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Root-Finding and Minimisation Functions

Source: R/rootfinding_and_minimisation.R
rootfinding_and_minimisation.Rd

Functions for root-finding and minimisation using various algorithms.

Usage

bisect(
  f,
  lower,
  upper,
  digits = .Machine$double.digits,
  max_iter = .Machine$integer.max
)

bracket_and_solve_root(
  f,
  guess,
  factor,
  rising,
  digits = .Machine$double.digits,
  max_iter = .Machine$integer.max
)

toms748_solve(
  f,
  lower,
  upper,
  digits = .Machine$double.digits,
  max_iter = .Machine$integer.max
)

newton_raphson_iterate(
  f,
  guess,
  lower,
  upper,
  digits = .Machine$double.digits,
  max_iter = .Machine$integer.max
)

halley_iterate(
  f,
  guess,
  lower,
  upper,
  digits = .Machine$double.digits,
  max_iter = .Machine$integer.max
)

schroder_iterate(
  f,
  guess,
  lower,
  upper,
  digits = .Machine$double.digits,
  max_iter = .Machine$integer.max
)

brent_find_minima(
  f,
  lower,
  upper,
  digits = .Machine$double.digits,
  max_iter = .Machine$integer.max
)

Arguments

f

A function to find the root of or to minimise. It should take and return a single numeric value for root-finding, or a numeric vector for minimisation.

lower

The lower bound of the interval to search for the root or minimum.

upper

The upper bound of the interval to search for the root or minimum.

digits

The number of significant digits to which the root or minimum should be found. Defaults to double precision.

max_iter

The maximum number of iterations to perform. Defaults to the maximum integer value.

guess

A numeric value that is a guess for the root or minimum.

factor

Size of steps to take when searching for the root.

rising

If TRUE, the function is assumed to be rising, otherwise it is assumed to be falling.

Value

A list containing the root or minimum value, the value of the function at that point, and the number of iterations performed.

Details

This package provides a set of functions for finding roots of equations and minimising functions using different numerical methods. The methods include bisection, bracket and solve, TOMS 748, Newton-Raphson, Halley's method, Schroder's method, and Brent's method. It also includes functions for finding roots of polynomials (quadratic, cubic, quartic) and computing minima.

See also

Boost Documentation for more details on the mathematical background.

Examples

f <- function(x) x^2 - 2
bisect(f, lower = 0, upper = 2)
#> $lower
#> [1] 1.414214
#> 
#> $upper
#> [1] 1.414214
#> 
#> $iterations
#> [1] 54
#> 
bracket_and_solve_root(f, guess = 1, factor = 0.1, rising = TRUE)
#> $lower
#> [1] 1.414214
#> 
#> $upper
#> [1] 1.414214
#> 
#> $iterations
#> [1] 91
#> 
toms748_solve(f, lower = 0, upper = 2)
#> $lower
#> [1] 1.414214
#> 
#> $upper
#> [1] 1.414214
#> 
#> $iterations
#> [1] 9
#> 
f <- function(x) c(x^2 - 2, 2 * x)
newton_raphson_iterate(f, guess = 1, lower = 0, upper = 2)
#> [1] 1.414214
#> attr(,"iterations")
#> [1] 6
f <- function(x) c(x^2 - 2, 2 * x, 2)
halley_iterate(f, guess = 1, lower = 0, upper = 2)
#> [1] 1.414214
#> attr(,"iterations")
#> [1] 4
schroder_iterate(f, guess = 1, lower = 0, upper = 2)
#> [1] 1.414214
#> attr(,"iterations")
#> [1] 5
f <- function(x) (x - 2)^2 + 1
brent_find_minima(f, lower = 0, upper = 4)
#> $minimum
#> [1] 2
#> 
#> $value
#> [1] 1
#> 
#> $iterations
#> [1] 5
#>