mirror of
https://github.com/saitohirga/WSJT-X.git
synced 2025-10-30 20:40:28 -04:00
356 lines
15 KiB
C++
356 lines
15 KiB
C++
|
// Copyright Paul A. Bristow 2010.
|
||
|
|
// Copyright John Maddock 2010.
|
||
|
|
|
||
|
|
// Use, modification and distribution are subject to 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)
|
||
|
|
|
||
|
|
#ifdef _MSC_VER
|
||
|
|
# pragma warning (disable : 4224) // nonstandard extension used : formal parameter 'type' was previously defined as a type
|
||
|
|
// in Boost.test and lexical_cast
|
||
|
|
# pragma warning (disable : 4310) // cast truncates constant value
|
||
|
|
# pragma warning (disable : 4512) // assignment operator could not be generated
|
||
|
|
|
||
|
|
#endif
|
||
|
|
|
||
|
|
//#include <pch.hpp> // include directory libs/math/src/tr1/ is needed.
|
||
|
|
|
||
|
|
#include <boost/math/tools/test.hpp>
|
||
|
|
#include <boost/math/concepts/real_concept.hpp> // for real_concept
|
||
|
|
#define BOOST_TEST_MAIN
|
||
|
|
#include <boost/test/unit_test.hpp> // Boost.Test
|
||
|
|
#include <boost/test/floating_point_comparison.hpp>
|
||
|
|
|
||
|
|
#include <boost/math/distributions/inverse_gaussian.hpp>
|
||
|
|
using boost::math::inverse_gaussian_distribution;
|
||
|
|
using boost::math::inverse_gaussian;
|
||
|
|
|
||
|
|
#include <boost/math/tools/test.hpp>
|
||
|
|
#include "test_out_of_range.hpp"
|
||
|
|
|
||
|
|
#include <iostream>
|
||
|
|
#include <iomanip>
|
||
|
|
using std::cout;
|
||
|
|
using std::endl;
|
||
|
|
using std::setprecision;
|
||
|
|
#include <limits>
|
||
|
|
using std::numeric_limits;
|
||
|
|
|
||
|
|
template <class RealType>
|
||
|
|
void check_inverse_gaussian(RealType mean, RealType scale, RealType x, RealType p, RealType q, RealType tol)
|
||
|
|
{
|
||
|
|
using boost::math::inverse_gaussian_distribution;
|
||
|
|
|
||
|
|
BOOST_CHECK_CLOSE_FRACTION(
|
||
|
|
::boost::math::cdf( // Check cdf
|
||
|
|
inverse_gaussian_distribution<RealType>(mean, scale), // distribution.
|
||
|
|
x), // random variable.
|
||
|
|
p, // probability.
|
||
|
|
tol); // tolerance.
|
||
|
|
BOOST_CHECK_CLOSE_FRACTION(
|
||
|
|
::boost::math::cdf( // Check cdf complement
|
||
|
|
complement(
|
||
|
|
inverse_gaussian_distribution<RealType>(mean, scale), // distribution.
|
||
|
|
x)), // random variable.
|
||
|
|
q, // probability complement.
|
||
|
|
tol); // %tolerance.
|
||
|
|
BOOST_CHECK_CLOSE_FRACTION(
|
||
|
|
::boost::math::quantile( // Check quantile
|
||
|
|
inverse_gaussian_distribution<RealType>(mean, scale), // distribution.
|
||
|
|
p), // probability.
|
||
|
|
x, // random variable.
|
||
|
|
tol); // tolerance.
|
||
|
|
BOOST_CHECK_CLOSE_FRACTION(
|
||
|
|
::boost::math::quantile( // Check quantile complement
|
||
|
|
complement(
|
||
|
|
inverse_gaussian_distribution<RealType>(mean, scale), // distribution.
|
||
|
|
q)), // probability complement.
|
||
|
|
x, // random variable.
|
||
|
|
tol); // tolerance.
