Lambert W function - MATLAB lambertw (2024)

Table of Contents
Syntax Description Examples Return Equation with Lambert W Function as Its Solution Lambert W Function for Numeric and Symbolic Arguments Plot Two Main Branches of Lambert W Function Lambert W Function Plot on Complex Plane Input Arguments x — Input number | vector | matrix | array | symbolic number | symbolic variable | symbolic array | symbolic function | symbolic expression k — Branch of Lambert W function integer | vector or matrix of integers | symbolic integer | symbolic vector or matrix of integers More About Lambert W Function References Version History See Also Functions MATLAB Command Americas Europe Asia Pacific References

Lambert W function

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Syntax

lambertw(x)

lambertw(k,x)

Description

example

lambertw(x) returns the principal branch of the Lambert W function. This syntax is equivalent to lambertw(0,x).

example

lambertw(k,x) is the kth branch of the Lambert W function. This syntax returns real values only if k = 0 or k = -1.

Examples

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Return Equation with Lambert W Function as Its Solution

The Lambert W function W(x) isa set of solutions of the equation x = W(x)eW(x).

Solve this equation. The solution is the Lambert W function.

syms x Weqn = x == W*exp(W);solve(eqn,W)
ans =lambertw(0, x)

Verify that branches of the Lambert W function are valid solutions of the equation x = W*eW:

k = -2:2;eqn = subs(eqn,W,lambertw(k,x));isAlways(eqn)
ans = 1×5 logical array 1 1 1 1 1

Depending on its arguments, lambertw canreturn floating-point or exact symbolic results.

Compute the Lambert W functions for these numbers. Because the numbers are not symbolic objects, you get floating-point results.

A = [0 -1/exp(1); pi i];lambertw(A)
ans = 0.0000 + 0.0000i -1.0000 + 0.0000i 1.0737 + 0.0000i 0.3747 + 0.5764i
lambertw(-1,A)
ans = -Inf + 0.0000i -1.0000 + 0.0000i -0.3910 - 4.6281i -1.0896 - 2.7664i

Compute the Lambert W functions for the numbers converted to symbolic objects. For most symbolic (exact) numbers, lambertw returns unresolved symbolic calls.

A = [0 -1/exp(sym(1)); pi i];W0 = lambertw(A)
W0 =[ 0, -1][ lambertw(0, pi), lambertw(0, 1i)]
Wmin1 = lambertw(-1,A)
Wmin1 =[ -Inf, -1][ lambertw(-1, pi), lambertw(-1, 1i)]

Convert symbolic results to double by using double.

double(W0)
ans = 0.0000 + 0.0000i -1.0000 + 0.0000i 1.0737 + 0.0000i 0.3747 + 0.5764i

Plot Two Main Branches of Lambert W Function

Open Live Script

Plot the two main branches, W0(x) and W-1(x), of the Lambert W function.

syms xfplot(lambertw(x))hold onfplot(lambertw(-1,x))hold offaxis([-0.5 4 -4 2])title('Lambert W function, two main branches')legend('k=0','k=1','Location','best')

Lambert W function - MATLAB lambertw (1)

Lambert W Function Plot on Complex Plane

Open Live Script

Plot the principal branch of the Lambert W function on the complex plane.

Plot the real value of the Lambert W function by using fmesh. Simultaneously plot the contours by setting 'ShowContours' to 'On'.

syms x yf = lambertw(x + 1i*y);interval = [-100 100 -100 100];fmesh(real(f),interval,'ShowContours','On')

Lambert W function - MATLAB lambertw (2)

Plot the imaginary value of the Lambert W function. The plot has a branch cut along the negative real axis. Plot the contours separately.

fmesh(imag(f),interval)

Lambert W function - MATLAB lambertw (3)

fcontour(imag(f),interval,'Fill','on')

Lambert W function - MATLAB lambertw (4)

Plot the absolute value of the Lambert W function.

fmesh(abs(f),interval,'ShowContours','On')

Lambert W function - MATLAB lambertw (5)

Input Arguments

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x — Input
number | vector | matrix | array | symbolic number | symbolic variable | symbolic array | symbolic function | symbolic expression

Input, specified as a number, vector, matrix, or array, or a symbolic number, variable, array, function, or expression.

At least one input argument must be a scalar, or both arguments must be vectors or matrices of the same size. If one input argument is a scalar and the other is a vector or matrix, lambertw expands the scalar into a vector or matrix of the same size as the other argument with all elements equal to that scalar.

k — Branch of Lambert W function
integer | vector or matrix of integers | symbolic integer | symbolic vector or matrix of integers

Branch of Lambert W function, specified as an integer, a vector or matrix of integers, a symbolic integer, or a symbolic vector or matrix of integers.

At least one input argument must be a scalar, or both arguments must be vectors or matrices of the same size. If one input argument is a scalar and the other is a vector or matrix, lambertw expands the scalar into a vector or matrix of the same size as the other argument with all elements equal to that scalar.

More About

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Lambert W Function

The Lambert W function W(x) representsthe solutions y of the equation yey=x for any complex number x.

  • For complex x, the equation hasan infinite number of solutions y = lambertW(k,x) where k rangesover all integers.

  • For all real x ≥ 0, the equation has exactly one real solution y = lambertW(x) = lambertW(0,x).

  • For real x where −e−1<x<0, the equation has exactly two real solutions. The larger solution is represented by y = lambertW(x) and the smaller solution by y = lambertW(–1,x).

  • For x=−e−1, the equation has exactly one real solution y = –1 = lambertW(0, –exp(–1)) = lambertW(–1, -exp(–1)).

References

[1] Corless, R.M., G.H. Gonnet, D.E.G. Hare, D.J. Jeffrey, and D.E. Knuth. "On the Lambert W Function." Advances in Computational Mathematics, Vol. 5, pp. 329–359, 1996.

Version History

Introduced before R2006a

See Also

Functions

  • wrightOmega

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                                                        Lambert W function - MATLAB lambertw (2024)
                        

                            
                        

                    

                                                            
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