1 // This file is part of Eigen, a lightweight C++ template library
2 // for linear algebra.
3 //
4 // Copyright (C) 2011, 2013 Jitse Niesen <jitse@maths.leeds.ac.uk>
5 //
6 // This Source Code Form is subject to the terms of the Mozilla
7 // Public License v. 2.0. If a copy of the MPL was not distributed
8 // with this file, You can obtain one at http://mozilla.org/MPL/2.0/.
9
10 #ifndef EIGEN_MATRIX_SQUARE_ROOT
11 #define EIGEN_MATRIX_SQUARE_ROOT
12
13 namespace Eigen {
14
15 namespace internal {
16
17 // pre: T.block(i,i,2,2) has complex conjugate eigenvalues
18 // post: sqrtT.block(i,i,2,2) is square root of T.block(i,i,2,2)
19 template <typename MatrixType, typename ResultType>
matrix_sqrt_quasi_triangular_2x2_diagonal_block(const MatrixType & T,typename MatrixType::Index i,ResultType & sqrtT)20 void matrix_sqrt_quasi_triangular_2x2_diagonal_block(const MatrixType& T, typename MatrixType::Index i, ResultType& sqrtT)
21 {
22 // TODO: This case (2-by-2 blocks with complex conjugate eigenvalues) is probably hidden somewhere
23 // in EigenSolver. If we expose it, we could call it directly from here.
24 typedef typename traits<MatrixType>::Scalar Scalar;
25 Matrix<Scalar,2,2> block = T.template block<2,2>(i,i);
26 EigenSolver<Matrix<Scalar,2,2> > es(block);
27 sqrtT.template block<2,2>(i,i)
28 = (es.eigenvectors() * es.eigenvalues().cwiseSqrt().asDiagonal() * es.eigenvectors().inverse()).real();
29 }
30
31 // pre: block structure of T is such that (i,j) is a 1x1 block,
32 // all blocks of sqrtT to left of and below (i,j) are correct
33 // post: sqrtT(i,j) has the correct value
34 template <typename MatrixType, typename ResultType>
matrix_sqrt_quasi_triangular_1x1_off_diagonal_block(const MatrixType & T,typename MatrixType::Index i,typename MatrixType::Index j,ResultType & sqrtT)35 void matrix_sqrt_quasi_triangular_1x1_off_diagonal_block(const MatrixType& T, typename MatrixType::Index i, typename MatrixType::Index j, ResultType& sqrtT)
36 {
37 typedef typename traits<MatrixType>::Scalar Scalar;
38 Scalar tmp = (sqrtT.row(i).segment(i+1,j-i-1) * sqrtT.col(j).segment(i+1,j-i-1)).value();
39 sqrtT.coeffRef(i,j) = (T.coeff(i,j) - tmp) / (sqrtT.coeff(i,i) + sqrtT.coeff(j,j));
40 }
41
42 // similar to compute1x1offDiagonalBlock()
43 template <typename MatrixType, typename ResultType>
matrix_sqrt_quasi_triangular_1x2_off_diagonal_block(const MatrixType & T,typename MatrixType::Index i,typename MatrixType::Index j,ResultType & sqrtT)44 void matrix_sqrt_quasi_triangular_1x2_off_diagonal_block(const MatrixType& T, typename MatrixType::Index i, typename MatrixType::Index j, ResultType& sqrtT)
45 {
46 typedef typename traits<MatrixType>::Scalar Scalar;
47 Matrix<Scalar,1,2> rhs = T.template block<1,2>(i,j);
48 if (j-i > 1)
49 rhs -= sqrtT.block(i, i+1, 1, j-i-1) * sqrtT.block(i+1, j, j-i-1, 2);
50 Matrix<Scalar,2,2> A = sqrtT.coeff(i,i) * Matrix<Scalar,2,2>::Identity();
