Barycentric Lagrange Interpolation
Barycentric Lagrange interpolation in one dimension.
barylag(xi, yi, x)
xi, yi |
x- and y-coordinates of supporting nodes. |
x |
x-coordinates of interpolation points. |
barylag interpolates the given data using the barycentric
Lagrange interpolation formula (vectorized to remove all loops).
Values of interpolated data at points x.
Barycentric interpolation is preferred because of its numerical stability.
Berrut, J.-P., and L. Nick Trefethen (2004). “Barycentric Lagrange Interpolation”. SIAM Review, Vol. 46(3), pp.501–517.
Lagrange or Newton interpolation.
## Generates an example with plot.
# Input:
# fun --- function that shall be 'approximated'
# a, b --- interval [a, b] to be used for the example
# n --- number of supporting nodes
# m --- number of interpolation points
# Output
# plot of function, interpolation, and nodes
# return value is NULL (invisible)
## Not run:
barycentricExample <- function(fun, a, b, n, m)
{
xi <- seq(a, b, len=n)
yi <- fun(xi)
x <- seq(a, b, len=m)
y <- barylag(xi, yi, x)
plot(xi, yi, col="red", xlab="x", ylab="y",
main="Example of barycentric interpolation")
lines(x, fun(x), col="yellow", lwd=2)
lines(x, y, col="darkred")
grid()
}
barycentricExample(sin, -pi, pi, 11, 101) # good interpolation
barycentricExample(runge, -1, 1, 21, 101) # bad interpolation
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