319 lines
11 KiB
C++
319 lines
11 KiB
C++
// This file is part of Eigen, a lightweight C++ template library
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// for linear algebra.
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//
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// Copyright (C) 2006-2008, 2010 Benoit Jacob <jacob.benoit.1@gmail.com>
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//
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// This Source Code Form is subject to the terms of the Mozilla
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// Public License v. 2.0. If a copy of the MPL was not distributed
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// with this file, You can obtain one at http://mozilla.org/MPL/2.0/.
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#ifndef EIGEN_DOT_H
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#define EIGEN_DOT_H
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namespace Eigen {
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namespace internal {
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// helper function for dot(). The problem is that if we put that in the body of dot(), then upon calling dot
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// with mismatched types, the compiler emits errors about failing to instantiate cwiseProduct BEFORE
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// looking at the static assertions. Thus this is a trick to get better compile errors.
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template<typename T, typename U,
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// the NeedToTranspose condition here is taken straight from Assign.h
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bool NeedToTranspose = T::IsVectorAtCompileTime
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&& U::IsVectorAtCompileTime
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&& ((int(T::RowsAtCompileTime) == 1 && int(U::ColsAtCompileTime) == 1)
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| // FIXME | instead of || to please GCC 4.4.0 stupid warning "suggest parentheses around &&".
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// revert to || as soon as not needed anymore.
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(int(T::ColsAtCompileTime) == 1 && int(U::RowsAtCompileTime) == 1))
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>
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struct dot_nocheck
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{
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typedef scalar_conj_product_op<typename traits<T>::Scalar,typename traits<U>::Scalar> conj_prod;
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typedef typename conj_prod::result_type ResScalar;
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EIGEN_DEVICE_FUNC
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EIGEN_STRONG_INLINE
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static ResScalar run(const MatrixBase<T>& a, const MatrixBase<U>& b)
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{
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return a.template binaryExpr<conj_prod>(b).sum();
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}
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};
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template<typename T, typename U>
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struct dot_nocheck<T, U, true>
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{
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typedef scalar_conj_product_op<typename traits<T>::Scalar,typename traits<U>::Scalar> conj_prod;
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typedef typename conj_prod::result_type ResScalar;
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EIGEN_DEVICE_FUNC
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EIGEN_STRONG_INLINE
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static ResScalar run(const MatrixBase<T>& a, const MatrixBase<U>& b)
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{
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return a.transpose().template binaryExpr<conj_prod>(b).sum();
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}
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};
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} // end namespace internal
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/** \fn MatrixBase::dot
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* \returns the dot product of *this with other.
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*
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* \only_for_vectors
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*
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* \note If the scalar type is complex numbers, then this function returns the hermitian
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* (sesquilinear) dot product, conjugate-linear in the first variable and linear in the
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* second variable.
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*
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* \sa squaredNorm(), norm()
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*/
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template<typename Derived>
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template<typename OtherDerived>
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EIGEN_DEVICE_FUNC
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EIGEN_STRONG_INLINE
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typename ScalarBinaryOpTraits<typename internal::traits<Derived>::Scalar,typename internal::traits<OtherDerived>::Scalar>::ReturnType
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MatrixBase<Derived>::dot(const MatrixBase<OtherDerived>& other) const
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{
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EIGEN_STATIC_ASSERT_VECTOR_ONLY(Derived)
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EIGEN_STATIC_ASSERT_VECTOR_ONLY(OtherDerived)
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EIGEN_STATIC_ASSERT_SAME_VECTOR_SIZE(Derived,OtherDerived)
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#if !(defined(EIGEN_NO_STATIC_ASSERT) && defined(EIGEN_NO_DEBUG))
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typedef internal::scalar_conj_product_op<Scalar,typename OtherDerived::Scalar> func;
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EIGEN_CHECK_BINARY_COMPATIBILIY(func,Scalar,typename OtherDerived::Scalar);
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#endif
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eigen_assert(size() == other.size());
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return internal::dot_nocheck<Derived,OtherDerived>::run(*this, other);
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}
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//---------- implementation of L2 norm and related functions ----------
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/** \returns, for vectors, the squared \em l2 norm of \c *this, and for matrices the squared Frobenius norm.
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* In both cases, it consists in the sum of the square of all the matrix entries.
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* For vectors, this is also equals to the dot product of \c *this with itself.
