Diff: tvmet/VectorFunctions.h

Revision:

25:143b19c1fb05

--- /dev/null	Thu Jan 01 00:00:00 1970 +0000
+++ b/tvmet/VectorFunctions.h	Wed Oct 17 22:22:28 2012 +0000
@@ -0,0 +1,883 @@
+/*
+ * 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 1.37 2007-06-23 15:58:58 opetzold Exp $
+ */
+
+#ifndef TVMET_VECTOR_FUNCTIONS_H
+#define TVMET_VECTOR_FUNCTIONS_H
+
+#include <tvmet/Extremum.h>
+
+namespace tvmet {
+
+
+/*********************************************************
+ * PART I: DECLARATION
+ *********************************************************/
+
+
+/*++++++++++++++++++++++++++++++++++++++++++++++++++++++++
+ * Vector arithmetic functions add, sub, mul and div
+ *+++++++++++++++++++++++++++++++++++++++++++++++++++++++*/
+
+
+/*
+ * function(Vector<T1, Sz>, Vector<T2, Sz>)
+ * function(Vector<T, Sz>, XprVector<E, Sz>)
+ * function(XprVector<E, Sz>, Vector<T, Sz>)
+ */
+#define TVMET_DECLARE_MACRO(NAME)				\
+template<class T1, class T2, std::size_t Sz>			\
+XprVector<							\
+  XprBinOp<							\
+    Fcnl_##NAME<T1, T2>,					\
+    VectorConstReference<T1, Sz>,				\
+    VectorConstReference<T2, Sz>				\
+  >,								\
+  Sz								\
+>								\
+NAME (const Vector<T1, Sz>& lhs,				\
+      const Vector<T2, Sz>& rhs) TVMET_CXX_ALWAYS_INLINE;	\
+								\
+template<class E, class T, std::size_t Sz>			\
+XprVector<							\
+  XprBinOp<							\
+    Fcnl_##NAME<typename E::value_type, T>,			\
+    XprVector<E, Sz>,						\
+    VectorConstReference<T, Sz>					\
+  >,								\
+  Sz								\
+>								\
+NAME (const XprVector<E, Sz>& lhs,				\
+      const Vector<T, Sz>& rhs) TVMET_CXX_ALWAYS_INLINE;	\
+								\
+template<class E, class T, std::size_t Sz>			\
+XprVector<							\
+  XprBinOp<							\
+    Fcnl_##NAME<T, typename E::value_type>,			\
+    VectorConstReference<T, Sz>,				\
+    XprVector<E, Sz>						\
+  >,								\
+  Sz								\
+>								\
+NAME (const Vector<T, Sz>& lhs,					\
+      const XprVector<E, 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(Vector<T, Sz>, POD)
+ * function(POD, Vector<T, Sz>)
+ * Note: - operations +,-,*,/ are per se element wise
+ */
+#define TVMET_DECLARE_MACRO(NAME, POD)				\
+template<class T, std::size_t Sz>				\
+XprVector<							\
+  XprBinOp<							\
+    Fcnl_##NAME< T, POD >,					\
+    VectorConstReference<T, Sz>,				\
+    XprLiteral< POD >						\
+  >,								\
+  Sz								\
+>								\
+NAME (const Vector<T, Sz>& lhs, 				\
+      POD rhs) TVMET_CXX_ALWAYS_INLINE;				\
+								\
+template<class T, std::size_t Sz>				\
+XprVector<							\
+  XprBinOp<							\
+    Fcnl_##NAME< POD, T>,					\
+    XprLiteral< POD >,						\
+    VectorConstReference<T, Sz>					\
+  >,								\
+  Sz								\
+>								\
+NAME (POD lhs, 							\
+      const Vector<T, 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(Vector<std::complex<T>, Sz>, std::complex<T>)
+ * function(std::complex<T>, Vector<std::complex<T>, Sz>)
+ * Note: per se element wise
+ * \todo type promotion
+ */
+#define TVMET_DECLARE_MACRO(NAME)					\
+template<class T, std::size_t Sz>					\
