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### Exponential Integral Ei

##### Synopsis

    #include <boost/math/special_functions/expint.hpp>

    namespace boost{ namespace math{

    template <class T>
    BOOST_MATH_GPU_ENABLED calculated-result-type expint(T z);

    template <class T, class Policy>
    BOOST_MATH_GPU_ENABLED calculated-result-type expint(T z, const Policy&);

    }} // namespaces

The return type of these functions is computed using the result type calculation rules: the return type is `double` if T is an integer type, and T otherwise.

The final Policy argument is optional and can be used to control the behaviour of the function: how it handles errors, what level of precision to use etc. Refer to the policy documentation for more details.

##### Description

    template <class T>
    BOOST_MATH_GPU_ENABLED calculated-result-type expint(T z);

    template <class T, class Policy>
    BOOST_MATH_GPU_ENABLED calculated-result-type expint(T z, const Policy&);

Returns the exponential integral of z:

##### Accuracy

The following table shows the peak errors (in units of epsilon) found on various platforms with various floating point types, along with comparisons to Cody's SPECFUN implementation and the GSL-1.9 library. Unless otherwise specified any floating point type that is narrower than the one shown will have effectively zero error.

Table 8.78. Error rates for expint (Ei)

| GNU C++ version 7.1.0 linux long double | GNU C++ version 7.1.0 linux double | Sun compiler version 0x5150 Sun Solaris long double | Microsoft Visual C++ version 14.1 Win32 double
--- | --- | --- | --- | ---
Exponential Integral Ei | Max = 5.05ε (Mean = 0.821ε) (<cmath>: Max = 14.1ε (Mean = 2.43ε) And other failures.) | Max = 0.994ε (Mean = 0.142ε) (GSL 2.1: Max = 8.96ε (Mean = 0.703ε)) | Max = 5.05ε (Mean = 0.835ε) | Max = 1.43ε (Mean = 0.54ε)
Exponential Integral Ei: double exponent range | Max = 1.72ε (Mean = 0.593ε) (<cmath>: Max = 3.11ε (Mean = 1.13ε)) | Max = 0.998ε (Mean = 0.156ε) (GSL 2.1: Max = 1.5ε (Mean = 0.612ε)) | Max = 1.72ε (Mean = 0.607ε) | Max = 1.7ε (Mean = 0.66ε)
Exponential Integral Ei: long exponent range | Max = 1.98ε (Mean = 0.595ε) (<cmath>: Max = 1.93ε (Mean = 0.855ε)) |  | Max = 1.98ε (Mean = 0.575ε) |

It should be noted that all three libraries tested above offer sub-epsilon precision over most of their range.

GSL has the greatest difficulty near the positive root of En, while Cody's SPECFUN along with this implementation increase their error rates very slightly over the range [4,6].

The following error plot are based on an exhaustive search of the functions domain, MSVC-15.5 at `double` precision, and GCC-7.1/Ubuntu for `long double` and `__float128`.

##### Testing

The tests for these functions come in two parts: basic sanity checks use spot values calculated using Mathworld's online evaluator, while accuracy checks use high-precision test values calculated at 1000-bit precision with NTL::RR and this implementation. Note that the generic and type-specific versions of these functions use differing implementations internally, so this gives us reasonably independent test data. Using our test data to test other "known good" implementations also provides an additional sanity check.

##### Implementation

For x < 0 this function just calls zeta(1, -x): which in turn is implemented in terms of rational approximations when the type of x has 113 or fewer bits of precision.

For x > 0 the generic version is implemented using the infinite series:

However, when the precision of the argument type is known at compile time and is 113 bits or less, then rational approximations devised by JM are used.

For 0 < z < 6 a root-preserving approximation of the form:

is used, where z0 is the positive root of the function, and R(z/3 - 1) is a minimax rational approximation rescaled so that it is evaluated over [-1,1]. Note that while the rational approximation over [0,6] converges rapidly to the minimax solution it is rather ill-conditioned in practice. Cody and Thacher [11] experienced the same issue and converted the polynomials into Chebeshev form to ensure stable computation. By experiment we found that the polynomials are just as stable in polynomial as Chebyshev form, provided they are computed over the interval [-1,1].

Over the a series of intervals [a, b] and [b, INF] the rational approximation takes the form:

where c is a constant, and R(t) is a minimax solution optimised for low absolute error compared to c. Variable t is `1/z` when the range in infinite and `2z/(b-a) - (2a/(b-a) + 1)` otherwise: this has the effect of scaling z to the interval [-1,1]. As before rational approximations over arbitrary intervals were found to be ill-conditioned: Cody and Thacher solved this issue by converting the polynomials to their J-Fraction equivalent. However, as long as the interval of evaluation was [-1,1] and the number of terms carefully chosen, it was found that the polynomials could be evaluated to suitable precision: error rates are typically 2 to 3 epsilon which is comparable to the error rate that Cody and Thacher achieved using J-Fractions, but marginally more efficient given that fewer divisions are involved.

[11] W. J. Cody and H. C. Thacher, Jr., Rational Chebyshev approximations for the exponential integral E1(x), Math. Comp. 22 (1968), 641-649, and W. J. Cody and H. C. Thacher, Jr., Chebyshev approximations for the exponential integral Ei(x), Math. Comp. 23 (1969), 289-303.