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* Update Jamfile.v2 * Update introduction.qbk * Update tutorial.qbk * Update tutorial_cpp_int.qbk * Update tutorial_gmp_int.qbk * Update tutorial_tommath.qbk * Update integer_examples.cpp * Update tutorial_cpp_bin_float.qbk * Update tutorial_cpp_dec_float.qbk * Update tutorial_gmp_float.qbk * Update tutorial_mpfr_float.qbk * Update tutorial_float128.qbk * Update tutorial_float_builtin_ctor.qbk * Update big_seventh.cpp * Update tutorial_float_eg.qbk * Update floating_point_examples.cpp * Update mpfr_precision.cpp * Update gauss_laguerre_quadrature.cpp * Update tutorial_interval_mpfi.qbk * Update tutorial_cpp_complex.qbk * Update tutorial_mpc_complex.qbk * Update tutorial_float128_complex.qbk * Update tutorial_complex_adaptor.qbk * Update tutorial_rational.qbk * Update tutorial_tommath_rational.qbk * Update tutorial_logged_adaptor.qbk * Update tutorial_debug_adaptor.qbk * Update tutorial_visualizers.qbk * Update tutorial_fwd.qbk * Update tutorial_conversions.qbk * Update tutorial_random.qbk * Update random_snips.cpp * Update tutorial_constexpr.qbk * Update tutorial_import_export.qbk * Update cpp_int_import_export.cpp * Update tutorial_mixed_precision.qbk * Update tutorial_variable_precision.qbk * Update scoped_precision_example.cpp * Update tutorial_numeric_limits.qbk * Update tutorial_numeric_limits.qbk * Update numeric_limits_snips.cpp * Update numeric_limits_snips.cpp * Update tutorial_numeric_limits.qbk * Update numeric_limits_snips.cpp * Update numeric_limits_snips.cpp * Update tutorial_io.qbk * Update reference_number.qbk * Update reference_cpp_bin_float.qbk * Update reference_cpp_double_fp_backend.qbk * Update reference_internal_support.qbk * Update reference_backend_requirements.qbk * Update performance.qbk * Update performance_overhead.qbk * Update performance_real_world.qbk * Update performance_integer_real_world.qbk * Update performance_rational_real_world.qbk * Update reference_number.qbk * Update tutorial_numeric_limits.qbk * Update reference_backend_requirements.qbk
138 lines
4.4 KiB
Plaintext
138 lines
4.4 KiB
Plaintext
[/
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Copyright 2011 - 2020 John Maddock.
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Copyright 2013 - 2019 Paul A. Bristow.
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Copyright 2013 Christopher Kormanyos.
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Distributed under the Boost Software License, Version 1.0.
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(See accompanying file LICENSE_1_0.txt or copy at
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http://www.boost.org/LICENSE_1_0.txt).
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]
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[section:fp_eg Examples]
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[include tutorial_float_builtin_ctor.qbk]
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[import ../example/floating_point_examples.cpp]
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[section:aos Area of Circle]
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[/I:\boost\libs\multiprecision\example\floating_point_examples.cpp]
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[AOS1]
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[AOS2]
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[AOS3]
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[endsect] [/section:aos Area of Circle]
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[section:caveats Drop-in Caveats]
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[import ../example/big_seventh.cpp]
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[big_seventh_example_1]
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[big_seventh_example_2]
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[big_seventh_example_3]
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[big_seventh_example_constexpr] [/ Hopefully a temporary restriction.]
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[big_seventh_example_4]
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Typical output is:
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[big_seventh_example_output]
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[endsect] [/section:caveats Caveats]
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[section:jel Defining a Special Function]
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[JEL]
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[endsect] [/section:jel Defining a Special Function]
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[section:nd Calculating a Derivative]
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[ND1]
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[ND2]
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[ND3]
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[endsect] [/section:nd Calculating a Derivative]
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[section:gi Calculating an Integral]
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[GI1]
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[GI2]
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[endsect] [/section:gi Calculating an Integral]
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[section:poly_eg Polynomial Evaluation]
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[POLY]
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[endsect] [/section:poly_eg Polynomial Evaluation]
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[section:variable_precision Variable-Precision Newton Evaluation]
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[mpfr_variable]
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[endsect] [/section:variable_precision Variable-Precision Newton Evaluation]
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[section:gauss_lagerre_quadrature Gauss-Laguerre quadrature]
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[import ../example/gauss_laguerre_quadrature.cpp]
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This example uses __Boost_Multiprecision to implement a high-precision Gauss-Laguerre quadrature integration.
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The quadrature is used to calculate the `airy_ai(x)` function for real-valued `x` on the positive axis with `x >= 1`.
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In this way, the integral representation could be seen as part of a scheme to calculate
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real-valued Airy functions on the positive axis for medium to large argument.
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A Taylor series or hypergeometric function (not part of this example) could be used for smaller arguments.
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This example has been tested with decimal digits counts ranging from 21...301, by adjusting the parameter
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`local::my_digits10` at compile time.
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The quadrature integral representaion of `airy_ai(x)` used in this example can be found in:
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A. Gil, J. Segura, N.M. Temme, "Numerical Methods for Special Functions"
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(SIAM Press 2007), ISBN 9780898717822, Sect. 5.3.3, in particular Eq. 5.110, page 145.
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Subsequently, Gil et al's book cites the another work:
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W. Gautschi, "Computation of Bessel and Airy functions and of related Gaussian quadrature formulae",
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BIT, 42 (2002), pp. 110-118, [@https://doi.org/10.1023/A:1021974203359] that is also available as
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[@https://www.cs.purdue.edu/homes/wxg/selected_works/section_02/169.pdf Gautschi_169.pdf].
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This Gauss-Laguerre quadrature is designed for `airy_ai(x)` with real-valued `x >= 1`.
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The example uses Gauss-Laguerre quadrature integration to compute `airy_ai(x / 7)`
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with `7 <= x <= 120` and where `x` is incremented in steps of 1.
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During development of this example, we have empirically found the numbers of Gauss-Laguerre coefficients
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needed for convergence when using various counts of base-10 digits.
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Let's calibrate, for instance, the number of coefficients needed at the point `x = 1`.
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Empirical data were used with __WolframAlphaGLCoefs :
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``
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Fit[{{21.0, 3.5}, {51.0, 11.1}, {101.0, 22.5}, {201.0, 46.8}}, {1, d, d^2}, d] FullSimplify[%]
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0.0000178915 d^2 + 0.235487 d - 1.28301
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or
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-1.28301 + (0.235487 + 0.0000178915 d) d
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``
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We need significantly more coefficients at smaller real values than are needed
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at larger real values because the slope derivative of `airy_ai(x)` gets more
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steep as x approaches zero. `laguerre_order` is calculated using this equation.
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Snippets from (copious) output from a progress bar during calculation of approximate root estimates
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followed by calculation of accurate abscissa and weights is:
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[gauss_laguerre_quadrature_output_1]
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Finally the result using Gauss-Laguerre quadrature is compared with a calculation using `cyl_bessel_k`,
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and both are listed, conclusively confirming that none differ by more than a small tolerance.
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[gauss_laguerre_quadrature_output_2]
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For more detail see comments in the source code for this example at [@../../example/gauss_laguerre_quadrature.cpp gauss_laguerre_quadrature.cpp].
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[endsect] [/section:gauss_lagerre_quadrature Gauss-Laguerre quadrature]
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[endsect] [/section:fp_eg Examples]
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