+XprVector<								\
+  XprBinOp<								\
+    Fcnl_##NAME< std::complex<T>, std::complex<T> >,			\
+    VectorConstReference< std::complex<T>, Sz>,				\
+    XprLiteral< std::complex<T> >					\
+  >,									\
+  Sz									\
+>									\
+NAME (const Vector<std::complex<T>, Sz>& lhs,				\
+      const std::complex<T>& rhs) TVMET_CXX_ALWAYS_INLINE;		\
+									\
+template<class T, std::size_t Sz>					\
+XprVector<								\
+  XprBinOp<								\
+    Fcnl_##NAME< std::complex<T>, std::complex<T> >,			\
+    XprLiteral< std::complex<T> >,					\
+    VectorConstReference< std::complex<T>, Sz>				\
+  >,									\
+  Sz									\
+>									\
+NAME (const std::complex<T>& lhs,					\
+      const Vector< std::complex<T>, 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 T, std::size_t Sz>
+typename NumericTraits<T>::sum_type
+sum(const Vector<T, Sz>& v) TVMET_CXX_ALWAYS_INLINE;
+
+
+template<class T, std::size_t Sz>
+typename NumericTraits<T>::sum_type
+product(const Vector<T, Sz>& v) TVMET_CXX_ALWAYS_INLINE;
+
+
+template<class T1, class T2, std::size_t Sz>
+typename PromoteTraits<T1, T2>::value_type
+dot(const Vector<T1, Sz>& lhs,
+    const Vector<T2, Sz>& rhs) TVMET_CXX_ALWAYS_INLINE;
+
+
+template<class T1, class T2>
+Vector<typename PromoteTraits<T1, T2>::value_type, 3>
+cross(const Vector<T1, 3>& lhs,
+      const Vector<T2, 3>& rhs) TVMET_CXX_ALWAYS_INLINE;
+
+
+template<class T, std::size_t Sz>
+typename NumericTraits<T>::sum_type
+norm1(const Vector<T, Sz>& v) TVMET_CXX_ALWAYS_INLINE;
+
+
+template<class T, std::size_t Sz>
+typename NumericTraits<T>::sum_type
+norm2(const Vector<T, Sz>& v) TVMET_CXX_ALWAYS_INLINE;
+
+
+template<class T, std::size_t Sz>
+XprVector<
+  XprBinOp<
+    Fcnl_div<T, T>,
+    VectorConstReference<T, Sz>,
+    XprLiteral< T >
+  >,
+  Sz
+>
+normalize(const Vector<T, Sz>& v) TVMET_CXX_ALWAYS_INLINE;
+
+
+/*++++++++++++++++++++++++++++++++++++++++++++++++++++++++
+ * min/max unary functions
+ *+++++++++++++++++++++++++++++++++++++++++++++++++++++++*/
+
+template<class E, std::size_t Sz>
+Extremum<typename E::value_type, std::size_t, vector_tag>
+maximum(const XprVector<E, Sz>& e); // NOT TVMET_CXX_ALWAYS_INLINE;
+
+
+template<class T, std::size_t Sz>
+Extremum<T, std::size_t, vector_tag>
+maximum(const Vector<T, Sz>& v) TVMET_CXX_ALWAYS_INLINE;
+
+
+template<class E, std::size_t Sz>
+Extremum<typename E::value_type, std::size_t, vector_tag>
+minimum(const XprVector<E, Sz>& e); // NOT TVMET_CXX_ALWAYS_INLINE;
+
+
+template<class T, std::size_t Sz>
+Extremum<T, std::size_t, vector_tag>
+minimum(const Vector<T, Sz>& v) TVMET_CXX_ALWAYS_INLINE;
+
+
+template<class E, std::size_t Sz>
+typename E::value_type
+max(const XprVector<E, Sz>& e); // NOT TVMET_CXX_ALWAYS_INLINE;
+
+
+template<class T, std::size_t Sz>
+T max(const Vector<T, Sz>& v) TVMET_CXX_ALWAYS_INLINE;
+
+
+template<class E, std::size_t Sz>
+typename E::value_type
+min(const XprVector<E, Sz>& e); // NOT TVMET_CXX_ALWAYS_INLINE;
+
+
+template<class T, std::size_t Sz>
+T min(const Vector<T, Sz>& v) TVMET_CXX_ALWAYS_INLINE;
