...one of the most highly
regarded and expertly designed C++ library projects in the
world.
— Herb Sutter and Andrei
Alexandrescu, C++
Coding Standards
#include <boost/math/special_functions/trigamma.hpp>
namespace boost{ namespace math{ template <class T> calculatedresulttype trigamma(T z); template <class T, class Policy> calculatedresulttype trigamma(T z, const Policy&); }} // namespaces
Returns the trigamma function of x. Trigamma is defined as the derivative of the digamma function:
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.
The return type of this function is computed using the result
type calculation rules: the result is of type double
when T is an integer type, and type
T otherwise.
The following table shows the peak errors (in units of epsilon) found on various platforms with various floating point types. Unless otherwise specified any floating point type that is narrower than the one shown will have effectively zero error.
Table 7.5. Error rates for trigamma
GNU C++ version 7.1.0 
GNU C++ version 7.1.0 
Sun compiler version 0x5150 
Microsoft Visual C++ version 14.1 


Mathematica Data 
Max = 0.998ε (Mean = 0.105ε) 
Max = 1.28ε (Mean = 0.449ε) 
Max = 1.28ε (Mean = 0.449ε) 
Max = 1ε (Mean = 0.382ε) 
As shown above, error rates are generally very low for built in types. For multiprecision types, error rates are typically in the order of a few epsilon.
The following error plot are based on an exhaustive search of the functions
domain, MSVC15.5 at double
precision, and GCC7.1/Ubuntu for long
double
and __float128
.
Testing is against Mathematica generated spot values to 35 digit precision.
The arbitrary precision version of this function simply calls polygamma.
For built in fixed precision types, negative arguments are first made positive via:
Then arguments in the range [0, 1) are shifted to >= 1 via:
Then evaluation is via one of a number of rational approximations, for small x these are of the form:
and for large x of the form: