numderiv: Richardson's Numerical Derivative

View source: R/numderiv.R

numderivR Documentation

Richardson's Numerical Derivative


Richardson's method applied to the computation of the numerical derivative.


numderiv(f, x0, maxiter = 16, h = 1/2, ..., tol = .Machine$double.eps)

numdiff(f, x, maxiter = 16, h = 1/2, ..., tol = .Machine$double.eps)



function to be differentiated.

x0, x

point(s) at which the derivative is to be computed.


maximum number of iterations.


starting step size, should be the default h=0.5.


relative tolerance.


variables to be passed to function f.


numderiv returns the derivative of f at x0, where x0 must be a single scalar in the domain of the function.

numdiff is a vectorized form of numderiv such that the derivatives will be returned at all points of the vector x.


Numeric scalar or vector of approximated derivatives.


See grad in the ‘numDeriv’ package for another implementation of Richardson's method in the context of numerical differentiation.


Mathews, J. H., and K. D. Fink (1999). Numerical Methods Using Matlab. Third Edition, Prentice Hall.

See Also

fderiv, complexstep


# Differentiate an anti-derivative function
f <- function(x) sin(x)*sqrt(1+sin(x))
F <- function(x)
        integrate(f, 0, x, rel.tol = 1e-12)$value
x0 <- 1
dF0 <- numderiv(F, x0, tol = 6.5e-15)   #=> 1.141882942715462
f(x0)                                   #   1.141882942715464 true value
# fderiv(F, x0)                         #   1.141882942704476
# numDeriv::grad(F, x0)                 #   1.141882942705797

# Compare over a whole period
x <- seq(0, 2*pi, length.out = 11)
max(abs(numdiff(sin, x) - cos(x)))          #=> 3.44e-15
# max(abs(numDeriv::grad(sin, x) - cos(x))) #   7.70e-12

# Example from complex step
f <- function(x) exp(x) / sqrt(sin(x)^3 + cos(x)^3)
x0 <- 1.5
numderiv(f, x0)                          #   4.05342789389876, error 0.5e-12
                                         #   4.053427893898621... true value

pracma documentation built on Nov. 7, 2023, 3:01 a.m.