+
+
+template<class T, std::size_t Sz>
+XprVector<
+  VectorConstReference<T, Sz>,
+  Sz
+>
+cvector_ref(const T* mem) TVMET_CXX_ALWAYS_INLINE;
+
+
+/*********************************************************
+ * PART II: IMPLEMENTATION
+ *********************************************************/
+
+
+/*++++++++++++++++++++++++++++++++++++++++++++++++++++++++
+ * Vector arithmetic functions add, sub, mul and div
+ *+++++++++++++++++++++++++++++++++++++++++++++++++++++++*/
+
+
+/*
+ * function(Vector<T1, Sz>, Vector<T2, Sz>)
+ * function(Vector<T, Sz>, XprVector<E, Sz>)
+ * function(XprVector<E, Sz>, Vector<T, Sz>)
+ */
+#define TVMET_IMPLEMENT_MACRO(NAME)					\
+template<class T1, class T2, std::size_t Sz>				\
+inline									\
+XprVector<								\
+  XprBinOp<								\
+    Fcnl_##NAME<T1, T2>,						\
+    VectorConstReference<T1, Sz>,					\
+    VectorConstReference<T2, Sz>					\
+  >,									\
+  Sz									\
+>									\
+NAME (const Vector<T1, Sz>& lhs, const Vector<T2, Sz>& rhs) {		\
+  typedef XprBinOp <							\
+    Fcnl_##NAME<T1, T2>,						\
+    VectorConstReference<T1, Sz>,					\
+    VectorConstReference<T2, Sz>					\
+  >							expr_type;	\
+  return XprVector<expr_type, Sz>(					\
+    expr_type(lhs.const_ref(), rhs.const_ref()));			\
+}									\
+									\
+template<class E, class T, std::size_t Sz>				\
+inline									\
+XprVector<								\
+  XprBinOp<								\
+    Fcnl_##NAME<typename E::value_type, T>,				\
+    XprVector<E, Sz>,							\
+    VectorConstReference<T, Sz>						\
+  >,									\
+  Sz									\
+>									\
+NAME (const XprVector<E, Sz>& lhs, const Vector<T, Sz>& rhs) {		\
+  typedef XprBinOp<							\
+     Fcnl_##NAME<typename E::value_type, T>,				\
+    XprVector<E, Sz>,							\
+    VectorConstReference<T, Sz>						\
+  > 							 expr_type;	\
+  return XprVector<expr_type, Sz>(					\
+    expr_type(lhs, rhs.const_ref()));					\
+}									\
+									\
+template<class E, class T, std::size_t Sz>				\
+inline									\
+XprVector<								\
+  XprBinOp<								\
+    Fcnl_##NAME<T, typename E::value_type>,				\
+    VectorConstReference<T, Sz>,					\
+    XprVector<E, Sz>							\
+  >,									\
+  Sz									\
+>									\
+NAME (const Vector<T, Sz>& lhs, const XprVector<E, Sz>& rhs) {		\
+  typedef XprBinOp<							\
+    Fcnl_##NAME<T, typename E::value_type>,				\
+    VectorConstReference<T, Sz>,					\
+    XprVector<E, Sz>							\
+  > 						 	expr_type;	\
+  return XprVector<expr_type, Sz>(					\
+    expr_type(lhs.const_ref(), 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(Vector<T, Sz>, POD)
+ * function(POD, Vector<T, Sz>)
+ * Note: - operations +,-,*,/ are per se element wise
+ */
+#define TVMET_IMPLEMENT_MACRO(NAME, POD)				\
+template<class T, std::size_t Sz>					\
+inline									\
+XprVector<								\
+  XprBinOp<								\
+    Fcnl_##NAME< T, POD >,						\
+    VectorConstReference<T, Sz>,					\
+    XprLiteral< POD >							\
+  >,									\
+  Sz									\
+>									\
+NAME (const Vector<T, Sz>& lhs, POD rhs) {				\
+  typedef XprBinOp<							\
