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Roots.jl

Documentation for Roots.jl

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Roots is a Julia package for finding zeros of continuous scalar functions of a single real variable using floating point numbers. That is solving $f(x)=0$ for $x$ adjusting for floating-point idiosyncrasies.

The find_zero function provides the primary interface. It supports various algorithms through the specification of a method. These include:

  • Bisection-like methods. For functions where a bracketing interval is known (one where $f(a)$ and $f(b)$ have alternate signs), there are several bracketing methods, including Bisection. For most floating point number types, bisection occurs in a manner exploiting floating point storage conventions leading to an exact zero or a bracketing interval as small as floating point computations allows. Other methods include A42, AlefeldPotraShi, Roots.Brent, Roots.Chandrapatlu, Roots.ITP, Roots.Ridders, and $12$-flavors of FalsePosition. The default bracketing method for the basic floating-point types is Bisection , as it is more robust to some inputs, but A42 and AlefeldPotraShi typically converge in a few iterations and are more performant.
  • Several derivative-free methods. These are specified through the methods Order0, Order1 (the secant method), Order2 (the Steffensen method), Order5, Order8, and Order16. The number indicates, roughly, the order of convergence. The Order0 method is the default, and the most robust, as it finishes off with a bracketing method when a bracket is encountered, The higher order methods promise higher order (faster) convergence, though don't always yield results with fewer function calls than Order1 or Order2. Roots.Sidi is a family of methods. The methods Roots.Order1B and Roots.Order2B are superlinear and quadratically converging methods independent of the multiplicity of the zero.
  • Methods requiring one or more derivatives: Roots.Newton, Roots.Halley are classical ones, Roots.QuadraticInverse, Roots.ChebyshevLike, Roots.SuperHalley are others. Roots.Schroder provides a quadratic method, like Newton's method, which is independent of the multiplicity of the zero. The Roots.ThukralXB, X=2, 3, 4, or 5 are also multiplicity free. The X denotes the number of derivatives that need specifying. The Roots.LithBoonkkampIJzerman{S,D} methods remember S steps and use D derivatives.