28

13

26

21

Linux: Windows:

# Root finding functions for Julia

This package contains simple routines for finding roots of continuous scalar functions of a single real variable. The `find_zero`function provides the primary interface. It supports various algorithms through the specification of an method. These include:

• Bisection-like algorithms. For functions where a bracketing interval is known (one where f(a) and f(b) have alternate signs), the `Bisection` method can be specified with a guaranteed convergence. For most floating point number types, bisection occurs in a manner exploiting floating point storage conventions. For others, an algorithm of Alefeld, Potra, and Shi is used.

For typically faster convergence -- though not guaranteed -- the `FalsePosition` method can be specified. This method has one of 12 implementations for a modified secant method to accelerate convergence.

• Several derivative-free methods are implemented. 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, but generally takes many more function calls. The higher order methods promise higer order convergence, though don't always yield results with fewer function calls than `Order1` or `Order2`.

• There are two historic methods that require a derivative: `Roots.Newton` and `Roots.Halley`. (Neither is currently exported.) If a derivative is not given, an automatic derivative is found using the `ForwardDiff` package.

Each method's documentation has additional detail.

Some examples:

``````f(x) = exp(x) - x^4

# a bisection method has the bracket specified with a tuple or vector
julia> find_zero(f, (8,9), Bisection())
8.613169456441398

julia> find_zero(f, (-10, 0))  # Bisection if x is a tuple and no method
-0.8155534188089606

julia> find_zero(f, (-10, 0), FalsePosition())  # just 11 function evaluations
-0.8155534188089607

## find_zero(f, x0::Number) will use Order0()
julia> find_zero(f, 3)         # default is Order0()
1.4296118247255556

julia> find_zero(f, 3, Order1()) # same answer, different method
1.4296118247255556

julia> find_zero(sin, BigFloat(3.0), Order16())
3.141592653589793238462643383279502884197169399375105820974944592307816406286198
``````

The `find_zero` function can be used with callable objects:

``````using SymEngine
@vars x
find_zero(x^5 - x - 1, 1.0)  # 1.1673039782614185
``````

Or,

``````using Polynomials
x = variable(Int)
fzero(x^5 - x - 1, 1.0)  # 1.1673039782614185
``````

The function should respect the units of the `Unitful` package:

``````using Unitful
s = u"s"; m = u"m"
g = 9.8*m/s^2
v0 = 10m/s
y0 = 16m
y(t) = -g*t^2 + v0*t + y0
find_zero(y, 1s)      # 1.886053370668014 s
``````

Newton's method can be used without taking derivatives:

``````f(x) = x^3 - 2x - 5
x0 = 2
find_zero(f, x0, Roots.Newton())   # 2.0945514815423265
``````

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])
end
fzero(D(M), .5) - mean(as)    # 0.0

## median
function m(x)
sum([abs(x-a) for a in as])

end
fzero(D(m), 0, 1)  - median(as) # 0.0
``````

### Multiple zeros

The `find_zeros` function can be used to search for all zeros in a specified interval. The basic algorithm splits the interval into many subintervals. For each, if there is a bracket a bracketing algorithm is used to identify a zero, otherwise a derivative free method is used to check. This algorithm can miss zeros for various reasons, so the results should be confirmed by other means.

``````f(x) = exp(x) - x^4
find_zeros(f, -10, 10)
``````

### Convergence

For most algorithms (besides the `Bisection` ones) convergence is decided when

• The value f(x_n) ≈ 0 with tolerances `atol` and `rtol` or

• the values x_n ≈ x_{n-1} with tolerances `xatol` and `xrtol` and f(x_n) ≈ 0 with a relaxed tolerance based on `atol` and `rtol`.

• an algorithm encounters an `NaN` or `Inf` and yet f(x_n) ≈ 0 with a relaxed tolerance based on `atol` and `rtol`.

There is no convergence if the number of iterations exceed `maxevals`, or the number of function calls exceeds `maxfnevals`.

The tolerances may need to be adjusted. To determine if convergence occurs due to f(x_n) ≈ 0, it is necessary to consider that even if `xstar` is the correct answer mathematically, due to floating point roundoff it is expected that f(xstar) ≈ f'(xstar) ⋅ xstar ⋅ ϵ. The relative error used accounts for the value of `x`, but the default tolerance may need adjustment if the derivative is large near the zero, as the default is a bit aggressive. On the other hand, the absolute tolerance might seem too relaxed.

To determine if convergence is determined as x_n ≈ x_{n-1} the check on f(x_n) ≈ 0 is done as algorithms can be fooled by asymptotes, or other areas where the tangent lines have large slopes.

The `Bisection` and `Roots.A42` methods will converge, so the tolerances are ignored.

## An alternate interface

For MATLAB users, this functionality is provided by the `fzero` function. `Roots` also provides this alternative interface:

• `fzero(f, a::Real, b::Real)` and ```fzero(f, bracket::Vector)``` call the `find_zero` algorithm with the `Bisection` method.

• `fzero(f, x0::Real; order::Int=0)` calls a derivative-free method. with the order specified matching one of `Order0`, `Order1`, etc.

• `fzeros(f, a::Real, b::Real; no_pts::Int=200)` will call `find_zeros`.

• The function `secant_method`, `newton`, and `halley` provide direct access to those methods.

## 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
``````

04/11/2013

10 days ago

233 commits