array.hpp

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/* -*- mode: c++; tab-width: 4; indent-tabs-mode: nil; c-basic-offset: 4 -*- */
 
/*
 Copyright (C) 2000, 2001, 2002, 2003 RiskMap srl
 Copyright (C) 2003, 2004, 2005, 2006, 2009 StatPro Italia srl
 Copyright (C) 2004 Ferdinando Ametrano
 
 This file is part of QuantLib, a free-software/open-source library
 for financial quantitative analysts and developers - http://quantlib.org/
 
 QuantLib is free software: you can redistribute it and/or modify it
 under the terms of the QuantLib license.  You should have received a
 copy of the license along with this program; if not, please email
 <quantlib-dev@lists.sf.net>. The license is also available online at
 <http://quantlib.org/license.shtml>.
 
 This program 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 license for more details.
*/
 
/*! \file array.hpp
    \brief 1-D array used in linear algebra.
*/
 
#ifndef quantlib_array_hpp
#define quantlib_array_hpp
 
#include <ql/types.hpp>
#include <ql/errors.hpp>
#include <ql/utilities/null.hpp>
#include <iterator>
#include <functional>
#include <algorithm>
#include <numeric>
#include <vector>
#include <initializer_list>
#include <iomanip>
#include <memory>
#include <type_traits>
 
namespace QuantLib {
 
    //! 1-D array used in linear algebra.
    /*! This class implements the concept of vector as used in linear
        algebra.
        As such, it is <b>not</b> meant to be used as a container -
        <tt>std::vector</tt> should be used instead.
 
        \test construction of arrays is checked in a number of cases
    */
    class Array {
      public:
        //! \name Constructors, destructor, and assignment
        //@{
        //! creates the array with size 0
        Array() : Array(static_cast<Size>(0)) {}
        //! creates the array with the given dimension
        explicit Array(Size size);
        //! creates the array and fills it with <tt>value</tt>
        Array(Size size, Real value);
        /*! \brief creates the array and fills it according to
            \f$ a_{0} = value, a_{i}=a_{i-1}+increment \f$
        */
        Array(Size size, Real value, Real increment);
        Array(const Array&);
        Array(Array&&) noexcept;
        Array(std::initializer_list<Real>);
        //! creates the array from an iterable sequence
        template <class ForwardIterator>
        Array(ForwardIterator begin, ForwardIterator end);
        ~Array() = default;
 
        Array& operator=(const Array&);
        Array& operator=(Array&&) noexcept;
 
        bool operator==(const Array&) const;
        bool operator!=(const Array&) const;
        //@}
        /*! \name Vector algebra
 
            <tt>v += x</tt> and similar operation involving a scalar value
            are shortcuts for \f$ \forall i : v_i = v_i + x \f$
 
            <tt>v *= w</tt> and similar operation involving two vectors are
            shortcuts for \f$ \forall i : v_i = v_i \times w_i \f$
 
            \pre all arrays involved in an algebraic expression must have
            the same size.
        */
        //@{
        const Array& operator+=(const Array&);
        const Array& operator+=(Real);
        const Array& operator-=(const Array&);
        const Array& operator-=(Real);
        const Array& operator*=(const Array&);
        const Array& operator*=(Real);
        const Array& operator/=(const Array&);
        const Array& operator/=(Real);
        //@}
        //! \name Element access
        //@{
        //! read-only
        Real operator[](Size) const;
        Real at(Size) const;
        Real front() const;
        Real back() const;
        //! read-write
        Real& operator[](Size);
        Real& at(Size);
        Real& front();
        Real& back();
        //@}
        //! \name Inspectors
        //@{
        //! dimension of the array
        Size size() const;
        //! whether the array is empty
        bool empty() const;
        //@}
        typedef Size size_type;
        typedef Real value_type;
        typedef Real* iterator;
        typedef const Real* const_iterator;
        typedef std::reverse_iterator<iterator> reverse_iterator;
        typedef std::reverse_iterator<const_iterator> const_reverse_iterator;
        //! \name Iterator access
        //@{
        const_iterator begin() const;
        iterator begin();
        const_iterator end() const;
        iterator end();
        const_reverse_iterator rbegin() const;
        reverse_iterator rbegin();
        const_reverse_iterator rend() const;
        reverse_iterator rend();
        //@}
        //! \name Utilities
        //@{
        void resize(Size n);
        void swap(Array&) noexcept;
        //@}
 
      private:
        std::unique_ptr<Real[]> data_;
        Size n_;
    };
 
