Adding upstream version 16.2.11+ds.upstream/16.2.11+dsupstream

Signed-off-by: Daniel Baumann <daniel.baumann@progress-linux.org>

1 files changed, 775 insertions, 0 deletions

diff --git a/src/boost/libs/math/test/octonion_test.cpp b/src/boost/libs/math/test/octonion_test.cpp
new file mode 100644
index 000000000..6c528b1fc
--- /dev/null
+++ b/src/boost/libs/math/test/octonion_test.cpp
@@ -0,0 +1,775 @@
+// test file for octonion.hpp
+
+//  (C) Copyright Hubert Holin 2001.
+//  Distributed under the Boost Software License, Version 1.0. (See
+//  accompanying file LICENSE_1_0.txt or copy at
+//  http://www.boost.org/LICENSE_1_0.txt)
+
+
+#include <iomanip>
+
+
+#include <boost/mpl/list.hpp>
+
+#include <boost/test/unit_test.hpp>
+#include <boost/test/unit_test_log.hpp>
+
+#include <boost/math/octonion.hpp>
+
+
+template<typename T>
+struct string_type_name;
+
+#define DEFINE_TYPE_NAME(Type)              \
+template<> struct string_type_name<Type>    \
+{                                           \
+    static char const * _()                 \
+    {                                       \
+        return #Type;                       \
+    }                                       \
+}
+
+DEFINE_TYPE_NAME(float);
+DEFINE_TYPE_NAME(double);
+#ifndef BOOST_MATH_NO_LONG_DOUBLE_MATH_FUNCTIONS
+DEFINE_TYPE_NAME(long double);
+#endif
+
+#ifndef BOOST_MATH_NO_LONG_DOUBLE_MATH_FUNCTIONS
+typedef boost::mpl::list<float,double,long double>  test_types;
+#else
+typedef boost::mpl::list<float,double>  test_types;
+#endif
+
+// Apple GCC 4.0 uses the "double double" format for its long double,
+// which means that epsilon is VERY small but useless for
+// comparisons. So, don't do those comparisons.
+#if (defined(__APPLE_CC__) && defined(__GNUC__) && __GNUC__ == 4) || defined(BOOST_MATH_NO_LONG_DOUBLE_MATH_FUNCTIONS)
+typedef boost::mpl::list<float,double>  near_eps_test_types;
+#else
+typedef boost::mpl::list<float,double,long double>  near_eps_test_types;
+#endif
+
+#if BOOST_WORKAROUND(__GNUC__, < 3)
+    // gcc 2.x ignores function scope using declarations,
+    // put them in the scope of the enclosing namespace instead:
+using   ::std::sqrt;
+using   ::std::atan;
+using   ::std::log;
+using   ::std::exp;
+using   ::std::cos;
+using   ::std::sin;
+using   ::std::tan;
+using   ::std::cosh;
+using   ::std::sinh;
+using   ::std::tanh;
+
+using   ::std::numeric_limits;
+
+using   ::boost::math::abs;
+#endif  /* BOOST_WORKAROUND(__GNUC__, < 3) */
+
+
+#ifdef  BOOST_NO_STDC_NAMESPACE
+using   ::sqrt;
+using   ::atan;
+using   ::log;
+using   ::exp;
+using   ::cos;
+using   ::sin;
+using   ::tan;
+using   ::cosh;
+using   ::sinh;
+using   ::tanh;
+#endif  /* BOOST_NO_STDC_NAMESPACE */
+
+#ifdef  BOOST_NO_ARGUMENT_DEPENDENT_LOOKUP
+using   ::boost::math::real;
+using   ::boost::math::unreal;
+using   ::boost::math::sup;
+using   ::boost::math::l1;
+using   ::boost::math::abs;
+using   ::boost::math::norm;
+using   ::boost::math::conj;
+using   ::boost::math::exp;
+using   ::boost::math::pow;
+using   ::boost::math::cos;
+using   ::boost::math::sin;
+using   ::boost::math::tan;
+using   ::boost::math::cosh;
+using   ::boost::math::sinh;
+using   ::boost::math::tanh;
+using   ::boost::math::sinc_pi;
+using   ::boost::math::sinhc_pi;
+#endif  /* BOOST_NO_ARGUMENT_DEPENDENT_LOOKUP */
+  
+// Provide standard floating point abs() overloads if older Microsoft
+// library is used with _MSC_EXTENSIONS defined. This code also works
+// for the Intel compiler using the Microsoft library.
