Implement b=-1 #111

DanielVandH · 2024-06-19T17:03:57Z

Need to get #109, #105, and JuliaArrays/LazyArrays.jl#320 (or something like it) over the line to get this completed. There could be a way to just force it all to work by putting ApplyArrays everywhere (though there are issues with those too, I've found) but getting those done first will probably be more useful in the long run. I'll mark this as ready to review once that's done. Tests did pass when I last worked on this four days ago, not sure where all these ambiguities have come from but I'll work on it eventually.

This PR implements $P_n^{t, (u, -1, w)}$, defined so that

$$P_n^{t, (u, -1, w)}(y) = (1 - y)P_{n-1}^{t, (u, 1, w)}(y), \quad n \geq 1,$$

with $P_0^{t, (u, -1, w)}(y) = 1$. The Jacobi matrix with this definition is defined by

$$yP_n^{t, (u, -, 1w)} = c_n^{t, (u, -1, w)}P_{n-1}^{t, (u, -1, w)} + a_n^{t, (u, -1, w)}P_n^{t, (u, -1, w)} + b_n^{t, (u, -1, w)}P_{n+1}^{t, (u, -1, w)},$$

with

$$\begin{align*} a_n^{t, (u, -1, w)} &= \begin{cases} 1 & n = 0, \\ a_{n-1}^{t, (u, 1, w)} & n \geq 1, \end{cases} \\\ b_n^{t, (u, -1, w)} &= \begin{cases} -1 & n = 0, \\ b_{n-1}^{t, (u, -1, w)} & n \geq 1, \end{cases} \\\ c_n^{t, (u, -1, w)} &= \begin{cases} 0 & n = 1, \\ c_{n-1}^{t, (u, 1, w)} & n \geq 2. \end{cases} \end{align*}$$

Differentiation is given by

$$\frac{\mathrm d}{\mathrm dy}\boldsymbol P^{t, (u, -1, w)} = \boldsymbol P^{t, (u+1, 0, w+1)} \boldsymbol D_{(u, -1, w)}^{t, (u+1,0,w+1)}, \quad \boldsymbol D_{(u, -1, w)}^{(t, (u+1,0,w+1)} = \begin{bmatrix} \boldsymbol 0 & \boldsymbol R_{(u+1,1,w+1)}^{t,(u+1,0,w+1)}\boldsymbol D_{(u,0,w)}^{t, (u+1,1,w+1)}\boldsymbol R_{\mathrm b, (u,1,w)}^{t,(u,0,w)} \end{bmatrix},$$

where

$$(1-y)\boldsymbol P^{t, (u, 1, w)} = \boldsymbol P^{t, (u, 0, w)}\boldsymbol R_{\mathrm b, (u, 1, w)}^{t, (u, 0, w)}$$

and

$$\frac{\mathrm d}{\mathrm dy}\boldsymbol P^{t, (u, 0, w)} = \boldsymbol P^{t, (u+1,1,w+1)} \boldsymbol D_{(u, 0, w)}^{t, (u+1,1,w+1)}.$$

Need to add weighted derivatives

dlfivefifty · 2024-06-21T09:06:52Z

I'm looking at this now, will try to get the tests to pass

DanielVandH · 2024-07-01T23:17:41Z

Will hopefully have this all fixed up soon after I get out of a whole rabbit hole of ambiguities. The expansion tests started failing after #112 for some reason

TSGut · 2024-07-02T00:02:17Z

What's the error? Or does it just give wrong values in the expansion without erroring? If it's a problem in that PR let me know.

DanielVandH · 2024-07-02T00:08:29Z

I think it's just issues with the layouts. Not really an issue with the pr, probably just that it's revealed some other issues with ambiguities. I'll give an example of the error in the morning

…

On Tue, 2 July 2024, 1:02 am Timon Salar Gutleb, ***@***.***> wrote: What's the error? Or does it just give wrong values in the expansion without erroring? If it's a problem in that PR let me know. — Reply to this email directly, view it on GitHub <#111 (comment)>, or unsubscribe <https://github.com/notifications/unsubscribe-auth/AWZPH4G3LWTVBYUA3RW6JOTZKHUZ5AVCNFSM6AAAAABJSNRMSOVHI2DSMVQWIX3LMV43OSLTON2WKQ3PNVWWK3TUHMZDEMBRGQ3TGOJXHA> . You are receiving this because you authored the thread.Message ID: <JuliaApproximation/SemiclassicalOrthogonalPolynomials. ***@***.***>

DanielVandH · 2024-07-02T07:52:43Z

@TSGut This is the issue:

julia> using SemiclassicalOrthogonalPolynomials, ClassicalOrthogonalPolynomials # make sure you're on this pr branch

julia> Ps = SemiclassicalJacobi.(2, -1//2:5//2, -1.0, -1//2:5//2)
4-element SemiclassicalOrthogonalPolynomials.SemiclassicalJacobiFamily{Float64, UnitRange{Rational{Int64}}, Float64, UnitRange{Rational{Int64}}}:
 SemiclassicalJacobi with weight x^-0.5 * (1-x)^-1.0 * (2.0-x)^-0.5 on 0..1
 SemiclassicalJacobi with weight x^0.5 * (1-x)^-1.0 * (2.0-x)^0.5 on 0..1
 SemiclassicalJacobi with weight x^1.5 * (1-x)^-1.0 * (2.0-x)^1.5 on 0..1
 SemiclassicalJacobi with weight x^2.5 * (1-x)^-1.0 * (2.0-x)^2.5 on 0..1

julia> g = x -> exp(x) + sin(x)
#11 (generic function with 1 method)

julia> expand(Ps[1], g) # works 
SemiclassicalJacobi with weight x^-0.5 * (1-x)^-1.0 * (2.0-x)^-0.5 on 0..1 * [3.559752813266927, -2.689556624759561, -0.1419489233276318, -0.00815710939820469, -0.0012292671637252207, -7.007691099937732e-5, -1.3644779541294862e-6, -3.3513086473469194e-8, -2.7275540904437935e-9, -8.832973058841338e-11  …  ]

julia> expand(Ps[2], g) # works 
SemiclassicalJacobi with weight x^0.5 * (1-x)^-1.0 * (2.0-x)^0.5 on 0..1 * [3.559752813266927, -2.813114026394983, -0.15330885488899992, -0.009528747519073118, -0.0013411342705156972, -7.427293498329727e-5, -1.4527735961828238e-6, -3.734131077723306e-8, -2.925216076248728e-9, -9.315982251717606e-11  …  ]

julia> expand(Ps[3], g) 
ERROR: MethodError: simplifiable(::ArrayLayouts.Mul{LazyArrays.ApplyLayout{typeof(\)}, ArrayLayouts.DiagonalLayout{ArrayLayouts.OnesLayout}, LazyArrays.ApplyMatrix{Float64, typeof(\), Tuple{LinearAlgebra.UpperTriangular{Float64, InfiniteLinearAlgebra.AdaptiveQRFactors{Float64, BandedMatrices.BandedMatrix{Float64, Matrix{Float64}, Base.OneTo{Int64}}, LazyBandedMatrices.SymTridiagonal{Float64, LazyArrays.BroadcastVector{Float64, typeof(-), Tuple{Float64, ClassicalOrthogonalPolynomials.LanczosJacobiBand{Float64}}}, LazyArrays.BroadcastVector{Float64, typeof(-), Tuple{ClassicalOrthogonalPolynomials.LanczosJacobiBand{Float64}}}}}}, LinearAlgebra.Diagonal{Float64, LazyArrays.BroadcastVector{Float64, typeof(*), Tuple{LazyArrays.BroadcastVector{Float64, typeof(sign), Tuple{SubArray{Float64, 1, LinearAlgebra.UpperTriangular{Float64, InfiniteLinearAlgebra.AdaptiveQRFactors{Float64, BandedMatrices.BandedMatrix{Float64, Matrix{Float64}, Base.OneTo{Int64}}, LazyBandedMatrices.SymTridiagonal{Float64, LazyArrays.BroadcastVector{Float64, typeof(-), Tuple{Float64, ClassicalOrthogonalPolynomials.LanczosJacobiBand{Float64}}}, LazyArrays.BroadcastVector{Float64, typeof(-), Tuple{ClassicalOrthogonalPolynomials.LanczosJacobiBand{Float64}}}}}}, Tuple{InfiniteLinearAlgebra.InfBandCartesianIndices}, false}}}, Float64}}}}}, LinearAlgebra.Diagonal{Float64, FillArrays.Ones{Float64, 1, Tuple{InfiniteArrays.OneToInf{Int64}}}}}) is ambiguous.

