Python: module moremath

moremath

Various bits of useful math not in the standard python library. Constants : - euler_gamma = 0.577215... - catalan = 0.915965... - golden_ratio = 1.618033... - bits_per_nat = log2(e) = 1/log(2) - sqrt_2pi = 2.50662... Special Functions : - gamma() -- Gamma function. - lngamma() -- Logarithm of the gamma function - factorial() -- The factorial function. - digamma() -- Digamma function (logarithmic derivative of gamma). - trigamma() -- Trigamma function (derivative of digamma). - entropy() -- The entropy of a probability vector - incomplete_gamma() -- The 'upper' incomplete gamma function. - normalized_incomplete_gamma() -- - lg() -- Base 2 logarithms. Vector Operations : - rmsd() -- Root mean squared deviation of two point vectors - minimize_rmsd() -- Find the rigid transformation that minimized the RMSD between two vectors of points. Minimization : - find_root() -- 1d root finding Probability Distributions : - Gamma - Dirichlet - Multinomial - Gaussian

Functions

argmax(alist): Return the index of the last occurance of the maximum value in the list.

argmin(alist): Return the index of the first occurance of the minimum value in the list.

digamma(z): The digamma function, the logarithmic derivative of the gamma function.
digamma(z) = d/dz ln( gamma(z) )

See: Eric W. Weisstein. "Digamma Function." From MathWorld--
A Wolfram Web Resource. http://mathworld.wolfram.com/DigammaFunction.html

entropy(pvec, base=2.718281828459045): The entropy S = -Sum_i p_i ln p_i
pvec is a frequency vector, not necessarily normalized.

factorial(z): The factorial function.
factorial(z) == gamma(z+1)

gamma(z): The gamma function. Returns exact results for small integers. Will
overflow for modest sized arguments. Use lngamma(z) instead.

See: Eric W. Weisstein. "Gamma Function." From MathWorld, A Wolfram Web Resource.
http://mathworld.wolfram.com/GammaFunction.html

incomplete_gamma(a, x): The 'upper' incomplete gamma function:

                        oo
                         -
                        |    -t  a-1
incomplete_gamma(a,x) = |   e   t   dt.
                        |
                       -
                        x

In Mathematica, Gamma[a,x].

Note that, very confusingly, the phrase 'incomplete gamma fucntion'
can also refer to the same integral between 0 and x, (the 'lower'
incomplete gamma function) or to the normalized versions,
normalized_incomplete_gamma() )

See: Eric W. Weisstein. "Gamma Function." From MathWorld, A Wolfram Web Resource.
     http://mathworld.wolfram.com/IncompleteGammaFunction.html

Bugs :
    This implentation is not very accurate for some arguments.

lngamma(z): The logarithm of the gamma function.

log2(x): Return the base 2 logarithm of x

normalized_incomplete_gamma(a, x): The upper, incomplete gamma function normalized so that the limiting
values are zero and one.

Q(a,x) = incomplete_gamma(a,x) / gamma(a)

See:
incomplete_gamma()
Bugs :
This implentation is not very accurate for some arguments.

trigamma(z): The trigamma function, the derivative of the digamma function.
trigamma(z) = d/dz digamma(z) = d/dz d/dz ln( gamma(z) )

See: Eric W. Weisstein. "Digamma Function." From MathWorld--
A Wolfram Web Resource. http://mathworld.wolfram.com/TrigammaFunction.html

Data
		__all__ = ('euler_gamma', 'catalan', 'golden_ratio', 'bits_per_nat', 'sqrt_2pi', 'gamma', 'lngamma', 'factorial', 'digamma', 'trigamma', 'entropy', 'log2', 'incomplete_gamma', 'normalized_incomplete_gamma', 'argmax', 'argmin') bits_per_nat = 1.4426950408889634 catalan = 0.915965594177219 euler_gamma = 0.5772156649015329 golden_ratio = 1.618033988749895 sqrt_2pi = 2.5066282746310007