```
## 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 0
```

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