ResultType>
75EIGEN_DEVICE_FUNC
76inline void compute_inverse_size2_helper(
77 const MatrixType& matrix, const typename ResultType::Scalar& invdet,
78 ResultType& result)
79{
80 result.coeffRef(0,0) = matrix.coeff(1,1) * invdet;
81 result.coeffRef(1,0) = -matrix.coeff(1,0) * invdet;
82 result.coeffRef(0,1) = -matrix.coeff(0,1) * invdet;
83 result.coeffRef(1,1) = matrix.coeff(0,0) * invdet;
84}
85
86template<typename MatrixType, typename ResultType>
87struct compute_inverse<MatrixType, ResultType, 2>
88{
89 EIGEN_DEVICE_FUNC
90 static inline void run(const MatrixType& matrix, ResultType& result)
91 {
92 typedef typename ResultType::Scalar Scalar;
93 const Scalar invdet = typename MatrixType::Scalar(1) / matrix.determinant();
94 compute_inverse_size2_helper(matrix, invdet, result);
95 }
96};
97
98template<typename MatrixType, typename ResultType>
99struct compute_inverse_and_det_with_check<MatrixType, ResultType, 2>
100{
101 EIGEN_DEVICE_FUNC
102 static inline void run(
103 const MatrixType& matrix,
104 const typename MatrixType::RealScalar& absDeterminantThreshold,
105 ResultType& inverse,
106 typename ResultType::Scalar& determinant,
107 bool& invertible
108 )
109 {
110 using std::abs;
111 typedef typename ResultType::Scalar Scalar;
112 determinant = matrix.determinant();
113 invertible = abs(determinant) > absDeterminantThreshold;
114 if(!invertible) return;
115 const Scalar invdet = Scalar(1) / determinant;
116 compute_inverse_size2_helper(matrix, invdet, inverse);
117 }
118};
119
120/****************************
121*** Size 3 implementation ***
122****************************/
123
124template<typename MatrixType, int i, int j>
125EIGEN_DEVICE_FUNC
126inline typename MatrixType::Scalar cofactor_3x3(const MatrixType& m)
127{
128 enum {
129 i1 = (i+1) % 3,
130 i2 = (i+2) % 3,
131 j1 = (j+1) % 3,
132 j2 = (j+2) % 3
133 };
134 return m.coeff(i1, j1) * m.coeff(i2, j2)
135 - m.coeff(i1, j2) * m.coeff(i2, j1);
136}
137
138template<typename MatrixType, typename ResultType>
139EIGEN_DEVICE_FUNC
140inline void compute_inverse_size3_helper(
141 const MatrixType& matrix,
142 const typename ResultType::Scalar& invdet,
143 const Matrix<typename ResultType::Scalar,3,1>& cofactors_col0,
144 ResultType& result)
145{
146 result.row(0) = cofactors_col0 * invdet;
147 result.coeffRef(1,0) = cofactor_3x3<MatrixType,0,1>(matrix) * invdet;
148 result.coeffRef(1,1) = cofactor_3x3<MatrixType,1,1>(matrix) * invdet;
149 result.coeffRef(1,2) = cofactor_3x3<MatrixType,2,1>(matrix) * invdet;
150 result.coeffRef(2,0) = cofactor_3x3<MatrixType,0,2>(matrix) * invdet;
151 result.coeffRef(2,1) = cofactor_3x3<MatrixType,1,2>(matrix) * invdet;
152 result.coeffRef(2,2) = cofactor_3x3<MatrixType,2,2>(matrix) * invdet;
153}
154
155template<typename MatrixType, typename ResultType>
156struct compute_inverse<MatrixType, ResultType, 3>
157{
158 EIGEN_DEVICE_FUNC
159 static inline void run(const MatrixType& matrix, ResultType& result)
160 {
161 typedef typename ResultType::Scalar Scalar;
162 Matrix<typename MatrixType::Scalar,3,1> cofactors_col0;