|
||
|
|
|
||
|
|
inverse_gaussian_distribution<RealType> dist (mean, scale);
|
||
|
|
|
||
|
|
if((p < 0.999) && (q < 0.999))
|
||
|
|
{ // We can only check this if P is not too close to 1,
|
||
|
|
// so that we can guarantee Q is accurate:
|
||
|
|
BOOST_CHECK_CLOSE_FRACTION(
|
||
|
|
cdf(complement(dist, x)), q, tol); // 1 - cdf
|
||
|
|
BOOST_CHECK_CLOSE_FRACTION(
|
||
|
|
quantile(dist, p), x, tol); // quantile(cdf) = x
|
||
|
|
BOOST_CHECK_CLOSE_FRACTION(
|
||
|
|
quantile(complement(dist, q)), x, tol); // quantile(complement(1 - cdf)) = x
|
||
|
|
}
|
||
|
|
}
|
||
|
|
|
||
|
|
template <class RealType>
|
||
|
|
void test_spots(RealType)
|
||
|
|
{
|
||
|
|
// Basic sanity checks
|
||
|
|
RealType tolerance = static_cast<RealType>(1e-4L); //
|
||
|
|
cout << "Tolerance for type " << typeid(RealType).name() << " is " << tolerance << endl;
|
||
|
|
|
||
|
|
// Check some bad parameters to the distribution,
|
||
|
|
#ifndef BOOST_NO_EXCEPTIONS
|
||
|
|
BOOST_MATH_CHECK_THROW(boost::math::inverse_gaussian_distribution<RealType> nbad1(0, 0), std::domain_error); // zero scale
|
||
|
|
BOOST_MATH_CHECK_THROW(boost::math::inverse_gaussian_distribution<RealType> nbad1(0, -1), std::domain_error); // negative scale
|
||
|
|
#else
|
||
|
|
BOOST_MATH_CHECK_THROW(boost::math::inverse_gaussian_distribution<RealType>(0, 0), std::domain_error); // zero scale
|
||
|
|
BOOST_MATH_CHECK_THROW(boost::math::inverse_gaussian_distribution<RealType>(0, -1), std::domain_error); // negative scale
|
||
|
|
#endif
|
||
|
|
|
||
|
|
inverse_gaussian_distribution<RealType> w11;
|
||
|
|
|
||
|
|
// Error tests:
|
||
|
|
check_out_of_range<inverse_gaussian_distribution<RealType> >(0.25, 1);
|
||
|
|
|
||
|
|
// Check complements.
|
||
|
|
|
||
|
|
BOOST_CHECK_CLOSE_FRACTION(
|
||
|
|
cdf(complement(w11, 1.)), static_cast<RealType>(1) - cdf(w11, 1.), tolerance); // cdf complement
|
||
|
|
// cdf(complement = 1 - cdf - but if cdf near unity, then loss of accuracy in cdf,
|
||
|
|
// but cdf complement is near zero but more accurate.
|
||
|
|
|
||
|
|
BOOST_CHECK_CLOSE_FRACTION( // quantile(complement p) == quantile(1 - p)
|
||
|
|
quantile(complement(w11, static_cast<RealType>(0.5))),
|
||
|
|
quantile(w11, 1 - static_cast<RealType>(0.5)),
|
||
|
|
tolerance); // cdf complement
|
||
|
|
|
||
|
|
check_inverse_gaussian(
|
||
|
|
static_cast<RealType>(2),
|
||
|
|
static_cast<RealType>(3),
|
||
|
|
static_cast<RealType>(1),
|
||
|
|
static_cast<RealType>(0.28738674440477374),
|
||
|
|
static_cast<RealType>(1 - 0.28738674440477374),
|
||
|
|
tolerance);
|
||
|
|
|
||
|
|
RealType tolfeweps = boost::math::tools::epsilon<RealType>() * 5;
|
||
|
|
|
||
|
|
inverse_gaussian_distribution<RealType> dist(2, 3);
|
||
|
|
|
||
|
|
using namespace std; // ADL of std names.
