1 // This file is part of Eigen, a lightweight C++ template library 2 // for linear algebra. 3 // 4 // Copyright (C) 2006-2008, 2010 Benoit Jacob <jacob.benoit.1 (at) gmail.com> 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_DOT_H 11 #define EIGEN_DOT_H 12 13 namespace Eigen { 14 15 namespace internal { 16 17 // helper function for dot(). The problem is that if we put that in the body of dot(), then upon calling dot 18 // with mismatched types, the compiler emits errors about failing to instantiate cwiseProduct BEFORE 19 // looking at the static assertions. Thus this is a trick to get better compile errors. 20 template<typename T, typename U, 21 // the NeedToTranspose condition here is taken straight from Assign.h 22 bool NeedToTranspose = T::IsVectorAtCompileTime 23 && U::IsVectorAtCompileTime 24 && ((int(T::RowsAtCompileTime) == 1 && int(U::ColsAtCompileTime) == 1) 25 | // FIXME | instead of || to please GCC 4.4.0 stupid warning "suggest parentheses around &&". 26 // revert to || as soon as not needed anymore. 27 (int(T::ColsAtCompileTime) == 1 && int(U::RowsAtCompileTime) == 1)) 28 > 29 struct dot_nocheck 30 { 31 typedef scalar_conj_product_op<typename traits<T>::Scalar,typename traits<U>::Scalar> conj_prod; 32 typedef typename conj_prod::result_type ResScalar; 33 EIGEN_DEVICE_FUNC 34 static inline ResScalar run(const MatrixBase<T>& a, const MatrixBase<U>& b) 35 { 36 return a.template binaryExpr<conj_prod>(b).sum(); 37 } 38 }; 39 40 template<typename T, typename U> 41 struct dot_nocheck<T, U, true> 42 { 43 typedef scalar_conj_product_op<typename traits<T>::Scalar,typename traits<U>::Scalar> conj_prod; 44 typedef typename conj_prod::result_type ResScalar; 45 EIGEN_DEVICE_FUNC 46 static inline ResScalar run(const MatrixBase<T>& a, const MatrixBase<U>& b) 47 { 48 return a.transpose().template binaryExpr<conj_prod>(b).sum(); 49 } 50 }; 51 52 } // end namespace internal 53 54 /** \fn MatrixBase::dot 55 * \returns the dot product of *this with other. 56 * 57 * \only_for_vectors 58 * 59 * \note If the scalar type is complex numbers, then this function returns the hermitian 60 * (sesquilinear) dot product, conjugate-linear in the first variable and linear in the 61 * second variable. 62 * 63 * \sa squaredNorm(), norm() 64 */ 65 template<typename Derived> 66 template<typename OtherDerived> 67 EIGEN_DEVICE_FUNC 68 typename ScalarBinaryOpTraits<typename internal::traits<Derived>::Scalar,typename internal::traits<OtherDerived>::Scalar>::ReturnType 69 MatrixBase<Derived>::dot(const MatrixBase<OtherDerived>& other) const 70 { 71 EIGEN_STATIC_ASSERT_VECTOR_ONLY(Derived) 72 EIGEN_STATIC_ASSERT_VECTOR_ONLY(OtherDerived) 73 EIGEN_STATIC_ASSERT_SAME_VECTOR_SIZE(Derived,OtherDerived) 74 #if !(defined(EIGEN_NO_STATIC_ASSERT) && defined(EIGEN_NO_DEBUG)) 75 typedef internal::scalar_conj_product_op<Scalar,typename OtherDerived::Scalar> func; 76 EIGEN_CHECK_BINARY_COMPATIBILIY(func,Scalar,typename OtherDerived::Scalar); 77 #endif 78 79 eigen_assert(size() == other.size()); 80 81 return internal::dot_nocheck<Derived,OtherDerived>::run(*this, other); 82 } 83 84 //---------- implementation of L2 norm and related functions ---------- 85 86 /** \returns, for vectors, the squared \em l2 norm of \c *this, and for matrices the Frobenius norm. 87 * In both cases, it consists in the sum of the square of all the matrix entries. 88 * For vectors, this is also equals to the dot product of \c *this with itself. 89 * 90 * \sa dot(), norm(), lpNorm() 91 */ 92 template<typename Derived> 93 EIGEN_STRONG_INLINE typename NumTraits<typename internal::traits<Derived>::Scalar>::Real MatrixBase<Derived>::squaredNorm() const 94 { 95 return numext::real((*this).cwiseAbs2().sum()); 96 } 97 98 /** \returns, for vectors, the \em l2 norm of \c *this, and for matrices the Frobenius norm. 99 * In both cases, it consists in the square root of the sum of the square of all the matrix entries. 