+    Fcnl_##NAME<T, POD >,						\
+    VectorConstReference<T, Sz>,					\
+    XprLiteral< POD >							\
+  >							expr_type;	\
+  return XprVector<expr_type, Sz>(					\
+    expr_type(lhs.const_ref(), XprLiteral< POD >(rhs)));		\
+}									\
+									\
+template<class T, std::size_t Sz>					\
+inline									\
+XprVector<								\
+  XprBinOp<								\
+    Fcnl_##NAME< POD, T>,						\
+    XprLiteral< POD >,							\
+    VectorConstReference<T, Sz>						\
+  >,									\
+  Sz									\
+>									\
+NAME (POD lhs, const Vector<T, Sz>& rhs) {				\
+  typedef XprBinOp<							\
+    Fcnl_##NAME< POD, T>,						\
+    XprLiteral< POD >,							\
+    VectorConstReference<T, Sz>						\
+  >							expr_type;	\
+  return XprVector<expr_type, Sz>(					\
+    expr_type(XprLiteral< POD >(lhs), rhs.const_ref()));		\
+}
+
+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(Vector<std::complex<T>, Sz>, std::complex<T>)
+ * function(std::complex<T>, Vector<std::complex<T>, Sz>)
+ * Note: per se element wise
+ * \todo type promotion
+ */
+#define TVMET_IMPLEMENT_MACRO(NAME)						\
+template<class T, std::size_t Sz>						\
+inline										\
+XprVector<									\
+  XprBinOp<									\
+    Fcnl_##NAME< std::complex<T>, std::complex<T> >,				\
+    VectorConstReference< std::complex<T>, Sz>,					\
+    XprLiteral< std::complex<T> >						\
+  >,										\
+  Sz										\
+>										\
+NAME (const Vector<std::complex<T>, Sz>& lhs, const std::complex<T>& rhs) {	\
+  typedef XprBinOp<								\
+    Fcnl_##NAME< std::complex<T>, std::complex<T> >,				\
+    VectorConstReference< std::complex<T>, Sz>,					\
+    XprLiteral< std::complex<T> >						\
+  >							expr_type;		\
+  return XprVector<expr_type, Sz>(						\
+    expr_type(lhs.const_ref(), XprLiteral< std::complex<T> >(rhs)));		\
+}										\
+										\
+template<class T, std::size_t Sz>						\
+inline										\
+XprVector<									\
+  XprBinOp<									\
+    Fcnl_##NAME< std::complex<T>, std::complex<T> >,				\
+    XprLiteral< std::complex<T> >,						\
+    VectorConstReference< std::complex<T>, Sz>					\
+  >,										\
+  Sz										\
+>										\
+NAME (const std::complex<T>& lhs, const Vector< std::complex<T>, Sz>& rhs) {	\
+  typedef XprBinOp<								\
+    Fcnl_##NAME< std::complex<T>, std::complex<T> >,				\
+    XprLiteral< std::complex<T> >,						\
+    VectorConstReference< std::complex<T>, Sz>					\
+  >							expr_type;		\
+  return XprVector<expr_type, Sz>(						\
+    expr_type(XprLiteral< std::complex<T> >(lhs), rhs.const_ref()));		\
+}
+
+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 Vector<T, Sz>& v)
+ * \brief Compute the sum of the vector.
+ * \ingroup _unary_function
+ *
+ * Simply compute the sum of the given vector as:
+ * \f[
+ * \sum_{i = 0}^{Sz-1} v[i]
+ * \f]
+ */
+template<class T, std::size_t Sz>
+inline
+typename NumericTraits<T>::sum_type
+sum(const Vector<T, Sz>& v) {
+  return meta::Vector<Sz>::sum(v);
+}
+
+
+/**
+ * \fn product(const Vector<T, Sz>& v)
+ * \brief Compute the product of the vector elements.
