Eigen libary for mbed
Diff: src/Core/MathFunctions.h
- Revision:
- 0:13a5d365ba16
--- /dev/null Thu Jan 01 00:00:00 1970 +0000
+++ b/src/Core/MathFunctions.h Thu Oct 13 04:07:23 2016 +0000
@@ -0,0 +1,768 @@
+// This file is part of Eigen, a lightweight C++ template library
+// for linear algebra.
+//
+// Copyright (C) 2006-2010 Benoit Jacob <jacob.benoit.1@gmail.com>
+//
+// This Source Code Form is subject to the terms of the Mozilla
+// Public License v. 2.0. If a copy of the MPL was not distributed
+// with this file, You can obtain one at http://mozilla.org/MPL/2.0/.
+
+#ifndef EIGEN_MATHFUNCTIONS_H
+#define EIGEN_MATHFUNCTIONS_H
+
+namespace Eigen {
+
+namespace internal {
+
+/** \internal \struct global_math_functions_filtering_base
+ *
+ * What it does:
+ * Defines a typedef 'type' as follows:
+ * - if type T has a member typedef Eigen_BaseClassForSpecializationOfGlobalMathFuncImpl, then
+ * global_math_functions_filtering_base<T>::type is a typedef for it.
+ * - otherwise, global_math_functions_filtering_base<T>::type is a typedef for T.
+ *
+ * How it's used:
+ * To allow to defined the global math functions (like sin...) in certain cases, like the Array expressions.
+ * When you do sin(array1+array2), the object array1+array2 has a complicated expression type, all what you want to know
+ * is that it inherits ArrayBase. So we implement a partial specialization of sin_impl for ArrayBase<Derived>.
+ * So we must make sure to use sin_impl<ArrayBase<Derived> > and not sin_impl<Derived>, otherwise our partial specialization
+ * won't be used. How does sin know that? That's exactly what global_math_functions_filtering_base tells it.
+ *
+ * How it's implemented:
+ * SFINAE in the style of enable_if. Highly susceptible of breaking compilers. With GCC, it sure does work, but if you replace
+ * the typename dummy by an integer template parameter, it doesn't work anymore!
+ */
+
+template<typename T, typename dummy = void>
+struct global_math_functions_filtering_base
+{
+ typedef T type;
+};
+
+template<typename T> struct always_void { typedef void type; };
+
+template<typename T>
+struct global_math_functions_filtering_base
+ <T,
+ typename always_void<typename T::Eigen_BaseClassForSpecializationOfGlobalMathFuncImpl>::type
+ >
+{
+ typedef typename T::Eigen_BaseClassForSpecializationOfGlobalMathFuncImpl type;
+};
+
+#define EIGEN_MATHFUNC_IMPL(func, scalar) Eigen::internal::func##_impl<typename Eigen::internal::global_math_functions_filtering_base<scalar>::type>
+#define EIGEN_MATHFUNC_RETVAL(func, scalar) typename Eigen::internal::func##_retval<typename Eigen::internal::global_math_functions_filtering_base<scalar>::type>::type
+
+/****************************************************************************
+* Implementation of real *
+****************************************************************************/
+
+template<typename Scalar, bool IsComplex = NumTraits<Scalar>::IsComplex>
+struct real_default_impl
+{
+ typedef typename NumTraits<Scalar>::Real RealScalar;
+ static inline RealScalar run(const Scalar& x)
+ {
+ return x;
+ }
+};
+
+template<typename Scalar>
+struct real_default_impl<Scalar,true>
+{
+ typedef typename NumTraits<Scalar>::Real RealScalar;
+ static inline RealScalar run(const Scalar& x)
+ {
+ using std::real;
+ return real(x);
+ }
+};
+
+template<typename Scalar> struct real_impl : real_default_impl<Scalar> {};
+
+template<typename Scalar>
+struct real_retval
+{
+ typedef typename NumTraits<Scalar>::Real type;
+};
+
+