51 A += sqrtT.template block<2,2>(j,j).transpose();
52 sqrtT.template block<1,2>(i,j).transpose() = A.fullPivLu().solve(rhs.transpose());
53 }
54
55 // similar to compute1x1offDiagonalBlock()
56 template <typename MatrixType, typename ResultType>
matrix_sqrt_quasi_triangular_2x1_off_diagonal_block(const MatrixType & T,typename MatrixType::Index i,typename MatrixType::Index j,ResultType & sqrtT)57 void matrix_sqrt_quasi_triangular_2x1_off_diagonal_block(const MatrixType& T, typename MatrixType::Index i, typename MatrixType::Index j, ResultType& sqrtT)
58 {
59 typedef typename traits<MatrixType>::Scalar Scalar;
60 Matrix<Scalar,2,1> rhs = T.template block<2,1>(i,j);
61 if (j-i > 2)
62 rhs -= sqrtT.block(i, i+2, 2, j-i-2) * sqrtT.block(i+2, j, j-i-2, 1);
63 Matrix<Scalar,2,2> A = sqrtT.coeff(j,j) * Matrix<Scalar,2,2>::Identity();
64 A += sqrtT.template block<2,2>(i,i);
65 sqrtT.template block<2,1>(i,j) = A.fullPivLu().solve(rhs);
66 }
67
68 // solves the equation A X + X B = C where all matrices are 2-by-2
69 template <typename MatrixType>
matrix_sqrt_quasi_triangular_solve_auxiliary_equation(MatrixType & X,const MatrixType & A,const MatrixType & B,const MatrixType & C)70 void matrix_sqrt_quasi_triangular_solve_auxiliary_equation(MatrixType& X, const MatrixType& A, const MatrixType& B, const MatrixType& C)
71 {
72 typedef typename traits<MatrixType>::Scalar Scalar;
73 Matrix<Scalar,4,4> coeffMatrix = Matrix<Scalar,4,4>::Zero();
74 coeffMatrix.coeffRef(0,0) = A.coeff(0,0) + B.coeff(0,0);
75 coeffMatrix.coeffRef(1,1) = A.coeff(0,0) + B.coeff(1,1);
76 coeffMatrix.coeffRef(2,2) = A.coeff(1,1) + B.coeff(0,0);
77 coeffMatrix.coeffRef(3,3) = A.coeff(1,1) + B.coeff(1,1);
78 coeffMatrix.coeffRef(0,1) = B.coeff(1,0);
79 coeffMatrix.coeffRef(0,2) = A.coeff(0,1);
80 coeffMatrix.coeffRef(1,0) = B.coeff(0,1);
81 coeffMatrix.coeffRef(1,3) = A.coeff(0,1);
82 coeffMatrix.coeffRef(2,0) = A.coeff(1,0);
83 coeffMatrix.coeffRef(2,3) = B.coeff(1,0);
84 coeffMatrix.coeffRef(3,1) = A.coeff(1,0);
85 coeffMatrix.coeffRef(3,2) = B.coeff(0,1);
86
87 Matrix<Scalar,4,1> rhs;
88 rhs.coeffRef(0) = C.coeff(0,0);
89 rhs.coeffRef(1) = C.coeff(0,1);
90 rhs.coeffRef(2) = C.coeff(1,0);
91 rhs.coeffRef(3) = C.coeff(1,1);
92
93 Matrix<Scalar,4,1> result;
94 result = coeffMatrix.fullPivLu().solve(rhs);
95
96 X.coeffRef(0,0) = result.coeff(0);
97 X.coeffRef(0,1) = result.coeff(1);
98 X.coeffRef(1,0) = result.coeff(2);
99 X.coeffRef(1,1) = result.coeff(3);
100 }
101
102 // similar to compute1x1offDiagonalBlock()
103 template <typename MatrixType, typename ResultType>
matrix_sqrt_quasi_triangular_2x2_off_diagonal_block(const MatrixType & T,typename MatrixType::Index i,typename MatrixType::Index j,ResultType & sqrtT)104 void matrix_sqrt_quasi_triangular_2x2_off_diagonal_block(const MatrixType& T, typename MatrixType::Index i, typename MatrixType::Index j, ResultType& sqrtT)
105 {
