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Root finding algorithms for nonlinear functions

Type definition
type solver =
  1. | Bisec
  2. | FalsePos
  3. | Ridder
  4. | Brent
    (*

    Type of root functions of univariate functions.

    *)
Core functions
val fzero : ?solver:solver -> ?max_iter:int -> ?xtol:float -> (float -> float) -> float -> float -> float

fzero ~solver f a b tries to find the root of univariate function f in the bracket [a, b] using method solver. This is the hub function of the individual root finding algorithms in the following sections. You can certainly call each individual ones.

Parameters: * solver: solver, default one is Brent method. * max_iter: maximum number of iterations, default value is 1000. * xtol: the tolerance of x on abscissa, default value is 1e-6. * f: the univariate scalar function to find root. * a: boundary of bracket. * b: boundary of bracket.

Root of univariate functions
val bisec : ?max_iter:int -> ?xtol:float -> (float -> float) -> float -> float -> float

bisec f a b tries to find the root of univariate function f in the bracket defined by [a, b].

Parameters: * max_iter: maximum number of iterations. * xtol: the tolerance of x on abscissa. * f: the univariate scalar function to find root. * a: boundary of bracket. * b: boundary of bracket.

val false_pos : ?max_iter:int -> ?xtol:float -> (float -> float) -> float -> float -> float

false_pos f a b tries to find the root of univariate function f in the bracket defined by [a, b].

Parameters: * max_iter: maximum number of iterations. * xtol: the tolerance of x on abscissa. * f: the univariate scalar function to find root. * a: boundary of bracket. * b: boundary of bracket.

val ridder : ?max_iter:int -> ?xtol:float -> (float -> float) -> float -> float -> float

ridder f a b tries to find the root of univariate function f in the bracket defined by [a, b].

Parameters: * max_iter: maximum number of iterations. * xtol: the tolerance of x on abscissa. * f: the univariate scalar function to find root. * a: boundary of bracket. * b: boundary of bracket.

val brent : ?max_iter:int -> ?xtol:float -> (float -> float) -> float -> float -> float

brent f a b tries to find the root of univariate function f in the bracket defined by [a, b].

Parameters: * max_iter: maximum number of iterations. * xtol: the tolerance of x on abscissa. * f: the univariate scalar function to find root. * a: boundary of bracket. * b: boundary of bracket.

Refer to :cite:`brent2013algorithms`

Helper functions
val bracket_expand : ?rate:float -> ?max_iter:int -> (float -> float) -> float -> float -> (float * float) option

bracket_expand f a b expands the bracket [a, b] for a given function f until it finds f a and f b have different signs.

Parameters: * max_iter: maximum number of iterations, the default is 100 * rate: rate of growth, the default is 1.6 * f: the univariate scalar function to find root. * a: initial boundary of bracket. * b: initial boundary of bracket.

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