Diff: xpr/VectorFunctions.h

Revision:

0:feb4117d16d8

--- /dev/null	Thu Jan 01 00:00:00 1970 +0000
+++ b/xpr/VectorFunctions.h	Wed Mar 28 15:53:45 2012 +0000
@@ -0,0 +1,684 @@
+/*
+ * Tiny Vector Matrix Library
+ * Dense Vector Matrix Libary of Tiny size using Expression Templates
+ *
+ * Copyright (C) 2001 - 2007 Olaf Petzold <opetzold@users.sourceforge.net>
+ *
+ * This library is free software; you can redistribute it and/or
+ * modify it under the terms of the GNU Lesser General Public
+ * License as published by the Free Software Foundation; either
+ * version 2.1 of the License, or (at your option) any later version.
+ *
+ * This library is distributed in the hope that it will be useful,
+ * but WITHOUT ANY WARRANTY; without even the implied warranty of
+ * MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE.  See the GNU
+ * Lesser General Public License for more details.
+ *
+ * You should have received a copy of the GNU Lesser General Public
+ * License along with this library; if not, write to the Free Software
+ * Foundation, Inc., 59 Temple Place, Suite 330, Boston, MA  02111-1307  USA
+ *
+ * $Id: VectorFunctions.h,v 1.21 2007-06-23 15:59:00 opetzold Exp $
+ */
+
+#ifndef TVMET_XPR_VECTOR_FUNCTIONS_H
+#define TVMET_XPR_VECTOR_FUNCTIONS_H
+
+namespace tvmet {
+
+
+/* forwards */
+template<class T, std::size_t Sz> class Vector;
+
+
+/*********************************************************
+ * PART I: DECLARATION
+ *********************************************************/
+
+
+/*++++++++++++++++++++++++++++++++++++++++++++++++++++++++
+ * Vector arithmetic functions add, sub, mul and div
+ *+++++++++++++++++++++++++++++++++++++++++++++++++++++++*/
+
+
+/*
+ * function(XprVector<E1, Sz>, XprVector<E2, Sz>)
+ */
+#define TVMET_DECLARE_MACRO(NAME)					\
+template<class E1, class E2, std::size_t Sz>				\
+XprVector<								\
+  XprBinOp<								\
+    Fcnl_##NAME<typename E1::value_type, typename E2::value_type>,	\
+    XprVector<E1, Sz>,							\
+    XprVector<E2, Sz>							\
+  >,									\
+  Sz									\
+>									\
+NAME (const XprVector<E1, Sz>& lhs,					\
+      const XprVector<E2, Sz>& rhs) TVMET_CXX_ALWAYS_INLINE;
+
+TVMET_DECLARE_MACRO(add)		// per se element wise
+TVMET_DECLARE_MACRO(sub)		// per se element wise
+TVMET_DECLARE_MACRO(mul)		// per se element wise
+namespace element_wise {
+  TVMET_DECLARE_MACRO(div)		// not defined for vectors
+}
+
+#undef TVMET_DECLARE_MACRO
+
+
+/*
+ * function(XprVector<E, Sz>, POD)
+ * function(POD, XprVector<E, Sz>)
+ * Note: - operations +,-,*,/ are per se element wise
+ */
+#define TVMET_DECLARE_MACRO(NAME, POD)				\
+template<class E, std::size_t Sz>				\
+XprVector<							\
+  XprBinOp<							\
+    Fcnl_##NAME< typename E::value_type, POD >,			\
+    XprVector<E, Sz>,						\
+    XprLiteral< POD >						\
+  >,								\
