deriv: Partial differentiation of spray objects

derivR Documentation

Partial differentiation of spray objects

Description

\loadmathjax

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 \mjeqn\partial^n S/\partial x_i^n = S^\left(x_i,...,x_i\right)d^n S/dx_i^n = S^(x_i...x_i).

Function aderiv() is the generalized derivative; if S is a spray of arity 3, then aderiv(S,c(i,j,k)) returns \mjeqn\frac\partial^i+j+k S\partial x_1^i\partial x_2^j\partial x_3^kd^(i+j+k)S/dx_1^i dx_2^j dx_3^k.

Value

Both functions return a spray object.

Author(s)

Robin K. S. Hankin

See Also

asum

Examples



(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)) 
)


spray documentation built on Aug. 10, 2023, 5:11 p.m.