+#if defined(_MSC_EXTENSIONS) && defined(_MSC_VER) && _MSC_VER < 1310
+inline float        abs(float v)
+{
+    return(fabs(v));
+}
+
+inline double        abs(double v)
+{
+    return(fabs(v));
+}
+
+inline long double    abs(long double v)
+{
+    return(fabs(v));
+}
+#endif /* BOOST_WORKAROUND(BOOST_MSVC) */
+
+
+// explicit (if ludicrous) instanciation
+#if !BOOST_WORKAROUND(__GNUC__, < 3)
+template    class ::boost::math::octonion<int>;
+#else
+// gcc doesn't like the absolutely-qualified namespace
+template class boost::math::octonion<int>;
+#endif /* !BOOST_WORKAROUND(__GNUC__) */
+
+
+namespace
+{
+    template<typename T>
+    ::boost::math::octonion<T>    index_i_element(int idx)
+    {
+        return(
+            ::boost::math::octonion<T>(
+                        (idx == 0) ?
+                            static_cast<T>(1) :
+                            static_cast<T>(0),
+                        (idx == 1) ?
+                            static_cast<T>(1) :
+                            static_cast<T>(0),
+                        (idx == 2) ?
+                            static_cast<T>(1) :
+                            static_cast<T>(0),
+                        (idx == 3) ?
+                            static_cast<T>(1) :
+                            static_cast<T>(0),
+                        (idx == 4) ?
+                            static_cast<T>(1) :
+                            static_cast<T>(0),
+                        (idx == 5) ?
+                            static_cast<T>(1) :
+                            static_cast<T>(0),
+                        (idx == 6) ?
+                            static_cast<T>(1) :
+                            static_cast<T>(0),
+                        (idx == 7) ?
+                            static_cast<T>(1) :
+                            static_cast<T>(0)
+            ));
+    }
+}
+
+
+
+void    octonion_manual_test()
+{
+    // tests for evaluation by humans
+    
+    
+    // using default constructor
+    ::boost::math::octonion<float>            o0;
+    
+    ::boost::math::octonion<float>            oa[2];
+    
+    // using constructor "O seen as R^8"
+    ::boost::math::octonion<float>            o1(1,2,3,4,5,6,7,8);
+    
+    ::std::complex<double>                    c0(9,10);
+    
+    // using constructor "O seen as C^4"
+    ::boost::math::octonion<double>            o2(c0);
+    
+    ::boost::math::quaternion<long double>    q0(11,12,13,14);
+    
+    // using constructor "O seen as H^2"
+    ::boost::math::octonion<long double>      o3(q0);
+    
+    // using UNtemplated copy constructor
+    ::boost::math::octonion<float>            o4(o1);
+    
+    // using templated copy constructor
+    ::boost::math::octonion<long double>      o5(o2);
+    
+    // using UNtemplated assignment operator
+    o5 = o3;
+    oa[0] = o0;
+    
+    // using templated assignment operator
+    o5 = o2;
+    oa[1] = o5;
+    
+    float                                     f0(15);
+    
+    // using converting assignment operator
+    o0 = f0;
+    
+    // using converting assignment operator
+    o2 = c0;
+    
+    // using converting assignment operator
+    o5 = q0;
+    
+    // using += (const T &)
+    o4 += f0;
+    
+    // using += (const ::std::complex<T> &)
+    o2 += c0;
+    