+  Sz								\
+>								\
+NAME (const XprVector<E, Sz>& lhs, 				\
+      POD rhs) TVMET_CXX_ALWAYS_INLINE;				\
+								\
+template<class E, std::size_t Sz>				\
+XprVector<							\
+  XprBinOp<							\
+    Fcnl_##NAME< POD, typename E::value_type>,			\
+    XprLiteral< POD >,						\
+    XprVector<E, Sz>						\
+  >,								\
+  Sz								\
+>								\
+NAME (POD lhs, 							\
+      const XprVector<E, Sz>& rhs) TVMET_CXX_ALWAYS_INLINE;
+
+TVMET_DECLARE_MACRO(add, int)
+TVMET_DECLARE_MACRO(sub, int)
+TVMET_DECLARE_MACRO(mul, int)
+TVMET_DECLARE_MACRO(div, int)
+
+#if defined(TVMET_HAVE_LONG_LONG)
+TVMET_DECLARE_MACRO(add, long long int)
+TVMET_DECLARE_MACRO(sub, long long int)
+TVMET_DECLARE_MACRO(mul, long long int)
+TVMET_DECLARE_MACRO(div, long long int)
+#endif
+
+TVMET_DECLARE_MACRO(add, float)
+TVMET_DECLARE_MACRO(sub, float)
+TVMET_DECLARE_MACRO(mul, float)
+TVMET_DECLARE_MACRO(div, float)
+
+TVMET_DECLARE_MACRO(add, double)
+TVMET_DECLARE_MACRO(sub, double)
+TVMET_DECLARE_MACRO(mul, double)
+TVMET_DECLARE_MACRO(div, double)
+
+#if defined(TVMET_HAVE_LONG_DOUBLE)
+TVMET_DECLARE_MACRO(add, long double)
+TVMET_DECLARE_MACRO(sub, long double)
+TVMET_DECLARE_MACRO(mul, long double)
+TVMET_DECLARE_MACRO(div, long double)
+#endif
+
+#undef TVMET_DECLARE_MACRO
+
+
+#if defined(TVMET_HAVE_COMPLEX)
+/*
+ * function(XprMatrix<E, Rows, Cols>, complex<T>)
+ * function(complex<T>, XprMatrix<E, Rows, Cols>)
+ * Note: - operations +,-,*,/ are per se element wise
+ * \todo type promotion
+ */
+#define TVMET_DECLARE_MACRO(NAME)				\
+template<class E, std::size_t Sz, class T>			\
+XprVector<							\
+  XprBinOp<							\
+    Fcnl_##NAME< typename E::value_type, std::complex<T> >,	\
+    XprVector<E, Sz>,						\
+    XprLiteral< std::complex<T> >				\
+  >,								\
+  Sz								\
+>								\
+NAME (const XprVector<E, Sz>& lhs,				\
+      const std::complex<T>& rhs) TVMET_CXX_ALWAYS_INLINE;	\
+								\
+template<class E, std::size_t Sz, class T>			\
+XprVector<							\
+  XprBinOp<							\
+    Fcnl_##NAME< std::complex<T>, typename E::value_type>,	\
+    XprLiteral< std::complex<T> >,				\
+    XprVector<E, Sz>						\
+  >,								\
+  Sz								\
+>								\
+NAME (const std::complex<T>& lhs, 				\
+      const XprVector<E, Sz>& rhs) TVMET_CXX_ALWAYS_INLINE;
+
+TVMET_DECLARE_MACRO(add)
+TVMET_DECLARE_MACRO(sub)
+TVMET_DECLARE_MACRO(mul)
+TVMET_DECLARE_MACRO(div)
+
+#undef TVMET_DECLARE_MACRO
+
+#endif // defined(TVMET_HAVE_COMPLEX)
+
+
+/*++++++++++++++++++++++++++++++++++++++++++++++++++++++++
+ * vector specific functions
+ *+++++++++++++++++++++++++++++++++++++++++++++++++++++++*/
+
+
+template<class E, std::size_t Sz>
+typename NumericTraits<typename E::value_type>::sum_type
+sum(const XprVector<E, Sz>& v) TVMET_CXX_ALWAYS_INLINE;
+
+
+template<class E, std::size_t Sz>
