ceph/src/boost/libs/math/test/legendre_stieltjes_test.cpp

   1 // Copyright Nick Thompson 2017.
   2 // Use, modification and distribution are subject to the
   3 // Boost Software License, Version 1.0.
   4 // (See accompanying file LICENSE_1_0.txt
   5 // or copy at http://www.boost.org/LICENSE_1_0.txt)
   6
   7 #define BOOST_TEST_MAIN
   8
   9 #include <boost/test/unit_test.hpp>
  10 #include <boost/math/special_functions/legendre.hpp>
  11 #include <boost/math/special_functions/legendre_stieltjes.hpp>
  12 #include <boost/math/constants/constants.hpp>
  13 #include <boost/multiprecision/cpp_bin_float.hpp>
  14
  15
  16 using boost::math::legendre_stieltjes;
  17 using boost::math::legendre_p;
  18 using boost::multiprecision::cpp_bin_float_quad;
  19
  20
  21 template<class Real>
  22 void test_legendre_stieltjes()
  23 {
  24     std::cout << std::setprecision(std::numeric_limits<Real>::digits10);
  25     using std::sqrt;
  26     using std::abs;
  27     using boost::math::constants::third;
  28     using boost::math::constants::half;
  29
  30     Real tol = std::numeric_limits<Real>::epsilon();
  31     legendre_stieltjes<Real> ls1(1);
  32     legendre_stieltjes<Real> ls2(2);
  33     legendre_stieltjes<Real> ls3(3);
  34     legendre_stieltjes<Real> ls4(4);
  35     legendre_stieltjes<Real> ls5(5);
  36     legendre_stieltjes<Real> ls8(8);
  37     Real x = -1;
  38     while(x <= 1)
  39     {
  40         BOOST_CHECK_CLOSE_FRACTION(ls1(x), x, tol);
  41         BOOST_CHECK_CLOSE_FRACTION(ls1.prime(x), 1, tol);
  42
  43         Real p2 = legendre_p(2, x);
  44         BOOST_CHECK_CLOSE_FRACTION(ls2(x), p2 - 2/static_cast<Real>(5), tol);
  45         BOOST_CHECK_CLOSE_FRACTION(ls2.prime(x), 3*x, tol);
  46
  47         Real p3 = legendre_p(3, x);
  48         BOOST_CHECK_CLOSE_FRACTION(ls3(x), p3 - 9*x/static_cast<Real>(14), 100*tol);
  49         BOOST_CHECK_CLOSE_FRACTION(ls3.prime(x), 15*x*x*half<Real>() -3*half<Real>()-9/static_cast<Real>(14), 100*tol);
  50
  51         Real p4 = legendre_p(4, x);
  52         //-20P_2(x)/27 + 14P_0(x)/891
  53         Real E4 = p4 - 20*p2/static_cast<Real>(27) + 14/static_cast<Real>(891);
  54         BOOST_CHECK_CLOSE_FRACTION(ls4(x), E4, 250*tol);
  55         BOOST_CHECK_CLOSE_FRACTION(ls4.prime(x), 35*x*(9*x*x -5)/static_cast<Real>(18), 250*tol);
  56
  57         Real p5 = legendre_p(5, x);
  58         Real E5 = p5 - 35*p3/static_cast<Real>(44) + 135*x/static_cast<Real>(12584);
  59         BOOST_CHECK_CLOSE_FRACTION(ls5(x), E5, 29000*tol);
  60         Real E5prime = (315*(123 + 143*x*x*(11*x*x-9)))/static_cast<Real>(12584);
  61         BOOST_CHECK_CLOSE_FRACTION(ls5.prime(x), E5prime, 29000*tol);
  62         x += 1/static_cast<Real>(1 << 9);
  63     }
  64
  65     // Test norm:
  66     // E_1 = x
  67     Real expected_norm_sq = 2*third<Real>();
  68     BOOST_CHECK_CLOSE_FRACTION(expected_norm_sq, ls1.norm_sq(), tol);
  69
  70     // E_2 = P[sub 2](x) - 2P[sup 0](x)/5
  71     expected_norm_sq = 2/static_cast<Real>(5) + 8/static_cast<Real>(25);
  72     BOOST_CHECK_CLOSE_FRACTION(expected_norm_sq, ls2.norm_sq(), tol);
  73
  74     // E_3 = P[sub 3](x) - 9P[sub 1]/14
  75     expected_norm_sq = 2/static_cast<Real>(7) + 9*9*2*third<Real>()/static_cast<Real>(14*14);
  76     BOOST_CHECK_CLOSE_FRACTION(expected_norm_sq, ls3.norm_sq(), tol);
  77
  78     // E_4 = P[sub 4](x) -20P[sub 2](x)/27 + 14P[sub 0](x)/891
  79     expected_norm_sq = static_cast<Real>(2)/static_cast<Real>(9) + static_cast<Real>(20*20*2)/static_cast<Real>(27*27*5) + 14*14*2/static_cast<Real>(891*891);
  80     BOOST_CHECK_CLOSE_FRACTION(expected_norm_sq, ls4.norm_sq(), tol);
  81
  82     // E_5 = P[sub 5](x) - 35P[sub 3](x)/44 + 135P[sub 1](x)/12584
  83     expected_norm_sq = 2/static_cast<Real>(11) + (35*35/static_cast<Real>(44*44))*(2/static_cast<Real>(7)) + (135*135/static_cast<Real>(12584*12584))*2*third<Real>();
  84     BOOST_CHECK_CLOSE_FRACTION(expected_norm_sq, ls5.norm_sq(), tol);
  85
  86     // Only zero of E1 is 0:
  87     std::vector<Real> zeros = ls1.zeros();
  88     BOOST_CHECK(zeros.size() == 1);
  89     BOOST_CHECK_SMALL(zeros[0], tol);
  90     BOOST_CHECK_SMALL(ls1(zeros[0]), tol);
  91
  92     zeros = ls2.zeros();
  93     BOOST_CHECK(zeros.size() == 1);
  94     BOOST_CHECK_CLOSE_FRACTION(zeros[0], sqrt(3/static_cast<Real>(5)), tol);
  95     BOOST_CHECK_SMALL(ls2(zeros[0]), tol);
  96
  97     zeros = ls3.zeros();
  98     BOOST_CHECK(zeros.size() == 2);
  99     BOOST_CHECK_SMALL(zeros[0], tol);
 100     BOOST_CHECK_CLOSE_FRACTION(zeros[1], sqrt(6/static_cast<Real>(7)), tol);
 101
 102
 103     zeros = ls4.zeros();
 104     BOOST_CHECK(zeros.size() == 2);
 105     Real expected = sqrt( (55 - 2*sqrt(static_cast<Real>(330)))/static_cast<Real>(11) )/static_cast<Real>(3);
 106     BOOST_CHECK_CLOSE_FRACTION(zeros[0], expected, tol);
 107
 108     expected = sqrt( (55 + 2*sqrt(static_cast<Real>(330)))/static_cast<Real>(11) )/static_cast<Real>(3);
 109     BOOST_CHECK_CLOSE_FRACTION(zeros[1], expected, 10*tol);
 110
 111
 112     zeros = ls5.zeros();
 113     BOOST_CHECK(zeros.size() == 3);
 114     BOOST_CHECK_SMALL(zeros[0], tol);
 115
 116     expected = sqrt( ( 195 - sqrt(static_cast<Real>(6045)) )/static_cast<Real>(286));
 117     BOOST_CHECK_CLOSE_FRACTION(zeros[1], expected, tol);
 118
 119     expected = sqrt( ( 195 + sqrt(static_cast<Real>(6045)) )/static_cast<Real>(286));
 120     BOOST_CHECK_CLOSE_FRACTION(zeros[2], expected, tol);
 121
 122
 123     for (size_t i = 6; i < 50; ++i)
 124     {
 125         legendre_stieltjes<Real> En(i);
 126         zeros = En.zeros();
 127         for(auto const & zero : zeros)
 128         {
 129             BOOST_CHECK_SMALL(En(zero), 50*tol);
 130         }
 131     }
 132 }
 133
 134
 135 BOOST_AUTO_TEST_CASE(LegendreStieltjesZeros)
 136 {
 137     test_legendre_stieltjes<double>();
 138     test_legendre_stieltjes<long double>();
 139     test_legendre_stieltjes<cpp_bin_float_quad>();
 140     //test_legendre_stieltjes<boost::multiprecision::cpp_bin_float_100>();
 141 }