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- New projects should use Boost.Multiprecision. + New projects should use Boost.Multiprecision.

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- New projects should use Boost.Multiprecision. + New projects should use Boost.Multiprecision.

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    - Christopher Kormanyos wrote some examples for using Boost.Multiprecision, + Christopher Kormanyos wrote some examples for using Boost.Multiprecision, and added methods for finding zeros of Bessel Functions.

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  • The edit distance between the two values: i.e. how many (binary) floating-point values are between two values a and b? - This is provided by the function Boost.Math + This is provided by the function Boost.Math float_distance, but is probably only useful when you know that the distance should be very small. This function is somewhat difficult to compute, and doesn't scale to values that are very far apart. In other @@ -198,7 +198,7 @@

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    - Boost.Multiprecision + Boost.Multiprecision can operate in both ways, but the most popular choice is likely to be about a hundred decimal digits, though examples of computing about a million digits have been demonstrated. @@ -106,7 +106,7 @@

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    - Boost.Multiprecision + Boost.Multiprecision was a development from the e_float (TOMS Algorithm 910) library by Christopher Kormanyos.

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    - As noted in the Boost.Multiprecision + As noted in the Boost.Multiprecision documentation, certain program constructs will not compile when using expression templates. One example that many users may encounter is Boost.Test (1.54 and earlier) when using macro BOOST_CHECK_CLOSE and BOOST_CHECK_CLOSE_FRACTION. @@ -139,7 +139,7 @@

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  • I need more decimal digits for values/computations. -
    You can use Boost.Math with Boost.Multiprecision: - typically cpp_dec_float +
    You can use Boost.Math with Boost.Multiprecision: + typically cpp_dec_float is a useful user-defined type to provide a fixed number of decimal digits, usually 50 or 100.
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  • - To calculate the edit distance between two floats use Boost.Math + To calculate the edit distance between two floats use Boost.Math float_distance.

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    @@ -38,10 +38,10 @@ User-defined types that meet the conceptual requirements can also be used. For example, with a thin wrapper class one of the types provided with NTL - (RR) can be used. But now that Boost.Multiprecision + (RR) can be used. But now that Boost.Multiprecision library is available, this has become the preferred real-number type, typically - cpp_dec_float - or cpp_bin_float. + cpp_dec_float + or cpp_bin_float.

    Submissions of binding to other extended precision types would also still be @@ -1014,7 +1014,7 @@

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    and, on platforms that provide it, a 128-bit - quad type. (See float128). + quad type. (See float128).

    For this optional include, the build should define the macro BOOST_HAVE_QUADMATH: @@ -452,7 +452,7 @@ If a higher precision than double (or long double if that is more precise) is required, then this is easily achieved using - Boost.Multiprecision + Boost.Multiprecision with some includes from 

    #include <boost/multiprecision/cpp_dec_float.hpp> // For decimal boost::multiprecision::cpp_dec_float_50.
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    In the table below, the cube root of 28 was computed for three fundamental - types floating-point types, and one Boost.Multiprecision - type cpp_bin_float + types floating-point types, and one Boost.Multiprecision + type cpp_bin_float using 50 decimal digit precision, using four algorithms.

    @@ -39,7 +39,7 @@ 

    cpp_bin_float_100 full_answer ("3.036588971875662519420809578505669635581453977248111123242141654169177268411884961770250390838097895");

    - Times were measured using Boost.Timer + Times were measured using Boost.Timer using class cpu_timer.

