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MatrixSquareRoot.h
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
13namespace Eigen {
14
15namespace 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)
19template <typename MatrixType, typename ResultType>
20void matrix_sqrt_quasi_triangular_2x2_diagonal_block(const MatrixType& T, 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
34template <typename MatrixType, typename ResultType>
35void matrix_sqrt_quasi_triangular_1x1_off_diagonal_block(const MatrixType& T, Index i, 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()
43template <typename MatrixType, typename ResultType>
44void matrix_sqrt_quasi_triangular_1x2_off_diagonal_block(const MatrixType& T, Index i, 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()
56template <typename MatrixType, typename ResultType>
57void matrix_sqrt_quasi_triangular_2x1_off_diagonal_block(const MatrixType& T, Index i, 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
69template <typename MatrixType>
70void 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()
103template <typename MatrixType, typename ResultType>
104void matrix_sqrt_quasi_triangular_2x2_off_diagonal_block(const MatrixType& T, Index i, 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
119template <typename MatrixType, typename ResultType>
120void 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.
138template <typename MatrixType, typename ResultType>
139void 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
179template <typename MatrixType, typename ResultType>
180void 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
203template <typename MatrixType, typename ResultType>
204void 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
228namespace internal {
229
237template <typename MatrixType, int IsComplex = NumTraits<typename internal::traits<MatrixType>::Scalar>::IsComplex>
238struct matrix_sqrt_compute
239{
247 template <typename ResultType> static void run(const MatrixType &arg, ResultType &result);
248};
249
250
251// ********** Partial specialization for real matrices **********
252
253template <typename MatrixType>
254struct matrix_sqrt_compute<MatrixType, 0>
255{
256 typedef typename MatrixType::PlainObject PlainType;
257 template <typename ResultType>
258 static void run(const MatrixType &arg, ResultType &result)
259 {
260 eigen_assert(arg.rows() == arg.cols());
261
262 // Compute Schur decomposition of arg
263 const RealSchur<PlainType> schurOfA(arg);
264 const PlainType& T = schurOfA.matrixT();
265 const PlainType& U = schurOfA.matrixU();
266
267 // Compute square root of T
268 PlainType sqrtT = PlainType::Zero(arg.rows(), arg.cols());
270
271 // Compute square root of arg
272 result = U * sqrtT * U.adjoint();
273 }
274};
275
276
277// ********** Partial specialization for complex matrices **********
278
279template <typename MatrixType>
280struct matrix_sqrt_compute<MatrixType, 1>
281{
282 typedef typename MatrixType::PlainObject PlainType;
283 template <typename ResultType>
284 static void run(const MatrixType &arg, ResultType &result)
285 {
286 eigen_assert(arg.rows() == arg.cols());
287
288 // Compute Schur decomposition of arg
289 const ComplexSchur<PlainType> schurOfA(arg);
290 const PlainType& T = schurOfA.matrixT();
291 const PlainType& U = schurOfA.matrixU();
292
293 // Compute square root of T
294 PlainType sqrtT;
295 matrix_sqrt_triangular(T, sqrtT);
296
297 // Compute square root of arg
298 result = U * (sqrtT.template triangularView<Upper>() * U.adjoint());
299 }
300};
301
302} // end namespace internal
303
316template<typename Derived> class MatrixSquareRootReturnValue
317: public ReturnByValue<MatrixSquareRootReturnValue<Derived> >
318{
319 protected:
320 typedef typename internal::ref_selector<Derived>::type DerivedNested;
321
322 public:
328 explicit MatrixSquareRootReturnValue(const Derived& src) : m_src(src) { }
329
335 template <typename ResultType>
336 inline void evalTo(ResultType& result) const
337 {
338 typedef typename internal::nested_eval<Derived, 10>::type DerivedEvalType;
339 typedef typename internal::remove_all<DerivedEvalType>::type DerivedEvalTypeClean;
340 DerivedEvalType tmp(m_src);
341 internal::matrix_sqrt_compute<DerivedEvalTypeClean>::run(tmp, result);
342 }
343
344 Index rows() const { return m_src.rows(); }
345 Index cols() const { return m_src.cols(); }
346
347 protected:
348 const DerivedNested m_src;
349};
350
351namespace internal {
352template<typename Derived>
353struct traits<MatrixSquareRootReturnValue<Derived> >
354{
355 typedef typename Derived::PlainObject ReturnType;
356};
357}
358
359template <typename Derived>
360const MatrixSquareRootReturnValue<Derived> MatrixBase<Derived>::sqrt() const
361{
362 eigen_assert(rows() == cols());
363 return MatrixSquareRootReturnValue<Derived>(derived());
364}
365
366} // end namespace Eigen
367
368#endif // EIGEN_MATRIX_FUNCTION
const MatrixSquareRootReturnValue< Derived > sqrt() const
Definition: MatrixSquareRoot.h:360
Proxy for the matrix square root of some matrix (expression).
Definition: MatrixSquareRoot.h:318
void evalTo(ResultType &result) const
Compute the matrix square root.
Definition: MatrixSquareRoot.h:336
MatrixSquareRootReturnValue(const Derived &src)
Constructor.
Definition: MatrixSquareRoot.h:328
void matrix_sqrt_quasi_triangular(const MatrixType &arg, ResultType &result)
Compute matrix square root of quasi-triangular matrix.
Definition: MatrixSquareRoot.h:180
void matrix_sqrt_triangular(const MatrixType &arg, ResultType &result)
Compute matrix square root of triangular matrix.
Definition: MatrixSquareRoot.h:204
Namespace containing all symbols from the Eigen library.
const Eigen::CwiseUnaryOp< Eigen::internal::scalar_sqrt_op< typename Derived::Scalar >, const Derived > sqrt(const Eigen::ArrayBase< Derived > &x)
const Eigen::CwiseUnaryOp< Eigen::internal::scalar_real_op< typename Derived::Scalar >, const Derived > real(const Eigen::ArrayBase< Derived > &x)
EIGEN_DEFAULT_DENSE_INDEX_TYPE Index
const Eigen::CwiseUnaryOp< Eigen::internal::scalar_arg_op< typename Derived::Scalar >, const Derived > arg(const Eigen::ArrayBase< Derived > &x)