Matrix functions

[MikaelSlevinsky]

After Julia 1.0 (JuliaLang/julia#23233), the following should "just work:"

julia> using ApproxFun

julia> Fun(exp)(rand(2,2))
ERROR: MethodError: no method matching isless(::Array{Float64,2}, ::Int64)
Closest candidates are:
  isless(::ApproxFun.Infinity{Bool}, ::Number) at /Users/Mikael/.julia/v0.6/ApproxFun/src/LinearAlgebra/helper.jl:517
  isless(::AbstractFloat, ::Real) at operators.jl:98
  isless(::Real, ::Real) at operators.jl:266
  ...
Stacktrace:
 [1] in(::Array{Float64,2}, ::ApproxFun.Segment{Float64}) at /Users/Mikael/.julia/v0.6/ApproxFun/src/Fun/Domain.jl:113
 [2] evaluate at /Users/Mikael/.julia/v0.6/ApproxFun/src/Spaces/PolynomialSpace.jl:19 [inlined]
 [3] evaluate(::ApproxFun.Fun{ApproxFun.Chebyshev{ApproxFun.Segment{Float64},Float64},Float64,Array{Float64,1}}, ::Array{Float64,2}) at /Users/Mikael/.julia/v0.6/ApproxFun/src/Fun/Fun.jl:189
 [4] (::ApproxFun.Fun{ApproxFun.Chebyshev{ApproxFun.Segment{Float64},Float64},Float64,Array{Float64,1}})(::Array{Float64,2}, ::Vararg{Array{Float64,2},N} where N) at /Users/Mikael/.julia/v0.6/ApproxFun/src/Fun/Fun.jl:193

Of course, it's a design decision whether or not it should be up to the user to decide if the Chebyshev series is a good approximation on an open superset of spectrum of the matrix.

[MikaelSlevinsky]

Reference on matrix polynomials in monomial and Chebyshev polynomial bases:

W. Liang, R. Baer, C. Saravanan, Y. Shao, A. T. Bell, and M. Head-Gordon. Fast methods for resumming matrix polynomials and Chebyshev matrix polynomials. J. Comp. Phys., 194:575–587, 2004.