+/****************************************************************************
+* Implementation of imag *
+****************************************************************************/
+
+template<typename Scalar, bool IsComplex = NumTraits<Scalar>::IsComplex>
+struct imag_default_impl
+{
+ typedef typename NumTraits<Scalar>::Real RealScalar;
+ static inline RealScalar run(const Scalar&)
+ {
+ return RealScalar(0);
+ }
+};
+
+template<typename Scalar>
+struct imag_default_impl<Scalar,true>
+{
+ typedef typename NumTraits<Scalar>::Real RealScalar;
+ static inline RealScalar run(const Scalar& x)
+ {
+ using std::imag;
+ return imag(x);
+ }
+};
+
+template<typename Scalar> struct imag_impl : imag_default_impl<Scalar> {};
+
+template<typename Scalar>
+struct imag_retval
+{
+ typedef typename NumTraits<Scalar>::Real type;
+};
+
+/****************************************************************************
+* Implementation of real_ref *
+****************************************************************************/
+
+template<typename Scalar>
+struct real_ref_impl
+{
+ typedef typename NumTraits<Scalar>::Real RealScalar;
+ static inline RealScalar& run(Scalar& x)
+ {
+ return reinterpret_cast<RealScalar*>(&x)[0];
+ }
+ static inline const RealScalar& run(const Scalar& x)
+ {
+ return reinterpret_cast<const RealScalar*>(&x)[0];
+ }
+};
+
+template<typename Scalar>
+struct real_ref_retval
+{
+ typedef typename NumTraits<Scalar>::Real & type;
+};
+
+/****************************************************************************
+* Implementation of imag_ref *
+****************************************************************************/
+
+template<typename Scalar, bool IsComplex>
+struct imag_ref_default_impl
+{
+ typedef typename NumTraits<Scalar>::Real RealScalar;
+ static inline RealScalar& run(Scalar& x)
+ {
+ return reinterpret_cast<RealScalar*>(&x)[1];
+ }
+ static inline const RealScalar& run(const Scalar& x)
+ {
+ return reinterpret_cast<RealScalar*>(&x)[1];
+ }
+};
+
+template<typename Scalar>
+struct imag_ref_default_impl<Scalar, false>
+{
+ static inline Scalar run(Scalar&)
+ {
+ return Scalar(0);
+ }
+ static inline const Scalar run(const Scalar&)
+ {
+ return Scalar(0);
+ }
+};
+
+template<typename Scalar>
+struct imag_ref_impl : imag_ref_default_impl<Scalar, NumTraits<Scalar>::IsComplex> {};
+
+template<typename Scalar>
+struct imag_ref_retval
+{
+ typedef typename NumTraits<Scalar>::Real & type;
+};
+
+/****************************************************************************
+* Implementation of conj *
+****************************************************************************/
+
+template<typename Scalar, bool IsComplex = NumTraits<Scalar>::IsComplex>
+struct conj_impl
+{
+ static inline Scalar run(const Scalar& x)
+ {
+ return x;
+ }
+};
+
+template<typename Scalar>
+struct conj_impl<Scalar,true>
+{
+ static inline Scalar run(const Scalar& x)
+ {
+ using std::conj;
+ return conj(x);
+ }
+};
+
+template<typename Scalar>
+struct conj_retval
+{
+ typedef Scalar type;
+};
+
+/****************************************************************************
+* Implementation of abs2 *
+****************************************************************************/
+
+template<typename Scalar>
+struct abs2_impl
+{
+ typedef typename NumTraits<Scalar>::Real RealScalar;
+ static inline RealScalar run(const Scalar& x)
+ {
+ return x*x;
+ }
+};
+
+template<typename RealScalar>
+struct abs2_impl<std::complex<RealScalar> >
+{
+ static inline RealScalar run(const std::complex<RealScalar>& x)
