fda (version 6.1.8)

quadset: Quadrature points and weights for Simpson's rule

Description

Set up quadrature points and weights for Simpson's rule.

Usage

quadset(nquad=5, basisobj=NULL, breaks)

Value

If is.null(basisobj), quadset returns a 'quadvals' matrix with columns quadpts and quadwts. Otherwise, it returns basisobj with the two components set as follows:

quadvals

cbind(quadpts=quadpts, quadwts=quadwts)

value

a list with two components containing eval.basis(quadpts, basisobj, ival-1) for ival=1, 2.

Arguments

nquad

an odd integer at least 5 giving the number of evenly spaced Simpson's rule quadrature points to use over each interval (breaks[i], breaks[i+1]).

basisobj

the basis object that will contain the quadrature points and weights

breaks

optional interval boundaries. If this is provided, the first value must be the initial point of the interval over which the basis is defined, and the final value must be the end point. If this is not supplied, and 'basisobj' is of type 'bspline', the knots are used as these values.

Details

Set up quadrature points and weights for Simpson's rule and store information in the output 'basisobj'. Simpson's rule is used to integrate a function between successive values in vector 'breaks'. That is, over each interval (breaks[i], breaks[i+1]). Simpson's rule uses 'nquad' equally spaced quadrature points over this interval, starting with the the left boundary and ending with the right boundary. The quadrature weights are the values delta*c(1,4,2,4,2,4,..., 2,4,1) where 'delta' is the difference between successive quadrature points, that is, delta = (breaks[i-1]-breaks[i])/(nquad-1).

References

Ramsay, James O., Hooker, Giles, and Graves, Spencer (2009), Functional data analysis with R and Matlab, Springer, New York.

Ramsay, James O., and Silverman, Bernard W. (2005), Functional Data Analysis, 2nd ed., Springer, New York.

Ramsay, James O., and Silverman, Bernard W. (2002), Applied Functional Data Analysis, Springer, New York.

See Also

create.bspline.basis eval.basis

Examples

(qs7.1 <- quadset(nquad=7, breaks=c(0, .3, 1)))
# cbind(quadpts=c(seq(0, 0.3, length=7),
#              seq(0.3, 1, length=7)),
#    quadwts=c((0.3/18)*c(1, 4, 2, 4, 2, 4, 1),
#              (0.7/18)*c(1, 4, 2, 4, 2, 4, 1) ) )

# The simplest basis currently available with this function:
bspl2.2 <- create.bspline.basis(norder=2, breaks=c(0,.5, 1))

bspl2.2a <- quadset(basisobj=bspl2.2)
bspl2.2a$quadvals
# cbind(quadpts=c((0:4)/8, .5+(0:4)/8),
#       quadwts=rep(c(1,4,2,4,1)/24, 2) )
bspl2.2a$values
# a list of length 2
# [[1]] = matrix of dimension c(10, 3) with the 3 basis
#      functions evaluated at the 10 quadrature points:
# values[[1]][, 1] = c(1, .75, .5, .25, rep(0, 6))
# values[[1]][, 2] = c(0, .25, .5, .75, 1, .75, .5, .25, 0)
# values[[1]][, 3] = values[10:1, 1]
#
# values[[2]] = matrix of dimension c(10, 3) with the
#     first derivative of values[[1]], being either
#    -2, 0, or 2.