+    // using += (const ::boost::math::quaternion<T> &)
+    o3 += q0;
+    
+    // using += (const quaternion<X> &)
+    o5 += o4;
+    
+    // using -= (const T &)
+    o1 -= f0;
+    
+    // using -= (const ::std::complex<T> &)
+    o2 -= c0;
+    
+    // using -= (const ::boost::math::quaternion<T> &)
+    o5 -= q0;
+    
+    // using -= (const octonion<X> &)
+    o3 -= o4;
+    
+    double                                    d0(16);
+    ::std::complex<double>                    c1(17,18);
+    ::boost::math::quaternion<double>         q1(19,20,21,22);
+    
+    // using *= (const T &)
+    o2 *= d0;
+    
+    // using *= (const ::std::complex<T> &)
+    o2 *= c1;
+    
+    // using *= (const ::boost::math::quaternion<T> &)
+    o2 *= q1;
+    
+    // using *= (const octonion<X> &)
+    o2 *= o4;
+    
+    long double                               l0(23);
+    ::std::complex<long double>               c2(24,25);
+    
+    // using /= (const T &)
+    o5 /= l0;
+    
+    // using /= (const ::std::complex<T> &)
+    o5 /= c2;
+    
+    // using /= (const quaternion<X> &)
+    o5 /= q0;
+    
+    // using /= (const octonion<X> &)
+    o5 /= o5;
+    
+    // using + (const T &, const octonion<T> &)
+    ::boost::math::octonion<float>            o6 = f0+o0;
+    
+    // using + (const octonion<T> &, const T &)
+    ::boost::math::octonion<float>            o7 = o0+f0;
+    
+    // using + (const ::std::complex<T> &, const quaternion<T> &)
+    ::boost::math::octonion<double>           o8 = c0+o2;
+    
+    // using + (const octonion<T> &, const ::std::complex<T> &)
+    ::boost::math::octonion<double>           o9 = o2+c0;
+    
+    // using + (const ::boost::math::quaternion<T>, const octonion<T> &)
+    ::boost::math::octonion<long double>      o10 = q0+o3;
+    
+    // using + (const octonion<T> &, const ::boost::math::quaternion<T> &)
+    ::boost::math::octonion<long double>      o11 = o3+q0;
+    
+    // using + (const quaternion<T> &,const quaternion<T> &)
+    ::boost::math::octonion<float>            o12 = o0+o4;
+    
+    // using - (const T &, const octonion<T> &)
+    o6 = f0-o0;
+    
+    // using - (const octonion<T> &, const T &)
+    o7 = o0-f0;
+    
+    // using - (const ::std::complex<T> &, const octonion<T> &)
+    o8 = c0-o2;
+    
+    // using - (const octonion<T> &, const ::std::complex<T> &)
+    o9 = o2-c0;
+    
+    // using - (const quaternion<T> &,const octonion<T> &)
+    o10 = q0-o3;
+    
+    // using - (const octonion<T> &,const quaternion<T> &)
+    o11 = o3-q0;
+    
+    // using - (const octonion<T> &,const octonion<T> &)
+    o12 = o0-o4;
+    
+    // using * (const T &, const octonion<T> &)
+    o6 = f0*o0;
+    
+    // using * (const octonion<T> &, const T &)
+    o7 = o0*f0;
+    
+    // using * (const ::std::complex<T> &, const octonion<T> &)
+    o8 = c0*o2;
+    
+    // using * (const octonion<T> &, const ::std::complex<T> &)
+    o9 = o2*c0;
+    
+    // using * (const quaternion<T> &,const octonion<T> &)
+    o10 = q0*o3;
+    
+    // using * (const octonion<T> &,const quaternion<T> &)
+    o11 = o3*q0;
+    
+    // using * (const octonion<T> &,const octonion<T> &)
+    o12 = o0*o4;
+    
+    // using / (const T &, const octonion<T> &)
+    o6 = f0/o0;
+    
+    // using / (const octonion<T> &, const T &)