+ {
+ return real(x)*real(x) + imag(x)*imag(x);
+ }
+};
+
+template<typename Scalar>
+struct abs2_retval
+{
+ typedef typename NumTraits<Scalar>::Real type;
+};
+
+/****************************************************************************
+* Implementation of norm1 *
+****************************************************************************/
+
+template<typename Scalar, bool IsComplex>
+struct norm1_default_impl
+{
+ typedef typename NumTraits<Scalar>::Real RealScalar;
+ static inline RealScalar run(const Scalar& x)
+ {
+ using std::abs;
+ return abs(real(x)) + abs(imag(x));
+ }
+};
+
+template<typename Scalar>
+struct norm1_default_impl<Scalar, false>
+{
+ static inline Scalar run(const Scalar& x)
+ {
+ using std::abs;
+ return abs(x);
+ }
+};
+
+template<typename Scalar>
+struct norm1_impl : norm1_default_impl<Scalar, NumTraits<Scalar>::IsComplex> {};
+
+template<typename Scalar>
+struct norm1_retval
+{
+ typedef typename NumTraits<Scalar>::Real type;
+};
+
+/****************************************************************************
+* Implementation of hypot *
+****************************************************************************/
+
+template<typename Scalar>
+struct hypot_impl
+{
+ typedef typename NumTraits<Scalar>::Real RealScalar;
+ static inline RealScalar run(const Scalar& x, const Scalar& y)
+ {
+ using std::max;
+ using std::min;
+ using std::abs;
+ using std::sqrt;
+ RealScalar _x = abs(x);
+ RealScalar _y = abs(y);
+ RealScalar p = (max)(_x, _y);
+ if(p==RealScalar(0)) return RealScalar(0);
+ RealScalar q = (min)(_x, _y);
+ RealScalar qp = q/p;
+ return p * sqrt(RealScalar(1) + qp*qp);
+ }
+};
+
+template<typename Scalar>
+struct hypot_retval
+{
+ typedef typename NumTraits<Scalar>::Real type;
+};
+
+/****************************************************************************
+* Implementation of cast *
+****************************************************************************/
+
+template<typename OldType, typename NewType>
+struct cast_impl
+{
+ static inline NewType run(const OldType& x)
+ {
+ return static_cast<NewType>(x);
+ }
+};
+
+// here, for once, we're plainly returning NewType: we don't want cast to do weird things.
+
+template<typename OldType, typename NewType>
+inline NewType cast(const OldType& x)
+{
+ return cast_impl<OldType, NewType>::run(x);
+}
+
+/****************************************************************************
+* Implementation of atanh2 *
+****************************************************************************/
+
+template<typename Scalar, bool IsInteger>
+struct atanh2_default_impl
+{
+ typedef Scalar retval;
+ typedef typename NumTraits<Scalar>::Real RealScalar;
+ static inline Scalar run(const Scalar& x, const Scalar& y)
+ {
+ using std::abs;
+ using std::log;
+ using std::sqrt;
+ Scalar z = x / y;
+ if (y == Scalar(0) || abs(z) > sqrt(NumTraits<RealScalar>::epsilon()))
+ return RealScalar(0.5) * log((y + x) / (y - x));
+ else
+ return z + z*z*z / RealScalar(3);
+ }
+};
+
+template<typename Scalar>
+struct atanh2_default_impl<Scalar, true>
+{
+ static inline Scalar run(const Scalar&, const Scalar&)
+ {
+ EIGEN_STATIC_ASSERT_NON_INTEGER(Scalar)
+ return Scalar(0);
+ }
+};
+
+template<typename Scalar>
+struct atanh2_impl : atanh2_default_impl<Scalar, NumTraits<Scalar>::IsInteger> {};
+
+template<typename Scalar>
+struct atanh2_retval
+{
+ typedef Scalar type;
+};
+
+/****************************************************************************
+* Implementation of pow *
