ellipke

Complete elliptic integrals of first and second kind

Syntax

K = ellipke(M)

[K,E] = ellipke(M)

[K,E] = ellipke(M,tol)

Description

K = ellipke(M) returns the complete elliptic integral of the first kind for each element in M.

[K,E] = ellipke(M) returns the complete elliptic integral of the first and second kind.

[K,E] = ellipke(M,tol) computes the complete elliptic integral to accuracy tol. The default value of tol is eps. Increase tol for a less accurate but more quickly computed answer.

Examples

Find Complete Elliptic Integrals of First and Second Kind

Find the complete elliptic integrals of the first and second kind for M = 0.5.

M = 0.5;
[K,E] = ellipke(M)

K = 1.8541

E = 1.3506

Plot Complete Elliptic Integrals of First and Second Kind

Plot the complete elliptic integrals of the first and second kind for the allowed range of M.

M = 0:0.01:1;
[K,E] = ellipke(M);
plot(M,K,M,E)
grid on
xlabel('M')
title('Complete Elliptic Integrals of First and Second Kind')
legend('First kind','Second kind')

Faster Calculations of the Complete Elliptic Integrals by Changing the Tolerance

The default value of tol is eps. Find the runtime with the default value for arbitrary M using tic and toc. Increase tol by a factor of thousand and find the runtime. Compare the runtimes.

tic
ellipke(0.904561)

ans = 2.6001

toc

Elapsed time is 0.412887 seconds.

tic
ellipke(0.904561,eps*1000)

ans = 2.6001

toc

Elapsed time is 0.028556 seconds.

ellipke runs significantly faster when tolerance is significantly increased.

Input Arguments

M — Input array

scalar | vector | matrix | multidimensional array

Input array, specified as a scalar, vector, matrix, or multidimensional array. M is limited to values 0≤m≤1.

Data Types: single | double

tol — Accuracy of result

eps (default) | nonnegative real number

Accuracy of result, specified as a nonnegative real number. The default value is eps.

Data Types: single | double | int8 | int16 | int32 | int64 | uint8 | uint16 | uint32 | uint64

Output Arguments

K — Complete elliptic integral of first kind

scalar | vector | matrix | multidimensional array

Complete elliptic integral of the first kind, returned as a scalar, vector, matrix, or multidimensional array.

E — Complete elliptic integral of second kind

scalar | vector | matrix | multidimensional array

Complete elliptic integral of the second kind, returned as a scalar, vector, matrix, or multidimensional array.

More About

Complete Elliptic Integrals of the First and Second Kind

The complete elliptic integral of the first kind is

$$[K(m)]={\displaystyle {\int }_0^1{[(1-t^2)(1-m{t}^2)]}^{-\frac{1}{2}}}dt.$$

where m is the first argument of ellipke.

The complete elliptic integral of the second kind is

$$E(m)={\displaystyle {\int }_0^1(}1-t^2{)}^{-\frac{1}{2}}{(1-m{t}^2)}^{\frac{1}{2}}dt.$$

Some definitions of the elliptic functions use the elliptical modulus k or modular angle α instead of the parameter m. They are related by

$$k^2=m={\sin }^2\alpha .$$

References

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

Extended Capabilities

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.

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