100 * For vectors, this is also equals to the square root of the dot product of \c *this with itself. 101 * 102 * \sa lpNorm(), dot(), squaredNorm() 103 */ 104 template<typename Derived> 105 inline typename NumTraits<typename internal::traits<Derived>::Scalar>::Real MatrixBase<Derived>::norm() const 106 { 107 return numext::sqrt(squaredNorm()); 108 } 109 110 /** \returns an expression of the quotient of \c *this by its own norm. 111 * 112 * \warning If the input vector is too small (i.e., this->norm()==0), 113 * then this function returns a copy of the input. 114 * 115 * \only_for_vectors 116 * 117 * \sa norm(), normalize() 118 */ 119 template<typename Derived> 120 inline const typename MatrixBase<Derived>::PlainObject 121 MatrixBase<Derived>::normalized() const 122 { 123 typedef typename internal::nested_eval<Derived,2>::type _Nested; 124 _Nested n(derived()); 125 RealScalar z = n.squaredNorm(); 126 // NOTE: after extensive benchmarking, this conditional does not impact performance, at least on recent x86 CPU 127 if(z>RealScalar(0)) 128 return n / numext::sqrt(z); 129 else 130 return n; 131 } 132 133 /** Normalizes the vector, i.e. divides it by its own norm. 134 * 135 * \only_for_vectors 136 * 137 * \warning If the input vector is too small (i.e., this->norm()==0), then \c *this is left unchanged. 138 * 139 * \sa norm(), normalized() 140 */ 141 template<typename Derived> 142 inline void MatrixBase<Derived>::normalize() 143 { 144 RealScalar z = squaredNorm(); 145 // NOTE: after extensive benchmarking, this conditional does not impact performance, at least on recent x86 CPU 146 if(z>RealScalar(0)) 147 derived() /= numext::sqrt(z); 148 } 149 150 /** \returns an expression of the quotient of \c *this by its own norm while avoiding underflow and overflow. 151 * 152 * \only_for_vectors 153 * 154 * This method is analogue to the normalized() method, but it reduces the risk of 155 * underflow and overflow when computing the norm. 156 * 157 * \warning If the input vector is too small (i.e., this->norm()==0), 158 * then this function returns a copy of the input. 159 * 160 * \sa stableNorm(), stableNormalize(), normalized() 161 */ 162 template<typename Derived> 163 inline const typename MatrixBase<Derived>::PlainObject 164 MatrixBase<Derived>::stableNormalized() const 165 { 166 typedef typename internal::nested_eval<Derived,3>::type _Nested; 167 _Nested n(derived()); 168 RealScalar w = n.cwiseAbs().maxCoeff(); 169 RealScalar z = (n/w).squaredNorm(); 170 if(z>RealScalar(0)) 171 return n / (numext::sqrt(z)*w); 172 else 173 return n; 174 } 175 176 /** Normalizes the vector while avoid underflow and overflow 177 * 178 * \only_for_vectors 179 * 180 * This method is analogue to the normalize() method, but it reduces the risk of 181 * underflow and overflow when computing the norm. 182 * 183 * \warning If the input vector is too small (i.e., this->norm()==0), then \c *this is left unchanged. 184 * 185 * \sa stableNorm(), stableNormalized(), normalize() 186 */ 187 template<typename Derived> 188 inline void MatrixBase<Derived>::stableNormalize() 189 { 190 RealScalar w = cwiseAbs().maxCoeff(); 191 RealScalar z = (derived()/w).squaredNorm(); 192 if(z>RealScalar(0)) 193 derived() /= numext::sqrt(z)*w; 194 } 195 196 //---------- implementation of other norms ---------- 197 198 namespace internal { 199 200 template<typename Derived, int p> 201 struct lpNorm_selector 202 { 203 typedef typename NumTraits<typename traits<Derived>::Scalar>::Real RealScalar; 204 EIGEN_DEVICE_FUNC 205 static inline RealScalar run(const MatrixBase<Derived>& m) 206 { 207 EIGEN_USING_STD_MATH(pow) 208 return pow(m.cwiseAbs().array().pow(p).sum(), RealScalar(1)/p); 209 } 210 }; 211 212 template<typename Derived> 213 struct lpNorm_selector<Derived, 1> 214 { 215 EIGEN_DEVICE_FUNC 216 static inline typename NumTraits<typename traits<Derived>::Scalar>::Real run(const MatrixBase<Derived>& m) 217 { 218 return