ApproxFun is a package for approximating functions. It is in a similar vein to the Matlab
package Chebfun
and the Mathematica package RHPackage
.
The ApproxFun Documentation
contains detailed information, or read on for a brief overview of the package.
The ApproxFun Examples
contains many examples of
using this package, in Jupyter notebooks and Julia scripts.
Take your two favourite functions on an interval and create approximations to them as simply as:
using LinearAlgebra, SpecialFunctions, Plots, ApproxFun
x = Fun(identity,0..10)
f = sin(x^2)
g = cos(x)
Evaluating f(.1)
will return a high
accuracy approximation to sin(0.01)
. All the algebraic manipulations of functions
are supported and more. For example, we can add f
and g^2
together and compute
the roots and extrema:
h = f + g^2
r = roots(h)
rp = roots(h')
plot(h; label="f + g^2")
scatter!(r, h.(r); label="roots")
scatter!(rp, h.(rp); label="extrema")
Notice from above that to find the extrema, we used '
overridden for the differentiate
function. Several other Julia
base functions are overridden for the purposes of calculus. Because the exponential is its own
derivative, the norm
is small:
f = Fun(x->exp(x), -1..1)
norm(f-f') # 4.4391656415701095e-14
Similarly, cumsum
defines an indefinite integration operator:
g = cumsum(f)
g = g + f(-1)
norm(f-g) # 3.4989733283850415e-15d
Algebraic and differential operations are also implemented where possible, and most of Julia's built-in functions are overridden to accept Fun
s:
x = Fun()
f = erf(x)
g = besselj(3,exp(f))
h = airyai(10asin(f)+2g)