gradient() R function from [calculus]

Numerical and Symbolic Gradient

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

in arbitrary orthogonal coordinate systems.


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

f %gradient% 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 gradient as a vector and not as an array for scalar-valued functions.

Returns

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

Details

The gradient of a scalar-valued function $F$ is the vector $(\nabla F)_i$ whose components are the partial derivatives of $F$

with respect to each variable $i$ . The gradient is computed in arbitrary orthogonal coordinate systems using the scale factors $h_i$ :

(\nabla F)_i = \frac{1}{h_i}\partial_iF

When the function $F$ is a tensor-valued function $F_{d_1,\dots,d_n}$ , the gradient is computed for each scalar component. In particular, it becomes the Jacobian matrix for vector-valued function.

(\nabla F_{d_1,\dots,d_n})_i = \frac{1}{h_i}\partial_iF_{d_1,\dots,d_n}

Functions

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

Examples


### symbolic gradient 
gradient("x*y*z", var = c("x", "y", "z"))

### numerical gradient in (x=1, y=2, z=3)
f <- function(x, y, z) x*y*z
gradient(f = f, var = c(x=1, y=2, z=3))

### vectorized interface
f <- function(x) x[1]*x[2]*x[3]
gradient(f = f, var = c(1, 2, 3))

### symbolic vector-valued functions
f <- c("y*sin(x)", "x*cos(y)")
gradient(f = f, var = c("x","y"))

### numerical vector-valued functions
f <- function(x) c(sum(x), prod(x))
gradient(f = f, var = c(0,0,0))

### binary operator
"x*y^2" %gradient% c(x=1, y=3)

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")

gradient function