238 lines
7.0 KiB
C++
238 lines
7.0 KiB
C++
// Use, modification and distribution are subject to the
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// Boost Software License, Version 1.0.
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// (See accompanying file LICENSE_1_0.txt
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// or copy at http://www.boost.org/LICENSE_1_0.txt)
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// Copyright Jeremy W. Murphy 2015.
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// This file is written to be included from a Quickbook .qbk document.
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// It can be compiled by the C++ compiler, and run. Any output can
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// also be added here as comment or included or pasted in elsewhere.
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// Caution: this file contains Quickbook markup as well as code
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// and comments: don't change any of the special comment markups!
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//[polynomial_arithmetic_0
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/*`First include the essential polynomial header (and others) to make the example:
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*/
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#include <boost/math/tools/polynomial.hpp>
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//] [polynomial_arithmetic_0
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#include <boost/array.hpp>
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#include <boost/lexical_cast.hpp>
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#include <boost/assert.hpp>
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#include <iostream>
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#include <stdexcept>
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#include <cmath>
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#include <string>
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#include <utility>
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//[polynomial_arithmetic_1
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/*`and some using statements are convenient:
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*/
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using std::string;
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using std::exception;
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using std::cout;
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using std::abs;
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using std::pair;
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using namespace boost::math;
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using namespace boost::math::tools; // for polynomial
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using boost::lexical_cast;
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//] [/polynomial_arithmetic_1]
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template <typename T>
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string sign_str(T const &x)
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{
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return x < 0 ? "-" : "+";
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}
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template <typename T>
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string inner_coefficient(T const &x)
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{
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string result(" " + sign_str(x) + " ");
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if (abs(x) != T(1))
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result += lexical_cast<string>(abs(x));
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return result;
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}
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/*! Output in formula format.
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For example: from a polynomial in Boost container storage [ 10, -6, -4, 3 ]
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show as human-friendly formula notation: 3x^3 - 4x^2 - 6x + 10.
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*/
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template <typename T>
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string formula_format(polynomial<T> const &a)
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{
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string result;
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if (a.size() == 0)
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result += lexical_cast<string>(T(0));
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else
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{
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// First one is a special case as it may need unary negate.
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unsigned i = a.size() - 1;
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if (a[i] < 0)
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result += "-";
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if (abs(a[i]) != T(1))
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result += lexical_cast<string>(abs(a[i]));
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if (i > 0)
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{
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result += "x";
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if (i > 1)
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{
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result += "^" + lexical_cast<string>(i);
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i--;
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for (; i != 1; i--)
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if (a[i])
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result += inner_coefficient(a[i]) + "x^" + lexical_cast<string>(i);
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if (a[i])
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result += inner_coefficient(a[i]) + "x";
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}
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i--;
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if (a[i])
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result += " " + sign_str(a[i]) + " " + lexical_cast<string>(abs(a[i]));
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}
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}
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return result;
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} // string formula_format(polynomial<T> const &a)
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int main()
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{
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cout << "Example: Polynomial arithmetic.\n\n";
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try
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{
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//[polynomial_arithmetic_2
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/*`Store the coefficients in a convenient way to access them,
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then create some polynomials using construction from an iterator range,
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and finally output in a 'pretty' formula format.
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[tip Although we might conventionally write a polynomial from left to right
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in descending order of degree, Boost.Math stores in [*ascending order of degree].]
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Read/write for humans: 3x^3 - 4x^2 - 6x + 10
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Boost polynomial storage: [ 10, -6, -4, 3 ]
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*/
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boost::array<double, 4> const d3a = {{10, -6, -4, 3}};
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polynomial<double> const a(d3a.begin(), d3a.end());
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// With C++11 and later, you can also use initializer_list construction.
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polynomial<double> const b{{-2.0, 1.0}};
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// formula_format() converts from Boost storage to human notation.