    #ifdef QL_NULL_AS_FUNCTIONS
 
    //! specialization of null template for this class
    template <>
    inline Array Null<Array>() {
        return {};
    }
 
    #else
 
    //! specialization of null template for this class
    template <>
    class Null<Array> {
      public:
        Null() = default;
        operator Array() const { return Array(); }
    };
 
    #endif
 
    /*! \relates Array */
    Real DotProduct(const Array&, const Array&);
 
    /*! \relates Array */
    Real Norm2(const Array&);
 
    // unary operators
    /*! \relates Array */
    Array operator+(const Array& v);
    /*! \relates Array */
    Array operator+(Array&& v);
    /*! \relates Array */
    Array operator-(const Array& v);
    /*! \relates Array */
    Array operator-(Array&& v);
 
    // binary operators
    /*! \relates Array */
    Array operator+(const Array&, const Array&);
    /*! \relates Array */
    Array operator+(const Array&, Array&&);
    /*! \relates Array */
    Array operator+(Array&&, const Array&);
    /*! \relates Array */
    Array operator+(Array&&, Array&&);
    /*! \relates Array */
    Array operator+(const Array&, Real);
    /*! \relates Array */
    Array operator+(Array&&, Real);
    /*! \relates Array */
    Array operator+(Real, const Array&);
    /*! \relates Array */
    Array operator+(Real, Array&&);
    /*! \relates Array */
    Array operator-(const Array&, const Array&);
    /*! \relates Array */
    Array operator-(const Array&, Array&&);
    /*! \relates Array */
    Array operator-(Array&&, const Array&);
    /*! \relates Array */
    Array operator-(Array&&, Array&&);
    /*! \relates Array */
    Array operator-(const Array&, Real);
    /*! \relates Array */
    Array operator-(Real, const Array&);
    /*! \relates Array */
    Array operator-(Array&&, Real);
    /*! \relates Array */
    Array operator-(Real, Array&&);
    /*! \relates Array */
    Array operator*(const Array&, const Array&);
    /*! \relates Array */
    Array operator*(const Array&, Array&&);
    /*! \relates Array */
    Array operator*(Array&&, const Array&);
    /*! \relates Array */
    Array operator*(Array&&, Array&&);
    /*! \relates Array */
    Array operator*(const Array&, Real);
    /*! \relates Array */
    Array operator*(Real, const Array&);
    /*! \relates Array */
    Array operator*(Array&&, Real);
    /*! \relates Array */
    Array operator*(Real, Array&&);
    /*! \relates Array */
    Array operator/(const Array&, const Array&);
    /*! \relates Array */
    Array operator/(const Array&, Array&&);
    /*! \relates Array */
    Array operator/(Array&&, const Array&);
    /*! \relates Array */
    Array operator/(Array&&, Array&&);
    /*! \relates Array */
    Array operator/(const Array&, Real);
    /*! \relates Array */
    Array operator/(Real, const Array&);
    /*! \relates Array */
    Array operator/(Array&&, Real);
    /*! \relates Array */
    Array operator/(Real, Array&&);
 
    // math functions
    /*! \relates Array */
    Array Abs(const Array&);
    /*! \relates Array */
    Array Abs(Array&&);
    /*! \relates Array */
    Array Sqrt(const Array&);
    /*! \relates Array */
    Array Sqrt(Array&&);
    /*! \relates Array */
    Array Log(const Array&);
    /*! \relates Array */
    Array Log(Array&&);
    /*! \relates Array */
    Array Exp(const Array&);
    /*! \relates Array */
    Array Exp(Array&&);
    /*! \relates Array */
    Array Pow(const Array&, Real);
    /*! \relates Array */
    Array Pow(Array&&, Real);
 
    // utilities
    /*! \relates Array */
    void swap(Array&, Array&) noexcept;
 
    // format
    /*! \relates Array */
    std::ostream& operator<<(std::ostream&, const Array&);
 
