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385 lines
11 KiB
Plaintext
385 lines
11 KiB
Plaintext
[/
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Copyright (c) 2012 John Maddock
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Use, modification and distribution are subject to the
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Boost Software License, Version 1.0. (See accompanying file
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LICENSE_1_0.txt or copy at http://www.boost.org/LICENSE_1_0.txt)
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]
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[section:airy Airy Functions]
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[section:ai Airy Ai Function]
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[heading Synopsis]
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``
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#include <boost/math/special_functions/airy.hpp>
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``
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namespace boost { namespace math {
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template <class T>
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``__sf_result`` airy_ai(T x);
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template <class T, class Policy>
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``__sf_result`` airy_ai(T x, const Policy&);
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}} // namespaces
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[heading Description]
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The function __airy_ai calculates the Airy function Ai which is the first solution
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to the differential equation:
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[equation airy]
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See Weisstein, Eric W. "Airy Functions." From MathWorld--A Wolfram Web Resource.
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[@http://mathworld.wolfram.com/AiryFunctions.html];
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[optional_policy]
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The following graph illustrates how this function changes as /x/ changes: for negative /x/
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the function is cyclic, while for positive /x/ the value tends to zero:
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[graph airy_ai]
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[heading Accuracy]
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This function is implemented entirely in terms of the Bessel functions
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__cyl_bessel_j and __cyl_bessel_k - refer to those functions for detailed accuracy information.
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In general though, the relative error is low (less than 100 [epsilon]) for /x > 0/ while
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only the absolute error is low for /x < 0/.
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[heading Testing]
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Since this function is implemented in terms of other special functions, there are only a few
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basic sanity checks, using test values from [@http://functions.wolfram.com/ Wolfram Airy Functions].
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[heading Implementation]
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This function is implemented in terms of the Bessel functions using the relations:
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[equation airy_ai]
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[endsect]
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[section:bi Airy Bi Function]
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[heading Synopsis]
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``
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#include <boost/math/special_functions/airy.hpp>
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``
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namespace boost { namespace math {
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template <class T>
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``__sf_result`` airy_bi(T x);
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template <class T, class Policy>
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``__sf_result`` airy_bi(T x, const Policy&);
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}} // namespaces
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[heading Description]
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The function __airy_bi calculates the Airy function Bi which is the second solution to the differential equation:
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[equation airy]
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[optional_policy]
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The following graph illustrates how this function changes as /x/ changes: for negative /x/
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the function is cyclic, while for positive /x/ the value tends to infinity:
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[graph airy_bi]
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[heading Accuracy]
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This function is implemented entirely in terms of the Bessel functions
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__cyl_bessel_i and __cyl_bessel_j - refer to those functions for detailed accuracy information.
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In general though, the relative error is low (less than 100 [epsilon]) for /x > 0/ while
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only the absolute error is low for /x < 0/.
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[heading Testing]
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Since this function is implemented in terms of other special functions, there are only a few
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basic sanity checks, using test values from [@http://functions.wolfram.com functions.wolfram.com].
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[heading Implementation]
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This function is implemented in terms of the Bessel functions using the relations:
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[equation airy_bi]
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[endsect]
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[section:aip Airy Ai' Function]
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[heading Synopsis]
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``
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#include <boost/math/special_functions/airy.hpp>
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``
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namespace boost { namespace math {
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template <class T>
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``__sf_result`` airy_ai_prime(T x);
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template <class T, class Policy>
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``__sf_result`` airy_ai_prime(T x, const Policy&);
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}} // namespaces
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[heading Description]
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The function __airy_ai_prime calculates the Airy function Ai' which is the derivative of the first solution to the differential equation:
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[equation airy]
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[optional_policy]
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The following graph illustrates how this function changes as /x/ changes: for negative /x/
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the function is cyclic, while for positive /x/ the value tends to zero:
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[graph airy_aip]
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[heading Accuracy]
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This function is implemented entirely in terms of the Bessel functions
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__cyl_bessel_j and __cyl_bessel_k - refer to those functions for detailed accuracy information.
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In general though, the relative error is low (less than 100 [epsilon]) for /x > 0/ while
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only the absolute error is low for /x < 0/.
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[heading Testing]
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Since this function is implemented in terms of other special functions, there are only a few
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basic sanity checks, using test values from [@http://functions.wolfram.com functions.wolfram.com].
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[heading Implementation]
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This function is implemented in terms of the Bessel functions using the relations:
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[equation airy_aip]
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[endsect]
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[section:bip Airy Bi' Function]
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[heading Synopsis]
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``
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#include <boost/math/special_functions/airy.hpp>
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``
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namespace boost { namespace math {
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template <class T>
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``__sf_result`` airy_bi_prime(T x);
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template <class T, class Policy>
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``__sf_result`` airy_bi_prime(T x, const Policy&);
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}} // namespaces
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[heading Description]
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The function __airy_bi_prime calculates the Airy function Bi' which is the derivative of the second solution to the differential equation:
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[equation airy]
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[optional_policy]
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The following graph illustrates how this function changes as /x/ changes: for negative /x/
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the function is cyclic, while for positive /x/ the value tends to infinity:
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[graph airy_bi]
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[heading Accuracy]
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This function is implemented entirely in terms of the Bessel functions
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__cyl_bessel_i and __cyl_bessel_j - refer to those functions for detailed accuracy information.
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In general though, the relative error is low (less than 100 [epsilon]) for /x > 0/ while
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only the absolute error is low for /x < 0/.
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[heading Testing]
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Since this function is implemented in terms of other special functions, there are only a few
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basic sanity checks, using test values from [@http://functions.wolfram.com functions.wolfram.com].
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[heading Implementation]
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This function is implemented in terms of the Bessel functions using the relations:
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[equation airy_bip]
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[endsect]
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[section:airy_root Finding Zeros of Airy Functions]
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[h4 Synopsis]
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`#include <boost/math/special_functions/airy.hpp>`
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Functions for obtaining both a single zero or root of the Airy functions,
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and placing multiple zeros into a container like `std::vector`
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by providing an output iterator.
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The signature of the single value functions are:
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template <class T>
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T airy_ai_zero(
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int m); // 1-based index of zero.
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template <class T>
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T airy_bi_zero(
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int m); // 1-based index of zero.
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and for multiple zeros:
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template <class T, class OutputIterator>
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OutputIterator airy_ai_zero(
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int start_index, // 1-based index of first zero.
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unsigned number_of_zeros, // How many zeros to generate.
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OutputIterator out_it); // Destination for zeros.
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template <class T, class OutputIterator>
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OutputIterator airy_bi_zero(
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int start_index, // 1-based index of zero.
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unsigned number_of_zeros, // How many zeros to generate
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OutputIterator out_it); // Destination for zeros.
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There are also versions which allow control of the __policy_section for error handling and precision.
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template <class T>
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T airy_ai_zero(
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int m, // 1-based index of zero.
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const Policy&); // Policy to use.
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template <class T>
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T airy_bi_zero(
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int m, // 1-based index of zero.
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const Policy&); // Policy to use.
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template <class T, class OutputIterator>
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OutputIterator airy_ai_zero(
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int start_index, // 1-based index of first zero.
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unsigned number_of_zeros, // How many zeros to generate.
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OutputIterator out_it, // Destination for zeros.
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const Policy& pol); // Policy to use.
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template <class T, class OutputIterator>
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OutputIterator airy_bi_zero(
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int start_index, // 1-based index of zero.
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unsigned number_of_zeros, // How many zeros to generate.
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OutputIterator out_it, // Destination for zeros.
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const Policy& pol); // Policy to use.
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[h4 Description]
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The Airy Ai and Bi functions have an infinite
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number of zeros on the negative real axis. The real zeros on the negative real
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axis can be found by solving for the roots of
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[emquad] ['Ai(x[sub m]) = 0]
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[emquad] ['Bi(y[sub m]) = 0]
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Here, ['x[sub m]] represents the ['m[super th]]
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root of the Airy Ai function,
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and ['y[sub m]] represents the ['m[super th]]
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root of the Airy Bi function.
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The zeros or roots (values of `x` where the function crosses the horizontal `y = 0` axis)
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of the Airy Ai and Bi functions are computed by two functions,
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`airy_ai_zero` and `airy_bi_zero`.
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In each case the index or rank of the zero
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returned is 1-based, which is to say:
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airy_ai_zero(1);
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returns the first zero of Ai.
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Passing an `start_index <= 0` results in a __domain_error being raised.
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The first few zeros returned by these functions have approximate values as follows:
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[table
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[[m][Ai][Bi]]
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[[1][-2.33811...][-1.17371...]]
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[[2][-4.08795...][-3.27109...]]
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[[3][-5.52056...][-4.83074...]]
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[[4][-6.78671...][-6.16985...]]
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[[5][-7.94413...][-7.37676...]]
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[[6][-9.02265...][-8.49195...]]
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]
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[graph airy_zeros]
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[h4 Examples of finding Airy Zeros]
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[import ../../example/airy_zeros_example.cpp]
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[airy_zeros_example_1]
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[airy_zeros_example_2]
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Produces the program output:
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[pre
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boost::math::airy_ai_zero<double>(1) = -2.33811
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boost::math::airy_ai_zero<double>(2) = -4.08795
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boost::math::airy_bi_zero<double>(3) = -4.83074
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airy_ai_zeros:
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-2.33811
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-4.08795
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-5.52056
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-6.78671
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-7.94413
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boost::math::airy_bi_zero<float_type>(1) = -2.3381074104597670384891972524467354406385401456711
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boost::math::airy_bi_zero<float_type>(2) = -4.0879494441309706166369887014573910602247646991085
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boost::math::airy_bi_zero<float_type>(7) = -9.5381943793462388866329885451560196208390720763825
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airy_ai_zeros:
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-2.3381074104597670384891972524467354406385401456711
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-4.0879494441309706166369887014573910602247646991085
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-5.5205598280955510591298555129312935737972142806175
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]
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The full code (and output) for this example is at
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[@../../example/airy_zeros_example.cpp airy_zeros_example.cpp],
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[h3 Implementation]
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Given the following function (A&S 10.4.105):
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[equation airy_zero_1]
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Then an initial estimate for the n[super th] zero a[sub n] of Ai is given by (A&S 10.4.94):
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[equation airy_zero_2]
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and an initial estimate for the n[super th] zero b[sub n] of Bi is given by (A&S 10.4.98):
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[equation airy_zero_3]
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Thereafter the roots are refined using Newton iteration.
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[h3 Testing]
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The precision of evaluation of zeros was tested at 50 decimal digits using `cpp_dec_float_50`
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and found identical with spot values computed by __WolframAlpha.
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[endsect] [/section:bessel Finding Zeros of Bessel Functions of the First and Second Kinds]
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[endsect]
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