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.

Value

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.

References

Guidotti E (2022). "calculus: High-Dimensional Numerical and Symbolic Calculus in R." Journal of Statistical Software, 104(5), 1-37. 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"))
#> [1] "3 * (2 * x) + 3 * (2 * y) + 3 * (2 * 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))
#> [1] 18

### vectorized interface
f <- function(x) sum(x^3)
laplacian(f = f, var = c(1, 1, 1))
#> [1] 18

### 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"))
#>      [,1] [,2]
#> [1,] "2"  "0" 
#> [2,] "0"  "2" 

### 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))
#>      [,1] [,2]
#> [1,]    2    0
#> [2,]    0    2

### binary operator
"x^3+y^3+z^3" %laplacian% c("x","y","z")
#> [1] "3 * (2 * x) + 3 * (2 * y) + 3 * (2 * z)"