 
    // inline definitions
 
    inline Array::Array(Size size)
    : data_(size != 0U ? new Real[size] : (Real*)nullptr), n_(size) {}
 
    inline Array::Array(Size size, Real value)
    : data_(size != 0U ? new Real[size] : (Real*)nullptr), n_(size) {
        std::fill(begin(),end(),value);
    }
 
    inline Array::Array(Size size, Real value, Real increment)
    : data_(size != 0U ? new Real[size] : (Real*)nullptr), n_(size) {
        for (iterator i=begin(); i!=end(); ++i, value+=increment)
            *i = value;
    }
 
    inline Array::Array(const Array& from)
    : data_(from.n_ != 0U ? new Real[from.n_] : (Real*)nullptr), n_(from.n_) {
        if (data_)
            std::copy(from.begin(),from.end(),begin());
    }
 
    inline Array::Array(Array&& from) noexcept
    : data_((Real*)nullptr), n_(0) {
        swap(from);
    }
 
    namespace detail {
 
        template <class I>
        inline void _fill_array_(Array& a,
                                 std::unique_ptr<Real[]>& data_,
                                 Size& n_,
                                 I begin, I end,
                                 const std::true_type&) {
            // we got redirected here from a call like Array(3, 4)
            // because it matched the constructor below exactly with
            // ForwardIterator = int.  What we wanted was fill an
            // Array with a given value, which we do here.
            Size n = begin;
            Real value = end;
            data_.reset(n ? new Real[n] : (Real*)nullptr);
            n_ = n;
            std::fill(a.begin(),a.end(),value);
        }
 
        template <class I>
        inline void _fill_array_(Array& a,
                                 std::unique_ptr<Real[]>& data_,
                                 Size& n_,
                                 I begin, I end,
                                 const std::false_type&) {
            // true iterators
            Size n = std::distance(begin, end);
            data_.reset(n ? new Real[n] : (Real*)nullptr);
            n_ = n;
            #if defined(QL_PATCH_MSVC) && defined(QL_DEBUG)
            if (n_)
            #endif
            std::copy(begin, end, a.begin());
        }
 
    }
 
    inline Array::Array(std::initializer_list<Real> init) {
        detail::_fill_array_(*this, data_, n_, init.begin(), init.end(),
                             std::false_type());
    }
 
    template <class ForwardIterator>
    inline Array::Array(ForwardIterator begin, ForwardIterator end) {
        // Unfortunately, calls such as Array(3, 4) match this constructor.
        // We have to detect integral types and dispatch.
        detail::_fill_array_(*this, data_, n_, begin, end,
                             std::is_integral<ForwardIterator>());
    }
 
    inline Array& Array::operator=(const Array& from) {
        // strong guarantee
        Array temp(from);
        swap(temp);
        return *this;
    }
 
    inline Array& Array::operator=(Array&& from) noexcept {
        swap(from);
        return *this;
    }
 
    inline bool Array::operator==(const Array& to) const {
        return (n_ == to.n_) && std::equal(begin(), end(), to.begin());
    }
 
    inline bool Array::operator!=(const Array& to) const {
        return !(this->operator==(to));
    }
 
    inline const Array& Array::operator+=(const Array& v) {
        QL_REQUIRE(n_ == v.n_,
                   "arrays with different sizes (" << n_ << ", "
                   << v.n_ << ") cannot be added");
        std::transform(begin(),end(),v.begin(),begin(), std::plus<>());
        return *this;
    }
 
 
    inline const Array& Array::operator+=(Real x) {
        std::transform(begin(), end(), begin(), [=](Real y) -> Real { return y + x; });
        return *this;
    }
 
    inline const Array& Array::operator-=(const Array& v) {
        QL_REQUIRE(n_ == v.n_,
                   "arrays with different sizes (" << n_ << ", "
                   << v.n_ << ") cannot be subtracted");
        std::transform(begin(), end(), v.begin(), begin(), std::minus<>());
        return *this;
    }
 
    inline const Array& Array::operator-=(Real x) {
        std::transform(begin(),end(),begin(), [=](Real y) -> Real { return y - x; });
        return *this;
    }
 
    inline const Array& Array::operator*=(const Array& v) {
        QL_REQUIRE(n_ == v.n_,
                   "arrays with different sizes (" << n_ << ", "
                   << v.n_ << ") cannot be multiplied");
        std::transform(begin(), end(), v.begin(), begin(), std::multiplies<>());
        return *this;
    }
 
    inline const Array& Array::operator*=(Real x) {
        std::transform(begin(), end(), begin(), [=](Real y) -> Real { return y * x; });
        return *this;
    }
 