+ * \ingroup _unary_function
+ *
+ * Simply computer the product of the given vector as:
+ * \f[
+ * \prod_{i = 0}^{Sz - 1} v[i]
+ * \f]
+ */
+template<class T, std::size_t Sz>
+inline
+typename NumericTraits<T>::sum_type
+product(const Vector<T, Sz>& v) {
+  return meta::Vector<Sz>::product(v);
+}
+
+
+/**
+ * \fn dot(const Vector<T1, Sz>& lhs, const Vector<T2, 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 T1, class T2, std::size_t Sz>
+inline
+typename PromoteTraits<T1, T2>::value_type
+dot(const Vector<T1, Sz>& lhs, const Vector<T2, Sz>& rhs) {
+  return meta::Vector<Sz>::dot(lhs, rhs);
+}
+
+
+/**
+ * \fn cross(const Vector<T1, 3>& lhs, const Vector<T2, 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 T2>
+inline
+Vector<typename PromoteTraits<T1, T2>::value_type, 3>
+cross(const Vector<T1, 3>& lhs, const Vector<T2, 3>& rhs) {
+  typedef typename PromoteTraits<T1, T2>::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 Vector<T, Sz>& v)
+ * \brief The \f$l_1\f$ norm of a vector v.
+ * \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 T, std::size_t Sz>
+inline
+typename NumericTraits<T>::sum_type
+norm1(const Vector<T, Sz>& v) {
+  return sum(abs(v));
+}
+
+
+/**
+ * \fn norm2(const Vector<T, Sz>& v)
+ * \brief The euklidian norm (or \f$l_2\f$ norm) of a vector v.
+ * \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 T, std::size_t Sz>
+inline
+typename NumericTraits<T>::sum_type
+norm2(const Vector<T, Sz>& v) {
+  return static_cast<T>( std::sqrt(static_cast<typename NumericTraits<T>::float_type>(dot(v, v))) );
+}
+
+
+/**
+ * \fn normalize(const Vector<T, Sz>& v)
+ * \brief Normalize the given vector.
+ * \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 T, std::size_t Sz>
+inline
+XprVector<
+  XprBinOp<
+    Fcnl_div<T, T>,
+    VectorConstReference<T, Sz>,
+    XprLiteral< T >
+  >,
+  Sz
+>
+normalize(const Vector<T, Sz>& v) {
+  typedef XprBinOp<
+    Fcnl_div<T, T>,
+    VectorConstReference<T, Sz>,
+    XprLiteral< T >
+  >							expr_type;
+  return XprVector<expr_type, Sz>(
+    expr_type(v.const_ref(), XprLiteral< T >(norm2(v))));
+}
+
+
+/*++++++++++++++++++++++++++++++++++++++++++++++++++++++++
+ * min/max unary functions
+ *+++++++++++++++++++++++++++++++++++++++++++++++++++++++*/
+
+
+/**
+ * \fn maximum(const XprVector<E, Sz>& e)
+ * \brief Find the maximum of a vector expression
+ * \ingroup _unary_function
+ */
+template<class E, std::size_t Sz>
+inline
+Extremum<typename E::value_type, std::size_t, vector_tag>
+maximum(const XprVector<E, Sz>& e) {
+  typedef typename E::value_type 			value_type;
+
+  value_type 						m_max(e(0));
+  std::size_t 						m_idx(0);
+
+  // this loop is faster than meta templates!
+  for(std::size_t i = 1; i != Sz; ++i) {
+    if(e(i) > m_max) {
+      m_max = e(i);
+      m_idx = i;
+    }
+  }
+
+  return Extremum<value_type, std::size_t, vector_tag>(m_max, m_idx);
+}
+
+
+/**
+ * \fn maximum(const Vector<T, Sz>& v)
+ * \brief Find the maximum of a vector
+ * \ingroup _unary_function
+ */
+template<class T, std::size_t Sz>
+inline
+Extremum<T, std::size_t, vector_tag>
+maximum(const Vector<T, Sz>& v) { return maximum(v.as_expr()); }
+
+
+/**
+ * \fn minimum(const XprVector<E, Sz>& e)
+ * \brief Find the minimum of a vector expression
+ * \ingroup _unary_function
+ */
+template<class E, std::size_t Sz>
+inline
+Extremum<typename E::value_type, std::size_t, vector_tag>
+minimum(const XprVector<E, Sz>& e) {
+  typedef typename E::value_type 			value_type;
+
+  value_type 						m_min(e(0));
+  std::size_t 						m_idx(0);
+
+  // this loop is faster than meta templates!
