# Specifying the coefficients for a 2D polynomial in ApproxFun.jl

**URL:** <https://discourse.julialang.org/t/specifying-the-coefficients-for-a-2d-polynomial-in-approxfun-jl/3782>\
**Category:** Numerics\
**Created:** [May 18, 2017, 1:08am UTC](https://discourse.julialang.org/t/specifying-the-coefficients-for-a-2d-polynomial-in-approxfun-jl/3782 "2017-05-18T01:08:09Z")\
**Posts on this page:** 5\
**Page:** 1

<div class="post-metadata">

**Author:** ![JackDevine](https://sea2.discourse-cdn.com/julialang/user_avatar/discourse.julialang.org/jackdevine/32/1048_2.png) [@JackDevine](https://discourse.julialang.org/u/JackDevine)\
**Post date:** [May 18, 2017, 1:08am UTC](https://discourse.julialang.org/t/specifying-the-coefficients-for-a-2d-polynomial-in-approxfun-jl/3782/1 "2017-05-18T01:08:09Z")

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I am trying to understand how `ApproxFun.jl` constructs 2D functions from their coefficients. In 1D, it all makes sense, for example:

```julia
julia> using ApproxFun

julia> f = Fun(Interval(-1, 1), [1,2,3])
Fun(Chebyshev(【-1.0,1.0】),[1.0,2.0,3.0])

julia> f(0.1)
-1.74

julia> 1 + 2*cos(acos(0.1)) + 3*cos(2acos(0.1))
-1.74

```

So, `ApproxFun` just uses the coefficients f\_k = [1,2,3] in the formula f(x) \approx \sum\_k f\_k \cos(k \acos(x)) (sorry about the bad formatting). You can see a similar discussion in [this](http://juliaapproximation.github.io/ApproxFun.jl/latest/usage/spaces.html) part of the docs, but what they don’t seem to discuss is what happens in 2D:

```julia
julia> f = Fun(Interval(-1, 1)^2, [1,2,3])
Fun(Chebyshev(【-1.0,1.0】)⊗Chebyshev(【-1.0,1.0】),[1.0,2.0,3.0])

julia> f(0.1, 0.1)
1.50

```

So how does `ApproxFun` calculate `f(x, y)` from `x`, `y` and the coefficients `[1,2,3]`?

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<div class="post-metadata">

**Author:** ![dlfivefifty](https://sea2.discourse-cdn.com/julialang/user_avatar/discourse.julialang.org/dlfivefifty/32/1959_2.png) [@dlfivefifty](https://discourse.julialang.org/u/dlfivefifty)\
**Post date:** [May 18, 2017, 10:36am UTC](https://discourse.julialang.org/t/specifying-the-coefficients-for-a-2d-polynomial-in-approxfun-jl/3782/2 "2017-05-18T10:36:55Z")

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It goes in order of polynomial degree lexigraphically, that is:

```julia
T_0(x)T_0(y) 
T_0(x)T_1(y) 
T_1(x)T_0(y)
T_0(x)T_2(y)
T_1(x)T_1(y)
T_2(x)T_0(y)
…

```

For example:

```julia
julia> f=Fun(Chebyshev()^2,[1.,2.,3.])
Fun(Chebyshev(【-1.0,1.0】)⊗Chebyshev(【-1.0,1.0】),[1.0, 2.0, 3.0])

julia> f(0.1,0.2)
1.7000000000000002

julia> 1 + 2*0.2 + 3*0.1
1.7

```

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<div class="post-metadata">

**Author:** ![JackDevine](https://sea2.discourse-cdn.com/julialang/user_avatar/discourse.julialang.org/jackdevine/32/1048_2.png) [@JackDevine](https://discourse.julialang.org/u/JackDevine)\
**Post date:** [May 18, 2017, 11:30am UTC](https://discourse.julialang.org/t/specifying-the-coefficients-for-a-2d-polynomial-in-approxfun-jl/3782/3 "2017-05-18T11:30:50Z")

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Thanks @dlfivefifty !

Just let me use one more example to make sure that I understand…

```julia
julia> using ApproxFun

julia> f = Fun((x, y) -> 1 + 1.2cos(2acos(x)) + 1.3cos(acos(y)), Interval(-1,1)^2)
Fun(Chebyshev(【-1.0,1.0】)⊗Chebyshev(【-1.0,1.0】),[1.0,1.3,3.31642e-17,4.11708e-17,0.0,1.2])

julia> f.coefficients
6-element Array{Float64,1}:
 1.0        
 1.3        
 3.31642e-17
 4.11708e-17
 0.0        
 1.2 

```

The first coefficient is for `T_0(x) T_0(y) == cos(0.0acos(x))*cos(0.0acos(y))`, the second coefficient is for `T_0(x) T_1(y) == cos(0.0acos(x))*cos(1.0acos(y))` and the last coefficient is for `T_2(x) T_0(y) == cos(2.0acos(x))*cos(0.0acos(y))`.

So a function that I could use to compute the polynomial from the coefficients is:

```julia
function compute_poly(c, x, y)
    ret = 0.0
    indeces = [[0,0],[0,1],[1,0],[0,2],[1,1],[2,0]]
    for (i, coeff) in enumerate(c)
        ret += coeff*cos(indeces[i][1]*acos(x))*cos(indeces[i][2]*acos(y))
    end
    return ret
end

```

I know that `ApproxFun` does things in a much more clever way, but I am just trying to understand the mathematics.

One thing that I noticed on my adventure trying to implement `compute_poly()` above is that the ordering of indeces is different from what you get with `Combinatorics.jl`:

```julia
julia> using Combinatorics

julia> a = multiset_combinations([0,0,1,1,2,2], 2) |> collect
  6-element Array{Array{Int64,1},1}:
 [0,0]
 [0,1]
 [0,2]
 [1,1]
 [1,2]
 [2,2]

```

So apparently people in combinatorics have different conventions from people in numerics?

One last question. Although your explanation makes sense now, it was not excactly intuitive to me… Do you think that this would be worth adding to the documentation?

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**Author:** ![dlfivefifty](https://sea2.discourse-cdn.com/julialang/user_avatar/discourse.julialang.org/dlfivefifty/32/1959_2.png) [@dlfivefifty](https://discourse.julialang.org/u/dlfivefifty)\
**Post date:** [May 18, 2017, 11:46am UTC](https://discourse.julialang.org/t/specifying-the-coefficients-for-a-2d-polynomial-in-approxfun-jl/3782/4 "2017-05-18T11:46:03Z")

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> So apparently people in combinatorics have different conventions from people in numerics?

I wouldn’t say this is a standard convention, it’s probably more common to just a matrix. The reason for this convention is that it means every coefficient is hit in a unique location (allowing most of the 1D code in ApproxFun to work also for 2D). The ordering from Combinatorics.jl wouldn’t work since the coefficients would be in different places if you went up to 3.

> Do you think that this would be worth adding to the documentation?

Yes of course 😉 Feel free to make a pull request, or even just an Issue.

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<div class="post-metadata">

**Author:** ![JackDevine](https://sea2.discourse-cdn.com/julialang/user_avatar/discourse.julialang.org/jackdevine/32/1048_2.png) [@JackDevine](https://discourse.julialang.org/u/JackDevine)\
**Post date:** [May 18, 2017, 8:13pm UTC](https://discourse.julialang.org/t/specifying-the-coefficients-for-a-2d-polynomial-in-approxfun-jl/3782/5 "2017-05-18T20:13:50Z")

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Great, thanks for all of your help!

I will try and make a PR asap.