laplacian {calculus}  R Documentation 
Numerical and Symbolic Laplacian
Description
Computes the numerical Laplacian of functions
or the symbolic Laplacian of characters
in arbitrary orthogonal coordinate systems.
Usage
laplacian(
f,
var,
params = list(),
coordinates = "cartesian",
accuracy = 4,
stepsize = NULL,
drop = TRUE
)
f %laplacian% 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 

coordinates 
coordinate system to use. One of: 
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
The Laplacian is a differential operator given by the divergence of the
gradient of a scalarvalued function F
, resulting in a scalar value giving
the flux density of the gradient flow of a function.
The laplacian
is computed in arbitrary orthogonal coordinate systems using
the scale factors h_i
:
\nabla^2F = \frac{1}{J}\sum_i\partial_i\Biggl(\frac{J}{h_i^2}\partial_iF\Biggl)
where J=\prod_ih_i
. When the function F
is a tensorvalued function
F_{d_1\dots d_n}
, the laplacian
is computed for each scalar component:
(\nabla^2F)_{d_1\dots d_n} = \frac{1}{J}\sum_i\partial_i\Biggl(\frac{J}{h_i^2}\partial_iF_{d_1\dots d_n}\Biggl)
Value
Scalar for scalarvalued functions when drop=TRUE
, array
otherwise.
Functions

f %laplacian% 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()
,
hessian()
,
jacobian()
Examples
### symbolic Laplacian
laplacian("x^3+y^3+z^3", var = c("x","y","z"))
### numerical Laplacian in (x=1, y=1, z=1)
f < function(x, y, z) x^3+y^3+z^3
laplacian(f = f, var = c(x=1, y=1, z=1))
### vectorized interface
f < function(x) sum(x^3)
laplacian(f = f, var = c(1, 1, 1))
### symbolic vectorvalued functions
f < array(c("x^2","x*y","x*y","y^2"), dim = c(2,2))
laplacian(f = f, var = c("x","y"))
### numerical vectorvalued functions
f < function(x, y) array(c(x^2,x*y,x*y,y^2), dim = c(2,2))
laplacian(f = f, var = c(x=0,y=0))
### binary operator
"x^3+y^3+z^3" %laplacian% c("x","y","z")