163 cofactors_col0.coeffRef(0) = cofactor_3x3<MatrixType,0,0>(matrix);
164 cofactors_col0.coeffRef(1) = cofactor_3x3<MatrixType,1,0>(matrix);
165 cofactors_col0.coeffRef(2) = cofactor_3x3<MatrixType,2,0>(matrix);
166 const Scalar det = (cofactors_col0.cwiseProduct(matrix.col(0))).sum();
167 const Scalar invdet = Scalar(1) / det;
168 compute_inverse_size3_helper(matrix, invdet, cofactors_col0, result);
169 }
170};
171
172template<typename MatrixType, typename ResultType>
173struct compute_inverse_and_det_with_check<MatrixType, ResultType, 3>
174{
175 EIGEN_DEVICE_FUNC
176 static inline void run(
177 const MatrixType& matrix,
178 const typename MatrixType::RealScalar& absDeterminantThreshold,
179 ResultType& inverse,
180 typename ResultType::Scalar& determinant,
181 bool& invertible
182 )
183 {
184 using std::abs;
185 typedef typename ResultType::Scalar Scalar;
186 Matrix<Scalar,3,1> cofactors_col0;
187 cofactors_col0.coeffRef(0) = cofactor_3x3<MatrixType,0,0>(matrix);
188 cofactors_col0.coeffRef(1) = cofactor_3x3<MatrixType,1,0>(matrix);
189 cofactors_col0.coeffRef(2) = cofactor_3x3<MatrixType,2,0>(matrix);
190 determinant = (cofactors_col0.cwiseProduct(matrix.col(0))).sum();
191 invertible = abs(determinant) > absDeterminantThreshold;
192 if(!invertible) return;
193 const Scalar invdet = Scalar(1) / determinant;
194 compute_inverse_size3_helper(matrix, invdet, cofactors_col0, inverse);
195 }
196};
197
198/****************************
199*** Size 4 implementation ***
200****************************/
201
202template<typename Derived>
203EIGEN_DEVICE_FUNC
204inline const typename Derived::Scalar general_det3_helper
205(const MatrixBase<Derived>& matrix, int i1, int i2, int i3, int j1, int j2, int j3)
206{
207 return matrix.coeff(i1,j1)
208 * (matrix.coeff(i2,j2) * matrix.coeff(i3,j3) - matrix.coeff(i2,j3) * matrix.coeff(i3,j2));
209}
210
211template<typename MatrixType, int i, int j>
212EIGEN_DEVICE_FUNC
213inline typename MatrixType::Scalar cofactor_4x4(const MatrixType& matrix)
214{
215 enum {
216 i1 = (i+1) % 4,
217 i2 = (i+2) % 4,
218 i3 = (i+3) % 4,
219 j1 = (j+1) % 4,
220 j2 = (j+2) % 4,
221 j3 = (j+3) % 4
222 };
223 return general_det3_helper(matrix, i1, i2, i3, j1, j2, j3)
224 + general_det3_helper(matrix, i2, i3, i1, j1, j2, j3)
225 + general_det3_helper(matrix, i3, i1, i2, j1, j2, j3);
226}
227
228template<int Arch, typename Scalar, typename MatrixType, typename ResultType>
229struct compute_inverse_size4
230{
231 EIGEN_DEVICE_FUNC
232 static void run(const MatrixType& matrix, ResultType& result)
233 {
234 result.coeffRef(0,0) = cofactor_4x4<MatrixType,0,0>(matrix);
235 result.coeffRef(1,0) = -cofactor_4x4<MatrixType,0,1>(matrix);
236 result.coeffRef(2,0) = cofactor_4x4<MatrixType,0,2>(matrix);
237 result.coeffRef(3,0) = -cofactor_4x4<MatrixType,0,3>(matrix);
238 result.coeffRef(0,2) = cofactor_4x4<MatrixType,2,0>(matrix);
239 result.coeffRef(1,2) = -cofactor_4x4<MatrixType,2,1>(matrix);
240 result.coeffRef(2,2) = cofactor_4x4<MatrixType,2,2>(matrix);
241 result.coeffRef(3,2) = -cofactor_4x4<MatrixType,2,3>(matrix);
242 result.coeffRef(0,1) = -cofactor_4x4<MatrixType,1,0>(matrix);
