libs/math/example/polynomial_arithmetic.cpp
// 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)
// Copyright Jeremy W. Murphy 2015.
// This file is written to be included from a Quickbook .qbk document.
// It can be compiled by the C++ compiler, and run. Any output can
// also be added here as comment or included or pasted in elsewhere.
// Caution: this file contains Quickbook markup as well as code
// and comments: don't change any of the special comment markups!
//[polynomial_arithmetic_0
/*`First include the essential polynomial header (and others) to make the example:
*/
#include <boost/math/tools/polynomial.hpp>
//] [polynomial_arithmetic_0
#include <boost/array.hpp>
#include <boost/lexical_cast.hpp>
#include <boost/math/tools/assert.hpp>
#include <iostream>
#include <stdexcept>
#include <cmath>
#include <string>
#include <utility>
//[polynomial_arithmetic_1
/*`and some using statements are convenient:
*/
using std::string;
using std::exception;
using std::cout;
using std::abs;
using std::pair;
using namespace boost::math;
using namespace boost::math::tools; // for polynomial
using boost::lexical_cast;
//] [/polynomial_arithmetic_1]
template <typename T>
string sign_str(T const &x)
{
return x < 0 ? "-" : "+";
}
template <typename T>
string inner_coefficient(T const &x)
{
string result(" " + sign_str(x) + " ");
if (abs(x) != T(1))
result += lexical_cast<string>(abs(x));
return result;
}
/*! Output in formula format.
For example: from a polynomial in Boost container storage [ 10, -6, -4, 3 ]
show as human-friendly formula notation: 3x^3 - 4x^2 - 6x + 10.
*/
template <typename T>
string formula_format(polynomial<T> const &a)
{
string result;
if (a.size() == 0)
result += lexical_cast<string>(T(0));
else
{
// First one is a special case as it may need unary negate.
unsigned i = a.size() - 1;
if (a[i] < 0)
result += "-";
if (abs(a[i]) != T(1))
result += lexical_cast<string>(abs(a[i]));
if (i > 0)
{
result += "x";
if (i > 1)
{
result += "^" + lexical_cast<string>(i);
i--;
for (; i != 1; i--)
if (a[i])
result += inner_coefficient(a[i]) + "x^" + lexical_cast<string>(i);
if (a[i])
result += inner_coefficient(a[i]) + "x";
}
i--;
if (a[i])
result += " " + sign_str(a[i]) + " " + lexical_cast<string>(abs(a[i]));
}
}
return result;
} // string formula_format(polynomial<T> const &a)
int main()
{
cout << "Example: Polynomial arithmetic.\n\n";
try
{
//[polynomial_arithmetic_2
/*`Store the coefficients in a convenient way to access them,
then create some polynomials using construction from an iterator range,
and finally output in a 'pretty' formula format.
[tip Although we might conventionally write a polynomial from left to right
in descending order of degree, Boost.Math stores in [*ascending order of degree].]
Read/write for humans: 3x^3 - 4x^2 - 6x + 10
Boost polynomial storage: [ 10, -6, -4, 3 ]
*/
std::array<double, 4> const d3a = {{10, -6, -4, 3}};
polynomial<double> const a(d3a.begin(), d3a.end());
// With C++11 and later, you can also use initializer_list construction.
polynomial<double> const b{{-2.0, 1.0}};
// formula_format() converts from Boost storage to human notation.
cout << "a = " << formula_format(a)
<< "\nb = " << formula_format(b) << "\n\n";
//] [/polynomial_arithmetic_2]
//[polynomial_arithmetic_3
// Now we can do arithmetic with the usual infix operators: + - * / and %.
polynomial<double> s = a + b;
cout << "a + b = " << formula_format(s) << "\n";
polynomial<double> d = a - b;
cout << "a - b = " << formula_format(d) << "\n";
polynomial<double> p = a * b;
cout << "a * b = " << formula_format(p) << "\n";
polynomial<double> q = a / b;
cout << "a / b = " << formula_format(q) << "\n";
polynomial<double> r = a % b;
cout << "a % b = " << formula_format(r) << "\n";
//] [/polynomial_arithmetic_3]
//[polynomial_arithmetic_4
/*`
Division is a special case where you can calculate two for the price of one.
Actually, quotient and remainder are always calculated together due to the nature
of the algorithm: the infix operators return one result and throw the other
away.
If you are doing a lot of division and want both the quotient and remainder, then
you don't want to do twice the work necessary.
In that case you can call the underlying function, [^quotient_remainder],
to get both results together as a pair.
*/
pair< polynomial<double>, polynomial<double> > result;
result = quotient_remainder(a, b);
// Reassure ourselves that the result is the same.
BOOST_MATH_ASSERT(result.first == q);
BOOST_MATH_ASSERT(result.second == r);
//] [/polynomial_arithmetic_4]
//[polynomial_arithmetic_5
/*
We can use the right and left shift operators to add and remove a factor of x.
This has the same semantics as left and right shift for integers where it is a
factor of 2. x is the smallest prime factor of a polynomial as is 2 for integers.
*/
cout << "Right and left shift operators.\n";
cout << "\n" << formula_format(p) << "\n";
cout << "... right shift by 1 ...\n";
p >>= 1;
cout << formula_format(p) << "\n";
cout << "... left shift by 2 ...\n";
p <<= 2;
cout << formula_format(p) << "\n";
/*
We can also give a meaning to odd and even for a polynomial that is consistent
with these operations: a polynomial is odd if it has a non-zero constant value,
even otherwise. That is:
x^2 + 1 odd
x^2 even
*/
cout << std::boolalpha;
cout << "\nPrint whether a polynomial is odd.\n";
cout << formula_format(s) << " odd? " << odd(s) << "\n";
// We cheekily use the internal details to subtract the constant, making it even.
s -= s.data().front();
cout << formula_format(s) << " odd? " << odd(s) << "\n";
// And of course you can check if it is even:
cout << formula_format(s) << " even? " << even(s) << "\n";
//] [/polynomial_arithmetic_5]
//[polynomial_arithmetic_6]
/* For performance and convenience, we can test whether a polynomial is zero
* by implicitly converting to bool with the same semantics as int. */
polynomial<double> zero; // Default construction is 0.
cout << "zero: " << (zero ? "not zero" : "zero") << "\n";
cout << "r: " << (r ? "not zero" : "zero") << "\n";
/* We can also set a polynomial to zero without needing a another zero
* polynomial to assign to it. */
r.set_zero();
cout << "r: " << (r ? "not zero" : "zero") << "\n";
//] [/polynomial_arithmetic_6]
}
catch (exception const &e)
{
cout << "\nMessage from thrown exception was:\n " << e.what() << "\n";
}
return 0;
} // int main()
/*
//[polynomial_output_1
a = 3x^3 - 4x^2 - 6x + 10
b = x - 2
//] [/polynomial_output_1]
//[polynomial_output_2
a + b = 3x^3 - 4x^2 - 5x + 8
a - b = 3x^3 - 4x^2 - 7x + 12
a * b = 3x^4 - 10x^3 + 2x^2 + 22x - 20
a / b = 3x^2 + 2x - 2
a % b = 6
//] [/polynomial_output_2]
*/