106 typedef typename traits<MatrixType>::Scalar Scalar;
107 Matrix<Scalar,2,2> A = sqrtT.template block<2,2>(i,i);
108 Matrix<Scalar,2,2> B = sqrtT.template block<2,2>(j,j);
109 Matrix<Scalar,2,2> C = T.template block<2,2>(i,j);
110 if (j-i > 2)
111 C -= sqrtT.block(i, i+2, 2, j-i-2) * sqrtT.block(i+2, j, j-i-2, 2);
112 Matrix<Scalar,2,2> X;
113 matrix_sqrt_quasi_triangular_solve_auxiliary_equation(X, A, B, C);
114 sqrtT.template block<2,2>(i,j) = X;
115 }
116
117 // pre: T is quasi-upper-triangular and sqrtT is a zero matrix of the same size
118 // post: the diagonal blocks of sqrtT are the square roots of the diagonal blocks of T
119 template <typename MatrixType, typename ResultType>
matrix_sqrt_quasi_triangular_diagonal(const MatrixType & T,ResultType & sqrtT)120 void matrix_sqrt_quasi_triangular_diagonal(const MatrixType& T, ResultType& sqrtT)
121 {
122 using std::sqrt;
123 const Index size = T.rows();
124 for (Index i = 0; i < size; i++) {
125 if (i == size - 1 || T.coeff(i+1, i) == 0) {
126 eigen_assert(T(i,i) >= 0);
127 sqrtT.coeffRef(i,i) = sqrt(T.coeff(i,i));
128 }
129 else {
130 matrix_sqrt_quasi_triangular_2x2_diagonal_block(T, i, sqrtT);
131 ++i;
132 }
133 }
134 }
135
136 // pre: T is quasi-upper-triangular and diagonal blocks of sqrtT are square root of diagonal blocks of T.
137 // post: sqrtT is the square root of T.
138 template <typename MatrixType, typename ResultType>
matrix_sqrt_quasi_triangular_off_diagonal(const MatrixType & T,ResultType & sqrtT)139 void matrix_sqrt_quasi_triangular_off_diagonal(const MatrixType& T, ResultType& sqrtT)
140 {
141 const Index size = T.rows();
142 for (Index j = 1; j < size; j++) {
143 if (T.coeff(j, j-1) != 0) // if T(j-1:j, j-1:j) is a 2-by-2 block
144 continue;
145 for (Index i = j-1; i >= 0; i--) {
146 if (i > 0 && T.coeff(i, i-1) != 0) // if T(i-1:i, i-1:i) is a 2-by-2 block
147 continue;
148 bool iBlockIs2x2 = (i < size - 1) && (T.coeff(i+1, i) != 0);
149 bool jBlockIs2x2 = (j < size - 1) && (T.coeff(j+1, j) != 0);
150 if (iBlockIs2x2 && jBlockIs2x2)
151 matrix_sqrt_quasi_triangular_2x2_off_diagonal_block(T, i, j, sqrtT);
152 else if (iBlockIs2x2 && !jBlockIs2x2)
153 matrix_sqrt_quasi_triangular_2x1_off_diagonal_block(T, i, j, sqrtT);
154 else if (!iBlockIs2x2 && jBlockIs2x2)
155 matrix_sqrt_quasi_triangular_1x2_off_diagonal_block(T, i, j, sqrtT);
156 else if (!iBlockIs2x2 && !jBlockIs2x2)
157 matrix_sqrt_quasi_triangular_1x1_off_diagonal_block(T, i, j, sqrtT);
158 }
159 }
160 }
161
162 } // end of namespace internal
163
164 /** \ingroup MatrixFunctions_Module
165 * \brief Compute matrix square root of quasi-triangular matrix.
166 *
167 * \tparam MatrixType type of \p arg, the argument of matrix square root,
168 * expected to be an instantiation of the Matrix class template.
169 * \tparam ResultType type of \p result, where result is to be stored.
170 * \param[in] arg argument of matrix square root.