+    o7 = o0/f0;
+    
+    // using / (const ::std::complex<T> &, const octonion<T> &)
+    o8 = c0/o2;
+    
+    // using / (const octonion<T> &, const ::std::complex<T> &)
+    o9 = o2/c0;
+    
+    // using / (const ::boost::math::quaternion<T> &, const octonion<T> &)
+    o10 = q0/o3;
+    
+    // using / (const octonion<T> &, const ::boost::math::quaternion<T> &)
+    o11 = o3/q0;
+    
+    // using / (const octonion<T> &,const octonion<T> &)
+    o12 = o0/o4;
+    
+    // using + (const octonion<T> &)
+    o4 = +o0;
+    
+    // using - (const octonion<T> &)
+    o0 = -o4;
+    
+    // using == (const T &, const octonion<T> &)
+    f0 == o0;
+    
+    // using == (const octonion<T> &, const T &)
+    o0 == f0;
+    
+    // using == (const ::std::complex<T> &, const octonion<T> &)
+    c0 == o2;
+    
+    // using == (const octonion<T> &, const ::std::complex<T> &)
+    o2 == c0;
+    
+    // using == (const ::boost::math::quaternion<T> &, const octonion<T> &)
+    q0 == o3;
+    
+    // using == (const octonion<T> &, const ::boost::math::quaternion<T> &)
+    o3 == q0;
+    
+    // using == (const octonion<T> &,const octonion<T> &)
+    o0 == o4;
+    
+    // using != (const T &, const octonion<T> &)
+    f0 != o0;
+    
+    // using != (const octonion<T> &, const T &)
+    o0 != f0;
+    
+    // using != (const ::std::complex<T> &, const octonion<T> &)
+    c0 != o2;
+    
+    // using != (const octonion<T> &, const ::std::complex<T> &)
+    o2 != c0;
+    
+    // using != (const ::boost::math::quaternion<T> &, const octonion<T> &)
+    q0 != o3;
+    
+    // using != (const octonion<T> &, const ::boost::math::quaternion<T> &)
+    o3 != q0;
+    
+    // using != (const octonion<T> &,const octonion<T> &)
+    o0 != o4;
+    
+    BOOST_TEST_MESSAGE("Please input an octonion...");
+    
+#ifdef BOOST_INTERACTIVE_TEST_INPUT_ITERATOR
+    ::std::cin >> o0;
+    
+    if    (::std::cin.fail())
+    {
+        BOOST_TEST_MESSAGE("You have entered nonsense!");
+    }
+    else
+    {
+        BOOST_TEST_MESSAGE("You have entered the octonion " << o0 << " .");
+    }
+#else
+    ::std::istringstream                bogus("(1,2,3,4,5,6,7,8)");
+    
+    bogus >> o0;
+    
+    BOOST_TEST_MESSAGE("You have entered the octonion " << o0 << " .");
+#endif
+    
+    BOOST_TEST_MESSAGE("For this octonion:");
+    
+    BOOST_TEST_MESSAGE( "the value of the real part is "
+                << real(o0));
+    
+    BOOST_TEST_MESSAGE( "the value of the unreal part is "
+                << unreal(o0));
+    
+    BOOST_TEST_MESSAGE( "the value of the sup norm is "
+                << sup(o0));
+    
+    BOOST_TEST_MESSAGE( "the value of the l1 norm is "
+                << l1(o0));
+    
+    BOOST_TEST_MESSAGE( "the value of the magnitude (Euclidean norm) is "
+                << abs(o0));
+    
+    BOOST_TEST_MESSAGE( "the value of the (Cayley) norm is "
+                << norm(o0));
+    
+    BOOST_TEST_MESSAGE( "the value of the conjugate is "
+                << conj(o0));
+    
+    BOOST_TEST_MESSAGE( "the value of the exponential is "
+                << exp(o0));
+    
+    BOOST_TEST_MESSAGE( "the value of the cube is "
+                << pow(o0,3));
+    
+    BOOST_TEST_MESSAGE( "the value of the cosinus is "