|
||
|
|
// mean:
|
||
|
|
BOOST_CHECK_CLOSE_FRACTION(mean(dist),
|
||
|
|
static_cast<RealType>(2), tolfeweps);
|
||
|
|
BOOST_CHECK_CLOSE_FRACTION(scale(dist),
|
||
|
|
static_cast<RealType>(3), tolfeweps);
|
||
|
|
|
||
|
|
// variance:
|
||
|
|
BOOST_CHECK_CLOSE_FRACTION(variance(dist),
|
||
|
|
static_cast<RealType>(2.6666666666666666666666666666666666666666666666666666666667L), 1000*tolfeweps);
|
||
|
|
// std deviation:
|
||
|
|
BOOST_CHECK_CLOSE_FRACTION(standard_deviation(dist),
|
||
|
|
static_cast<RealType>(1.632993L), 1000 * tolerance);
|
||
|
|
//// hazard:
|
||
|
|
//BOOST_CHECK_CLOSE_FRACTION(hazard(dist, x),
|
||
|
|
// pdf(dist, x) / cdf(complement(dist, x)), tolerance);
|
||
|
|
//// cumulative hazard:
|
||
|
|
//BOOST_CHECK_CLOSE_FRACTION(chf(dist, x),
|
||
|
|
// -log(cdf(complement(dist, x))), tolerance);
|
||
|
|
// coefficient_of_variation:
|
||
|
|
BOOST_CHECK_CLOSE_FRACTION(coefficient_of_variation(dist),
|
||
|
|
standard_deviation(dist) / mean(dist), tolerance);
|
||
|
|
// mode:
|
||
|
|
BOOST_CHECK_CLOSE_FRACTION(mode(dist),
|
||
|
|
static_cast<RealType>(0.8284271L), tolerance);
|
||
|
|
|
||
|
|
// median
|
||
|
|
BOOST_CHECK_CLOSE_FRACTION(median(dist),
|
||
|
|
static_cast<RealType>(1.5122506636053668L), tolerance);
|
||
|
|
// Fails for real_concept - because std::numeric_limits<RealType>::digits = 0
|
||
|
|
|
||
|
|
// skewness:
|
||
|
|
BOOST_CHECK_CLOSE_FRACTION(skewness(dist),
|
||
|
|
static_cast<RealType>(2.449490L), tolerance);
|
||
|
|
// kurtosis:
|
||
|
|
BOOST_CHECK_CLOSE_FRACTION(kurtosis(dist),
|
||
|
|
static_cast<RealType>(10-3), tolerance);
|
||
|
|
BOOST_CHECK_CLOSE_FRACTION(kurtosis_excess(dist),
|
||
|
|
static_cast<RealType>(10), tolerance);
|
||
|
|
} // template <class RealType>void test_spots(RealType)
|
||
|
|
|
||
|
|
BOOST_AUTO_TEST_CASE( test_main )
|
||
|
|
{
|
||
|
|
using boost::math::inverse_gaussian;
|
||
|
|
using boost::math::inverse_gaussian_distribution;
|
||
|
|
|
||
|
|
//int precision = 17; // std::numeric_limits<double::max_digits10;
|
||
|
|
double tolfeweps = numeric_limits<double>::epsilon() * 5;
|
||
|
|
//double tol6decdigits = numeric_limits<float>::epsilon() * 2;
|
||
|
|
// Check that can generate inverse_gaussian distribution using the two convenience methods:
|
||
|
|
boost::math::inverse_gaussian w12(1., 2); // Using typedef
|
||
|
|
inverse_gaussian_distribution<> w23(2., 3); // Using default RealType double.
|
||
|
|
boost::math::inverse_gaussian w11; // Use default unity values for mean and scale.
|
||
|
|
// Note NOT myn01() as the compiler will interpret as a function!
|
||
|
|
BOOST_CHECK_EQUAL(w11.mean(), 1);
|
||
|
|
BOOST_CHECK_EQUAL(w11.scale(), 1);
|
||
|
|
BOOST_CHECK_EQUAL(w23.mean(), 2);
|
||
|
|
BOOST_CHECK_EQUAL(w23.scale(), 3);
|
||
|
|
BOOST_CHECK_EQUAL(w23.shape(), 1.5L);
|
||
|
|
|
||
|
|
// Check the synonyms, provided to allow generic use of find_location and find_scale.
|
||
|
|
BOOST_CHECK_EQUAL(w11.mean(), w11.location());
|
||
|
|
BOOST_CHECK_EQUAL(w11.scale(), w11.scale());
|
||
|
|
|
||
|
|
BOOST_CHECK_CLOSE_FRACTION(mean(w11), static_cast<double>(1), tolfeweps); // Default mean == unity
|
||
|
|
BOOST_CHECK_CLOSE_FRACTION(scale(w11), static_cast<double>(1), tolfeweps); // Default mean == unity
|
||
|
|
|
||
|
|
// median
|
||
|
|
// (test double because fails for real_concept because numeric_limits<real_concept>::digits = 0)
|
||
|
|
BOOST_CHECK_CLOSE_FRACTION(median(w11),
|
||
|
|
static_cast<double>(0.67584130569523893), tolfeweps);
|
||
|
|
BOOST_CHECK_CLOSE_FRACTION(median(w23),
|
||
|
|
static_cast<double>(1.5122506636053668), tolfeweps);
|
||
|
|
|
||
|
|
// Initial spot tests using double values from R.
|
||
|
|
// library(SuppDists)
|
||
|
|
// formatC(SuppDists::dinverse_gaussian(1, 1, 1), digits=17) ...
|
||
|
|
BOOST_CHECK_CLOSE_FRACTION( // x = 1
|
||
|
|
pdf(w11, 1.), static_cast<double>(0.3989422804014327), tolfeweps); // pdf
|
||
|
|
BOOST_CHECK_CLOSE_FRACTION(
|
||
|
|
cdf(w11, 1.), static_cast<double>(0.66810200122317065), 10 * tolfeweps); // cdf
|
||
|
|
|
||
|
|
BOOST_CHECK_CLOSE_FRACTION(
|
||
|
|
pdf(w11, 0.1), static_cast<double>(0.21979480031862672), tolfeweps); // pdf
|
||
|
|
BOOST_CHECK_CLOSE_FRACTION(
|
||
|
|
cdf(w11, 0.1), static_cast<double>(0.0040761113207110162), 10 * tolfeweps); // cdf
|
||
|
|
|
||
|
|
BOOST_CHECK_CLOSE_FRACTION( // small x
|
||
|
|
pdf(w11, 0.01), static_cast<double>(2.0811768202028392e-19), tolfeweps); // pdf
|
||
|
|
BOOST_CHECK_CLOSE_FRACTION(
|
||
|
|
cdf(w11, 0.01), static_cast<double>(4.122313403318778e-23), 10 * tolfeweps); // cdf
|
||
|
|
|
||
|
|
BOOST_CHECK_CLOSE_FRACTION( // smaller x
|
||
|
|
pdf(w11, 0.001), static_cast<double>(2.4420044378793562e-213), tolfeweps); // pdf
|
||
|
|
BOOST_CHECK_CLOSE_FRACTION(
|
||
|
|
cdf(w11, 0.001), static_cast<double>(4.8791443010851493e-219), 1000 * tolfeweps); // cdf
|
||
|
|
// 4.8791443010859224e-219 versus 4.8791443010851493e-219 so still 14 decimal digits.
|
||
|
|
|
||
|
|
BOOST_CHECK_CLOSE_FRACTION(
|
||
|
|
quantile(w11, 0.66810200122317065), static_cast<double>(1.), 1 * tolfeweps); // cdf
|
||
|
|
BOOST_CHECK_CLOSE_FRACTION(
|
||
|
|
quantile(w11, 0.0040761113207110162), static_cast<double>(0.1), 1 * tolfeweps); // cdf
|
||
|
|
BOOST_CHECK_CLOSE_FRACTION(
|
||
|
|
quantile(w11, 4.122313403318778e-23), 0.01, 1 * tolfeweps); // quantile
|
||
|
|
BOOST_CHECK_CLOSE_FRACTION(
|
||
|
|
quantile(w11, 2.4420044378793562e-213), 0.001, 0.03); // quantile
|
||
|
|
// quantile 0.001026926242348481 compared to expected 0.001, so much less accurate,
|
||
|
|
// but better than R that gives up completely!