+typename NumericTraits<typename E::value_type>::sum_type
+product(const XprVector<E, Sz>& v) TVMET_CXX_ALWAYS_INLINE;
+
+
+template<class E1, class E2, std::size_t Sz>
+typename PromoteTraits<
+  typename E1::value_type,
+  typename E2::value_type
+>::value_type
+dot(const XprVector<E1, Sz>& lhs,
+    const XprVector<E2, Sz>& rhs) TVMET_CXX_ALWAYS_INLINE;
+
+
+template<class T, class E, std::size_t Sz>
+typename PromoteTraits<T, typename E::value_type>::value_type
+dot(const Vector<T, Sz>& lhs,
+    const XprVector<E, Sz>& rhs) TVMET_CXX_ALWAYS_INLINE;
+
+
+template<class E, class T, std::size_t Sz>
+typename PromoteTraits<T, typename E::value_type>::value_type
+dot(const XprVector<E, Sz>& lhs,
+    const Vector<T, Sz>& rhs) TVMET_CXX_ALWAYS_INLINE;
+
+
+template<class E1, class E2>
+Vector<
+  typename PromoteTraits<
+    typename E1::value_type,
+    typename E2::value_type
+  >::value_type,
+  3
+>
+cross(const XprVector<E1, 3>& lhs,
+      const XprVector<E2, 3>& rhs) TVMET_CXX_ALWAYS_INLINE;
+
+
+template<class T, class E>
+Vector<
+  typename PromoteTraits<T, typename E::value_type>::value_type, 3>
+cross(const Vector<T, 3>& lhs,
+      const XprVector<E, 3>& rhs) TVMET_CXX_ALWAYS_INLINE;
+
+
+template<class E, class T>
+Vector<
+  typename PromoteTraits<T, typename E::value_type>::value_type, 3>
+cross(const XprVector<E, 3>& lhs,
+      const Vector<T, 3>& rhs) TVMET_CXX_ALWAYS_INLINE;
+
+
+template<class E, std::size_t Sz>
+typename NumericTraits<typename E::value_type>::sum_type
+norm1(const XprVector<E, Sz>& v) TVMET_CXX_ALWAYS_INLINE;
+
+
+template<class E, std::size_t Sz>
+typename NumericTraits<typename E::value_type>::sum_type
+norm2(const XprVector<E, Sz>& v) TVMET_CXX_ALWAYS_INLINE;
+
+
+template<class E, std::size_t Sz>
+XprVector<
+  XprBinOp<
+    Fcnl_div<typename E::value_type, typename E::value_type>,
+    XprVector<E, Sz>,
+    XprLiteral<typename E::value_type>
+  >,
+  Sz
+>
+normalize(const XprVector<E, Sz>& v) TVMET_CXX_ALWAYS_INLINE;
+
+
+/*********************************************************
+ * PART II: IMPLEMENTATION
+ *********************************************************/
+
+
+/*
+ * function(XprVector<E1, Sz>, XprVector<E2, Sz>)
+ */
+#define TVMET_IMPLEMENT_MACRO(NAME)					\
+template<class E1, class E2, std::size_t Sz>				\
+inline									\
+XprVector<								\
+  XprBinOp<								\
+    Fcnl_##NAME<typename E1::value_type, typename E2::value_type>,	\
+    XprVector<E1, Sz>,							\
+    XprVector<E2, Sz>							\
+  >,									\
+  Sz									\
+>									\
+NAME (const XprVector<E1, Sz>& lhs, const XprVector<E2, Sz>& rhs) {	\
+  typedef XprBinOp<							\
+    Fcnl_##NAME<typename E1::value_type, typename E2::value_type>,	\
+    XprVector<E1, Sz>,							\
+    XprVector<E2, Sz>							\
+  > 							 expr_type;	\
+  return XprVector<expr_type, Sz>(expr_type(lhs, rhs));			\
