deriv: Partial differentiation of spray objects

Description

Partial differentiation of spray objects interpreted as multivariate polynomials

Usage

# S3 method for spray
deriv(expr, i , derivative = 1, ...)
aderiv(S,orders)

Arguments

expr

A spray object, interpreted as a multivariate polynomial

Dimension to differentiate with respect to

derivative

How many times to differentiate

...

Further arguments, currently ignored

spray object

orders

The orders of the differentials

Value

Both functions return a spray object.

Details

Function deriv.spray() is the method for generic spray(); if S is a spray object, then spray(S,i,n) returns \(\partial^n S/\partial x_i^n = S^{\left(x_i,\ldots,x_i\right)}\).

Function aderiv() is the generalized derivative; if S is a spray of arity 3, then aderiv(S,c(i,j,k)) returns \(\frac{\partial^{i+j+k} S}{\partial x_1^i\partial x_2^j\partial x_3^k}\).

Examples

Run this code

# NOT RUN {

S <- spray(matrix(sample(-2:2,15,replace=TRUE),ncol=3),addrepeats=TRUE)

deriv(S,1)
deriv(S,2,2)

# differentiation is invariant under order:
aderiv(S,1:3) == deriv(deriv(deriv(S,1,1),2,2),3,3)

# Leibniz's rule:
S1 <- spray(matrix(sample(0:3,replace=TRUE,21),ncol=3),sample(7),addrepeats=TRUE)
S2 <- spray(matrix(sample(0:3,replace=TRUE,15),ncol=3),sample(5),addrepeats=TRUE)

S1*deriv(S2,1) + deriv(S1,1)*S2 == deriv(S1*S2,1)

# Generalized Leibniz:
aderiv(S1*S2,c(1,1,0)) == (
aderiv(S1,c(0,0,0))*aderiv(S2,c(1,1,0)) +
aderiv(S1,c(0,1,0))*aderiv(S2,c(1,0,0)) +
aderiv(S1,c(1,0,0))*aderiv(S2,c(0,1,0)) +
aderiv(S1,c(1,1,0))*aderiv(S2,c(0,0,0)) 
)



# }