trigamma

Purpose

Computes the trigamma function, which is the second derivative of the log of the gamma function. Commonly used in Newton-Raphson iterations for maximum likelihood estimation of gamma and Dirichlet distribution parameters.

Format

y = trigamma(x)

Parameters:

x (MxN matrix or N-dimensional array) – values at which to evaluate the trigamma function. All elements must be positive.

Returns:

y (MxN matrix or N-dimensional array) – the trigamma function evaluated element-wise at each value of x.

Remarks

The trigamma() function is the second derivative of the log of the gamma function with respect to its argument:

\[\psi_1(x) = \frac{d^2}{dx^2} \ln \Gamma(x)\]

It is the derivative of the digamma() function. The trigamma function is defined for positive real numbers and approaches zero as x increases.

Examples

Example 1: Basic evaluation

// Evaluate trigamma at several points
x = { 0.5, 1, 2, 5, 10 };
y = trigamma(x);
print (x~y);

produces:

0.50000000        4.9348022
 1.0000000        1.6449341
 2.0000000       0.64493407
 5.0000000       0.22132296
 10.000000       0.10516634

Note that trigamma(1) equals \(\pi^2/6 \approx 1.6449\), and the values decrease toward zero for larger x.

Example 2: Use in Fisher information

// For a Gamma(alpha, beta) distribution, the Fisher information
// for alpha involves the trigamma function:
// I(alpha) = trigamma(alpha)

alpha = 3;
fisher_info = trigamma(alpha);
print "Fisher information for alpha = " alpha;
print "I(alpha) = " fisher_info;

produces:

Fisher information for alpha =        3.0000000
I(alpha) =       0.39493407

See also

Functions digamma(), gamma(), lnfact()