|
||
|
|
// R Error in SuppDists::qinverse_gaussian(4.87914430108515e-219, 1, 1) : Infinite value in NewtonRoot()
|
||
|
|
|
||
|
|
BOOST_CHECK_CLOSE_FRACTION(
|
||
|
|
pdf(w11, 0.5), static_cast<double>(0.87878257893544476), tolfeweps); // pdf
|
||
|
|
BOOST_CHECK_CLOSE_FRACTION(
|
||
|
|
cdf(w11, 0.5), static_cast<double>(0.3649755481729598), tolfeweps); // cdf
|
||
|
|
|
||
|
|
BOOST_CHECK_CLOSE_FRACTION(
|
||
|
|
pdf(w11, 2), static_cast<double>(0.10984782236693059), tolfeweps); // pdf
|
||
|
|
BOOST_CHECK_CLOSE_FRACTION(
|
||
|
|
cdf(w11, 2), static_cast<double>(.88547542598600637), tolfeweps); // cdf
|
||
|
|
|
||
|
|
BOOST_CHECK_CLOSE_FRACTION(
|
||
|
|
pdf(w11, 10), static_cast<double>(0.00021979480031862676), tolfeweps); // pdf
|
||
|
|
BOOST_CHECK_CLOSE_FRACTION(
|
||
|
|
cdf(w11, 10), static_cast<double>(0.99964958546279115), tolfeweps); // cdf
|
||
|
|
|
||
|
|
BOOST_CHECK_CLOSE_FRACTION(
|
||
|
|
pdf(w11, 100), static_cast<double>(2.0811768202028246e-25), tolfeweps); // pdf
|
||
|
|
BOOST_CHECK_CLOSE_FRACTION(
|
||
|
|
cdf(w11, 100), static_cast<double>(1), tolfeweps); // cdf
|
||
|
|
BOOST_CHECK_CLOSE_FRACTION(
|
||
|
|
pdf(w11, 1000), static_cast<double>(2.4420044378793564e-222), 10 * tolfeweps); // pdf
|
||
|
|
BOOST_CHECK_CLOSE_FRACTION(
|
||
|
|
cdf(w11, 1000), static_cast<double>(1.), tolfeweps); // cdf
|
||
|
|
|
||
|
|
// A few more misc tests, probably not very useful.
|
||
|
|
BOOST_CHECK_CLOSE_FRACTION(
|
||
|
|
cdf(w11, 1.), static_cast<double>(0.66810200122317065), tolfeweps); // cdf
|
||
|
|
BOOST_CHECK_CLOSE_FRACTION(
|
||
|
|
cdf(w11, 0.1), static_cast<double>(0.0040761113207110162), tolfeweps * 5); // cdf
|
||
|
|
// 0.0040761113207110162 0.0040761113207110362
|
||
|
|
BOOST_CHECK_CLOSE_FRACTION(
|
||
|
|
cdf(w11, 0.2), static_cast<double>(0.063753567519976254), tolfeweps * 5); // cdf
|
||
|
|
BOOST_CHECK_CLOSE_FRACTION(
|
||
|
|
cdf(w11, 0.5), static_cast<double>(0.3649755481729598), tolfeweps); // cdf
|
||
|
|
|
||
|
|
BOOST_CHECK_CLOSE_FRACTION(
|
||
|
|
cdf(w11, 0.9), static_cast<double>(0.62502320258649202), tolfeweps); // cdf
|
||
|
|
BOOST_CHECK_CLOSE_FRACTION(
|
||
|
|
cdf(w11, 0.99), static_cast<double>(0.66408247396139031), tolfeweps); // cdf
|
||
|
|
BOOST_CHECK_CLOSE_FRACTION(
|
||
|
|
cdf(w11, 0.999), static_cast<double>(0.66770275955311675), tolfeweps); // cdf
|
||
|
|
BOOST_CHECK_CLOSE_FRACTION(
|
||
|
|
cdf(w11, 10.), static_cast<double>(0.99964958546279115), tolfeweps); // cdf
|
||
|
|
BOOST_CHECK_CLOSE_FRACTION(
|
||
|
|
cdf(w11, 50.), static_cast<double>(0.99999999999992029), tolfeweps); // cdf
|
||
|
|
|
||
|
|
BOOST_CHECK_CLOSE_FRACTION(
|
||
|
|
quantile(w11, 0.3649755481729598), static_cast<double>(0.5), tolfeweps); // quantile
|
||
|
|
BOOST_CHECK_CLOSE_FRACTION(
|
||
|
|
quantile(w11, 0.62502320258649202), static_cast<double>(0.9), tolfeweps); // quantile
|
||
|
|
BOOST_CHECK_CLOSE_FRACTION(
|
||
|
|
quantile(w11, 0.0040761113207110162), static_cast<double>(0.1), tolfeweps); // quantile
|
||
|
|
|
||
|
|
// Wald(2,3) tests
|
||
|
|
// ===================
|
||
|
|
BOOST_CHECK_CLOSE_FRACTION( // formatC(SuppDists::dinvGauss(1, 2, 3), digits=17) "0.47490884963330904"
|
||
|
|
pdf(w23, 1.), static_cast<double>(0.47490884963330904), tolfeweps ); // pdf
|
||
|
|
|
||
|
|
BOOST_CHECK_CLOSE_FRACTION(
|
||
|
|
pdf(w23, 0.1), static_cast<double>(2.8854207087665401e-05), tolfeweps * 2); // pdf
|
||
|
|
//2.8854207087665452e-005 2.8854207087665401e-005
|
||
|
|
BOOST_CHECK_CLOSE_FRACTION(
|
||
|
|
pdf(w23, 10.), static_cast<double>(0.0019822751498574636), tolfeweps); // pdf
|
||
|
|
BOOST_CHECK_CLOSE_FRACTION(
|
||
|
|
pdf(w23, 10.), static_cast<double>(0.0019822751498574636), tolfeweps); // pdf
|
||
|
|
|
||
|
|
// Bigger changes in mean and scale.