+}
+
+TVMET_IMPLEMENT_MACRO(add)		// per se element wise
+TVMET_IMPLEMENT_MACRO(sub)		// per se element wise
+TVMET_IMPLEMENT_MACRO(mul)		// per se element wise
+namespace element_wise {
+  TVMET_IMPLEMENT_MACRO(div)		// not defined for vectors
+}
+
+#undef TVMET_IMPLEMENT_MACRO
+
+
+/*
+ * function(XprVector<E, Sz>, POD)
+ * function(POD, XprVector<E, Sz>)
+ * Note: - operations +,-,*,/ are per se element wise
+ */
+#define TVMET_IMPLEMENT_MACRO(NAME, POD)				\
+template<class E, std::size_t Sz>					\
+inline									\
+XprVector<								\
+  XprBinOp<								\
+    Fcnl_##NAME< typename E::value_type, POD >,				\
+    XprVector<E, Sz>,							\
+    XprLiteral< POD >							\
+  >,									\
+  Sz									\
+>									\
+NAME (const XprVector<E, Sz>& lhs, POD rhs) {				\
+  typedef XprBinOp<							\
+    Fcnl_##NAME< typename E::value_type, POD >,				\
+    XprVector<E, Sz>,							\
+    XprLiteral< POD >							\
+  >							expr_type;	\
+  return XprVector<expr_type, Sz>(					\
+    expr_type(lhs, XprLiteral< POD >(rhs)));				\
+}									\
+									\
+template<class E, std::size_t Sz>					\
+inline									\
+XprVector<								\
+  XprBinOp<								\
+    Fcnl_##NAME< POD, typename E::value_type>,				\
+    XprLiteral< POD >,							\
+    XprVector<E, Sz>							\
+  >,									\
+  Sz									\
+>									\
+NAME (POD lhs, const XprVector<E, Sz>& rhs) {				\
+  typedef XprBinOp<							\
+    Fcnl_##NAME< POD, typename E::value_type>,				\
+    XprLiteral< POD >,							\
+    XprVector<E, Sz>							\
+  >							expr_type;	\
+  return XprVector<expr_type, Sz>(					\
+    expr_type(XprLiteral< POD >(lhs), rhs));				\
+}
+
+TVMET_IMPLEMENT_MACRO(add, int)
+TVMET_IMPLEMENT_MACRO(sub, int)
+TVMET_IMPLEMENT_MACRO(mul, int)
+TVMET_IMPLEMENT_MACRO(div, int)
+
+#if defined(TVMET_HAVE_LONG_LONG)
+TVMET_IMPLEMENT_MACRO(add, long long int)
+TVMET_IMPLEMENT_MACRO(sub, long long int)
+TVMET_IMPLEMENT_MACRO(mul, long long int)
+TVMET_IMPLEMENT_MACRO(div, long long int)
+#endif
+
+TVMET_IMPLEMENT_MACRO(add, float)
+TVMET_IMPLEMENT_MACRO(sub, float)
+TVMET_IMPLEMENT_MACRO(mul, float)
+TVMET_IMPLEMENT_MACRO(div, float)
+
+TVMET_IMPLEMENT_MACRO(add, double)
+TVMET_IMPLEMENT_MACRO(sub, double)
+TVMET_IMPLEMENT_MACRO(mul, double)
+TVMET_IMPLEMENT_MACRO(div, double)
+
+#if defined(TVMET_HAVE_LONG_DOUBLE)
+TVMET_IMPLEMENT_MACRO(add, long double)
+TVMET_IMPLEMENT_MACRO(sub, long double)
+TVMET_IMPLEMENT_MACRO(mul, long double)
+TVMET_IMPLEMENT_MACRO(div, long double)
+#endif
+
+#undef TVMET_IMPLEMENT_MACRO
+
+
+#if defined(TVMET_HAVE_COMPLEX)
+/*
+ * function(XprMatrix<E, Rows, Cols>, complex<T>)
+ * function(complex<T>, XprMatrix<E, Rows, Cols>)
+ * Note: - operations +,-,*,/ are per se element wise
+ * \todo type promotion
+ */
+#define TVMET_IMPLEMENT_MACRO(NAME)				   \
+template<class E, std::size_t Sz, class T>				\