+  for(std::size_t i = 1; i != Sz; ++i) {
+    if(e(i) < m_min) {
+      m_min = e(i);
+      m_idx = i;
+    }
+  }
+
+  return Extremum<value_type, std::size_t, vector_tag>(m_min, m_idx);
+}
+
+
+/**
+ * \fn minimum(const Vector<T, Sz>& v)
+ * \brief Find the minimum of a vector
+ * \ingroup _unary_function
+ */
+template<class T, std::size_t Sz>
+inline
+Extremum<T, std::size_t, vector_tag>
+minimum(const Vector<T, Sz>& v) { return minimum(v.as_expr()); }
+
+
+/**
+ * \fn max(const XprVector<E, Sz>& e)
+ * \brief Find the maximum of a vector expression
+ * \ingroup _unary_function
+ */
+template<class E, std::size_t Sz>
+inline
+typename E::value_type
+max(const XprVector<E, Sz>& e) {
+  typedef typename E::value_type 			value_type;
+
+  value_type 						m_max(e(0));
+
+  // this loop is faster than meta templates!
+  for(std::size_t i = 1; i != Sz; ++i)
+    if(e(i) > m_max)
+      m_max = e(i);
+
+  return m_max;
+}
+
+
+/**
+ * \fn max(const Vector<T, Sz>& v)
+ * \brief Find the maximum of a vector
+ * \ingroup _unary_function
+ */
+template<class T, std::size_t Sz>
+inline
+T max(const Vector<T, Sz>& v) {
+  typedef T			 			value_type;
+  typedef typename Vector<T, Sz>::const_iterator	const_iterator;
+
+  const_iterator					iter(v.begin());
+  const_iterator					last(v.end());
+  value_type 						temp(*iter);
+
+  for( ; iter != last; ++iter)
+    if(*iter > temp)
+      temp = *iter;
+
+  return temp;
+}
+
+
+/**
+ * \fn min(const XprVector<E, Sz>& e)
+ * \brief Find the minimum of a vector expression
+ * \ingroup _unary_function
+ */
+template<class E, std::size_t Sz>
+inline
+typename E::value_type
+min(const XprVector<E, Sz>& e) {
+  typedef typename E::value_type 			value_type;
+
+  value_type 						m_min(e(0));
+
+  // this loop is faster than meta templates!
+  for(std::size_t i = 1; i != Sz; ++i)
+    if(e(i) < m_min)
+      m_min = e(i);
+
+  return m_min;
+}
+
+
+/**
+ * \fn min(const Vector<T, Sz>& v)
+ * \brief Find the minimum of a vector
+ * \ingroup _unary_function
+ */
+template<class T, std::size_t Sz>
+inline
+T min(const Vector<T, Sz>& v) {
+  typedef T			 			value_type;
+  typedef typename Vector<T, Sz>::const_iterator	const_iterator;
+
+  const_iterator					iter(v.begin());
+  const_iterator					last(v.end());
+  value_type 						temp(*iter);
+
+  for( ; iter != last; ++iter)
+    if(*iter < temp)
+      temp = *iter;
+
+  return temp;
+}
+
+
+/**
+ * \fn cvector_ref(const T* mem)
+ * \brief Creates an expression wrapper for a C like vector arrays.
+ * \ingroup _unary_function
+ *
+ * This is like creating a vector of external data, as described
+ * at \ref construct. With this function you wrap an expression
+ * around a C style vector array and you can operate directly with it
+ * as usual.
+ *
+ * \par Example:
+ * \code
+ * static float vertices[N][3] = {
+ *   {-1,  0,  1}, { 1,  0,  1}, ...
+ * };
+ * ...
+ * typedef Vector<float, 3>			vector_type;
+ * ...
+ * vector_type V( cross(cvector_ref<float, 3>(&vertices[0][0]),
+ *                      cvector_ref<float, 3>(&vertices[1][0])) );
+ * \endcode
+ *
+ * \since release 1.6.0
+ */
+template<class T, std::size_t Sz>
+inline
+XprVector<
+  VectorConstReference<T, Sz>,
+  Sz
+>
+cvector_ref(const T* mem) {
+  typedef VectorConstReference<T, Sz>		expr_type;
+
+  return XprVector<expr_type, Sz>(expr_type(mem));
+}
+
+
+} // namespace tvmet
+
+#endif // TVMET_VECTOR_FUNCTIONS_H
+
+// Local Variables:
+// mode:C++
+// tab-width:8
+// End: