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150 lines
3.7 KiB
Plaintext
150 lines
3.7 KiB
Plaintext
[def __effects [*Effects: ]]
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[def __formula [*Formula: ]]
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[def __exm1 '''<code>e<superscript>x</superscript> - 1</code>''']
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[def __ex '''<code>e<superscript>x</superscript></code>''']
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[def __te '''2ε''']
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[mathpart inverse_complex..Complex Number Functions]
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The following complex number algorithms are the inverses of trigonometric functions currently
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present in the C++ standard. Equivalents to these functions are part of the C99 standard, and
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are part of the [tr1].
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[section:complex_implementation Implementation and Accuracy]
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Although there are deceptively simple formulae available for all of these functions, a naive
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implementation that used these formulae would fail catastrophically for some input
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values. The Boost versions of these functions have been implemented using the methodology
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described in "Implementing the Complex Arcsine and Arccosine Functions Using Exception Handling"
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by T. E. Hull Thomas F. Fairgrieve and Ping Tak Peter Tang, ACM Transactions on Mathematical Software,
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Vol. 23, No. 3, September 1997. This means that the functions are well defined over the entire
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complex number range, and produce accurate values even at the extremes of that range, where as a naive
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formula would cause overflow or underflow to occur during the calculation, even though the result is
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actually a representable value. The maximum theoretical relative error for all of these functions
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is less than 9.5[epsilon] for every machine-representable point in the complex plane. Please refer to
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comments in the header files themselves and to the above mentioned paper for more information
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on the implementation methodology.
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[endsect]
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[section:asin asin]
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[h4 Header:]
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#include <boost/math/complex/asin.hpp>
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[h4 Synopsis:]
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template<class T>
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std::complex<T> asin(const std::complex<T>& z);
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__effects returns the inverse sine of the complex number z.
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__formula [$../images/asin.png]
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[endsect]
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[section:acos acos]
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[h4 Header:]
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#include <boost/math/complex/acos.hpp>
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[h4 Synopsis:]
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template<class T>
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std::complex<T> acos(const std::complex<T>& z);
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__effects returns the inverse cosine of the complex number z.
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__formula [$../images/acos.png]
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[endsect]
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[section:atan atan]
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[h4 Header:]
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#include <boost/math/complex/atan.hpp>
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[h4 Synopsis:]
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template<class T>
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std::complex<T> atan(const std::complex<T>& z);
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__effects returns the inverse tangent of the complex number z.
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__formula [$../images/atan.png]
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[endsect]
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[section:asinh asinh]
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[h4 Header:]
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#include <boost/math/complex/asinh.hpp>
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[h4 Synopsis:]
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template<class T>
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std::complex<T> asinh(const std::complex<T>& z);
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__effects returns the inverse hyperbolic sine of the complex number z.
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__formula [$../images/asinh.png]
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[endsect]
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[section:acosh acosh]
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[h4 Header:]
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#include <boost/math/complex/acosh.hpp>
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[h4 Synopsis:]
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template<class T>
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std::complex<T> acosh(const std::complex<T>& z);
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__effects returns the inverse hyperbolic cosine of the complex number z.
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__formula [$../images/acosh.png]
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[endsect]
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[section:atanh atanh]
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[h4 Header:]
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#include <boost/math/complex/atanh.hpp>
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[h4 Synopsis:]
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template<class T>
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std::complex<T> atanh(const std::complex<T>& z);
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__effects returns the inverse hyperbolic tangent of the complex number z.
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__formula [$../images/atanh.png]
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[endsect]
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[section:complex_history History]
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* 2005/12/17: Added support for platforms with no meaningful numeric_limits<>::infinity().
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* 2005/12/01: Initial version, added as part of the TR1 library.
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[endsect]
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[endmathpart]
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[/
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Copyright 2008, 2009 John Maddock and Paul A. Bristow.
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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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