+                << cos(o0));
+    
+    BOOST_TEST_MESSAGE( "the value of the sinus is "
+                << sin(o0));
+    
+    BOOST_TEST_MESSAGE( "the value of the tangent is "
+                << tan(o0));
+    
+    BOOST_TEST_MESSAGE( "the value of the hyperbolic cosinus is "
+                << cosh(o0));
+    
+    BOOST_TEST_MESSAGE( "the value of the hyperbolic sinus is "
+                << sinh(o0));
+    
+    BOOST_TEST_MESSAGE( "the value of the hyperbolic tangent is "
+                << tanh(o0));
+    
+#ifdef    BOOST_NO_TEMPLATE_TEMPLATES
+    BOOST_TEST_MESSAGE( "no template templates, can't compute cardinal functions");
+#else    /* BOOST_NO_TEMPLATE_TEMPLATES */
+    BOOST_TEST_MESSAGE( "the value of the Sinus Cardinal (of index pi) is "
+                << sinc_pi(o0));
+    
+    BOOST_TEST_MESSAGE( "the value of "
+                << "the Hyperbolic Sinus Cardinal (of index pi) is "
+                << sinhc_pi(o0));
+#endif    /* BOOST_NO_TEMPLATE_TEMPLATES */
+    
+    BOOST_TEST_MESSAGE(" ");
+    
+    float                            rho = ::std::sqrt(4096.0f);
+    float                            theta = ::std::atan(1.0f);
+    float                            phi1 = ::std::atan(1.0f);
+    float                            phi2 = ::std::atan(1.0f);
+    float                            phi3 = ::std::atan(1.0f);
+    float                            phi4 = ::std::atan(1.0f);
+    float                            phi5 = ::std::atan(1.0f);
+    float                            phi6 = ::std::atan(1.0f);
+    
+    BOOST_TEST_MESSAGE( "The value of the octonion represented "
+                << "in spherical form by "
+                << "rho = " << rho << " , theta = " << theta
+                << " , phi1 = " << phi1 << " , phi2 = " << phi2
+                << " , phi3 = " << phi3 << " , phi4 = " << phi4
+                << " , phi5 = " << phi5 << " , phi6 = " << phi6
+                << " is "
+                << ::boost::math::spherical(rho, theta,
+                        phi1, phi2, phi3, phi4, phi5, phi6));
+    
+    float                            rho1 = 1;
+    float                            rho2 = 2;
+    float                            rho3 = ::std::sqrt(2.0f);
+    float                            rho4 = ::std::sqrt(8.0f);
+    float                            theta1 = 0;
+    float                            theta2 = ::std::atan(1.0f)*2;
+    float                            theta3 = ::std::atan(1.0f);
+    float                            theta4 = ::std::atan(::std::sqrt(3.0f));
+    
+    BOOST_TEST_MESSAGE( "The value of the octonion represented "
+                << "in multipolar form by "
+                << "rho1 = " << rho1 << " , theta1 = " << theta1
+                << " , rho2 = " << rho2 << " , theta2 = " << theta2
+                << "rho3 = " << rho3 << " , theta3 = " << theta3
+                << " , rho4 = " << rho4 << " , theta4 = " << theta4
+                << " is "
+                << ::boost::math::multipolar(rho1, theta1, rho2, theta2,
+                        rho3, theta3, rho4, theta4));
+    
+    float                            r = ::std::sqrt(2.0f);
+    float                            angle = ::std::atan(1.0f);
+    float                            h1 = 3;