+inline									\
+XprVector<								\
+  XprBinOp<								\
+    Fcnl_##NAME< typename E::value_type, std::complex<T> >,		\
+    XprVector<E, Sz>,							\
+    XprLiteral< std::complex<T> >					\
+  >,									\
+  Sz									\
+>									\
+NAME (const XprVector<E, Sz>& lhs, const std::complex<T>& rhs) {	\
+  typedef XprBinOp<							\
+    Fcnl_##NAME< typename E::value_type, std::complex<T> >,		\
+    XprVector<E, Sz>,							\
+    XprLiteral< std::complex<T> >					\
+  >							expr_type;	\
+  return XprVector<expr_type, Sz>(					\
+    expr_type(lhs, XprLiteral< std::complex<T> >(rhs)));		\
+}									\
+									\
+template<class E, std::size_t Sz, class T>				\
+inline									\
+XprVector<								\
+  XprBinOp<								\
+    Fcnl_##NAME< std::complex<T>, typename E::value_type>,		\
+    XprLiteral< std::complex<T> >,					\
+    XprVector<E, Sz>							\
+  >,									\
+  Sz									\
+>									\
+NAME (const std::complex<T>& lhs, const XprVector<E, Sz>& rhs) {	\
+  typedef XprBinOp<							\
+    Fcnl_##NAME< std::complex<T>, typename E::value_type>,		\
+    XprLiteral< std::complex<T> >,					\
+    XprVector<E, Sz>							\
+  >							expr_type;	\
+  return XprVector<expr_type, Sz>(					\
+    expr_type(XprLiteral< std::complex<T> >(lhs), rhs));		\
+}
+
+TVMET_IMPLEMENT_MACRO(add)
+TVMET_IMPLEMENT_MACRO(sub)
+TVMET_IMPLEMENT_MACRO(mul)
+TVMET_IMPLEMENT_MACRO(div)
+
+#undef TVMET_IMPLEMENT_MACRO
+
+#endif // defined(TVMET_HAVE_COMPLEX)
+
+
+/*++++++++++++++++++++++++++++++++++++++++++++++++++++++++
+ * vector specific functions
+ *+++++++++++++++++++++++++++++++++++++++++++++++++++++++*/
+
+
+/**
+ * \fn sum(const XprVector<E, Sz>& v)
+ * \brief Compute the sum of the vector expression.
+ * \ingroup _unary_function
+ *
+ * Simply compute the sum of the given vector as:
+ * \f[
+ * \sum_{i = 0}^{Sz-1} v[i]
+ * \f]
+ */
+template<class E, std::size_t Sz>
+inline
+typename NumericTraits<typename E::value_type>::sum_type
+sum(const XprVector<E, Sz>& v) {
+  return meta::Vector<Sz>::sum(v);
+}
+
+
+/**
+ * \fn product(const XprVector<E, Sz>& v)
+ * \brief Compute the product of the vector elements.
+ * \ingroup _unary_function
+ *
+ * Simply computer the product of the given vector expression as:
+ * \f[
+ * \prod_{i = 0}^{Sz - 1} v[i]
+ * \f]
+ */
+template<class E, std::size_t Sz>
+inline
+typename NumericTraits<typename E::value_type>::sum_type
+product(const XprVector<E, Sz>& v) {
+  return meta::Vector<Sz>::product(v);
+}
+
+
+/**
+ * \fn dot(const XprVector<E1, Sz>& lhs, const XprVector<E2, Sz>& rhs)
+ * \brief Compute the dot/inner product
+ * \ingroup _binary_function
+ *
+ * Compute the dot product as:
+ * \f[
+ * \sum_{i = 0}^{Sz - 1} ( lhs[i] * rhs[i] )
+ * \f]
+ * where lhs is a column vector and rhs is a row vector, both vectors
+ * have the same dimension.