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cout << "a = " << formula_format(a)
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<< "\nb = " << formula_format(b) << "\n\n";
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//] [/polynomial_arithmetic_2]
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//[polynomial_arithmetic_3
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// Now we can do arithmetic with the usual infix operators: + - * / and %.
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polynomial<double> s = a + b;
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cout << "a + b = " << formula_format(s) << "\n";
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polynomial<double> d = a - b;
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cout << "a - b = " << formula_format(d) << "\n";
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polynomial<double> p = a * b;
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cout << "a * b = " << formula_format(p) << "\n";
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polynomial<double> q = a / b;
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cout << "a / b = " << formula_format(q) << "\n";
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polynomial<double> r = a % b;
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cout << "a % b = " << formula_format(r) << "\n";
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//] [/polynomial_arithmetic_3]
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//[polynomial_arithmetic_4
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/*`
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Division is a special case where you can calculate two for the price of one.
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Actually, quotient and remainder are always calculated together due to the nature
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of the algorithm: the infix operators return one result and throw the other
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away.
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If you are doing a lot of division and want both the quotient and remainder, then
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you don't want to do twice the work necessary.
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In that case you can call the underlying function, [^quotient_remainder],
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to get both results together as a pair.
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*/
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pair< polynomial<double>, polynomial<double> > result;
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result = quotient_remainder(a, b);
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// Reassure ourselves that the result is the same.
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BOOST_ASSERT(result.first == q);
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BOOST_ASSERT(result.second == r);
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//] [/polynomial_arithmetic_4]
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//[polynomial_arithmetic_5
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/*
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We can use the right and left shift operators to add and remove a factor of x.
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This has the same semantics as left and right shift for integers where it is a
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factor of 2. x is the smallest prime factor of a polynomial as is 2 for integers.
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*/
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cout << "Right and left shift operators.\n";
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cout << "\n" << formula_format(p) << "\n";
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cout << "... right shift by 1 ...\n";
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p >>= 1;
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cout << formula_format(p) << "\n";
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cout << "... left shift by 2 ...\n";
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p <<= 2;
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cout << formula_format(p) << "\n";
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/*
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We can also give a meaning to odd and even for a polynomial that is consistent
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with these operations: a polynomial is odd if it has a non-zero constant value,
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even otherwise. That is:
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x^2 + 1 odd
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x^2 even
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*/
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cout << std::boolalpha;
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cout << "\nPrint whether a polynomial is odd.\n";
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cout << formula_format(s) << " odd? " << odd(s) << "\n";
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// We cheekily use the internal details to subtract the constant, making it even.
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s -= s.data().front();
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cout << formula_format(s) << " odd? " << odd(s) << "\n";
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// And of course you can check if it is even:
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cout << formula_format(s) << " even? " << even(s) << "\n";
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//] [/polynomial_arithmetic_5]
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//[polynomial_arithmetic_6]
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/* For performance and convenience, we can test whether a polynomial is zero
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* by implicitly converting to bool with the same semantics as int. */
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polynomial<double> zero; // Default construction is 0.
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cout << "zero: " << (zero ? "not zero" : "zero") << "\n";
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cout << "r: " << (r ? "not zero" : "zero") << "\n";
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/* We can also set a polynomial to zero without needing a another zero
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* polynomial to assign to it. */
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r.set_zero();
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cout << "r: " << (r ? "not zero" : "zero") << "\n";
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//] [/polynomial_arithmetic_6]
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}
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catch (exception const &e)
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{
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cout << "\nMessage from thrown exception was:\n " << e.what() << "\n";
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}
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} // int main()
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/*
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//[polynomial_output_1
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a = 3x^3 - 4x^2 - 6x + 10
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b = x - 2
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//] [/polynomial_output_1]
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//[polynomial_output_2
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a + b = 3x^3 - 4x^2 - 5x + 8
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a - b = 3x^3 - 4x^2 - 7x + 12
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a * b = 3x^4 - 10x^3 + 2x^2 + 22x - 20
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a / b = 3x^2 + 2x - 2
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a % b = 6
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//] [/polynomial_output_2]
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*/
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