    inline const Array& Array::operator/=(const Array& v) {
        QL_REQUIRE(n_ == v.n_,
                   "arrays with different sizes (" << n_ << ", "
                   << v.n_ << ") cannot be divided");
        std::transform(begin(), end(), v.begin(), begin(), std::divides<>());
        return *this;
    }
 
    inline const Array& Array::operator/=(Real x) {
        #if defined(QL_EXTRA_SAFETY_CHECKS)
        QL_REQUIRE(x != 0.0, "division by zero");
        #endif
        std::transform(begin(), end(), begin(), [=](Real y) -> Real { return y / x; });
        return *this;
    }
 
    inline Real Array::operator[](Size i) const {
        #if defined(QL_EXTRA_SAFETY_CHECKS)
        QL_REQUIRE(i<n_,
                   "index (" << i << ") must be less than " << n_ <<
                   ": array access out of range");
        #endif
        return data_.get()[i];
    }
 
    inline Real Array::at(Size i) const {
        QL_REQUIRE(i<n_,
                   "index (" << i << ") must be less than " << n_ <<
                   ": array access out of range");
        return data_.get()[i];
    }
 
    inline Real Array::front() const {
        #if defined(QL_EXTRA_SAFETY_CHECKS)
        QL_REQUIRE(n_>0, "null Array: array access out of range");
        #endif
        return data_.get()[0];
    }
 
    inline Real Array::back() const {
        #if defined(QL_EXTRA_SAFETY_CHECKS)
        QL_REQUIRE(n_>0, "null Array: array access out of range");
        #endif
        return data_.get()[n_-1];
    }
 
    inline Real& Array::operator[](Size i) {
        #if defined(QL_EXTRA_SAFETY_CHECKS)
        QL_REQUIRE(i<n_,
                   "index (" << i << ") must be less than " << n_ <<
                   ": array access out of range");
        #endif
        return data_.get()[i];
    }
 
    inline Real& Array::at(Size i) {
        QL_REQUIRE(i<n_,
                   "index (" << i << ") must be less than " << n_ <<
                   ": array access out of range");
        return data_.get()[i];
    }
 
    inline Real& Array::front() {
        #if defined(QL_EXTRA_SAFETY_CHECKS)
        QL_REQUIRE(n_>0, "null Array: array access out of range");
        #endif
        return data_.get()[0];
    }
 
    inline Real& Array::back() {
        #if defined(QL_EXTRA_SAFETY_CHECKS)
        QL_REQUIRE(n_>0, "null Array: array access out of range");
        #endif
        return data_.get()[n_-1];
    }
 
    inline Size Array::size() const {
        return n_;
    }
 
    inline bool Array::empty() const {
        return n_ == 0;
    }
 
    inline Array::const_iterator Array::begin() const {
        return data_.get();
    }
 
    inline Array::iterator Array::begin() {
        return data_.get();
    }
 
    inline Array::const_iterator Array::end() const {
        return data_.get()+n_;
    }
 
    inline Array::iterator Array::end() {
        return data_.get()+n_;
    }
 
    inline Array::const_reverse_iterator Array::rbegin() const {
        return const_reverse_iterator(end());
    }
 
    inline Array::reverse_iterator Array::rbegin() {
        return reverse_iterator(end());
    }
 
    inline Array::const_reverse_iterator Array::rend() const {
        return const_reverse_iterator(begin());
    }
 
    inline Array::reverse_iterator Array::rend() {
        return reverse_iterator(begin());
    }
 
    inline void Array::resize(Size n) {
        if (n > n_) {
            Array swp(n);
            std::copy(begin(), end(), swp.begin());
            swap(swp);
        }
        else if (n < n_) {
            n_ = n;
        }
    }
 
    inline void Array::swap(Array& from) noexcept {
        data_.swap(from.data_);
        std::swap(n_, from.n_);
    }
 
    // dot product and norm
 
    inline Real DotProduct(const Array& v1, const Array& v2) {
        QL_REQUIRE(v1.size() == v2.size(),
                   "arrays with different sizes (" << v1.size() << ", "
                   << v2.size() << ") cannot be multiplied");
        return std::inner_product(v1.begin(),v1.end(),v2.begin(),Real(0.0));
    }
 
    inline Real Norm2(const Array& v) {
        return std::sqrt(DotProduct(v, v));
    }
 
    // overloaded operators
 
    // unary
 
    inline Array operator+(const Array& v) {
        Array result = v;
        return result;
    }
 
    inline Array operator+(Array&& v) {
        return std::move(v);
    }
 
    inline Array operator-(const Array& v) {
        Array result(v.size());
        std::transform(v.begin(), v.end(), result.begin(), std::negate<>());
        return result;
    }
 
    inline Array operator-(Array&& v) {
        Array result = std::move(v);
        std::transform(result.begin(), result.end(), result.begin(), std::negate<>());
        return result;
    }
 
    // binary operators
 
    inline Array operator+(const Array& v1, const Array& v2) {
        QL_REQUIRE(v1.size() == v2.size(),
                   "arrays with different sizes (" << v1.size() << ", "
                   << v2.size() << ") cannot be added");
        Array result(v1.size());
        std::transform(v1.begin(),v1.end(),v2.begin(),result.begin(), std::plus<>());
        return result;
    }
 
    inline Array operator+(const Array& v1, Array&& v2) {
        QL_REQUIRE(v1.size() == v2.size(),
                   "arrays with different sizes (" << v1.size() << ", "
                   << v2.size() << ") cannot be added");
        Array result = std::move(v2);
        std::transform(v1.begin(), v1.end(), result.begin(), result.begin(), std::plus<>());
        return result;
    }
 
    inline Array operator+(Array&& v1, const Array& v2) {
        QL_REQUIRE(v1.size() == v2.size(),
                   "arrays with different sizes (" << v1.size() << ", "
                   << v2.size() << ") cannot be added");
        Array result = std::move(v1);
        std::transform(result.begin(), result.end(), v2.begin(), result.begin(), std::plus<>());
        return result;
    }
 
    inline Array operator+(Array&& v1, Array&& v2) { // NOLINT(cppcoreguidelines-rvalue-reference-param-not-moved)
        QL_REQUIRE(v1.size() == v2.size(),
                   "arrays with different sizes (" << v1.size() << ", "
                   << v2.size() << ") cannot be added");
        Array result = std::move(v2);
        std::transform(v1.begin(), v1.end(), result.begin(), result.begin(), std::plus<>());
        return result;
    }
 
    inline Array operator+(const Array& v1, Real a) {
        Array result(v1.size());
        std::transform(v1.begin(), v1.end(), result.begin(), [=](Real y) -> Real { return y + a; });
        return result;
    }
 
    inline Array operator+(Array&& v1, Real a) {
        Array result = std::move(v1);
        std::transform(result.begin(), result.end(), result.begin(), [=](Real y) -> Real { return y + a; });
        return result;
    }
 
    inline Array operator+(Real a, const Array& v2) {
        Array result(v2.size());
        std::transform(v2.begin(),v2.end(),result.begin(), [=](Real y) -> Real { return a + y; });
        return result;
    }
 
    inline Array operator+(Real a, Array&& v2) {
        Array result = std::move(v2);
        std::transform(result.begin(), result.end(), result.begin(), [=](Real y) -> Real { return a + y; });
        return result;
    }
 
    inline Array operator-(const Array& v1, const Array& v2) {
        QL_REQUIRE(v1.size() == v2.size(),
                   "arrays with different sizes (" << v1.size() << ", "
                   << v2.size() << ") cannot be subtracted");
        Array result(v1.size());
        std::transform(v1.begin(), v1.end(), v2.begin(), result.begin(), std::minus<>());
        return result;
    }
 
    inline Array operator-(const Array& v1, Array&& v2) {
        QL_REQUIRE(v1.size() == v2.size(),
                   "arrays with different sizes (" << v1.size() << ", "
                   << v2.size() << ") cannot be subtracted");
        Array result = std::move(v2);
        std::transform(v1.begin(), v1.end(), result.begin(), result.begin(), std::minus<>());
        return result;
    }
 
    inline Array operator-(Array&& v1, const Array& v2) {
        QL_REQUIRE(v1.size() == v2.size(),
                   "arrays with different sizes (" << v1.size() << ", "
                   << v2.size() << ") cannot be subtracted");
        Array result = std::move(v1);
        std::transform(result.begin(), result.end(), v2.begin(), result.begin(), std::minus<>());
        return result;
    }
 
    inline Array operator-(Array&& v1, Array&& v2) { // NOLINT(cppcoreguidelines-rvalue-reference-param-not-moved)
        QL_REQUIRE(v1.size() == v2.size(),
                   "arrays with different sizes (" << v1.size() << ", "
                   << v2.size() << ") cannot be subtracted");
        Array result = std::move(v2);
        std::transform(v1.begin(), v1.end(), result.begin(), result.begin(), std::minus<>());
        return result;
    }
 
