Complex Step Derivatives
Complex step derivatives of real-valued functions, including gradients, Jacobians, and Hessians.
complexstep(f, x0, h = 1e-20, ...) grad_csd(f, x0, h = 1e-20, ...) jacobian_csd(f, x0, h = 1e-20, ...) hessian_csd(f, x0, h = 1e-20, ...) laplacian_csd(f, x0, h = 1e-20, ...)
f |
Function that is to be differentiated. |
x0 |
Point at which to differentiate the function. |
h |
Step size to be applied; shall be very small. |
... |
Additional variables to be passed to |
Complex step derivation is a fast and highly exact way of numerically differentiating a function. If the following conditions are satisfied, there will be no loss of accuracy between computing a function value and computing the derivative at a certain point.
f must have an analytical (i.e., complex differentiable)
continuation into an open neighborhood of x0.
x0 and f(x0) must be real.
h is real and very small: 0 < h << 1.
complexstep handles differentiation of univariate functions, while
grad_csd and jacobian_csd compute gradients and Jacobians by
applying the complex step approach iteratively. Please understand that these
functions are not vectorized, but complexstep is.
As complex step cannot be applied twice (the first derivative does not
fullfil the conditions), hessian_csd works differently. For the
first derivation, complex step is used, to the one time derived function
Richardson's method is applied. The same applies to lapalacian_csd.
complexstep(f, x0) returns the derivative f'(x_0) of f
at x_0. The function is vectorized in x0.
This surprising approach can be easily deduced from the complex-analytic Taylor formula.
HwB <hwborchers@googlemail.com>
Martins, J. R. R. A., P. Sturdza, and J. J. Alonso (2003). The Complex-step Derivative Approximation. ACM Transactions on Mathematical Software, Vol. 29, No. 3, pp. 245–262.
## Example from Martins et al.
f <- function(x) exp(x)/sqrt(sin(x)^3 + cos(x)^3) # derivative at x0 = 1.5
# central diff formula # 4.05342789402801, error 1e-10
# numDeriv::grad(f, 1.5) # 4.05342789388197, error 1e-12 Richardson
# pracma::numderiv # 4.05342789389868, error 5e-14 Richardson
complexstep(f, 1.5) # 4.05342789389862, error 1e-15
# Symbolic calculation: # 4.05342789389862
jacobian_csd(f, 1.5)
f1 <- function(x) sum(sin(x))
grad_csd(f1, rep(2*pi, 3))
## [1] 1 1 1
laplacian_csd(f1, rep(pi/2, 3))
## [1] -3
f2 <- function(x) c(sin(x[1]) * exp(-x[2]))
hessian_csd(f2, c(0.1, 0.5, 0.9))
## [,1] [,2] [,3]
## [1,] -0.06055203 -0.60350053 0
## [2,] -0.60350053 0.06055203 0
## [3,] 0.00000000 0.00000000 0
f3 <- function(u) {
x <- u[1]; y <- u[2]; z <- u[3]
matrix(c(exp(x^+y^2), sin(x+y), sin(x)*cos(y), x^2 - y^2), 2, 2)
}
jacobian_csd(f3, c(1,1,1))
## [,1] [,2] [,3]
## [1,] 2.7182818 0.0000000 0
## [2,] -0.4161468 -0.4161468 0
## [3,] 0.2919266 -0.7080734 0
## [4,] 2.0000000 -2.0000000 0Please choose more modern alternatives, such as Google Chrome or Mozilla Firefox.