hessian {calculus}  R Documentation 
Numerical and Symbolic Hessian
Description
Computes the numerical Hessian of functions
or the symbolic Hessian of characters
.
Usage
hessian(f, var, params = list(), accuracy = 4, stepsize = NULL, drop = TRUE)
f %hessian% var
Arguments
f 
array of 
var 
vector giving the variable names with respect to which the derivatives are to be computed and/or the point where the derivatives are to be evaluated. See 
params 

accuracy 
degree of accuracy for numerical derivatives. 
stepsize 
finite differences stepsize for numerical derivatives. It is based on the precision of the machine by default. 
drop 
if 
Details
In Cartesian coordinates, the Hessian of a scalarvalued function F
is the
square matrix of secondorder partial derivatives:
(H(F))_{ij} = \partial_{ij}F
When the function F
is a tensorvalued function F_{d_1,\dots,d_n}
,
the hessian
is computed for each scalar component.
(H(F))_{d_1\dots d_n,ij} = \partial_{ij}F_{d_1\dots d_n}
It might be tempting to apply the definition of the Hessian as the Jacobian of the gradient to write it in arbitrary orthogonal coordinate systems. However, this results in a Hessian matrix that is not symmetric and ignores the distinction between vector and covectors in tensor analysis. The generalization to arbitrary coordinate system is not currently supported.
Value
Hessian matrix for scalarvalued functions when drop=TRUE
, array
otherwise.
Functions

f %hessian% var
: binary operator with default parameters.
References
Guidotti E (2022). "calculus: HighDimensional Numerical and Symbolic Calculus in R." Journal of Statistical Software, 104(5), 137. doi:10.18637/jss.v104.i05
See Also
Other differential operators:
curl()
,
derivative()
,
divergence()
,
gradient()
,
jacobian()
,
laplacian()
Examples
### symbolic Hessian
hessian("x*y*z", var = c("x", "y", "z"))
### numerical Hessian in (x=1, y=2, z=3)
f < function(x, y, z) x*y*z
hessian(f = f, var = c(x=1, y=2, z=3))
### vectorized interface
f < function(x) x[1]*x[2]*x[3]
hessian(f = f, var = c(1, 2, 3))
### symbolic vectorvalued functions
f < c("y*sin(x)", "x*cos(y)")
hessian(f = f, var = c("x","y"))
### numerical vectorvalued functions
f < function(x) c(sum(x), prod(x))
hessian(f = f, var = c(0,0,0))
### binary operator
"x*y^2" %hessian% c(x=1, y=3)