+****************************************************************************/
+
+template<typename Scalar, bool IsInteger>
+struct pow_default_impl
+{
+ typedef Scalar retval;
+ static inline Scalar run(const Scalar& x, const Scalar& y)
+ {
+ using std::pow;
+ return pow(x, y);
+ }
+};
+
+template<typename Scalar>
+struct pow_default_impl<Scalar, true>
+{
+ static inline Scalar run(Scalar x, Scalar y)
+ {
+ Scalar res(1);
+ eigen_assert(!NumTraits<Scalar>::IsSigned || y >= 0);
+ if(y & 1) res *= x;
+ y >>= 1;
+ while(y)
+ {
+ x *= x;
+ if(y&1) res *= x;
+ y >>= 1;
+ }
+ return res;
+ }
+};
+
+template<typename Scalar>
+struct pow_impl : pow_default_impl<Scalar, NumTraits<Scalar>::IsInteger> {};
+
+template<typename Scalar>
+struct pow_retval
+{
+ typedef Scalar type;
+};
+
+/****************************************************************************
+* Implementation of random *
+****************************************************************************/
+
+template<typename Scalar,
+ bool IsComplex,
+ bool IsInteger>
+struct random_default_impl {};
+
+template<typename Scalar>
+struct random_impl : random_default_impl<Scalar, NumTraits<Scalar>::IsComplex, NumTraits<Scalar>::IsInteger> {};
+
+template<typename Scalar>
+struct random_retval
+{
+ typedef Scalar type;
+};
+
+template<typename Scalar> inline EIGEN_MATHFUNC_RETVAL(random, Scalar) random(const Scalar& x, const Scalar& y);
+template<typename Scalar> inline EIGEN_MATHFUNC_RETVAL(random, Scalar) random();
+
+template<typename Scalar>
+struct random_default_impl<Scalar, false, false>
+{
+ static inline Scalar run(const Scalar& x, const Scalar& y)
+ {
+ return x + (y-x) * Scalar(std::rand()) / Scalar(RAND_MAX);
+ }
+ static inline Scalar run()
+ {
+ return run(Scalar(NumTraits<Scalar>::IsSigned ? -1 : 0), Scalar(1));
+ }
+};
+
+enum {
+ floor_log2_terminate,
+ floor_log2_move_up,
+ floor_log2_move_down,
+ floor_log2_bogus
+};
+
+template<unsigned int n, int lower, int upper> struct floor_log2_selector
+{
+ enum { middle = (lower + upper) / 2,
+ value = (upper <= lower + 1) ? int(floor_log2_terminate)
+ : (n < (1 << middle)) ? int(floor_log2_move_down)
+ : (n==0) ? int(floor_log2_bogus)
+ : int(floor_log2_move_up)
+ };
+};
+
+template<unsigned int n,
+ int lower = 0,
+ int upper = sizeof(unsigned int) * CHAR_BIT - 1,
+ int selector = floor_log2_selector<n, lower, upper>::value>
+struct floor_log2 {};
+
+template<unsigned int n, int lower, int upper>
+struct floor_log2<n, lower, upper, floor_log2_move_down>
+{
+ enum { value = floor_log2<n, lower, floor_log2_selector<n, lower, upper>::middle>::value };
+};
+
+template<unsigned int n, int lower, int upper>
+struct floor_log2<n, lower, upper, floor_log2_move_up>
+{
+ enum { value = floor_log2<n, floor_log2_selector<n, lower, upper>::middle, upper>::value };
+};
+
+template<unsigned int n, int lower, int upper>
+struct floor_log2<n, lower, upper, floor_log2_terminate>
+{
+ enum { value = (n >= ((unsigned int)(1) << (lower+1))) ? lower+1 : lower };
+};
+
+template<unsigned int n, int lower, int upper>
+struct floor_log2<n, lower, upper, floor_log2_bogus>
+{
+ // no value, error at compile time
+};
+
+template<typename Scalar>
+struct random_default_impl<Scalar, false, true>
+{
+ typedef typename NumTraits<Scalar>::NonInteger NonInteger;
+
+ static inline Scalar run(const Scalar& x, const Scalar& y)
+ {
+ return x + Scalar((NonInteger(y)-x+1) * std::rand() / (RAND_MAX + NonInteger(1)));
+ }
+
+ static inline Scalar run()
+ {
+#ifdef EIGEN_MAKING_DOCS