+ */
+template<class E1, class E2, std::size_t Sz>
+inline
+typename PromoteTraits<
+  typename E1::value_type,
+  typename E2::value_type
+>::value_type
+dot(const XprVector<E1, Sz>& lhs, const XprVector<E2, Sz>& rhs) {
+  return meta::Vector<Sz>::dot(lhs, rhs);
+}
+
+
+/**
+ * \fn dot(const Vector<T, Sz>& lhs, const XprVector<E, Sz>& rhs)
+ * \brief Compute the dot/inner product
+ * \ingroup _binary_function
+ *
+ * Compute the dot product as:
+ * \f[
+ * \sum_{i = 0}^{Sz - 1} ( lhs[i] * rhs[i] )
+ * \f]
+ * where lhs is a column vector and rhs is a row vector, both vectors
+ * have the same dimension.
+ */
+template<class T, class E, std::size_t Sz>
+inline
+typename PromoteTraits<T, typename E::value_type>::value_type
+dot(const Vector<T, Sz>& lhs, const XprVector<E, Sz>& rhs) {
+  return meta::Vector<Sz>::dot(lhs, rhs);
+}
+
+
+/**
+ * \fn dot(const XprVector<E, Sz>& lhs, const Vector<T, Sz>& rhs)
+ * \brief Compute the dot/inner product
+ * \ingroup _binary_function
+ *
+ * Compute the dot product as:
+ * \f[
+ * \sum_{i = 0}^{Sz - 1} ( lhs[i] * rhs[i] )
+ * \f]
+ * where lhs is a column vector and rhs is a row vector, both vectors
+ * have the same dimension.
+ */
+template<class E, class T, std::size_t Sz>
+inline
+typename PromoteTraits<T, typename E::value_type>::value_type
+dot(const XprVector<E, Sz>& lhs, const Vector<T, Sz>& rhs) {
+  return meta::Vector<Sz>::dot(lhs, rhs);
+}
+
+
+/**
+ * \fn cross(const XprVector<E1, 3>& lhs, const XprVector<E2, 3>& rhs)
+ * \brief Compute the cross/outer product
+ * \ingroup _binary_function
+ * \note working only for vectors of size = 3
+ * \todo Implement vector outer product as ET and MT, returning a XprVector
+ */
+template<class E1, class E2>
+inline
+Vector<
+  typename PromoteTraits<
+    typename E1::value_type,
+    typename E2::value_type
+  >::value_type,
+  3
+>
+cross(const XprVector<E1, 3>& lhs, const XprVector<E2, 3>& rhs) {
+  typedef typename PromoteTraits<
+    typename E1::value_type,
+    typename E2::value_type
+  >::value_type						value_type;
+  return Vector<value_type, 3>(lhs(1)*rhs(2) - rhs(1)*lhs(2),
+			       rhs(0)*lhs(2) - lhs(0)*rhs(2),
+			       lhs(0)*rhs(1) - rhs(0)*lhs(1));
+}
+
+
+/**
+ * \fn cross(const XprVector<E, 3>& lhs, const Vector<T, 3>& rhs)
+ * \brief Compute the cross/outer product
+ * \ingroup _binary_function
+ * \note working only for vectors of size = 3
+ * \todo Implement vector outer product as ET and MT, returning a XprVector
+ */
+template<class E, class T>
+inline
+Vector<
+  typename PromoteTraits<T, typename E::value_type>::value_type, 3>
+cross(const XprVector<E, 3>& lhs, const Vector<T, 3>& rhs) {
+  typedef typename PromoteTraits<
+    typename E::value_type, T>::value_type 		value_type;
+  return Vector<value_type, 3>(lhs(1)*rhs(2) - rhs(1)*lhs(2),
+			       rhs(0)*lhs(2) - lhs(0)*rhs(2),
