integrateSpherePolynomial function

Integration of polynomials over sphere or ball.

Exact integration of polynomial over sphere or ball in n-dimensions.

integrateSpherePolynomial(p, valueOnly = TRUE)
integrateBallPolynomial(p, R = c(0, 1))

Arguments

• p: a list specifying the coefficients and powers of the polynomial. See details below
• valueOnly: boolean saying whether to return only the value of the integral, or return both the value and a intermediate terms. These intermediate terms are used by integrateBallPolynomial( ).
• R: inner and outer radius of the annular region: R[1] <= radius <= R[2].

Details

Compute the exact integral over the sphere in n dimensions of a polynomial

p(x[1],...,x[n])=sum (coef[i] * x[1]^k[i,1] * ... * x[n]^k[i,n]),

where the sum is over i=1,...,m. The polynomial is specified as a list p

with fields

• coef, an m-vector of doubles
• k, an (m x n) matrix of integers

m and n are given implicitly in the sizes of these arrays output is normally just a number, the value of the integral. If integrateSpherePolynomial is called with valueOnly=FALSE, a list with two fields:

• integral, a double containing the value of the integral
• term, a vector of length m of values used in function IntegratePolynomialBall( )

Returns

integrateSpherePolynomial() normally just returns a value of the integral, but if valueOnly=FALSE, it will return a list containing the value and intermediate terms. These intermediate terms correspond to the integral of each monomial term in the polynomial; they are used by integrateBallPolynomial().

integrateBallPolynomial( ) returns just the value of the integral.

References

Method is from How to Integrate a Polynomial over a Sphere, by G. Folland (2001), MAA Monthly 108, pg. 446-448.

Examples

n <- 3
# specify the polynomial p(x) = 1.0 * x[1]^2 * x[2]^0 * x[3]^0 + 7.0 * x[1]^0 * x[2]^3 * x[3]
p <- list(coef=c(1.0,7.0),k=matrix( c(2L,0L,0L,0L,3L,0L), byrow=TRUE, nrow=2) )
integrateSpherePolynomial( p )
integrateBallPolynomial( p )

# compare to adaptive integration
f4 <- function( x ) { return( x[1]^2 + 7*x[2]^2*x[3] ) }
adaptIntegrateSpherePolar( f4, n )$integral
adaptIntegrateBallPolar( f4, n )$integral