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*
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* \sa dot(), norm(), lpNorm()
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*/
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template<typename Derived>
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EIGEN_DEVICE_FUNC EIGEN_STRONG_INLINE typename NumTraits<typename internal::traits<Derived>::Scalar>::Real MatrixBase<Derived>::squaredNorm() const
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{
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return numext::real((*this).cwiseAbs2().sum());
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}
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/** \returns, for vectors, the \em l2 norm of \c *this, and for matrices the Frobenius norm.
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* In both cases, it consists in the square root of the sum of the square of all the matrix entries.
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* For vectors, this is also equals to the square root of the dot product of \c *this with itself.
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*
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* \sa lpNorm(), dot(), squaredNorm()
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*/
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template<typename Derived>
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EIGEN_DEVICE_FUNC EIGEN_STRONG_INLINE typename NumTraits<typename internal::traits<Derived>::Scalar>::Real MatrixBase<Derived>::norm() const
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{
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return numext::sqrt(squaredNorm());
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}
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/** \returns an expression of the quotient of \c *this by its own norm.
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*
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* \warning If the input vector is too small (i.e., this->norm()==0),
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* then this function returns a copy of the input.
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*
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* \only_for_vectors
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*
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* \sa norm(), normalize()
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*/
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template<typename Derived>
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EIGEN_DEVICE_FUNC EIGEN_STRONG_INLINE const typename MatrixBase<Derived>::PlainObject
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MatrixBase<Derived>::normalized() const
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{
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typedef typename internal::nested_eval<Derived,2>::type _Nested;
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_Nested n(derived());
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RealScalar z = n.squaredNorm();
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// NOTE: after extensive benchmarking, this conditional does not impact performance, at least on recent x86 CPU
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if(z>RealScalar(0))
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return n / numext::sqrt(z);
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else
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return n;
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}
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/** Normalizes the vector, i.e. divides it by its own norm.
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*
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* \only_for_vectors
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*
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* \warning If the input vector is too small (i.e., this->norm()==0), then \c *this is left unchanged.
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*
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* \sa norm(), normalized()
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*/
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template<typename Derived>
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EIGEN_DEVICE_FUNC EIGEN_STRONG_INLINE void MatrixBase<Derived>::normalize()
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{
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RealScalar z = squaredNorm();
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// NOTE: after extensive benchmarking, this conditional does not impact performance, at least on recent x86 CPU
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if(z>RealScalar(0))
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derived() /= numext::sqrt(z);
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}
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/** \returns an expression of the quotient of \c *this by its own norm while avoiding underflow and overflow.
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*
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* \only_for_vectors
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*
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* This method is analogue to the normalized() method, but it reduces the risk of
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* underflow and overflow when computing the norm.
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*
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* \warning If the input vector is too small (i.e., this->norm()==0),
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* then this function returns a copy of the input.
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*
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* \sa stableNorm(), stableNormalize(), normalized()
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*/
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template<typename Derived>
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EIGEN_DEVICE_FUNC EIGEN_STRONG_INLINE const typename MatrixBase<Derived>::PlainObject
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MatrixBase<Derived>::stableNormalized() const
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{
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typedef typename internal::nested_eval<Derived,3>::type _Nested;
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_Nested n(derived());
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RealScalar w = n.cwiseAbs().maxCoeff();
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RealScalar z = (n/w).squaredNorm();
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if(z>RealScalar(0))
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return n / (numext::sqrt(z)*w);
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else
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return n;
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}
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/** Normalizes the vector while avoid underflow and overflow
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*
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* \only_for_vectors
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*
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* This method is analogue to the normalize() method, but it reduces the risk of
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* underflow and overflow when computing the norm.
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*
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* \warning If the input vector is too small (i.e., this->norm()==0), then \c *this is left unchanged.
