psi

Digamma and polygamma functions

Syntax

Y = psi(X)

Y = psi(k,X)

Description

Y = psi(X) evaluates the digamma function for each element of array X, which must be real and nonnegative.

Y = psi(k,X) evaluates the polygamma function of X, which is the kth derivative of the digamma function at X. Thus, psi(0,X) is the digamma function, psi(1,X) is the trigamma function, psi(2,X) is the tetragamma function, and so on.

Examples

Evaluate Euler-Mascheroni Constant

Use the psi function to evaluate the Euler-Mascheroni constant $$\gamma$$, also known as Euler's constant.

format long
Y = -psi(1)

Y = 
   0.577215664901532

Evaluate Trigamma Function

Evaluate the trigamma function of 2.

format long
Y1 = psi(1,2)

Y1 = 
   0.644934066848226

Check that the result is equal to $${\pi }^2/6-1$$.

Y2 = pi^2/6 - 1

Y2 = 
   0.644934066848226

isequal(Y1,Y2)

ans = logical
   1

Plot Digamma and Polygamma Functions

Define the domain.

X = 0:0.05:5;

Calculate the digamma and the next three polygamma functions.

Y = zeros(4,101);
for i = 0:3
    Y(i+1,:) = psi(i,X);
end

Plot the digamma and the next three polygamma functions.

plot(X,Y)
axis([0 5 -10 10])
legend('\psi','\psi_1','\psi_2','\psi_3','Location','Best')
title('Digamma and The Next Three Polygamma Functions','interpreter','latex')
xlabel('$x$','interpreter','latex')
ylabel('$\psi_k(x)$','interpreter','latex')

Input Arguments

X — Input

scalar | vector | matrix | multidimensional array

Input, specified as a scalar, vector, matrix, or multidimensional array of nonnegative real numbers. X cannot be sparse.

Data Types: single | double

k — Order of derivative

nonnegative integer scalar

Order of derivative, specified as a nonnegative integer scalar. k must be smaller than 2³¹-1.

Data Types: single | double

More About

Digamma Function

The digamma function is the first derivative of the logarithm of the gamma function:

$$\psi \left(x\right)=\frac{d}{dx}\ln \Gamma \left(x\right)=\frac{{\Gamma }^{\prime }\left(x\right)}{\Gamma \left(x\right)}.$$

Polygamma Function

The polygamma function of the order k is the (k + 1)th derivative of the logarithm of the gamma function:

$${\psi }^{\left(k\right)}\left(x\right)=\frac{d^{k+1}}{d{x}^{k+1}}\ln \Gamma \left(x\right)=\frac{d^k}{d{x}^k}\psi \left(x\right).$$

References

[1] Abramowitz, M. and I. A. Stegun, Handbook of Mathematical Functions, Dover Publications, 1965, Sections 6.3 and 6.4.

Extended Capabilities

Tall Arrays

Calculate with arrays that have more rows than fit in memory.

This function supports tall arrays with the limitation:

For the syntax Y = psi(k,X), the order of derivative k must be a non-tall scalar.

For more information, see Tall Arrays for Out-of-Memory Data.

C/C++ Code Generation

Generate C and C++ code using MATLAB® Coder™.

Usage notes and limitations:

• Strict single-precision calculations are not supported. In the generated code, single-precision inputs produce single-precision outputs. However, variables inside the function might be double-precision.

Thread-Based Environment

Run code in the background using MATLAB® backgroundPool or accelerate code with Parallel Computing Toolbox™ ThreadPool.

This function fully supports thread-based environments. For more information, see Run MATLAB Functions in Thread-Based Environment.

GPU Arrays

Accelerate code by running on a graphics processing unit (GPU) using Parallel Computing Toolbox™.

The psi function fully supports GPU arrays. To run the function on a GPU, specify the input data as a gpuArray (Parallel Computing Toolbox). For more information, see Run MATLAB Functions on a GPU (Parallel Computing Toolbox).

Distributed Arrays

Partition large arrays across the combined memory of your cluster using Parallel Computing Toolbox™.

This function fully supports distributed arrays. For more information, see Run MATLAB Functions with Distributed Arrays (Parallel Computing Toolbox).

Version History

Introduced before R2006a