      @@ -57,7 +57,7 @@
    • Dis is the distance from the nearest representation of the 'exact' root in bits. Distance from the 'exact' answer is measured - by using function Boost.Math + by using function Boost.Math float_distance. One or two bits distance means that all results are effectively 'correct'. Zero means 'exact' - the nearest representable value for the floating-point type. @@ -152,7 +152,7 @@
    • The C++ Standard cube root function std::cbrt is only defined for built-in or fundamental types, so cannot be used - with any User-Defined floating-point types like Boost.Multiprecision. + with any User-Defined floating-point types like Boost.Multiprecision. This, and that the cube function is so impeccably-behaved, allows the implementer to use many tricks to achieve a fast computation. On some platforms,std::cbrt appeared several times as quick @@ -175,7 +175,7 @@
    • boost::math::cbrt allows use with any - user-defined floating-point type, conveniently Boost.Multiprecision. + user-defined floating-point type, conveniently Boost.Multiprecision. It too can take some advantage of the good-behaviour of the cube function, compared to the more general implementation in the nth root-finding examples. For example, it uses a polynomial approximation to generate @@ -203,7 +203,7 @@
    • For fundamental types, there was little to choose between the three derivative - methods, but for cpp_bin_float, + methods, but for cpp_bin_float, Newton-Raphson iteration was twice as fast. Note that the cube-root is an extreme test case as the cost of calling the functor is so cheap that the runtimes @@ -1583,7 +1583,7 @@
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    @@ -32,7 +32,7 @@ second radius of an ellipse with first radius 28 and arc length 300, for four floating-point types, float, double, long - double and a Boost.Multiprecision + double and a Boost.Multiprecision type cpp_bin_float_50.

    @@ -1865,7 +1865,7 @@

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    @@ -31,7 +31,7 @@ A second example compares four generalized nth-root finding algorithms for various n-th roots (5, 7 and 13) of a single value 28.0, for four floating-point types, float, double, long - double and a Boost.Multiprecision + double and a Boost.Multiprecision type cpp_bin_float_50. In each case the target accuracy was set using our "recomended" accuracy limits (or at least limits that make a good starting point - which @@ -5219,14 +5219,14 @@ by the complexity of the root finding method.
  • - For a Boost.Multiprecision + For a Boost.Multiprecision floating-point type, the Newton-Raphson iteration is a clear winner with a several-fold gain over TOMS Algorithm 748: enclosing zeros of continuous functions, and again no improvement from the second-derivative algorithms.
  • - The run-time of 50 decimal-digit Boost.Multiprecision + The run-time of 50 decimal-digit Boost.Multiprecision is about 30-fold greater than double.
  • @@ -5236,7 +5236,7 @@ Algorithm 748: enclosing zeros of continuous functions.
  • - The goodness of the 'guess' is especially crucial for Boost.Multiprecision. + The goodness of the 'guess' is especially crucial for Boost.Multiprecision. Separate experiments show that evaluating the 'guess' using double allows convergence to the final exact result in one or two iterations. So in this contrived example, crudely dividing the exponent by N for a 'guess', it would be far better @@ -5252,7 +5252,7 @@
  • Using MSVC, there was some improvement using 64-bit, markedly for - Boost.Multiprecision. + Boost.Multiprecision.
  • The GCC compiler 4.9.1 using 64-bit was at least five-folder faster @@ -5263,7 +5263,7 @@ Clearly, your mileage will vary, but in summary, Newton-Raphson iteration seems the first choice of algorithm, and effort to find - a good 'guess' the first speed-up target, especially for Boost.Multiprecision. + a good 'guess' the first speed-up target, especially for Boost.Multiprecision. And of course, compiler optimisation is crucial for speed.

    • @@ -5281,7 +5281,7 @@
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    @@ -58,7 +58,7 @@ (But these examples are only a demonstration and do not try to make the ultimate improvements of an 'industrial-strength' implementation, for example, of boost::math::cbrt, mainly by using a better computed - initial 'guess' at cbrt.hpp). + initial 'guess' at cbrt.hpp).

    Then we show how a higher root (fifth @@ -85,7 +85,7 @@

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    - The result of newton_raphson_iterate function + The result of newton_raphson_iterate function is a single value.

    - +
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    @@ -432,7 +432,7 @@ and reusing it, omits error handling, and does not handle negative values 

    cbrt_2deriv(28) = 3.0365889718756627

    - which we can compare with the boost::math::cbrt + which we can compare with the boost::math::cbrt 

    cbrt(28)              = 3.0365889718756627
    @@ -481,7 +481,7 @@ and reusing it, omits error handling, and does not handle negative values
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    @@ -108,14 +108,14 @@ a more efficient built-in quad floating-point type like __float128.

