!12366 Organize the formulars used in math.py

From: @peixu_ren
Reviewed-by: @sunnybeike,@zichun_ye
Signed-off-by: @zichun_ye
pull/12366/MERGE
mindspore-ci-bot 4 years ago committed by Gitee
commit 6b5d62cc86

@ -193,7 +193,7 @@ class LGamma(Cell):
Thus, the behaviour of LGamma follows:
when x > 0.5, return log(Gamma(x))
when x < 0.5 and is not an interger, return the real part of Log(Gamma(x)) where Log is the complex logarithm
when x < 0.5 and is not an integer, return the real part of Log(Gamma(x)) where Log is the complex logarithm
when x is an integer less or equal to 0, return +inf
when x = +/- inf, return +inf
@ -302,13 +302,12 @@ class DiGamma(Cell):
The algorithm is:
.. math::
digamma(z + 1) = log(t(z)) + A'(z) / A(z) - kLanczosGamma / t(z)
t(z) = z + kLanczosGamma + 1/2
A(z) = kBaseLanczosCoeff + \sum_{k=1}^n \frac{kLanczosCoefficients[i]}{z + k}
A'(z) = \sum_{k=1}^n \frac{kLanczosCoefficients[i]}{{z + k}^2}
\begin{array}{ll} \\
digamma(z + 1) = log(t(z)) + A'(z) / A(z) - kLanczosGamma / t(z) \\
t(z) = z + kLanczosGamma + 1/2 \\
A(z) = kBaseLanczosCoeff + \sum_{k=1}^n \frac{kLanczosCoefficients[i]}{z + k} \\
A'(z) = \sum_{k=1}^n \frac{kLanczosCoefficients[i]}{{z + k}^2}
\end{array}
However, if the input is less than 0.5 use Euler's reflection formula:
@ -659,7 +658,10 @@ class IGamma(Cell):
class LBeta(Cell):
r"""
This is semantically equal to lgamma(x) + lgamma(y) - lgamma(x + y).
This is semantically equal to
.. math::
P(x, y) = lgamma(x) + lgamma(y) - lgamma(x + y).
The method is more accurate for arguments above 8. The reason for accuracy loss in the naive computation
is catastrophic cancellation between the lgammas. This method avoids the numeric cancellation by explicitly

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