mirror of
https://github.com/saitohirga/WSJT-X.git
synced 2025-11-03 21:40:52 -05:00
177 lines
6.9 KiB
Plaintext
177 lines
6.9 KiB
Plaintext
[section:rational Polynomial and Rational Function Evaluation]
|
|
|
|
[h4 Synopsis]
|
|
|
|
``
|
|
#include <boost/math/tools/rational.hpp>
|
|
``
|
|
|
|
// Polynomials:
|
|
template <std::size_t N, class T, class V>
|
|
V evaluate_polynomial(const T(&poly)[N], const V& val);
|
|
|
|
template <std::size_t N, class T, class V>
|
|
V evaluate_polynomial(const boost::array<T,N>& poly, const V& val);
|
|
|
|
template <class T, class U>
|
|
U evaluate_polynomial(const T* poly, U z, std::size_t count);
|
|
|
|
// Even polynomials:
|
|
template <std::size_t N, class T, class V>
|
|
V evaluate_even_polynomial(const T(&poly)[N], const V& z);
|
|
|
|
template <std::size_t N, class T, class V>
|
|
V evaluate_even_polynomial(const boost::array<T,N>& poly, const V& z);
|
|
|
|
template <class T, class U>
|
|
U evaluate_even_polynomial(const T* poly, U z, std::size_t count);
|
|
|
|
// Odd polynomials
|
|
template <std::size_t N, class T, class V>
|
|
V evaluate_odd_polynomial(const T(&a)[N], const V& z);
|
|
|
|
template <std::size_t N, class T, class V>
|
|
V evaluate_odd_polynomial(const boost::array<T,N>& a, const V& z);
|
|
|
|
template <class T, class U>
|
|
U evaluate_odd_polynomial(const T* poly, U z, std::size_t count);
|
|
|
|
// Rational Functions:
|
|
template <std::size_t N, class T, class V>
|
|
V evaluate_rational(const T(&a)[N], const T(&b)[N], const V& z);
|
|
|
|
template <std::size_t N, class T, class V>
|
|
V evaluate_rational(const boost::array<T,N>& a, const boost::array<T,N>& b, const V& z);
|
|
|
|
template <class T, class U, class V>
|
|
V evaluate_rational(const T* num, const U* denom, V z, unsigned count);
|
|
|
|
[h4 Description]
|
|
|
|
Each of the functions come in three variants: a pair of overloaded functions
|
|
where the order of the polynomial or rational function is evaluated at
|
|
compile time, and an overload that accepts a runtime variable for the size
|
|
of the coefficient array. Generally speaking, compile time evaluation of the
|
|
array size results in better type safety, is less prone to programmer errors,
|
|
and may result in better optimised code. The polynomial evaluation functions
|
|
in particular, are specialised for various array sizes, allowing for
|
|
loop unrolling, and one hopes, optimal inline expansion.
|
|
|
|
template <std::size_t N, class T, class V>
|
|
V evaluate_polynomial(const T(&poly)[N], const V& val);
|
|
|
|
template <std::size_t N, class T, class V>
|
|
V evaluate_polynomial(const boost::array<T,N>& poly, const V& val);
|
|
|
|
template <class T, class U>
|
|
U evaluate_polynomial(const T* poly, U z, std::size_t count);
|
|
|
|
Evaluates the [@http://en.wikipedia.org/wiki/Polynomial polynomial] described by
|
|
the coefficients stored in /poly/.
|
|
|
|
If the size of the array is specified at runtime, then the polynomial
|
|
most have order /count-1/ with /count/ coefficients. Otherwise it has
|
|
order /N-1/ with /N/ coefficients.
|
|
|
|
Coefficients should be stored such that the coefficients for the x[super i] terms
|
|
are in poly[i].
|
|
|
|
The types of the coefficients and of variable
|
|
/z/ may differ as long as /*poly/ is convertible to type /U/.
|
|
This allows, for example, for the coefficient table
|
|
to be a table of integers if this is appropriate.
|
|
|
|
template <std::size_t N, class T, class V>
|
|
V evaluate_even_polynomial(const T(&poly)[N], const V& z);
|
|
|
|
template <std::size_t N, class T, class V>
|
|
V evaluate_even_polynomial(const boost::array<T,N>& poly, const V& z);
|
|
|
|
template <class T, class U>
|
|
U evaluate_even_polynomial(const T* poly, U z, std::size_t count);
|
|
|
|
As above, but evaluates an even polynomial: one where all the powers
|
|
of /z/ are even numbers. Equivalent to calling
|
|
`evaluate_polynomial(poly, z*z, count)`.
|
|
|
|
template <std::size_t N, class T, class V>
|
|
V evaluate_odd_polynomial(const T(&a)[N], const V& z);
|
|
|
|
template <std::size_t N, class T, class V>
|
|
V evaluate_odd_polynomial(const boost::array<T,N>& a, const V& z);
|
|
|
|
template <class T, class U>
|
|
U evaluate_odd_polynomial(const T* poly, U z, std::size_t count);
|
|
|
|
As above but evaluates a polynomial where all the powers are odd numbers.
|
|
Equivalent to `evaluate_polynomial(poly+1, z*z, count-1) * z + poly[0]`.
|
|
|
|
template <std::size_t N, class T, class U, class V>
|
|
V evaluate_rational(const T(&num)[N], const U(&denom)[N], const V& z);
|
|
|
|
template <std::size_t N, class T, class U, class V>
|
|
V evaluate_rational(const boost::array<T,N>& num, const boost::array<U,N>& denom, const V& z);
|
|
|
|
template <class T, class U, class V>
|
|
V evaluate_rational(const T* num, const U* denom, V z, unsigned count);
|
|
|
|
Evaluates the rational function (the ratio of two polynomials) described by
|
|
the coefficients stored in /num/ and /demom/.
|
|
|
|
If the size of the array is specified at runtime then both
|
|
polynomials most have order /count-1/ with /count/ coefficients.
|
|
Otherwise both polynomials have order /N-1/ with /N/ coefficients.
|
|
|
|
Array /num/ describes the numerator, and /demon/ the denominator.
|
|
|
|
Coefficients should be stored such that the coefficients for the x[super i ] terms
|
|
are in num[i] and denom[i].
|
|
|
|
The types of the coefficients and of variable
|
|
/v/ may differ as long as /*num/ and /*denom/ are convertible to type /V/.
|
|
This allows, for example, for one or both of the coefficient tables
|
|
to be a table of integers if this is appropriate.
|
|
|
|
These functions are designed to safely evaluate the result, even when the value
|
|
/z/ is very large. As such they do not take advantage of compile time array
|
|
sizes to make any optimisations. These functions are best reserved for situations
|
|
where /z/ may be large: if you can be sure that numerical overflow will not occur
|
|
then polynomial evaluation with compile-time array sizes may offer slightly
|
|
better performance.
|
|
|
|
[h4 Implementation]
|
|
|
|
Polynomials are evaluated by
|
|
[@http://en.wikipedia.org/wiki/Horner_algorithm Horners method].
|
|
If the array size is known at
|
|
compile time then the functions dispatch to size-specific implementations
|
|
that unroll the evaluation loop.
|
|
|
|
Rational evaluation is by
|
|
[@http://en.wikipedia.org/wiki/Horner_algorithm Horners method]:
|
|
with the two polynomials being evaluated
|
|
in parallel to make the most of the processors floating-point pipeline.
|
|
If /v/ is greater than one, then the polynomials are evaluated in reverse
|
|
order as polynomials in ['1\/v]: this avoids unnecessary numerical overflow when the
|
|
coefficients are large.
|
|
|
|
Both the polynomial and rational function evaluation algorithms can be
|
|
tuned using various configuration macros to provide optimal performance
|
|
for a particular combination of compiler and platform. This includes
|
|
support for second-order Horner's methods. The various options are
|
|
[link math_toolkit.tuning documented here]. However, the performance
|
|
benefits to be gained from these are marginal on most current hardware,
|
|
consequently it's best to run the
|
|
[link math_toolkit.perf_test_app performance test application] before
|
|
changing the default settings.
|
|
|
|
[endsect] [/section:rational Polynomial and Rational Function Evaluation]
|
|
|
|
[/
|
|
Copyright 2006 John Maddock and Paul A. Bristow.
|
|
Distributed under 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).
|
|
]
|
|
|