+    float                            h2 = 4;
+    float                            h3 = 5;
+    float                            h4 = 6;
+    float                            h5 = 7;
+    float                            h6 = 8;
+    
+    BOOST_TEST_MESSAGE( "The value of the octonion represented "
+                << "in cylindrical form by "
+                << "r = " << r << " , angle = " << angle
+                << " , h1 = " << h1 << " , h2 = " << h2
+                << " , h3 = " << h3 << " , h4 = " << h4
+                << " , h5 = " << h5 << " , h6 = " << h6
+                << " is " << ::boost::math::cylindrical(r, angle,
+                        h1, h2, h3, h4, h5, h6));
+    
+    double                               real_1(1);
+    ::std::complex<double>               complex_1(1);
+    ::std::complex<double>               complex_i(0,1);
+    ::boost::math::quaternion<double>    quaternion_1(1);
+    ::boost::math::quaternion<double>    quaternion_i(0,1);
+    ::boost::math::quaternion<double>    quaternion_j(0,0,1);
+    ::boost::math::quaternion<double>    quaternion_k(0,0,0,1);
+    ::boost::math::octonion<double>      octonion_1(1);
+    ::boost::math::octonion<double>      octonion_i(0,1);
+    ::boost::math::octonion<double>      octonion_j(0,0,1);
+    ::boost::math::octonion<double>      octonion_k(0,0,0,1);
+    ::boost::math::octonion<double>      octonion_e_prime(0,0,0,0,1);
+    ::boost::math::octonion<double>      octonion_i_prime(0,0,0,0,0,1);
+    ::boost::math::octonion<double>      octonion_j_prime(0,0,0,0,0,0,1);
+    ::boost::math::octonion<double>      octonion_k_prime(0,0,0,0,0,0,0,1);
+    
+    
+    BOOST_TEST_MESSAGE(" ");
+    
+    BOOST_TEST_MESSAGE( "Real 1: " << real_1
+                << " ; Complex 1: " << complex_1
+                << " ; Quaternion 1: " << quaternion_1
+                << " ; Octonion 1: " << octonion_1 << " .");
+                
+    BOOST_TEST_MESSAGE( "Complex i: " << complex_i
+                << " ; Quaternion i: " << quaternion_i
+                << " ; Octonion i : " << octonion_i << " .");
+                
+    BOOST_TEST_MESSAGE( "Quaternion j: " << quaternion_j
+                << " ; Octonion j: " << octonion_j << " .");
+    
+    BOOST_TEST_MESSAGE( "Quaternion k: " << quaternion_k
+                << " ; Octonion k: " << octonion_k << " .");
+    
+    BOOST_TEST_MESSAGE( "Quaternion e\': " << octonion_e_prime << " .");
+    
+    BOOST_TEST_MESSAGE( "Quaternion i\': " << octonion_i_prime << " .");
+    
+    BOOST_TEST_MESSAGE( "Quaternion j\': " << octonion_j_prime << " .");
+    
+    BOOST_TEST_MESSAGE( "Quaternion k\': " << octonion_k_prime << " .");
+    
+    BOOST_TEST_MESSAGE(" ");
+    
+    BOOST_TEST_MESSAGE( octonion_1*octonion_1 << " ; "
+                << octonion_1*octonion_i << " ; "
+                << octonion_1*octonion_j << " ; "
+                << octonion_1*octonion_k << " ; "
+                << octonion_1*octonion_e_prime << " ; "
+                << octonion_1*octonion_i_prime << " ; "
+                << octonion_1*octonion_j_prime << " ; "
+                << octonion_1*octonion_k_prime << " ; ");
+    
+    BOOST_TEST_MESSAGE( octonion_i*octonion_1 << " ; "
+                << octonion_i*octonion_i << " ; "
+                << octonion_i*octonion_j << " ; "
+                << octonion_i*octonion_k << " ; "
+                << octonion_i*octonion_e_prime << " ; "
+                << octonion_i*octonion_i_prime << " ; "
+                << octonion_i*octonion_j_prime << " ; "
+                << octonion_i*octonion_k_prime << " ; ");
+    
+    BOOST_TEST_MESSAGE( octonion_j*octonion_1 << " ; "
+                << octonion_j*octonion_i << " ; "
+                << octonion_j*octonion_j << " ; "
+                << octonion_j*octonion_k << " ; "
+                << octonion_j*octonion_e_prime << " ; "
+                << octonion_j*octonion_i_prime << " ; "
+                << octonion_j*octonion_j_prime << " ; "
+                << octonion_j*octonion_k_prime << " ; ");
+    
+    BOOST_TEST_MESSAGE( octonion_k*octonion_1 << " ; "
+                << octonion_k*octonion_i << " ; "
+                << octonion_k*octonion_j << " ; "
+                << octonion_k*octonion_k << " ; "
+                << octonion_k*octonion_e_prime << " ; "
+                << octonion_k*octonion_i_prime << " ; "
+                << octonion_k*octonion_j_prime << " ; "
+                << octonion_k*octonion_k_prime << " ; ");
+    
+    BOOST_TEST_MESSAGE( octonion_e_prime*octonion_1 << " ; "
+                << octonion_e_prime*octonion_i << " ; "
+                << octonion_e_prime*octonion_j << " ; "
+                << octonion_e_prime*octonion_k << " ; "
+                << octonion_e_prime*octonion_e_prime << " ; "
+                << octonion_e_prime*octonion_i_prime << " ; "
+                << octonion_e_prime*octonion_j_prime << " ; "
+                << octonion_e_prime*octonion_k_prime << " ; ");
+    
+    BOOST_TEST_MESSAGE( octonion_i_prime*octonion_1 << " ; "
+                << octonion_i_prime*octonion_i << " ; "
+                << octonion_i_prime*octonion_j << " ; "
+                << octonion_i_prime*octonion_k << " ; "
+                << octonion_i_prime*octonion_e_prime << " ; "
+                << octonion_i_prime*octonion_i_prime << " ; "
+                << octonion_i_prime*octonion_j_prime << " ; "
+                << octonion_i_prime*octonion_k_prime << " ; ");
+    
+    BOOST_TEST_MESSAGE( octonion_j_prime*octonion_1 << " ; "
+                << octonion_j_prime*octonion_i << " ; "
+                << octonion_j_prime*octonion_j << " ; "
+                << octonion_j_prime*octonion_k << " ; "
+                << octonion_j_prime*octonion_e_prime << " ; "
+                << octonion_j_prime*octonion_i_prime << " ; "
+                << octonion_j_prime*octonion_j_prime << " ; "
+                << octonion_j_prime*octonion_k_prime << " ; ");
+    
+    BOOST_TEST_MESSAGE( octonion_k_prime*octonion_1 << " ; "
+                << octonion_k_prime*octonion_i << " ; "
+                << octonion_k_prime*octonion_j << " ; "
+                << octonion_k_prime*octonion_k << " ; "
+                << octonion_k_prime*octonion_e_prime << " ; "
+                << octonion_k_prime*octonion_i_prime << " ; "
+                << octonion_k_prime*octonion_j_prime << " ; "
+                << octonion_k_prime*octonion_k_prime << " ; ");
+    
+    BOOST_TEST_MESSAGE(" ");
+    
+    BOOST_TEST_MESSAGE("i\'*(e\'*j) : "
+    << octonion_i_prime*(octonion_e_prime*octonion_j) << " ;");
+    
+    BOOST_TEST_MESSAGE("(i\'*e\')*j : "
+    << (octonion_i_prime*octonion_e_prime)*octonion_j << " ;");
+    
+    BOOST_TEST_MESSAGE(" ");
+}
+
+
+BOOST_TEST_CASE_TEMPLATE_FUNCTION(multiplication_test, T)
+{
+#if     BOOST_WORKAROUND(__GNUC__, < 3)
+#else   /* BOOST_WORKAROUND(__GNUC__, < 3) */
+    using ::std::numeric_limits;
+    
+    using ::boost::math::abs;
+#endif /* BOOST_WORKAROUND(__GNUC__, < 3) */
+    
+    
+    BOOST_TEST_MESSAGE("Testing multiplication for "
+        << string_type_name<T>::_() << ".");
+    
+    BOOST_REQUIRE_PREDICATE(::std::less_equal<T>(),
+        (abs(::boost::math::octonion<T>(1,0,0,0,0,0,0,0)*
+             ::boost::math::octonion<T>(1,0,0,0,0,0,0,0)-
+             static_cast<T>(1)))
+        (numeric_limits<T>::epsilon()));
+    
+    for    (int idx = 1; idx < 8; ++idx)
+    {
+        ::boost::math::octonion<T>    toto = index_i_element<T>(idx);
+        
+        BOOST_REQUIRE_PREDICATE(::std::less_equal<T>(),
+            (abs(toto*toto+static_cast<T>(1)))
+            (numeric_limits<T>::epsilon()));
+    }
+}
+
+
+BOOST_TEST_CASE_TEMPLATE_FUNCTION(exp_test, T)
+{
+#if     BOOST_WORKAROUND(__GNUC__, < 3)
+#else   /* BOOST_WORKAROUND(__GNUC__, < 3) */
+    using ::std::numeric_limits;
+    
+    using ::std::atan;
+    
+    using ::boost::math::abs;
+#endif  /* BOOST_WORKAROUND(__GNUC__, < 3) */
+    
+    
+    BOOST_TEST_MESSAGE("Testing exp for "
+        << string_type_name<T>::_() << ".");
+    
+    for    (int idx = 1; idx < 8; ++idx)
+    {
+        ::boost::math::octonion<T>    toto =
+            static_cast<T>(4)*atan(static_cast<T>(1))*index_i_element<T>(idx);
+            
+        BOOST_CHECK_PREDICATE(::std::less_equal<T>(),
+            (abs(exp(toto)+static_cast<T>(1)))
+            (2*numeric_limits<T>::epsilon()));
+    }
+}
+
+
+boost::unit_test::test_suite *    init_unit_test_suite(int, char *[])
+{
+    ::boost::unit_test::unit_test_log.
+        set_threshold_level(::boost::unit_test::log_messages);
+    
+    boost::unit_test::test_suite *    test =
+        BOOST_TEST_SUITE("octonion_test");
+    
+    BOOST_TEST_MESSAGE("Results of octonion test.");
+    BOOST_TEST_MESSAGE(" ");
+    BOOST_TEST_MESSAGE("(C) Copyright Hubert Holin 2003-2005.");
+    BOOST_TEST_MESSAGE("Distributed under the Boost Software License, Version 1.0.");
+    BOOST_TEST_MESSAGE("(See accompanying file LICENSE_1_0.txt or copy at");
+    BOOST_TEST_MESSAGE("http://www.boost.org/LICENSE_1_0.txt)");
+    BOOST_TEST_MESSAGE(" ");
+    
+#define    BOOST_OCTONION_COMMON_GENERATOR(fct) \
+    test->add(BOOST_TEST_CASE_TEMPLATE(fct##_test, test_types));
+    
+#define    BOOST_OCTONION_COMMON_GENERATOR_NEAR_EPS(fct) \
+    test->add(BOOST_TEST_CASE_TEMPLATE(fct##_test, near_eps_test_types));
+    
+    
+#define    BOOST_OCTONION_TEST                      \
+    BOOST_OCTONION_COMMON_GENERATOR(multiplication) \
+    BOOST_OCTONION_COMMON_GENERATOR_NEAR_EPS(exp)
+    
+    
+    BOOST_OCTONION_TEST
+    
+    
+#undef    BOOST_OCTONION_TEST
+    
+#undef    BOOST_OCTONION_COMMON_GENERATOR
+#undef BOOST_OCTONION_COMMON_GENERATOR_NEAR_EPS
+    
+#ifdef BOOST_OCTONION_TEST_VERBOSE
+    
+    test->add(BOOST_TEST_CASE(octonion_manual_test));
+    
+#endif    /* BOOST_OCTONION_TEST_VERBOSE */
+    
+    return test;
+}
+
+#undef DEFINE_TYPE_NAME