171 * \param[out] result matrix square root of upper Hessenberg part of \p arg.
172 *
173 * This function computes the square root of the upper quasi-triangular matrix stored in the upper
174 * Hessenberg part of \p arg. Only the upper Hessenberg part of \p result is updated, the rest is
175 * not touched. See MatrixBase::sqrt() for details on how this computation is implemented.
176 *
177 * \sa MatrixSquareRoot, MatrixSquareRootQuasiTriangular
178 */
179 template <typename MatrixType, typename ResultType>
matrix_sqrt_quasi_triangular(const MatrixType & arg,ResultType & result)180 void matrix_sqrt_quasi_triangular(const MatrixType &arg, ResultType &result)
181 {
182 eigen_assert(arg.rows() == arg.cols());
183 result.resize(arg.rows(), arg.cols());
184 internal::matrix_sqrt_quasi_triangular_diagonal(arg, result);
185 internal::matrix_sqrt_quasi_triangular_off_diagonal(arg, result);
186 }
187
188
189 /** \ingroup MatrixFunctions_Module
190 * \brief Compute matrix square root of triangular matrix.
191 *
192 * \tparam MatrixType type of \p arg, the argument of matrix square root,
193 * expected to be an instantiation of the Matrix class template.
194 * \tparam ResultType type of \p result, where result is to be stored.
195 * \param[in] arg argument of matrix square root.
196 * \param[out] result matrix square root of upper triangular part of \p arg.
197 *
198 * Only the upper triangular part (including the diagonal) of \p result is updated, the rest is not
199 * touched. See MatrixBase::sqrt() for details on how this computation is implemented.
200 *
201 * \sa MatrixSquareRoot, MatrixSquareRootQuasiTriangular
202 */
203 template <typename MatrixType, typename ResultType>
matrix_sqrt_triangular(const MatrixType & arg,ResultType & result)204 void matrix_sqrt_triangular(const MatrixType &arg, ResultType &result)
205 {
206 using std::sqrt;
207 typedef typename MatrixType::Scalar Scalar;
208
209 eigen_assert(arg.rows() == arg.cols());
210
211 // Compute square root of arg and store it in upper triangular part of result
212 // This uses that the square root of triangular matrices can be computed directly.
213 result.resize(arg.rows(), arg.cols());
214 for (Index i = 0; i < arg.rows(); i++) {
215 result.coeffRef(i,i) = sqrt(arg.coeff(i,i));
216 }
217 for (Index j = 1; j < arg.cols(); j++) {
218 for (Index i = j-1; i >= 0; i--) {
219 // if i = j-1, then segment has length 0 so tmp = 0
220 Scalar tmp = (result.row(i).segment(i+1,j-i-1) * result.col(j).segment(i+1,j-i-1)).value();
221 // denominator may be zero if original matrix is singular
222 result.coeffRef(i,j) = (arg.coeff(i,j) - tmp) / (result.coeff(i,i) + result.coeff(j,j));
223 }
224 }
225 }
226
227
228 namespace internal {
229
230 /** \ingroup MatrixFunctions_Module
231 * \brief Helper struct for computing matrix square roots of general matrices.
232 * \tparam MatrixType type of the argument of the matrix square root,
233 * expected to be an instantiation of the Matrix class template.
234 *
235 * \sa MatrixSquareRootTriangular, MatrixSquareRootQuasiTriangular, MatrixBase::sqrt()
236 */
237 template <typename MatrixType, int IsComplex = NumTraits<typename internal::traits<MatrixType>::Scalar>::IsComplex>
238 struct matrix_sqrt_compute
239 {
240 /** \brief Compute the matrix square root
241 *
242 * \param[in] arg matrix whose square root is to be computed.
243 * \param[out] result square root of \p arg.
244 *
245 * See MatrixBase::sqrt() for details on how this computation is implemented.
