![]() ![]() In quantum electrodynamics, polylogarithms of positive integer order arise in the calculation of processes represented by higher-order Feynman diagrams. In quantum statistics, the polylogarithm function appears as the closed form of integrals of the Fermi–Dirac distribution and the Bose–Einstein distribution, and is also known as the Fermi–Dirac integral or the Bose–Einstein integral. Only for special values of s does the polylogarithm reduce to an elementary function such as the natural logarithm or a rational function. In mathematics, the polylogarithm (also known as Jonquière's function, for Alfred Jonquière) is a special function Li s( z) of order s and argument z. We can only use something like uniroot.all and hope it works.Not to be confused with polylogarithmic function or logarithmic integral function. But there are still many more functions which cannot be transformed into polynomials by any stretch. Some functions, like sums of powers of for example, can be transformed by a trigonometric substitution into a complex polynomial using Euler’s famous formula. The main limitation of this approach is that not all functions can be made into polynomials. Call uniroot on each interval in the above partition.Partition domain into intervals each containing one of the transformed roots.Transform the remaining ones back to the original domain.Eliminate infeasible roots that fall outside the domain/range of the substitution rule.Use polyroot to find zeroes of transformed function.This was the case for the functions that came up in my research, and I settled on the following scheme which was much more stable and accurate. This error can be much larger if the substitution rule has a large derivative near one of the zeroes. Careful readers may have noticed the above roots have some different digits depending on which function computed them. As long as we carefully select the correct polynomial roots and transform back to the trigonometric scale, we’ll get all the roots of the original function: acos(Re(roots))Īnother word of caution regarding substitutions: they might be numerically unstable. The complex roots correspond to roots of the polynomial which are not achieved by the trigonometric version because. Since the polynomial had degree 4, there are 4 roots. If we call polyroot with the coefficients (in order of increasing degree) it will return all roots, including complex valued ones. Our running example happens to be a trigonometric polynomial (what a coincidence!). Note that the “poly” here does not stand for multiple, but rather for polynomial. And this brings us to another one of R’s great root finding functions, polyroot. Luckily for me, I was able to transform the functions I was working with into trigonometric polynomials. I could not manually plot them or check to see if uniroot.all was using a sufficiently fine grid to capture all the roots. ![]() I was recently working on an algorithm that required finding multiple roots of many functions. However, this approach is not guaranteed to work unless the number of subintervals is sufficiently large (which depends on how close together or to the boundary the roots are). ![]() It divides the search interval into many subintervals (the default is 100) and uses uniroot on those intervals which have a sign change. The package rootSolve contains a function uniroot.all which attempts to automate this procedure. If you need to find all the roots, you must call uniroot multiple times and each time specify an interval containing only one root. This is a serious problem arising from the underlying mathematical difficulty of finding multiple roots. In the case when there are an odd number of roots on the interval, uniroot will only find and return one of them. uniroot(f, c(0,2))į() values at end points not of opposite sign However, this function has major limitations if there is more than one root (as implied by its name!). Perhaps the most widely used root finding function in R is uniroot. Here the function is plotted and we can see it has two roots on this interval. As an example consider the following function f <- function(x) cos(x)^4 - 4*cos(x)^3 + 8*cos(x)^2 - 5*cos(x) + 1/2 In this post I describe some methods for root finding in R and the limitations of these when there is more than one root. ![]()
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