Universita` degli Studi di Perugia Facolta` di Scienze Matematiche, Fisiche e Naturali Corso di Laurea Triennale in Informatica The Lambert W function on matrices Candidato Relatore MassimilianoFasi BrunoIannazzo Contents Preface iii 1 The Lambert W function 1 1.1 Definitions............................. 1 1.2 Branches.............................. 2 1.3 Seriesexpansions ......................... 10 1.3.1 Taylor series and the Lagrange Inversion Theorem. 10 1.3.2 Asymptoticexpansions. 13 2 Lambert W function for scalar values 15 2.1 Iterativeroot-findingmethods. 16 2.1.1 Newton’smethod. 17 2.1.2 Halley’smethod . 18 2.1.3 K¨onig’s family of iterative methods . 20 2.2 Computing W ........................... 22 2.2.1 Choiceoftheinitialvalue . 23 2.2.2 Iteration.......................... 26 3 Lambert W function for matrices 29 3.1 Iterativeroot-findingmethods. 29 3.1.1 Newton’smethod. 31 3.2 Computing W ........................... 34 3.2.1 Computing W (A)trougheigenvectors . 34 3.2.2 Computing W (A) trough an iterative method . 36 A Complex numbers 45 A.1 Definitionandrepresentations. 45 B Functions of matrices 47 B.1 Definitions............................. 47 i ii CONTENTS C Source code 51 C.1 mixW(<branch>, <argument>) ................. 51 C.2 blockW(<branch>, <argument>, <guess>) .......... 52 C.3 matW(<branch>, <argument>) ................. 53 Preface Main aim of the present work was learning something about a not- so-widely known special function, that we will formally call Lambert W function. This function has many useful applications, although its presence goes sometimes unrecognised, in mathematics and in physics as well, and we found some of them very curious and amusing. One of the strangest situation in which it comes out is in writing in a simpler form the function .. z. h(z)= zz whenever it makes sense, that has been proven to be equal, wherever it converges, to the more elegant form W ( log(z)) − . log(z) − The most interesting aspect of our function is probably that there does not exists an explicit expression for it, but its inverse has an easy and elegant definition, within very good regularity properties. The fact makes the question challenging and, then, quite fun. And we had a lot of fun indeed, studying the scalar case, creating and drawing fractals, looking at them and performing a wide variety of numerical tests. We ended the first part of our work developing a new algorithm, slightly faster than the one we studied, that is implemented by the function lambertw(b,a) of Octave. We did not get a complete satisfaction, since we did not manage in proving that our algorithm and the one we started from were conver- gent, as we – and many other people – suspected, but we got happy iii iv CONTENTS enough when we found that experimental results confirmed our hy- pothesis. Since we were having too much fun, we turned our attention to the matrix case, more enveloped and then more suitable to rack our brains. We had indeed hard times, fighting against unstable and non- convergent algorithms. Nevertheless, at a certain point things changed, and we found a stable and convergent, even though slow, algorithm for computing the Lambert W function of matrix argument. We showed also that the currently used matrix algorithm had some defects, that the one we proposed did not have, even though is still less accurate in some cases. At the end of our work we felt a little bit more familiar with W , so that now among us we call it, very informally, the Lambert. Chapter 1 The Lambert W function 1.1 Definitions The Lambert W function is a multivalued complex function defined, for each x C, as the solution of the equation ∈ W (x)eW (x) = x, x C, (1.1) ∈ or, in a certain sense, as the inverse of the function f : C C, defined by x xex. → 7→ It should be noted that Equation (1.1), assuming x = 0, can be rewritten as 6 W (x) + log(W (x)) = log(x), x C (1.2) ∈ where we consider log(x) multivalued and fix the value of log(W (x)) by cutting the z-plane as we will see later. Note that (1.2) is the defining equation of the so-called Wright ω function [GLM99]. We will use the letter W for this function, following E. M. Wright usage (e.g. [Wri59]), that became a standard after the publication of [CGH+96], and will call it the Lambert W function because it is the logarithm of a special case of Euler’s version of Lambert’s series solution of the trinomial equation x = q + xm, where q R and m is a positive integer. The relationships between that serie∈s expansion and our function have been deeply investigated by Corless et al. in [CGH+96]. 