|
||
|
|
|
||
|
|
inverse_gaussian w012(0.1, 2);
|
||
|
|
BOOST_CHECK_CLOSE_FRACTION(
|
||
|
|
pdf(w012, 1.), static_cast<double>(3.7460367141230404e-36), tolfeweps ); // pdf
|
||
|
|
BOOST_CHECK_CLOSE_FRACTION(
|
||
|
|
cdf(w012, 1.), static_cast<double>(1), tolfeweps ); // pdf
|
||
|
|
|
||
|
|
inverse_gaussian w0110(0.1, 10);
|
||
|
|
BOOST_CHECK_CLOSE_FRACTION(
|
||
|
|
pdf(w0110, 1.), static_cast<double>(1.6279643678071011e-176), 100 * tolfeweps ); // pdf
|
||
|
|
BOOST_CHECK_CLOSE_FRACTION(
|
||
|
|
cdf(w0110, 1.), static_cast<double>(1), tolfeweps ); // cdf
|
||
|
|
BOOST_CHECK_CLOSE_FRACTION(
|
||
|
|
cdf(complement(w0110, 1.)), static_cast<double>(3.2787685715328683e-179), 1e6 * tolfeweps ); // cdf complement
|
||
|
|
// Differs because of loss of accuracy.
|
||
|
|
|
||
|
|
BOOST_CHECK_CLOSE_FRACTION(
|
||
|
|
pdf(w0110, 0.1), static_cast<double>(39.894228040143268), tolfeweps ); // pdf
|
||
|
|
BOOST_CHECK_CLOSE_FRACTION(
|
||
|
|
cdf(w0110, 0.1), static_cast<double>(0.51989761564832704), 10 * tolfeweps ); // cdf
|
||
|
|
|
||
|
|
// Basic sanity-check spot values for all floating-point types..
|
||
|
|
// (Parameter value, arbitrarily zero, only communicates the floating point type).
|
||
|
|
test_spots(0.0F); // Test float. OK at decdigits = 0 tolerance = 0.0001 %
|
||
|
|
test_spots(0.0); // Test double. OK at decdigits 7, tolerance = 1e07 %
|
||
|
|
#ifndef BOOST_MATH_NO_LONG_DOUBLE_MATH_FUNCTIONS
|
||
|
|
test_spots(0.0L); // Test long double.
|
||
|
|
#ifndef BOOST_MATH_NO_REAL_CONCEPT_TESTS
|
||
|
|
test_spots(boost::math::concepts::real_concept(0.)); // Test real concept.
|
||
|
|
#endif
|
||
|
|
#else
|
||
|
|
std::cout << "<note>The long double tests have been disabled on this platform "
|
||
|
|
"either because the long double overloads of the usual math functions are "
|
||
|
|
"not available at all, or because they are too inaccurate for these tests "
|
||
|
|
"to pass.</note>" << std::endl;
|
||
|
|
#endif
|
||
|
|
/* */
|
||
|
|
|
||
|
|
} // BOOST_AUTO_TEST_CASE( test_main )
|
||
|
|
|
||
|
|
/*
|
||
|
|
|
||
|
|
Output:
|
||
|
|
|
||
|
|
|
||
|
|
*/
|
||
|
|
|
||
|
|
|