| weightsGJ {rjacobi} | R Documentation |
Calculates the weights of the Q point Gauss-Jacobi quadrature.
z = weightsGJ(z, a=0.0, b=0.0) z = weightsGLJ(z, a=0.0, b=0.0) z = weightsGRJM(z, a=0.0, b=0.0) z = weightsGRJP(z, a=0.0, b=0.0)
z |
Quadrature node calculated with zerosGJ
according to quadrature type |
a |
α parameter of Jacobi polynomial. Defaults to 0 |
b |
β parameter of Jacobi polynomial. Defaults to 0 |
Different types of quadrature are possible:
GJ
GLJ, includes both end
points
GRJM, includes the -1 end
point
GRJP, includes the +1 end
pointThese functions find the quadrature nodes that will be used in subsequent calculations.
With these weights, integrals can be calculated according to the following expression:
int_{-1}^{1} f(x)dx approx sum_{i=1}{Q}w_i f(x_i)
A vector containing the weights of the quadrature
Abramowitz, Milton and Stegun, Irene (editors); "Handbook of Mathematical functions", Dover Publications 1965.
Karniadakis, George Em and Sherwin, Spencer; "Spectral/hp Element Methods for Computational Fluid Dynamics", Oxford Science Publications, 2nd edition, 2005.
### This example will shoud the exponential convergence when integrating
### smooth functions.
Q <- 2:15
f <- function(x)cos(5*x)
n <- length(Q)
integr <- double(n)
for (i in 1:n){
z <- zerosGJ(Q[i])
w <- weightsGJ(z)
integr[i] <- sum(w * f(z))
}
err <- abs(integr - 2*sin(5)/5)
plot(Q, err, ty='b', log='y', main="Error of integral of cos(5*x)",
xlab="No Quadrature points", ylab="Error")