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)

Arguments

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 h=0.5.

tol

relative tolerance.

...

variables to be passed to function f.

Details

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.

Value

Numeric scalar or vector of approximated derivatives.

References

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

Note

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

See also

fderiv, complexstep

Examples

# 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
#> [1] 1.141883
# 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
#> [1] 1.887379e-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
#> $df
#> [1] 4.053428
#> 
#> $rel.err
#> [1] 2.037796e-14
#> 
#> $niter
#> [1] 8
#> 
                                         #   4.053427893898621... true value