comphy (version 1.0.5)

roots_newton: Newton method for roots

Description

Find the zero of a function of one variable, \(f(x)\), given a starting value close to the zero, using Newton method.

Usage

roots_newton(
  f0,
  f1,
  x0 = 0,
  tol = 1e-09,
  imax = 1e+06,
  ftol = NULL,
  message = TRUE,
  logg = FALSE,
  ...
)

Value

A numeric value, the zero of the function (or, equivalently, the root of the equation \(f(x)=0\)).

Arguments

f0

A function of one variable. If the function includes variables, these will have to be passed as additional variables, using the same names in the function's definition (see examples).

f1

A function equal to the first derivative of `f0`. Parameters that are potentially included in `f0`, must be also included in `f1`.

x0

A numeric variable. The initial guess starting Newton's algorithm.

tol

A real small number. The smallest difference between the new zero's approximation and the previous one, above which the algorithm keeps working. As soon as the difference is less than `tol`, the algorithm stops and the current approximation is returned as the final approximation to the function's root. Default value is 1e-9.

imax

A positive integer. The maximum number of iterations of the algorithm. The default value is 1e6, although convergence is normally obtained with a number of iterations much smaller than imax. imax is important to stop search in those cases in which the algorithm gets stuck in endless loops (non-convergence).

ftol

A real small number. When `ftol` is not NULL (default value), Newton's algorithm stops when \(|f(x)| < ftol\). This parameter essentially introduces a different stopping criterion.

message

A logical variable to state whether messages about the root and the error have to be printed. The default is for the messages to be printed (`message=TRUE`).

logg

A logical variable to state whether information on the series of approximating roots is printed (TRUE) or not (FALSE). Default is for such information not to be printed (FALSE).

...

Parameters passed to the two functions `f0` and `f1`, if any.

Details

Finding the zero of \(f(x)\) is equivalent to finding the roots of the equation: $$ f(x) = 0 $$ The algorithm used is based on Newton method that needs an initial guess, x0, and the analytic expression of the function's first derivative. The method has a much faster convergence rate than both the bisection and secant methods, but it does not converge when the initial guess or any other subsequent approximations accidentally coincide with an optimal point of the function, i.e. a point at which the first derivative is zero. The algorithm can also potentially be stuck in an endless loop of repeating values for special combinations of functions and initial guess.

Examples

# The quadratic equation x^2-5*x+6=0 has two roots, 2 and 3
f0 <- function(x) return(x^2-5*x+6)

# First derivative
f1 <- function(x) return(2*x-5)

# Find root 2, starting from a single point close to 2
x0 <- 1
x <- roots_newton(f0,f1,x0=1)
print(x)

# Find root 3 (no message printing)
x <- roots_newton(f0,f1,x0=4,message=FALSE)
print(x)

# Function with a parameter f(x) = exp(kx) - 2
f0 <- function(x,k=2) return(exp(k*x)-2)

# First derivative (it includes the parameter)
f1 <- function(x,k=2) return(k*exp(k*x))

# Solution of exp(2x)-2=0 is log(2)/2
x <- roots_newton(f0,f1,k=2)
print(log(2)/2)