243 result.coeffRef(1,1) = cofactor_4x4<MatrixType,1,1>(matrix);
244 result.coeffRef(2,1) = -cofactor_4x4<MatrixType,1,2>(matrix);
245 result.coeffRef(3,1) = cofactor_4x4<MatrixType,1,3>(matrix);
246 result.coeffRef(0,3) = -cofactor_4x4<MatrixType,3,0>(matrix);
247 result.coeffRef(1,3) = cofactor_4x4<MatrixType,3,1>(matrix);
248 result.coeffRef(2,3) = -cofactor_4x4<MatrixType,3,2>(matrix);
249 result.coeffRef(3,3) = cofactor_4x4<MatrixType,3,3>(matrix);
250 result /= (matrix.col(0).cwiseProduct(result.row(0).transpose())).sum();
251 }
252};
253
254template<typename MatrixType, typename ResultType>
255struct compute_inverse<MatrixType, ResultType, 4>
256 : compute_inverse_size4<Architecture::Target, typename MatrixType::Scalar,
257 MatrixType, ResultType>
258{
259};
260
261template<typename MatrixType, typename ResultType>
262struct compute_inverse_and_det_with_check<MatrixType, ResultType, 4>
263{
264 EIGEN_DEVICE_FUNC
265 static inline void run(
266 const MatrixType& matrix,
267 const typename MatrixType::RealScalar& absDeterminantThreshold,
268 ResultType& inverse,
269 typename ResultType::Scalar& determinant,
270 bool& invertible
271 )
272 {
273 using std::abs;
274 determinant = matrix.determinant();
275 invertible = abs(determinant) > absDeterminantThreshold;
276 if(invertible) compute_inverse<MatrixType, ResultType>::run(matrix, inverse);
277 }
278};
279
280/*************************
281*** MatrixBase methods ***
282*************************/
283
284} // end namespace internal
285
286namespace internal {
287
288// Specialization for "dense = dense_xpr.inverse()"
289template<typename DstXprType, typename XprType>
290struct Assignment<DstXprType, Inverse<XprType>, internal::assign_op<typename DstXprType::Scalar,typename XprType::Scalar>, Dense2Dense>
291{
292 typedef Inverse<XprType> SrcXprType;
293 static void run(DstXprType &dst, const SrcXprType &src, const internal::assign_op<typename DstXprType::Scalar,typename XprType::Scalar> &)
294 {
295 Index dstRows = src.rows();
296 Index dstCols = src.cols();
297 if((dst.rows()!=dstRows) || (dst.cols()!=dstCols))
298 dst.resize(dstRows, dstCols);
299
300 const int Size = EIGEN_PLAIN_ENUM_MIN(XprType::ColsAtCompileTime,DstXprType::ColsAtCompileTime);
301 EIGEN_ONLY_USED_FOR_DEBUG(Size);
302 eigen_assert(( (Size<=1) || (Size>4) || (extract_data(src.nestedExpression())!=extract_data(dst)))
303 && "Aliasing problem detected in inverse(), you need to do inverse().eval() here.");
304
305 typedef typename internal::nested_eval<XprType,XprType::ColsAtCompileTime>::type ActualXprType;
306 typedef typename internal::remove_all<ActualXprType>::type ActualXprTypeCleanded;
307
308 ActualXprType actual_xpr(src.nestedExpression());
309
310 compute_inverse<ActualXprTypeCleanded, DstXprType>::run(actual_xpr, dst);
311 }
312};
313
314
315} // end namespace internal
316
317/** \lu_module
318 *
319 * \returns the matrix inverse of this matrix.
320 *
321 * For small fixed sizes up to 4x4, this method uses cofactors.
322 * In the general case, this method uses class PartialPivLU.
