xwGauss: Integration of Gauss-type

In enricoschumann/NMOF: Numerical Methods and Optimization in Finance

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. \Sexpr[results=rd]{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). 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))

enricoschumann/NMOF documentation built on April 13, 2024, 12:16 p.m.