root1d 0.5 · OCaml Package

1D Root finding algorithms.

version 0.5

val brent : ?tol:float -> (float -> float) -> float -> float -> float

brent f a b returns an approximation x of a root of f in the interval [a,b] with absolute accuracy 6. *. epsilon_float *. abs_float(x) +. tol.

raises Invalid_argument
if f(a) *. f(b) > 0..

parameter tol
desired length of the interval of uncertainty of the final result (must be >= 0). Default: sqrt epsilon_float.
Ref.: Brent, R. (1973) Algorithms for Minimization without Derivatives. Englewood Cliffs, NJ: Prentice-Hall.

val bisection : ?eps:float -> (float -> float) -> float -> float -> float

bisection f a b find an approximation of a root in the interval [a,b] using the bisection algorithm.

raises Invalid_argument
if f(a) *. f(b) > 0. or eps <= 0.

parameter eps
is the desired relative error on the solution. More precisely, it terminates when the interval [a,b] verifies |a-b| ≤ eps max(|a|, |b|). Default: sqrt epsilon_float.

val illinois : ?eps:float -> (float -> float) -> float -> float -> float

illinois f a b find an approximation of a root in the interval [a,b] using the Illinois algorithm (which is the Regula Falsi method with a small twist). Order of convergence: ³√3 ≈ 1.442.

raises Invalid_argument
if f(a) *. f(b) > 0. or eps <= 0.

parameter eps
is the desired relative error on the solution. More precisely, it terminates when the interval [a,b] verifies |a-b| ≤ eps max(|a|, |b|). Default: sqrt epsilon_float.
Ref.: M. Dowell, P. Jarrat, A modified Regula Falsi method for computing the root of anequation, BIT 11 (1971), 168–174.

val newton : 
  ?good_enough:(float -> float -> float -> bool) ->
  (float -> float * float) ->
  float ->
  float

newton f_f' x0 returns an approximate root of f close to the initial guess x0 using Newton's method. f_f' is a function such that f_f' x returns the couple (f x, f' x) where f' x is the derivative of f at x.

raises Failure
if the derivative vanishes during the computations.

parameter good_enough
takes as arguments the current approximation x, the previous approximation xprev, and f(x) and returns whether x is a good enough approximation. Default: abs_float(f x) < sqrt epsilon_float.

val brent2 : ?tol:float -> (float -> float * int) -> float -> float -> float

brent2 f a b finds a zero of the function f in the same way brent f a b does except that f x returns the couple (y, z) for the number y * 2**z. Thus underflow and overflow can be avoided for a function with large range.

Ref.: Brent, R. (1973) Algorithms for Minimization without Derivatives. Englewood Cliffs, NJ: Prentice-Hall.

package root1d