323 *
324 * \note This matrix must be invertible, otherwise the result is undefined. If you need an
325 * invertibility check, do the following:
326 * \li for fixed sizes up to 4x4, use computeInverseAndDetWithCheck().
327 * \li for the general case, use class FullPivLU.
328 *
329 * Example: \include MatrixBase_inverse.cpp
330 * Output: \verbinclude MatrixBase_inverse.out
331 *
332 * \sa computeInverseAndDetWithCheck()
333 */
334template<typename Derived>
335inline const Inverse<Derived> MatrixBase<Derived>::inverse() const
336{
337 EIGEN_STATIC_ASSERT(!NumTraits<Scalar>::IsInteger,THIS_FUNCTION_IS_NOT_FOR_INTEGER_NUMERIC_TYPES)
338 eigen_assert(rows() == cols());
339 return Inverse<Derived>(derived());
340}
341
342/** \lu_module
343 *
344 * Computation of matrix inverse and determinant, with invertibility check.
345 *
346 * This is only for fixed-size square matrices of size up to 4x4.
347 *
348 * \param inverse Reference to the matrix in which to store the inverse.
349 * \param determinant Reference to the variable in which to store the determinant.
350 * \param invertible Reference to the bool variable in which to store whether the matrix is invertible.
351 * \param absDeterminantThreshold Optional parameter controlling the invertibility check.
352 * The matrix will be declared invertible if the absolute value of its
353 * determinant is greater than this threshold.
354 *
355 * Example: \include MatrixBase_computeInverseAndDetWithCheck.cpp
356 * Output: \verbinclude MatrixBase_computeInverseAndDetWithCheck.out
357 *
358 * \sa inverse(), computeInverseWithCheck()
359 */
360template<typename Derived>
361template<typename ResultType>
362inline void MatrixBase<Derived>::computeInverseAndDetWithCheck(
363 ResultType& inverse,
364 typename ResultType::Scalar& determinant,
365 bool& invertible,
366 const RealScalar& absDeterminantThreshold
367 ) const
368{
369 // i'd love to put some static assertions there, but SFINAE means that they have no effect...
370 eigen_assert(rows() == cols());
371 // for 2x2, it's worth giving a chance to avoid evaluating.
372 // for larger sizes, evaluating has negligible cost and limits code size.
373 typedef typename internal::conditional<
374 RowsAtCompileTime == 2,
375 typename internal::remove_all<typename internal::nested_eval<Derived, 2>::type>::type,
376 PlainObject
377 >::type MatrixType;
378 internal::compute_inverse_and_det_with_check<MatrixType, ResultType>::run
379 (derived(), absDeterminantThreshold, inverse, determinant, invertible);
380}
381
382/** \lu_module
383 *
384 * Computation of matrix inverse, with invertibility check.
385 *
386 * This is only for fixed-size square matrices of size up to 4x4.
387 *
388 * \param inverse Reference to the matrix in which to store the inverse.
389 * \param invertible Reference to the bool variable in which to store whether the matrix is invertible.
390 * \param absDeterminantThreshold Optional parameter controlling the invertibility check.
391 * The matrix will be declared invertible if the absolute value of its
392 * determinant is greater than this threshold.
393 *
394 * Example: \include MatrixBase_computeInverseWithCheck.cpp
395 * Output: \verbinclude MatrixBase_computeInverseWithCheck.out
396 *
397 * \sa inverse(), computeInverseAndDetWithCheck()
398 */
399template<typename Derived>
400template<typename ResultType>
401inline void MatrixBase<Derived>::computeInverseWithCheck(
402 ResultType& inverse,
403 bool& invertible,
404 const RealScalar& absDeterminantThreshold
405 ) const
406{
407 Scalar determinant;
408 // i'd love to put some static assertions there, but SFINAE means that they have no effect...
409 eigen_assert(rows() == cols());
410 computeInverseAndDetWithCheck(inverse,determinant,invertible,absDeterminantThreshold);
411}
412
413} // end namespace Eigen
414
415#endif // EIGEN_INVERSE_IMPL_H
416