xwGauss function

Integration of Gauss-type

Compute nodes and weights for Gauss integration.

xwGauss(n, method = "legendre")
changeInterval(nodes, weights, oldmin, oldmax, newmin, newmax)

Arguments

• n: number of nodes

• method: character. default is "legendre"; also possible are "laguerre" and "hermite"

• nodes: the nodes (a numeric vector)

• weights: the weights (a numeric vector)

• oldmin: the minimum of the interval (typically as tabulated)

• oldmax: the maximum of the interval (typically as tabulated)

• newmin: the desired minimum of the interval

• newmax: the desired maximum of the interval

Details

xwGauss computes nodes and weights for integration for the interval -1 to 1. It uses the method of Golub and Welsch (1969).

changeInterval is a utility that transforms nodes and weights to an arbitrary interval.

Returns

a list with two elements - weights: a numeric vector

• nodes: a numeric vector

References

Gilli, M., Maringer, D. and Schumann, E. (2019) Numerical Methods and Optimization in Finance. 2nd edition. Elsevier. tools:::Rd_expr_doi("10.1016/C2017-0-01621-X")

Golub, G.H. and Welsch, J.H. (1969). Calculation of Gauss Quadrature Rules. Mathematics of Computation, 23 (106), pp. 221--230+s1--s10.

Schumann, E. (2023) Financial Optimisation with R (NMOF Manual). https://enricoschumann.net/NMOF.htm#NMOFmanual

Author(s)

Enrico Schumann

See Also

callHestoncf

Examples

## examples from Gilli/Maringer/Schumann (2019), ch. 17

## a test function
f1 <- function(x) exp(-x)
m <- 5; a <- 0; b <- 5
h <- (b - a)/m

## rectangular rule -- left
w <- h; k <- 0:(m-1); x <- a + k * h
sum(w * f1(x))

## rectangular rule -- right
w <- h; k <- 1:m ; x <- a + k * h
sum(w * f1(x))

## midpoint rule
w <- h; k <- 0:(m-1); x <- a + (k + 0.5)*h
sum(w * f1(x))

## trapezoidal rule
w <- h
k <- 1:(m-1)
x <- c(a, a + k*h, b)
aux <- w * f1(x)
sum(aux) - (aux[1] + aux[length(aux)])/2

## R's integrate (from package stats)
integrate(f1, lower = a,upper = b)

## Gauss--Legendre
temp <- xwGauss(m)
temp <- changeInterval(temp$nodes, temp$weights,
                       oldmin = -1, oldmax = 1, newmin =  a, newmax = b)
x <- temp$nodes; w <- temp$weights
sum(w * f1(x))