1 2 CHAPTER 1. THE LAMBERT W FUNCTION 1.2 Branches Since the Lambert W function is multivalued, choosing a convention for naming branches and branch points is mandatory, hence we will present the notation we will use in this work, recalling some basic concepts about multivalued functions, branches and branch points. Let us consider a function f : C C. We can create two planes, a z-plane for the domain space and a→w-plane for the range one. Thus we can view f(z) as a mapping from the z-plane to the w-plane, and in order to understand how such mapping works, we analyse how various geometric curves in the z-plane are mapped in the w-plane by w = f(z). To get a deeper insight, let us consider one of the simplest non- trivial complex functions, the p-th root function defined by w = zp, p> 0. (1.3) From the polar decomposition of a complex number we get that w = zp = zpepiθ (1.4) and then w = z p, (1.5) | | | | arg(w)= p arg(z). (1.6) Equation (1.5) shows that the circle z = ρ0 in the z-plane is mapped to the circle w = ρp, while equation| | (1.6) that a ray arg(z) = θ | | 0 0 issuing fom the origin in the z-plane is mapped to a ray arg(w)= pθ0 in the w-plane. In other words, as in the z-plane z moves in the positive direction at constant angular velocity around the circle of a radius of ρ, w moves in the w-plane around the circle of a radius of ρp, in the same direction but at p times the angular velocity (see Figure 1.1). Note that as in the z-plane z traverses the ray from 0 to at a constant speed, instead, w traverses the image ray in the same∞ direction but at an increasing speed. The positive real axis in the z-plane, that is a ray with angle 0, is mapped to the positive axis of the w-plane by the usual rule x xp. Turning back to the problem of finding an inverse function for7→ w = zp, we want to remark that every point w = 0 is hit by exactly p distinct 6 1.2. BRANCHES 3 (a) z-plane (b) w-plane Figure 1.1: The mapping of two vectors under w = zp for p = 2. values of z, the p p-th roots of z, then, in order to define an inverse function, we must restrict the domain in the z-plane so that each value w is hit by only one value of z: there are several ways of doing this, so we will proceed somewhat arbitrarily. Note that as rays sweep out an open sector of 2π/p of the z-plane, with the angle increasing from π/p to π/p, the image rays sweep out the entire w-plane, except for the− negative real axis, with the angle of the rays increasing from π to π. Thus we can draw a branch cut in the w-plane along the negative− real axis, from to 0, and define in that range an inverse function we will call the −∞principal branch of the p-th root function, and whose value is the unique p-th root lying in the aforementioned sector. The function we have just described is not the only continuous inverse function of w = zp that we can define, since for each sector Sk of the circle swept out by θ, where π π (2k 1) <θ< (2k + 1), k N, k<p, (1.7) p − p ∈ we can define a continuous inverse function from the w-plane to Sk . We refer to each of those p determinations of the inverse function as a branch of the p-th root function. The situation of the W function is not too much different, even thought it should be stressed that things are a little bit more com- plicated in that case. The Lambert W function has, indeed, infinite branches with different kind of boundaries. Let us proceed step by step. 4 CHAPTER 1. THE LAMBERT W FUNCTION First of all, we put w = W (z) and z = wew, and then specify the boundary curves that partition the w-plane and their mapping to the z-plane. If we put w = ξ + iη, (1.8) z = x + iy, (1.9) by equating (1.9) and z = wew, we get x =eξ(ξ cos η η sin η), (1.10) − y =eξ(η cos η + ξ sin η). (1.11) Note that in that case the w-plane maps onto the z-plane, while for the p-th root it was the z-plane that mapped onto the w-plane. That change of notation is because of the standard usage, and is not our arbitrary choice.