Barycentric Lagrange Interpolation

Barycentric Lagrange interpolation in one dimension.

barylag(xi, yi, x)

Arguments

xi, yi

x- and y-coordinates of supporting nodes.

x

x-coordinates of interpolation points.

Details

barylag interpolates the given data using the barycentric Lagrange interpolation formula (vectorized to remove all loops).

Value

Values of interpolated data at points x.

References

Berrut, J.-P., and L. Nick Trefethen (2004). “Barycentric Lagrange Interpolation”. SIAM Review, Vol. 46(3), pp.501–517.

Note

Barycentric interpolation is preferred because of its numerical stability.

See also

Lagrange or Newton interpolation.

Examples

##  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)
if (FALSE) { # \dontrun{
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
} # }