xwGauss {NMOF}

Integration of Gauss-type

Description

Compute nodes and weights for Gauss integration.

Usage

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.

Value

a list with two elements

weights

a numeric vector

nodes

a numeric vector

Author(s)

Enrico Schumann

References

Gilli, M., Maringer, D. and Schumann, E. (2019) Numerical Methods and Optimization in Finance. 2nd edition. Elsevier. 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). http://enricoschumann.net/NMOF.htm#NMOFmanual

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))