    - Almost all platforms can easily use Boost.Multiprecision, - for example, cpp_dec_float - or a binary type cpp_bin_float + Almost all platforms can easily use Boost.Multiprecision, + for example, cpp_dec_float + or a binary type cpp_bin_float types, to compute values at very much higher precision.

    - +
    [Note][Note] Note

    @@ -131,7 +131,7 @@

    Since the functors and functions used above are templated on the value - type, we can very simply use them with any of the Boost.Multiprecision + type, we can very simply use them with any of the Boost.Multiprecision types. As a reminder, here's our toy cube root function using 2 derivatives and C++11 lambda functions to find the root:

    @@ -237,7 +237,7 @@ value = 2, cube root =1.2599210498948731647672106072782283505702514647015
    - +
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    @@ -438,8 +438,8 @@ A macro BOOST_DEFINE_MATH_CONSTANT in constants.hpp is used to provide high accuracy constants to mathematical functions and distributions, since it is important to provide values uniformly for both built-in float, double and long - double types, and for User Defined types in Boost.Multiprecision - like cpp_dec_float. + double types, and for User Defined types in Boost.Multiprecision + like cpp_dec_float. and others like NTL::quad_float and NTL::RR.

    @@ -510,7 +510,7 @@ double are all thread safe.

    - For User_defined types, for example, cpp_dec_float, + For User_defined types, for example, cpp_dec_float, the Boost.Math should also be thread-safe, (thought we are unsure how to rigorously prove this).

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  • Calls to standard library functions should be made unqualified (this allows argument dependent lookup to find standard library functions for - user-defined floating point types such as those from Boost.Multiprecision). + user-defined floating point types such as those from Boost.Multiprecision). In addition, the macro BOOST_MATH_STD_USING should appear at the start of the function (note no semi-colon afterwards!) so that all the math functions in namespace @@ -385,7 +385,7 @@
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    @@ -328,7 +328,7 @@ error C3861: 'mybetad0': identifier not found

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    diff --git a/doc/html/math_toolkit/stat_tut/weg/f_eg.html b/doc/html/math_toolkit/stat_tut/weg/f_eg.html index 1ed59d1d9..d1564e5a5 100644 --- a/doc/html/math_toolkit/stat_tut/weg/f_eg.html +++ b/doc/html/math_toolkit/stat_tut/weg/f_eg.html @@ -11,16 +11,16 @@ - - - + + + - +
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    @@ -324,7 +324,7 @@ Standard deviation 1 is greater than standard deviation 2 ACCEPTED

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    @@ -185,7 +185,7 @@ Normal distribution with mean = 0.355146 has fraction > 2 = 0.05
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    diff --git a/doc/html/math_toolkit/stat_tut/weg/find_eg/find_mean_and_sd_eg.html b/doc/html/math_toolkit/stat_tut/weg/find_eg/find_mean_and_sd_eg.html index 7481876c3..a1091f8f4 100644 --- a/doc/html/math_toolkit/stat_tut/weg/find_eg/find_mean_and_sd_eg.html +++ b/doc/html/math_toolkit/stat_tut/weg/find_eg/find_mean_and_sd_eg.html @@ -11,16 +11,16 @@ - - - + + + - +
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    diff --git a/doc/html/math_toolkit/stat_tut/weg/neg_binom_eg.html b/doc/html/math_toolkit/stat_tut/weg/neg_binom_eg.html index cec52e5f9..dc37df8f0 100644 --- a/doc/html/math_toolkit/stat_tut/weg/neg_binom_eg.html +++ b/doc/html/math_toolkit/stat_tut/weg/neg_binom_eg.html @@ -11,16 +11,16 @@ - - - + + + - +
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    The most common example of a high-precision user-defined type will probably - be Boost.Multiprecision. + be Boost.Multiprecision.

    The syntax for using the function-call constants with user-defined types @@ -126,7 +126,7 @@

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