    inline Array operator-(const Array& v1, Real a) {
        Array result(v1.size());
        std::transform(v1.begin(),v1.end(),result.begin(), [=](Real y) -> Real { return y - a; });
        return result;
    }
 
    inline Array operator-(Array&& v1, Real a) {
        Array result = std::move(v1);
        std::transform(result.begin(), result.end(), result.begin(), [=](Real y) -> Real { return y - a; });
        return result;
    }
 
    inline Array operator-(Real a, const Array& v2) {
        Array result(v2.size());
        std::transform(v2.begin(),v2.end(),result.begin(), [=](Real y) -> Real { return a - y; });
        return result;
    }
 
    inline Array operator-(Real a, Array&& v2) {
        Array result = std::move(v2);
        std::transform(result.begin(), result.end(), result.begin(), [=](Real y) -> Real { return a - y; });
        return result;
    }
 
    inline Array operator*(const Array& v1, const Array& v2) {
        QL_REQUIRE(v1.size() == v2.size(),
                   "arrays with different sizes (" << v1.size() << ", "
                   << v2.size() << ") cannot be multiplied");
        Array result(v1.size());
        std::transform(v1.begin(), v1.end(), v2.begin(), result.begin(), std::multiplies<>());
        return result;
    }
 
    inline Array operator*(const Array& v1, Array&& v2) {
        QL_REQUIRE(v1.size() == v2.size(),
                   "arrays with different sizes (" << v1.size() << ", "
                   << v2.size() << ") cannot be multiplied");
        Array result = std::move(v2);
        std::transform(v1.begin(), v1.end(), result.begin(), result.begin(), std::multiplies<>());
        return result;
    }
 
    inline Array operator*(Array&& v1, const Array& v2) {
        QL_REQUIRE(v1.size() == v2.size(),
                   "arrays with different sizes (" << v1.size() << ", "
                   << v2.size() << ") cannot be multiplied");
        Array result = std::move(v1);
        std::transform(result.begin(), result.end(), v2.begin(), result.begin(), std::multiplies<>());
        return result;
    }
 
    inline Array operator*(Array&& v1, Array&& v2) { // NOLINT(cppcoreguidelines-rvalue-reference-param-not-moved)
        QL_REQUIRE(v1.size() == v2.size(),
                   "arrays with different sizes (" << v1.size() << ", "
                   << v2.size() << ") cannot be multiplied");
        Array result = std::move(v2);
        std::transform(v1.begin(), v1.end(), result.begin(), result.begin(), std::multiplies<>());
        return result;
    }
 
    inline Array operator*(const Array& v1, Real a) {
        Array result(v1.size());
        std::transform(v1.begin(),v1.end(),result.begin(), [=](Real y) -> Real { return y * a; });
        return result;
    }
 
    inline Array operator*(Array&& v1, Real a) {
        Array result = std::move(v1);
        std::transform(result.begin(), result.end(), result.begin(), [=](Real y) -> Real { return y * a; });
        return result;
    }
 
    inline Array operator*(Real a, const Array& v2) {
        Array result(v2.size());
        std::transform(v2.begin(),v2.end(),result.begin(), [=](Real y) -> Real { return a * y; });
        return result;
    }
 
    inline Array operator*(Real a, Array&& v2) {
        Array result = std::move(v2);
        std::transform(result.begin(), result.end(), result.begin(), [=](Real y) -> Real { return a * y; });
        return result;
    }
 
    inline Array operator/(const Array& v1, const Array& v2) {
        QL_REQUIRE(v1.size() == v2.size(),
                   "arrays with different sizes (" << v1.size() << ", "
                   << v2.size() << ") cannot be divided");
        Array result(v1.size());
        std::transform(v1.begin(), v1.end(), v2.begin(), result.begin(), std::divides<>());
        return result;
    }
 
    inline Array operator/(const Array& v1, Array&& v2) {
        QL_REQUIRE(v1.size() == v2.size(),
                   "arrays with different sizes (" << v1.size() << ", "
                   << v2.size() << ") cannot be divided");
        Array result = std::move(v2);
        std::transform(v1.begin(), v1.end(), result.begin(), result.begin(), std::divides<>());
        return result;
    }
 