Candidates:
  simplifiable(M::ArrayLayouts.Mul{LazyArrays.ApplyLayout{typeof(\)}})
    @ LazyArrays C:\Users\djv23\.julia\packages\LazyArrays\QwVhu\src\linalg\inv.jl:140
  simplifiable(M::ArrayLayouts.Mul{<:Any, <:ArrayLayouts.DiagonalLayout{<:ArrayLayouts.AbstractFillLayout}})      
    @ LazyArrays C:\Users\djv23\.julia\packages\LazyArrays\QwVhu\src\linalg\mul.jl:375

Possible fix, define
  simplifiable(::ArrayLayouts.Mul{LazyArrays.ApplyLayout{…}, <:ArrayLayouts.DiagonalLayout{…}})

Stacktrace:
  [1] simplifiable
    @ C:\Users\djv23\.julia\packages\LazyArrays\QwVhu\src\linalg\mul.jl:339 [inlined]
  [2] _most_simplifiable
    @ C:\Users\djv23\.julia\packages\LazyArrays\QwVhu\src\linalg\mul.jl:342 [inlined]
  [3] simplifiable(::typeof(*), ::LazyArrays.ApplyMatrix{…}, ::LazyArrays.ApplyMatrix{…}, ::LinearAlgebra.Diagonal{…})
    @ LazyArrays C:\Users\djv23\.julia\packages\LazyArrays\QwVhu\src\linalg\mul.jl:340
  [4] _simplify(::typeof(*), ::LazyArrays.ApplyMatrix{…}, ::LazyArrays.ApplyMatrix{…}, ::LinearAlgebra.Diagonal{…}, ::Vararg{…})
    @ LazyArrays C:\Users\djv23\.julia\packages\LazyArrays\QwVhu\src\linalg\mul.jl:349
  [5] simplify
    @ C:\Users\djv23\.julia\packages\LazyArrays\QwVhu\src\linalg\mul.jl:345 [inlined]
  [6] simplify
    @ C:\Users\djv23\.julia\packages\LazyArrays\QwVhu\src\linalg\mul.jl:355 [inlined]
  [7] copy
    @ C:\Users\djv23\.julia\packages\LazyArrays\QwVhu\ext\LazyArraysBandedMatricesExt.jl:576 [inlined]
  [8] materialize
    @ C:\Users\djv23\.julia\packages\ArrayLayouts\70nvu\src\mul.jl:137 [inlined]
  [9] mul
    @ C:\Users\djv23\.julia\packages\ArrayLayouts\70nvu\src\mul.jl:138 [inlined]
 [10] *(A::LazyArrays.ApplyArray{…}, B::LazyBandedMatrices.Bidiagonal{…})
    @ ArrayLayouts C:\Users\djv23\.julia\packages\ArrayLayouts\70nvu\src\mul.jl:223
 [11] \(A::SemiclassicalJacobi{Float64}, B::SemiclassicalJacobi{Float64})
    @ SemiclassicalOrthogonalPolynomials c:\Users\djv23\.julia\dev\SemiclassicalOrthogonalPolynomials.jl\src\SemiclassicalOrthogonalPolynomials.jl:446
 [12] \(A::SemiclassicalJacobi{Float64}, B::SemiclassicalJacobi{Float64})
    @ SemiclassicalOrthogonalPolynomials c:\Users\djv23\.julia\dev\SemiclassicalOrthogonalPolynomials.jl\src\SemiclassicalOrthogonalPolynomials.jl:435
 [13] semijacobi_ldiv(P::SemiclassicalJacobi{…}, Q::LanczosPolynomial{…})
    @ SemiclassicalOrthogonalPolynomials c:\Users\djv23\.julia\dev\SemiclassicalOrthogonalPolynomials.jl\src\SemiclassicalOrthogonalPolynomials.jl:278