+ return run(Scalar(NumTraits<Scalar>::IsSigned ? -10 : 0), Scalar(10));
+#else
+ enum { rand_bits = floor_log2<(unsigned int)(RAND_MAX)+1>::value,
+ scalar_bits = sizeof(Scalar) * CHAR_BIT,
+ shift = EIGEN_PLAIN_ENUM_MAX(0, int(rand_bits) - int(scalar_bits)),
+ offset = NumTraits<Scalar>::IsSigned ? (1 << (EIGEN_PLAIN_ENUM_MIN(rand_bits,scalar_bits)-1)) : 0
+ };
+ return Scalar((std::rand() >> shift) - offset);
+#endif
+ }
+};
+
+template<typename Scalar>
+struct random_default_impl<Scalar, true, false>
+{
+ static inline Scalar run(const Scalar& x, const Scalar& y)
+ {
+ return Scalar(random(real(x), real(y)),
+ random(imag(x), imag(y)));
+ }
+ static inline Scalar run()
+ {
+ typedef typename NumTraits<Scalar>::Real RealScalar;
+ return Scalar(random<RealScalar>(), random<RealScalar>());
+ }
+};
+
+template<typename Scalar>
+inline EIGEN_MATHFUNC_RETVAL(random, Scalar) random(const Scalar& x, const Scalar& y)
+{
+ return EIGEN_MATHFUNC_IMPL(random, Scalar)::run(x, y);
+}
+
+template<typename Scalar>
+inline EIGEN_MATHFUNC_RETVAL(random, Scalar) random()
+{
+ return EIGEN_MATHFUNC_IMPL(random, Scalar)::run();
+}
+
+} // end namespace internal
+
+/****************************************************************************
+* Generic math function *
+****************************************************************************/
+
+namespace numext {
+
+template<typename Scalar>
+inline EIGEN_MATHFUNC_RETVAL(real, Scalar) real(const Scalar& x)
+{
+ return EIGEN_MATHFUNC_IMPL(real, Scalar)::run(x);
+}
+
+template<typename Scalar>
+inline typename internal::add_const_on_value_type< EIGEN_MATHFUNC_RETVAL(real_ref, Scalar) >::type real_ref(const Scalar& x)
+{
+ return internal::real_ref_impl<Scalar>::run(x);
+}
+
+template<typename Scalar>
+inline EIGEN_MATHFUNC_RETVAL(real_ref, Scalar) real_ref(Scalar& x)
+{
+ return EIGEN_MATHFUNC_IMPL(real_ref, Scalar)::run(x);
+}
+
+template<typename Scalar>
+inline EIGEN_MATHFUNC_RETVAL(imag, Scalar) imag(const Scalar& x)
+{
+ return EIGEN_MATHFUNC_IMPL(imag, Scalar)::run(x);
+}
+
+template<typename Scalar>
+inline typename internal::add_const_on_value_type< EIGEN_MATHFUNC_RETVAL(imag_ref, Scalar) >::type imag_ref(const Scalar& x)
+{
+ return internal::imag_ref_impl<Scalar>::run(x);
+}
+
+template<typename Scalar>
+inline EIGEN_MATHFUNC_RETVAL(imag_ref, Scalar) imag_ref(Scalar& x)
+{
+ return EIGEN_MATHFUNC_IMPL(imag_ref, Scalar)::run(x);
+}
+
+template<typename Scalar>
+inline EIGEN_MATHFUNC_RETVAL(conj, Scalar) conj(const Scalar& x)
+{
+ return EIGEN_MATHFUNC_IMPL(conj, Scalar)::run(x);
+}
+
+template<typename Scalar>
+inline EIGEN_MATHFUNC_RETVAL(abs2, Scalar) abs2(const Scalar& x)
+{
+ return EIGEN_MATHFUNC_IMPL(abs2, Scalar)::run(x);
+}
+
+template<typename Scalar>
+inline EIGEN_MATHFUNC_RETVAL(norm1, Scalar) norm1(const Scalar& x)
+{
+ return EIGEN_MATHFUNC_IMPL(norm1, Scalar)::run(x);
+}
+
+template<typename Scalar>
+inline EIGEN_MATHFUNC_RETVAL(hypot, Scalar) hypot(const Scalar& x, const Scalar& y)
+{
+ return EIGEN_MATHFUNC_IMPL(hypot, Scalar)::run(x, y);
+}
+
+template<typename Scalar>
+inline EIGEN_MATHFUNC_RETVAL(atanh2, Scalar) atanh2(const Scalar& x, const Scalar& y)
+{
+ return EIGEN_MATHFUNC_IMPL(atanh2, Scalar)::run(x, y);
+}
+
+template<typename Scalar>
+inline EIGEN_MATHFUNC_RETVAL(pow, Scalar) pow(const Scalar& x, const Scalar& y)
+{
+ return EIGEN_MATHFUNC_IMPL(pow, Scalar)::run(x, y);
+}
+
+// std::isfinite is non standard, so let's define our own version,
+// even though it is not very efficient.