246 */
247 template <typename ResultType> static void run(const MatrixType &arg, ResultType &result);
248 };
249
250
251 // ********** Partial specialization for real matrices **********
252
253 template <typename MatrixType>
254 struct matrix_sqrt_compute<MatrixType, 0>
255 {
256 template <typename ResultType>
257 static void run(const MatrixType &arg, ResultType &result)
258 {
259 eigen_assert(arg.rows() == arg.cols());
260
261 // Compute Schur decomposition of arg
262 const RealSchur<MatrixType> schurOfA(arg);
263 const MatrixType& T = schurOfA.matrixT();
264 const MatrixType& U = schurOfA.matrixU();
265
266 // Compute square root of T
267 MatrixType sqrtT = MatrixType::Zero(arg.rows(), arg.cols());
268 matrix_sqrt_quasi_triangular(T, sqrtT);
269
270 // Compute square root of arg
271 result = U * sqrtT * U.adjoint();
272 }
273 };
274
275
276 // ********** Partial specialization for complex matrices **********
277
278 template <typename MatrixType>
279 struct matrix_sqrt_compute<MatrixType, 1>
280 {
281 template <typename ResultType>
282 static void run(const MatrixType &arg, ResultType &result)
283 {
284 eigen_assert(arg.rows() == arg.cols());
285
286 // Compute Schur decomposition of arg
287 const ComplexSchur<MatrixType> schurOfA(arg);
288 const MatrixType& T = schurOfA.matrixT();
289 const MatrixType& U = schurOfA.matrixU();
290
291 // Compute square root of T
292 MatrixType sqrtT;
293 matrix_sqrt_triangular(T, sqrtT);
294
295 // Compute square root of arg
296 result = U * (sqrtT.template triangularView<Upper>() * U.adjoint());
297 }
298 };
299
300 } // end namespace internal
301
302 /** \ingroup MatrixFunctions_Module
303 *
304 * \brief Proxy for the matrix square root of some matrix (expression).
305 *
306 * \tparam Derived Type of the argument to the matrix square root.
307 *
308 * This class holds the argument to the matrix square root until it
309 * is assigned or evaluated for some other reason (so the argument
310 * should not be changed in the meantime). It is the return type of
311 * MatrixBase::sqrt() and most of the time this is the only way it is
312 * used.
313 */
314 template<typename Derived> class MatrixSquareRootReturnValue
315 : public ReturnByValue<MatrixSquareRootReturnValue<Derived> >
316 {
317 protected:
318 typedef typename internal::ref_selector<Derived>::type DerivedNested;
319
320 public:
321 /** \brief Constructor.
322 *
323 * \param[in] src %Matrix (expression) forming the argument of the
324 * matrix square root.
325 */
326 explicit MatrixSquareRootReturnValue(const Derived& src) : m_src(src) { }
327
328 /** \brief Compute the matrix square root.
329 *
330 * \param[out] result the matrix square root of \p src in the
331 * constructor.
332 */
333 template <typename ResultType>
334 inline void evalTo(ResultType& result) const
335 {
336 typedef typename internal::nested_eval<Derived, 10>::type DerivedEvalType;
337 typedef typename internal::remove_all<DerivedEvalType>::type DerivedEvalTypeClean;
338 DerivedEvalType tmp(m_src);
339 internal::matrix_sqrt_compute<DerivedEvalTypeClean>::run(tmp, result);
340 }
341
342 Index rows() const { return m_src.rows(); }
343 Index cols() const { return m_src.cols(); }
344
345 protected:
346 const DerivedNested m_src;
347 };
348
349 namespace internal {
350 template<typename Derived>
351 struct traits<MatrixSquareRootReturnValue<Derived> >
352 {
353 typedef typename Derived::PlainObject ReturnType;
354 };
355 }
356
357 template <typename Derived>
358 const MatrixSquareRootReturnValue<Derived> MatrixBase<Derived>::sqrt() const
359 {
360 eigen_assert(rows() == cols());
361 return MatrixSquareRootReturnValue<Derived>(derived());
362 }
363
364 } // end namespace Eigen
365
366 #endif // EIGEN_MATRIX_FUNCTION
367