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.

example

Examples

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Evaluate Euler-Mascheroni Constant

Open Live Script

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

format long
Y = -psi(1)

Y = 
   0.577215664901532

Evaluate Trigamma Function

Open Live Script

Evaluate the trigamma function of 2.

format long
Y1 = psi(1,2)

Y1 = 
   0.644934066848226

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

Y2 = pi^2/6 - 1

Y2 = 
   0.644934066848226

isequal(Y1,Y2)

ans = logical
   1

Plot Digamma and Polygamma Functions

Open Live Script

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')

$Figure contains an axes object. The axes object with title Digamma and The Next Three Polygamma Functions, xlabel $x$, ylabel psi indexOf k baseline leftParenthesis x rightParenthesis contains 4 objects of type line. These objects represent \psi, \psi_1, \psi_2, \psi_3.$

Input Arguments

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`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

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Digamma Function

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

$ψ (x) = \frac{d}{d x} \ln Γ (x) = \frac{Γ^{'} (x)}{Γ (x)} .$

Polygamma Function

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

$ψ^{(k)} (x) = \frac{d^{k + 1}}{d x^{k + 1}} \ln Γ (x) = \frac{d^{k}}{d x^{k}} ψ (x) .$

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

psi

Syntax

Description

Examples

Evaluate Euler-Mascheroni Constant

Evaluate Trigamma Function

Plot Digamma and Polygamma Functions

Input Arguments

X — Input scalar | vector | matrix | multidimensional array

k — Order of derivative nonnegative integer scalar

More About

Digamma Function

Polygamma Function

References

Extended Capabilities

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

C/C++ Code Generation Generate C and C++ code using MATLAB® Coder™.

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

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

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

Version History

See Also

`X` — Input
scalar | vector | matrix | multidimensional array

`k` — Order of derivative
nonnegative integer scalar

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

C/C++ Code Generation
Generate C and C++ code using MATLAB® Coder™.

Thread-Based Environment
Run code in the background using MATLAB® `backgroundPool` or accelerate code with Parallel Computing Toolbox™ `ThreadPool`.

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

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