    inline Array operator/(Array&& v1, const Array& v2) {
        QL_REQUIRE(v1.size() == v2.size(),
                   "arrays with different sizes (" << v1.size() << ", "
                   << v2.size() << ") cannot be divided");
        Array result = std::move(v1);
        std::transform(result.begin(), result.end(), v2.begin(), result.begin(), std::divides<>());
        return result;
    }
 
    inline Array operator/(Array&& v1, Array&& v2) { // NOLINT(cppcoreguidelines-rvalue-reference-param-not-moved)
        QL_REQUIRE(v1.size() == v2.size(),
                   "arrays with different sizes (" << v1.size() << ", "
                   << v2.size() << ") cannot be divided");
        Array result = std::move(v2);
        std::transform(v1.begin(), v1.end(), result.begin(), result.begin(), std::divides<>());
        return result;
    }
 
    inline Array operator/(const Array& v1, Real a) {
        Array result(v1.size());
        std::transform(v1.begin(),v1.end(),result.begin(), [=](Real y) -> Real { return y / a; });
        return result;
    }
 
    inline Array operator/(Array&& v1, Real a) {
        Array result = std::move(v1);
        std::transform(result.begin(), result.end(), result.begin(), [=](Real y) -> Real { return y / a; });
        return result;
    }
 
    inline Array operator/(Real a, const Array& v2) {
        Array result(v2.size());
        std::transform(v2.begin(),v2.end(),result.begin(), [=](Real y) -> Real { return a / y; });
        return result;
    }
 
    inline Array operator/(Real a, Array&& v2) {
        Array result = std::move(v2);
        std::transform(result.begin(), result.end(), result.begin(), [=](Real y) -> Real { return a / y; });
        return result;
    }
 
    // functions
 
    inline Array Abs(const Array& v) {
        Array result(v.size());
        std::transform(v.begin(), v.end(), result.begin(),
                       [](Real x) -> Real { return std::fabs(x); });
        return result;
    }
 
    inline Array Abs(Array&& v) {
        Array result = std::move(v);
        std::transform(result.begin(), result.end(), result.begin(),
                       [](Real x) -> Real { return std::fabs(x); });
        return result;
    }
 
    inline Array Sqrt(const Array& v) {
        Array result(v.size());
        std::transform(v.begin(),v.end(),result.begin(),
                       [](Real x) -> Real { return std::sqrt(x); });
        return result;
    }
 
    inline Array Sqrt(Array&& v) {
        Array result = std::move(v);
        std::transform(result.begin(), result.end(), result.begin(),
                       [](Real x) -> Real { return std::sqrt(x); });
        return result;
    }
 
    inline Array Log(const Array& v) {
        Array result(v.size());
        std::transform(v.begin(),v.end(),result.begin(),
                       [](Real x) -> Real { return std::log(x); });
        return result;
    }
 
    inline Array Log(Array&& v) {
        Array result = std::move(v);
        std::transform(result.begin(), result.end(), result.begin(),
                       [](Real x) -> Real { return std::log(x); });
        return result;
    }
 
    inline Array Exp(const Array& v) {
        Array result(v.size());
        std::transform(v.begin(), v.end(), result.begin(),
                       [](Real x) -> Real { return std::exp(x); });
        return result;
    }
 
    inline Array Exp(Array&& v) {
        Array result = std::move(v);
        std::transform(result.begin(), result.end(), result.begin(),
                       [](Real x) -> Real { return std::exp(x); });
        return result;
    }
 
    inline Array Pow(const Array& v, Real alpha) {
        Array result(v.size());
        std::transform(v.begin(), v.end(), result.begin(),
                       [=](Real x) -> Real { return std::pow(x, alpha); });
        return result;
    }
 
    inline Array Pow(Array&& v, Real alpha) {
        Array result = std::move(v);
        std::transform(result.begin(), result.end(), result.begin(),
                       [=](Real x) -> Real { return std::pow(x, alpha); });
        return result;
    }
 
    inline void swap(Array& v, Array& w) noexcept {
        v.swap(w);
    }
 
    inline std::ostream& operator<<(std::ostream& out, const Array& a) {
        std::streamsize width = out.width();
        out << "[ ";
        if (!a.empty()) {
            for (Size n=0; n<a.size()-1; ++n)
                out << std::setw(int(width)) << a[n] << "; ";
            out << std::setw(int(width)) << a.back();
        }
        out << " ]";
        return out;
    }
 
}
 
 
#endif