Wolfram Research, Inc.

Problem 5

What is the largest zero of the 1000 Laguerre polynomial to 12 significant digits?

Result
3943.24739485...

Method 1: Find the root numerically.

A well-known upper bound for the largest root of the Laguerre polynomial is 4v + 2 (see [4, 5]). The Laguerre polynomials satisfy the well-known differential equation .

Transforming the equation to Liouville normal form using , we get . When , the solution will oscillate and will therefore have zeros. When , the solution will increase or decrease monotonically. The larger zero,, approximates the largest zero of .

This gives us a starting point ) for the numerical root finding procedure. We slide down the curve on the strictly concave side toward the root.

We define the function f to be the Laguerre polynomial and the gradient g of f. For faster and more reliable numerical calculation we avoid evaluating these functions as explicit polynomials.

The secant method gives the same result.

We check numerically to see that this is really the largest root.

Using real root isolation techniques from the package Algebra`RootIsolation`, we can prove that this is the largest root. We can use the same package to find an interval with rational endpoints containing the root to the required precision.

Method 2: Calculate all roots one after another.

We start at the origin and step toward the largest root by explicitly calculating all the roots on the way.

This is the largest root.

Here is the cumulative root density as well as the spacing between the roots.