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)
f |
function to be differentiated. |
x0, x |
point(s) at which the derivative is to be computed. |
maxiter |
maximum number of iterations. |
h |
starting step size, should be the default |
tol |
relative tolerance. |
... |
variables to be passed to function |
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.
# 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 valuePlease choose more modern alternatives, such as Google Chrome or Mozilla Firefox.