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*
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* \sa stableNorm(), stableNormalized(), normalize()
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*/
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template<typename Derived>
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EIGEN_DEVICE_FUNC EIGEN_STRONG_INLINE void MatrixBase<Derived>::stableNormalize()
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{
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RealScalar w = cwiseAbs().maxCoeff();
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RealScalar z = (derived()/w).squaredNorm();
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if(z>RealScalar(0))
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derived() /= numext::sqrt(z)*w;
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}
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//---------- implementation of other norms ----------
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namespace internal {
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template<typename Derived, int p>
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struct lpNorm_selector
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{
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typedef typename NumTraits<typename traits<Derived>::Scalar>::Real RealScalar;
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EIGEN_DEVICE_FUNC
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static inline RealScalar run(const MatrixBase<Derived>& m)
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{
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EIGEN_USING_STD(pow)
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return pow(m.cwiseAbs().array().pow(p).sum(), RealScalar(1)/p);
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}
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};
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template<typename Derived>
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struct lpNorm_selector<Derived, 1>
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{
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EIGEN_DEVICE_FUNC
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static inline typename NumTraits<typename traits<Derived>::Scalar>::Real run(const MatrixBase<Derived>& m)
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{
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return m.cwiseAbs().sum();
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}
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};
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template<typename Derived>
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struct lpNorm_selector<Derived, 2>
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{
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EIGEN_DEVICE_FUNC
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static inline typename NumTraits<typename traits<Derived>::Scalar>::Real run(const MatrixBase<Derived>& m)
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{
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return m.norm();
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}
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};
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template<typename Derived>
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struct lpNorm_selector<Derived, Infinity>
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{
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typedef typename NumTraits<typename traits<Derived>::Scalar>::Real RealScalar;
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EIGEN_DEVICE_FUNC
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static inline RealScalar run(const MatrixBase<Derived>& m)
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{
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if(Derived::SizeAtCompileTime==0 || (Derived::SizeAtCompileTime==Dynamic && m.size()==0))
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return RealScalar(0);
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return m.cwiseAbs().maxCoeff();
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}
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};
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} // end namespace internal
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/** \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
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* of the coefficients of \c *this. If \a p is the special value \a Eigen::Infinity, this function returns the \f$ \ell^\infty \f$
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* norm, that is the maximum of the absolute values of the coefficients of \c *this.
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*
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* In all cases, if \c *this is empty, then the value 0 is returned.
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*
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* \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.
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*
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* \sa norm()
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*/
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template<typename Derived>
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template<int p>
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#ifndef EIGEN_PARSED_BY_DOXYGEN
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EIGEN_DEVICE_FUNC inline typename NumTraits<typename internal::traits<Derived>::Scalar>::Real
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#else
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EIGEN_DEVICE_FUNC MatrixBase<Derived>::RealScalar
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#endif
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MatrixBase<Derived>::lpNorm() const
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{
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return internal::lpNorm_selector<Derived, p>::run(*this);
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}
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//---------- implementation of isOrthogonal / isUnitary ----------
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/** \returns true if *this is approximately orthogonal to \a other,
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* within the precision given by \a prec.
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*
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* Example: \include MatrixBase_isOrthogonal.cpp
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* Output: \verbinclude MatrixBase_isOrthogonal.out
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*/
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template<typename Derived>
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template<typename OtherDerived>
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bool MatrixBase<Derived>::isOrthogonal
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(const MatrixBase<OtherDerived>& other, const RealScalar& prec) const
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{
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typename internal::nested_eval<Derived,2>::type nested(derived());
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typename internal::nested_eval<OtherDerived,2>::type otherNested(other.derived());
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return numext::abs2(nested.dot(otherNested)) <= prec * prec * nested.squaredNorm() * otherNested.squaredNorm();
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}
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/** \returns true if *this is approximately an unitary matrix,
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* within the precision given by \a prec. In the case where the \a Scalar
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* type is real numbers, a unitary matrix is an orthogonal matrix, whence the name.
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*
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* \note This can be used to check whether a family of vectors forms an orthonormal basis.
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* Indeed, \c m.isUnitary() returns true if and only if the columns (equivalently, the rows) of m form an
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* orthonormal basis.
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*
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* Example: \include MatrixBase_isUnitary.cpp
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* Output: \verbinclude MatrixBase_isUnitary.out
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*/
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template<typename Derived>
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bool MatrixBase<Derived>::isUnitary(const RealScalar& prec) const
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{
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typename internal::nested_eval<Derived,1>::type self(derived());
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for(Index i = 0; i < cols(); ++i)
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{
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if(!internal::isApprox(self.col(i).squaredNorm(), static_cast<RealScalar>(1), prec))
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return false;
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for(Index j = 0; j < i; ++j)
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if(!internal::isMuchSmallerThan(self.col(i).dot(self.col(j)), static_cast<Scalar>(1), prec))
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return false;
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}
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return true;
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}
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} // end namespace Eigen
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#endif // EIGEN_DOT_H
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