Partial differentiation of spray objects

Partial differentiation of spray objects interpreted as multivariate polynomials

Usage

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

Arguments

expr: A spray object, interpreted as a multivariate polynomial
i: Dimension to differentiate with respect to
derivative: How many times to differentiate
...: Further arguments, currently ignored
S: spray object
orders: The orders of the differentials

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 \(\frac{\partial^{i+j+k} S}{\partial x_1^i\partial x_2^j\partial x_3^k}\).

Value

Both functions return a spray object.

Author

Robin K. S. Hankin

Examples



(S <- spray(matrix(sample(-2:2,15,replace=TRUE),ncol=3),addrepeats=TRUE))
#>               val
#>   0  0  2  =    1
#>   2  1  1  =    1
#>  -2 -1  1  =    1
#>  -1 -1 -1  =    1
#>   2  1 -1  =    1

deriv(S,1)
#>               val
#>   1  1  1  =    2
#>  -3 -1  1  =   -2
#>  -2 -1 -1  =   -1
#>   1  1 -1  =    2
deriv(S,2,2)
#>               val
#>  -2 -3  1  =    2
#>  -1 -3 -1  =    2

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

# 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)
#> [1] TRUE

# 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)) 
)
#> [1] TRUE