m.cwiseAbs().sum(); 219 } 220 }; 221 222 template<typename Derived> 223 struct lpNorm_selector<Derived, 2> 224 { 225 EIGEN_DEVICE_FUNC 226 static inline typename NumTraits<typename traits<Derived>::Scalar>::Real run(const MatrixBase<Derived>& m) 227 { 228 return m.norm(); 229 } 230 }; 231 232 template<typename Derived> 233 struct lpNorm_selector<Derived, Infinity> 234 { 235 typedef typename NumTraits<typename traits<Derived>::Scalar>::Real RealScalar; 236 EIGEN_DEVICE_FUNC 237 static inline RealScalar run(const MatrixBase<Derived>& m) 238 { 239 if(Derived::SizeAtCompileTime==0 || (Derived::SizeAtCompileTime==Dynamic && m.size()==0)) 240 return RealScalar(0); 241 return m.cwiseAbs().maxCoeff(); 242 } 243 }; 244 245 } // end namespace internal 246 247 /** \returns the \b coefficient-wise \f$ \ell^p \f$ norm of \c *this, that is, returns the p-th root of the sum of the p-th powers of the absolute values 248 * of the coefficients of \c *this. If \a p is the special value \a Eigen::Infinity, this function returns the \f$ \ell^\infty \f$ 249 * norm, that is the maximum of the absolute values of the coefficients of \c *this. 250 * 251 * In all cases, if \c *this is empty, then the value 0 is returned. 252 * 253 * \note For matrices, this function does not compute the <a href="https://en.wikipedia.org/wiki/Operator_norm">operator-norm</a>. That is, if \c *this is a matrix, then its coefficients are interpreted as a 1D vector. Nonetheless, you can easily compute the 1-norm and \f$\infty\f$-norm matrix operator norms using \link TutorialReductionsVisitorsBroadcastingReductionsNorm partial reductions \endlink. 254 * 255 * \sa norm() 256 */ 257 template<typename Derived> 258 template<int p> 259 #ifndef EIGEN_PARSED_BY_DOXYGEN 260 inline typename NumTraits<typename internal::traits<Derived>::Scalar>::Real 261 #else 262 MatrixBase<Derived>::RealScalar 263 #endif 264 MatrixBase<Derived>::lpNorm() const 265 { 266 return internal::lpNorm_selector<Derived, p>::run(*this); 267 } 268 269 //---------- implementation of isOrthogonal / isUnitary ---------- 270 271 /** \returns true if *this is approximately orthogonal to \a other, 272 * within the precision given by \a prec. 273 * 274 * Example: \include MatrixBase_isOrthogonal.cpp 275 * Output: \verbinclude MatrixBase_isOrthogonal.out 276 */ 277 template<typename Derived> 278 template<typename OtherDerived> 279 bool MatrixBase<Derived>::isOrthogonal 280 (const MatrixBase<OtherDerived>& other, const RealScalar& prec) const 281 { 282 typename internal::nested_eval<Derived,2>::type nested(derived()); 283 typename internal::nested_eval<OtherDerived,2>::type otherNested(other.derived()); 284 return numext::abs2(nested.dot(otherNested)) <= prec * prec * nested.squaredNorm() * otherNested.squaredNorm(); 285 } 286 287 /** \returns true if *this is approximately an unitary matrix, 288 * within the precision given by \a prec. In the case where the \a Scalar 289 * type is real numbers, a unitary matrix is an orthogonal matrix, whence the name. 290 * 291 * \note This can be used to check whether a family of vectors forms an orthonormal basis. 292 * Indeed, \c m.isUnitary() returns true if and only if the columns (equivalently, the rows) of m form an 293 * orthonormal basis. 294 * 295 * Example: \include MatrixBase_isUnitary.cpp 296 * Output: \verbinclude MatrixBase_isUnitary.out 297 */ 298 template<typename Derived> 299 bool MatrixBase<Derived>::isUnitary(const RealScalar& prec) const 300 { 301 typename internal::nested_eval<Derived,1>::type self(derived()); 302 for(Index i = 0; i < cols(); ++i) 303 { 304 if(!internal::isApprox(self.col(i).squaredNorm(), static_cast<RealScalar>(1), prec)) 305 return false; 306 for(Index j = 0; j < i; ++j) 307 if(!internal::isMuchSmallerThan(self.col(i).dot(self.col(j)), static_cast<Scalar>(1), prec)) 308 return false; 309 } 310 return true; 311 } 312 313 } // end namespace Eigen 314 315 #endif // EIGEN_DOT_H 316