+template<typename T> bool (isfinite)(const T& x)
+{
+ return x<NumTraits<T>::highest() && x>NumTraits<T>::lowest();
+}
+
+} // end namespace numext
+
+namespace internal {
+
+/****************************************************************************
+* Implementation of fuzzy comparisons *
+****************************************************************************/
+
+template<typename Scalar,
+ bool IsComplex,
+ bool IsInteger>
+struct scalar_fuzzy_default_impl {};
+
+template<typename Scalar>
+struct scalar_fuzzy_default_impl<Scalar, false, false>
+{
+ typedef typename NumTraits<Scalar>::Real RealScalar;
+ template<typename OtherScalar>
+ static inline bool isMuchSmallerThan(const Scalar& x, const OtherScalar& y, const RealScalar& prec)
+ {
+ using std::abs;
+ return abs(x) <= abs(y) * prec;
+ }
+ static inline bool isApprox(const Scalar& x, const Scalar& y, const RealScalar& prec)
+ {
+ using std::min;
+ using std::abs;
+ return abs(x - y) <= (min)(abs(x), abs(y)) * prec;
+ }
+ static inline bool isApproxOrLessThan(const Scalar& x, const Scalar& y, const RealScalar& prec)
+ {
+ return x <= y || isApprox(x, y, prec);
+ }
+};
+
+template<typename Scalar>
+struct scalar_fuzzy_default_impl<Scalar, false, true>
+{
+ typedef typename NumTraits<Scalar>::Real RealScalar;
+ template<typename OtherScalar>
+ static inline bool isMuchSmallerThan(const Scalar& x, const Scalar&, const RealScalar&)
+ {
+ return x == Scalar(0);
+ }
+ static inline bool isApprox(const Scalar& x, const Scalar& y, const RealScalar&)
+ {
+ return x == y;
+ }
+ static inline bool isApproxOrLessThan(const Scalar& x, const Scalar& y, const RealScalar&)
+ {
+ return x <= y;
+ }
+};
+
+template<typename Scalar>
+struct scalar_fuzzy_default_impl<Scalar, true, false>
+{
+ typedef typename NumTraits<Scalar>::Real RealScalar;
+ template<typename OtherScalar>
+ static inline bool isMuchSmallerThan(const Scalar& x, const OtherScalar& y, const RealScalar& prec)
+ {
+ return numext::abs2(x) <= numext::abs2(y) * prec * prec;
+ }
+ static inline bool isApprox(const Scalar& x, const Scalar& y, const RealScalar& prec)
+ {
+ using std::min;
+ return numext::abs2(x - y) <= (min)(numext::abs2(x), numext::abs2(y)) * prec * prec;
+ }
+};
+
+template<typename Scalar>
+struct scalar_fuzzy_impl : scalar_fuzzy_default_impl<Scalar, NumTraits<Scalar>::IsComplex, NumTraits<Scalar>::IsInteger> {};
+
+template<typename Scalar, typename OtherScalar>
+inline bool isMuchSmallerThan(const Scalar& x, const OtherScalar& y,
+ const typename NumTraits<Scalar>::Real &precision = NumTraits<Scalar>::dummy_precision())
+{
+ return scalar_fuzzy_impl<Scalar>::template isMuchSmallerThan<OtherScalar>(x, y, precision);
+}
+
+template<typename Scalar>
+inline bool isApprox(const Scalar& x, const Scalar& y,
+ const typename NumTraits<Scalar>::Real &precision = NumTraits<Scalar>::dummy_precision())
+{
+ return scalar_fuzzy_impl<Scalar>::isApprox(x, y, precision);
+}
+
+template<typename Scalar>
+inline bool isApproxOrLessThan(const Scalar& x, const Scalar& y,
+ const typename NumTraits<Scalar>::Real &precision = NumTraits<Scalar>::dummy_precision())
+{
+ return scalar_fuzzy_impl<Scalar>::isApproxOrLessThan(x, y, precision);
+}
+
+/******************************************
+*** The special case of the bool type ***
+******************************************/
+
+template<> struct random_impl<bool>
+{
+ static inline bool run()
+ {
+ return random<int>(0,1)==0 ? false : true;
+ }
+};
+
+template<> struct scalar_fuzzy_impl<bool>
+{
+ typedef bool RealScalar;
+
+ template<typename OtherScalar>
+ static inline bool isMuchSmallerThan(const bool& x, const bool&, const bool&)
+ {
+ return !x;
+ }
+
+ static inline bool isApprox(bool x, bool y, bool)
+ {
+ return x == y;
+ }
+
+ static inline bool isApproxOrLessThan(const bool& x, const bool& y, const bool&)
+ {
+ return (!x) || y;
+ }
+
+};
+
+
+} // end namespace internal
+
+} // end namespace Eigen
+
+#endif // EIGEN_MATHFUNCTIONS_H
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