+			       lhs(0)*rhs(1) - rhs(0)*lhs(1));
+}
+
+
+/**
+ * \fn cross(const Vector<T, 3>& lhs, const XprVector<E, 3>& rhs)
+ * \brief Compute the cross/outer product
+ * \ingroup _binary_function
+ * \note working only for vectors of size = 3
+ * \todo Implement vector outer product as ET and MT, returning a XprVector
+ */
+template<class T1, class E2>
+inline
+Vector<
+  typename PromoteTraits<T1, typename E2::value_type>::value_type, 3>
+cross(const Vector<T1, 3>& lhs, const XprVector<E2, 3>& rhs) {
+  typedef typename PromoteTraits<
+    typename E2::value_type, T1>::value_type 		value_type;
+  return Vector<value_type, 3>(lhs(1)*rhs(2) - rhs(1)*lhs(2),
+			       rhs(0)*lhs(2) - lhs(0)*rhs(2),
+			       lhs(0)*rhs(1) - rhs(0)*lhs(1));
+}
+
+
+/**
+ * \fn norm1(const XprVector<E, Sz>& v)
+ * \brief The \f$l_1\f$ norm of a vector expression.
+ * \ingroup _unary_function
+ * The norm of any vector is just the square root of the dot product of
+ * a vector with itself, or
+ *
+ * \f[
+ * |Vector<T, Sz> v| = |v| = \sum_{i=0}^{Sz-1}\,|v[i]|
+ * \f]
+ */
+template<class E, std::size_t Sz>
+inline
+typename NumericTraits<typename E::value_type>::sum_type
+norm1(const XprVector<E, Sz>& v) {
+  return sum(abs(v));
+}
+
+
+/**
+ * \fn norm2(const XprVector<E, Sz>& v)
+ * \brief The euklidian norm (or \f$l_2\f$ norm) of a vector expression.
+ * \ingroup _unary_function
+ * The norm of any vector is just the square root of the dot product of
+ * a vector with itself, or
+ *
+ * \f[
+ * |Vector<T, Sz> v| = |v| = \sqrt{ \sum_{i=0}^{Sz-1}\,v[i]^2 }
+ * \f]
+ *
+ * \note The internal cast for Vector<int> avoids warnings on sqrt.
+ */
+template<class E, std::size_t Sz>
+inline
+typename NumericTraits<typename E::value_type>::sum_type
+norm2(const XprVector<E, Sz>& v) {
+  typedef typename E::value_type			value_type;
+  return static_cast<value_type>( std::sqrt(static_cast<value_type>(dot(v, v))) );
+}
+
+
+/**
+ * \fn normalize(const XprVector<E, Sz>& v)
+ * \brief Normalize the given vector expression.
+ * \ingroup _unary_function
+ * \sa norm2
+ *
+ * using the equation:
+ * \f[
+ * \frac{Vector<T, Sz> v}{\sqrt{ \sum_{i=0}^{Sz-1}\,v[i]^2 }}
+ * \f]
+ */
+template<class E, std::size_t Sz>
+inline
+XprVector<
+  XprBinOp<
+    Fcnl_div<typename E::value_type, typename E::value_type>,
+    XprVector<E, Sz>,
+    XprLiteral<typename E::value_type>
+  >,
+  Sz
+>
+normalize(const XprVector<E, Sz>& v) {
+  typedef typename E::value_type			value_type;
+  typedef XprBinOp<
+    Fcnl_div<value_type, value_type>,
+    XprVector<E, Sz>,
+    XprLiteral<value_type>
+  >							expr_type;
+  return XprVector<expr_type, Sz>(
+    expr_type(v, XprLiteral< value_type >(norm2(v))));
+}
+
+
+} // namespace tvmet
+
+#endif // TVMET_XPR_VECTOR_FUNCTIONS_H
+
+// Local Variables:
+// mode:C++
+// tab-width:8
+// End: