1 // Copyright Paul A. Bristow 2014, 2015.
2
3 // Use, modification and distribution are subject to the
4 // Boost Software License, Version 1.0.
5 // (See accompanying file LICENSE_1_0.txt
6 // or copy at http://www.boost.org/LICENSE_1_0.txt)
7
8 // Note that this file contains Quickbook mark-up as well as code
9 // and comments, don't change any of the special comment mark-ups!
10
11 // Example of finding nth root using 1st and 2nd derivatives of x^n.
12
13 #include <boost/math/tools/roots.hpp>
14 //using boost::math::policies::policy;
15 //using boost::math::tools::newton_raphson_iterate;
16 //using boost::math::tools::halley_iterate;
17 //using boost::math::tools::eps_tolerance; // Binary functor for specified number of bits.
18 //using boost::math::tools::bracket_and_solve_root;
19 //using boost::math::tools::toms748_solve;
20
21 #include <boost/math/special_functions/next.hpp>
22 #include <boost/multiprecision/cpp_dec_float.hpp>
23 #include <boost/math/special_functions/pow.hpp>
24 #include <boost/math/constants/constants.hpp>
25
26 #include <boost/multiprecision/cpp_dec_float.hpp> // For cpp_dec_float_50.
27 #include <boost/multiprecision/cpp_bin_float.hpp> // using boost::multiprecision::cpp_bin_float_50;
28 #ifndef _MSC_VER // float128 is not yet supported by Microsoft compiler at 2013.
29 # include <boost/multiprecision/float128.hpp>
30 #endif
31
32 #include <iostream>
33 // using std::cout; using std::endl;
34 #include <iomanip>
35 // using std::setw; using std::setprecision;
36 #include <limits>
37 using std::numeric_limits;
38 #include <tuple>
39 #include <utility> // pair, make_pair
40
41
42 //[root_finding_nth_functor_2deriv
43 template <int N, class T = double>
44 struct nth_functor_2deriv
45 { // Functor returning both 1st and 2nd derivatives.
46 BOOST_STATIC_ASSERT_MSG(boost::is_integral<T>::value == false, "Only floating-point type types can be used!");
47 BOOST_STATIC_ASSERT_MSG((N > 0) == true, "root N must be > 0!");
48
nth_functor_2derivnth_functor_2deriv49 nth_functor_2deriv(T const& to_find_root_of) : a(to_find_root_of)
50 { /* Constructor stores value a to find root of, for example: */ }
51
52 // using boost::math::tuple; // to return three values.
operator ()nth_functor_2deriv53 std::tuple<T, T, T> operator()(T const& x)
54 {
55 // Return f(x), f'(x) and f''(x).
56 using boost::math::pow;
57 T fx = pow<N>(x) - a; // Difference (estimate x^n - a).
58 T dx = N * pow<N - 1>(x); // 1st derivative f'(x).
59 T d2x = N * (N - 1) * pow<N - 2 >(x); // 2nd derivative f''(x).
60
61 return std::make_tuple(fx, dx, d2x); // 'return' fx, dx and d2x.
62 }
63 private:
64 T a; // to be 'nth_rooted'.
65 };
66
67 //] [/root_finding_nth_functor_2deriv]
68
69 /*
70 To show the progress, one might use this before the return statement above?
71 #ifdef BOOST_MATH_ROOT_DIAGNOSTIC
72 std::cout << " x = " << x << ", fx = " << fx << ", dx = " << dx << ", dx2 = " << d2x << std::endl;
73 #endif
74 */
75
76 // If T is a floating-point type, might be quicker to compute the guess using a built-in type,
77 // probably quickest using double, but perhaps with float or long double, T.
78
79 // If T is a type for which frexp and ldexp are not defined,
80 // then it is necessary to compute the guess using a built-in type,
81 // probably quickest (but limited range) using double,
82 // but perhaps with float or long double, or a multiprecision T for the full range of T.
83 // typedef double guess_type; is used to specify the this.
84
85 //[root_finding_nth_function_2deriv
86
87 template <int N, class T = double>
nth_2deriv(T x)88 T nth_2deriv(T x)
89 { // return nth root of x using 1st and 2nd derivatives and Halley.
90
91 using namespace std; // Help ADL of std functions.
92 using namespace boost::math::tools; // For halley_iterate.
93
94 BOOST_STATIC_ASSERT_MSG(boost::is_integral<T>::value == false, "Only floating-point type types can be used!");
95 BOOST_STATIC_ASSERT_MSG((N > 0) == true, "root N must be > 0!");
96 BOOST_STATIC_ASSERT_MSG((N > 1000) == false, "root N is too big!");
97
98 typedef double guess_type; // double may restrict (exponent) range for a multiprecision T?
99
100 int exponent;
101 frexp(static_cast<guess_type>(x), &exponent); // Get exponent of z (ignore mantissa).
102 T guess = ldexp(static_cast<guess_type>(1.), exponent / N); // Rough guess is to divide the exponent by n.
103 T min = ldexp(static_cast<guess_type>(1.) / 2, exponent / N); // Minimum possible value is half our guess.
104 T max = ldexp(static_cast<guess_type>(2.), exponent / N); // Maximum possible value is twice our guess.
105
106 int digits = std::numeric_limits<T>::digits * 0.4; // Accuracy triples with each step, so stop when
107 // slightly more than one third of the digits are correct.
108 const boost::uintmax_t maxit = 20;
109 boost::uintmax_t it = maxit;
110 T result = halley_iterate(nth_functor_2deriv<N, T>(x), guess, min, max, digits, it);
111 return result;
112 }
113
114 //] [/root_finding_nth_function_2deriv]
115
116
117 template <int N, typename T = double>
show_nth_root(T value)118 T show_nth_root(T value)
119 { // Demonstrate by printing the nth root using all possibly significant digits.
120 //std::cout.precision(std::numeric_limits<T>::max_digits10);
121 // or use cout.precision(max_digits10 = 2 + std::numeric_limits<double>::digits * 3010/10000);
122 // Or guaranteed significant digits:
123 std::cout.precision(std::numeric_limits<T>::digits10);
124
125 T r = nth_2deriv<N>(value);
126 std::cout << "Type " << typeid(T).name() << " value = " << value << ", " << N << "th root = " << r << std::endl;
127 return r;
128 } // print_nth_root
129
130
main()131 int main()
132 {
133 std::cout << "nth Root finding Example." << std::endl;
134 using boost::multiprecision::cpp_dec_float_50; // decimal.
135 using boost::multiprecision::cpp_bin_float_50; // binary.
136 #ifndef _MSC_VER // Not supported by Microsoft compiler.
137 using boost::multiprecision::float128; // Requires libquadmath
138 #endif
139 try
140 { // Always use try'n'catch blocks with Boost.Math to get any error messages.
141
142 //[root_finding_n_example_1
143 double r1 = nth_2deriv<5, double>(2); // Integral value converted to double.
144
145 // double r2 = nth_2deriv<5>(2); // Only floating-point type types can be used!
146
147 //] [/root_finding_n_example_1
148
149 //show_nth_root<5, float>(2); // Integral value converted to float.
150 //show_nth_root<5, float>(2.F); // 'initializing' : conversion from 'double' to 'float', possible loss of data
151
152 //[root_finding_n_example_2
153
154
155 show_nth_root<5, double>(2.);
156 show_nth_root<5, long double>(2.);
157 #ifndef _MSC_VER // float128 is not supported by Microsoft compiler 2013.
158 show_nth_root<5, float128>(2);
159 #endif
160 show_nth_root<5, cpp_dec_float_50>(2); // dec
161 show_nth_root<5, cpp_bin_float_50>(2); // bin
162 //] [/root_finding_n_example_2
163
164 // show_nth_root<1000000>(2.); // Type double value = 2, 555th root = 1.00124969405651
165 // Type double value = 2, 1000th root = 1.00069338746258
166 // Type double value = 2, 1000000th root = 1.00000069314783
167 }
168 catch (const std::exception& e)
169 { // Always useful to include try & catch blocks because default policies
170 // are to throw exceptions on arguments that cause errors like underflow, overflow.
171 // Lacking try & catch blocks, the program will abort without a message below,
172 // which may give some helpful clues as to the cause of the exception.
173 std::cout <<
174 "\n""Message from thrown exception was:\n " << e.what() << std::endl;
175 }
176 return 0;
177 } // int main()
178
179
180 /*
181 //[root_finding_example_output_1
182 Using MSVC 2013
183
184 nth Root finding Example.
185 Type double value = 2, 5th root = 1.14869835499704
186 Type long double value = 2, 5th root = 1.14869835499704
187 Type class boost::multiprecision::number<class boost::multiprecision::backends::cpp_dec_float<50,int,void>,1> value = 2,
188 5th root = 1.1486983549970350067986269467779275894438508890978
189 Type class boost::multiprecision::number<class boost::multiprecision::backends::cpp_bin_float<50,10,void,int,0,0>,0> value = 2,
190 5th root = 1.1486983549970350067986269467779275894438508890978
191
192 //] [/root_finding_example_output_1]
193
194 //[root_finding_example_output_2
195
196 Using GCC 4.91 (includes float_128 type)
197
198 nth Root finding Example.
199 Type d value = 2, 5th root = 1.14869835499704
200 Type e value = 2, 5th root = 1.14869835499703501
201 Type N5boost14multiprecision6numberINS0_8backends16float128_backendELNS0_26expression_template_optionE0EEE value = 2, 5th root = 1.148698354997035006798626946777928
202 Type N5boost14multiprecision6numberINS0_8backends13cpp_dec_floatILj50EivEELNS0_26expression_template_optionE1EEE value = 2, 5th root = 1.1486983549970350067986269467779275894438508890978
203 Type N5boost14multiprecision6numberINS0_8backends13cpp_bin_floatILj50ELNS2_15digit_base_typeE10EviLi0ELi0EEELNS0_26expression_template_optionE0EEE value = 2, 5th root = 1.1486983549970350067986269467779275894438508890978
204
205 RUN SUCCESSFUL (total time: 63ms)
206
207 //] [/root_finding_example_output_2]
208 */
209
210 /*
211 Throw out of range using GCC release mode :-(
212
213 */
214