# Lambert W function Computes the Lambert W function, giving efficient solutions to the equation x*exp(x)==z ```r lambertW_base(z, b = 0, maxiter = 10, eps = .Machine$double.eps, min.imag = 1e-09) lWasymp(z,logz) lambertW(z,...) ``` ## Arguments - `z`: (complex) vector of values for which to compute the function - `logz`: (complex (?)) vector of $log(z)$ values (to be specified by name instead of `z`) - `b`: (integer) b=0 specifies the principal branch, 0 and -1 are the ones that can take non-complex values - `maxiter`: maximum numbers of iterations for convergence - `eps`: convergence tolerance - `min.imag`: maximum magnitude of imaginary part to chop when returning solutions - ``...``: arguments to pass to `lambertW_base` ## Details Compute the Lambert W function of z. This function satisfies $W(z)*exp(W(z)) = z$, and can thus be used to express solutions of transcendental equations involving exponentials or logarithms. For $z>1e307$, an asymptotic formula (from Corless et al by way of [http://mathworld.wolfram.com/LambertW-Function.html](http://mathworld.wolfram.com/LambertW-Function.html)) is used: `lambertW` is a wrapper that automatically selects the asymptotic formula where appropriate. * In ecology, the Lambert W can be used to solve the so-called "Rogers equation" for predator functional response with depletion. * In epidemiology, the Lambert W function solves the final-size equation of a simple SIR epidemic model. ## Returns Complex or real vector of solutions. ## References Corless, Gonnet, Hare, Jeffrey, and Knuth (1996), "On the Lambert W Function", Advances in Computational Mathematics 5(4):329-359 ## Author(s) Nici Schraudolph (original version (c) 1998), Ben Bolker (R translation) ## See Also `?Lambert` in the `gsl` package by Robin Hankin, which uses Gnu Scientific Library code; also `?lambertW` in the `VGAM` and `pracma` packages, and the `lambertW` package ## Note This implementation should return values within 2.5*eps of its counterpart in Maple V, release 3 or later. Please report any discrepancies to the author or translator. The derivative of the `lambertW` function is `plogis(-lambertW)`. ## Examples ```r curve(lambertW(x),from=0,to=10) pvec <- seq(-1,1,length=40) m <- outer(pvec,pvec,function(x,y)Re(lambertW(x+y*1i))) persp(pvec,pvec,m, theta=290,shade=0.5,zlab="lambertW") num1 <- uniroot(function(x) {x*exp(x)-1},lower=0,upper=1,tol=1e-9) abs(lambertW(1)-num1$root)<1e-9 ### ## Rogers random predator equation: rogers.pred <- function(N0,a,h,T) { N0 - lambertW(a*h*N0*exp(-a*(T-h*N0)))/(a*h) } holling2.pred <- function(N0,a,h) { a*N0/(1+a*h*N0) } curve(rogers.pred(x,a=1,h=0.2,T=1),from=0,to=60, ylab="Number eaten/unit time",xlab="Initial number",ylim=c(0,5), main="Predation: a=1, h=0.2") curve(rogers.pred(x,a=1,h=0.2,T=5)/5,add=TRUE,lty=2,from=0) curve(rogers.pred(x,a=1,h=0.2,T=0.2)*5,add=TRUE,lty=3,from=0) curve(rogers.pred(x,a=1,h=0.2,T=10)/10,add=TRUE,lty=4,from=0) curve(holling2.pred(x,a=1,h=0.2),add=TRUE,lty=1,lwd=2,from=0) abline(h=5) legend(30,2, c(paste("Rogers, T=",c(0.2,1,5,10),sep=""), "Holling type II"),lwd=c(rep(1,4),2),lty=c(3,1,2,4,1)) ## final size of an epidemic finalsize <- function(R0) { 1+1/R0*lambertW(-R0*exp(-R0)) } curve(finalsize,from=1,to=10,xlab=expression(R[0]),ylab="Final size") ## comparison of asymptotic results tmpf <- function(x) { L0 <- lambertW_base(10^x) L1 <- lWasymp(logz=x*log(10)) (L1-L0)/L0 } curve(tmpf,from=1,to=307,log="y") ## derivative ## don't run (avoid numDeriv dependency) ## require(numDeriv) ## grad(lambertW(1)) ## plogis(-lambertW(1)) ```

lambertW function