laplacian() R function from [calculus]

Numerical and Symbolic Laplacian

Computes the numerical Laplacian of functions or the symbolic Laplacian of characters

in arbitrary orthogonal coordinate systems.


laplacian(
  f,
  var,
  params = list(),
  coordinates = "cartesian",
  accuracy = 4,
  stepsize = NULL,
  drop = TRUE
)

f %laplacian% var

Arguments

f: array of characters or a function returning a numeric array.
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 derivative.
params: list of additional parameters passed to f.
coordinates: coordinate system to use. One of: cartesian, polar, spherical, cylindrical, parabolic, parabolic-cylindrical or a vector of scale factors for each varibale.
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 TRUE, return the Laplacian as a scalar and not as an array for scalar-valued functions.

Returns

Scalar for scalar-valued functions when drop=TRUE, array otherwise.

Details

The Laplacian is a differential operator given by the divergence of the gradient of a scalar-valued 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 tensor-valued 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)

Functions

f %laplacian% var: binary operator with default parameters.

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 vector-valued functions
f <- array(c("x^2","x*y","x*y","y^2"), dim = c(2,2))
laplacian(f = f, var = c("x","y"))

### numerical vector-valued 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")

References

Guidotti E (2022). "calculus: High-Dimensional Numerical and Symbolic Calculus in R." Journal of Statistical Software, 104(5), 1-37. tools:::Rd_expr_doi("10.18637/jss.v104.i05")

laplacian function