Root finding functions for Julia


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Root finding functions for Julia

This package contains simple routines for finding roots of continuous scalar functions of a single real variable. The basic interface is through the function fzero which dispatches to an appropriate algorithm based on its argument(s):

  • fzero(f, a::Real, b::Real) and fzero(f, bracket::Vector) call the find_zero algorithm to find a root within the bracket [a,b]. When a bracket is used with Float64 arguments, the algorithm is guaranteed to converge to a value x with either f(x) == 0 or at least one of f(prevfloat(x))*f(x) < 0 or f(x)*f(nextfloat(x)) < 0. (The function need not be continuous to apply the algorithm, as the last condition can still hold.)

  • fzero(f, x0::Real; order::Int=0) calls a derivative-free method. The default method is a bit plodding but more robust to the quality of the initial guess than some others. For faster convergence and fewer function calls, an order can be specified. Possible values are 1, 2, 5, 8, and 16. The order 2 Steffensen method can be the fastest, but is in need of a good initial guess. The order 8 method is more robust and often as fast. The higher-order methods may be faster when using Big values.

  • fzero(f, x0::Real, bracket::Vector) calls a derivative-free algorithm with initial guess x0 with steps constrained to remain in the specified bracket.

  • fzeros(f, a::Real, b::Real; no_pts::Int=200) will split the interval [a,b] into many subintervals and search for zeros in each using a bracketing method if possible. This naive algorithm may miss double zeros that lie within the same subinterval and zeros where there is no crossing of the x-axis.

For historical purposes, there are implementations of Newton's method (newton), Halley's method (halley), and the secant method (secant_method). For the first two, if derivatives are not specified, they will be computed using the ForwardDiff package.

Usage examples

f(x) = exp(x) - x^4
## bracketing
fzero(f, 8, 9)                # 8.613169456441398
fzero(f, -10, 0)              # -0.8155534188089606
fzeros(f, -10, 10)            # -0.815553, 1.42961  and 8.61317 

## use a derivative free method
fzero(f, 3)                   # 1.4296118247255558

## use a different order
fzero(sin, 3, order=16)       # 3.141592653589793

## BigFloat values yield more precision
fzero(sin, BigFloat(3.0))     # 3.1415926535897932384...with 256 bits of precision

The fzero function can be used with callable objects:

using SymEngine; @vars x
fzero(x^5 - x - 1, 1.0)


using Polynomials; x = variable(Int)
fzero(x^5 - x - 1, 1.0)

The well-known methods can be used with or without supplied derivatives. If not specified, the ForwardDiff package is used for automatic differentiation.

f(x) = exp(x) - x^4
fp(x) = exp(x) - 4x^3
fpp(x) = exp(x) - 12x^2
newton(f, fp, 8)              # 8.613169456441398
newton(f, 8)    
halley(f, fp, fpp, 8)
halley(f, 8)
secant_method(f, 8, 8.5)

The automatic derivatives allow for easy solutions to finding critical points of a function.

## mean
as = rand(5)
function M(x) 
  sum([(x-a)^2 for a in as])
fzero(D(M), .5) - mean(as)    # 0.0

## median
function m(x) 
  sum([abs(x-a) for a in as])
fzero(D(m), 0, 1)  - median(as) # 0.0

Alternate interface

As an alternative interface to the MATLAB-inherited one through fzero, the function find_zero can be used. For this, a type is used to specify the method. For example,

find_zero(sin, 3.0, Order0())
find_zero(x -> x^5 - x- 1, 1.0, Order1())  # also Order2(), Order5(), Order8(), Order16()

And bracketing methods:

find_zero(sin, (3, 4), Bisection())
find_zero(x -> x^5 - x - 1, (1,2), FalsePosition())

Some additional documentation can be read here.

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