 [14] \(A::SemiclassicalJacobi{Float64}, B::LanczosPolynomial{Float64, SemiclassicalJacobiWeight{…}, Normalized{…}})
    @ SemiclassicalOrthogonalPolynomials c:\Users\djv23\.julia\dev\SemiclassicalOrthogonalPolynomials.jl\src\SemiclassicalOrthogonalPolynomials.jl:452
 [15] ldiv(Q::SemiclassicalJacobi{Float64}, f::QuasiArrays.BroadcastQuasiVector{Float64, var"#11#12", Tuple{Inclusion{…}}})
    @ SemiclassicalOrthogonalPolynomials c:\Users\djv23\.julia\dev\SemiclassicalOrthogonalPolynomials.jl\src\SemiclassicalOrthogonalPolynomials.jl:519
 [16] \(A::SemiclassicalJacobi{Float64}, B::QuasiArrays.BroadcastQuasiVector{Float64, var"#11#12", Tuple{Inclusion{…}}})
    @ QuasiArrays C:\Users\djv23\.julia\packages\QuasiArrays\Jo8rC\src\matmul.jl:34
 [17] transform(A::SemiclassicalJacobi{Float64}, f::Function)
    @ ContinuumArrays C:\Users\djv23\.julia\packages\ContinuumArrays\d6Ead\src\bases\bases.jl:291
 [18] expand(A::SemiclassicalJacobi{Float64}, f::Function)
    @ ContinuumArrays C:\Users\djv23\.julia\packages\ContinuumArrays\d6Ead\src\bases\bases.jl:302
 [19] top-level scope
    @ c:\Users\djv23\.julia\dev\SemiclassicalOrthogonalPolynomials.jl\test\test_neg1b.jl:80
Some type information was truncated. Use `show(err)` to see complete types.

Part of the issue is probably what I've had to do here

https://github.com/DanielVandH/SemiclassicalOrthogonalPolynomials.jl/blob/1bdaa2bcde28b21481a94ce02bf71c2768d7ec7e/src/SemiclassicalOrthogonalPolynomials.jl#L422-L435

to fix a new colsupport issue. Maybe you have some insight into what might've changed to have broken this?

TSGut · 2024-07-03T02:14:56Z

So replacing the line return Rₐ₀ᵪᴬ * Rᵦₐ₋₁ᵪᵗᵃ⁰ᶜ with return ApplyArray(*,UpperTriangular(Rₐ₀ᵪᴬ), Rᵦₐ₋₁ᵪᵗᵃ⁰ᶜ) works, so it's definitely layout shenanigans. That's probably not a great long term solution but short term it may let you finish what you want to do here and unless I missed something it's not all that different from what would be done internally anyway, so it may not matter performance wise (may be missing something, I only glanced at the methods mentioned below).

The long term fix would be to disambiguate in LazyArrays.jl, the error message is pretty spot on, the issue is these two options are ambiguous for our multiplication chain of Rₐ₀ᵪᴬ * Rᵦₐ₋₁ᵪᵗᵃ⁰ᶜ:

Candidates:
  simplifiable(M::ArrayLayouts.Mul{LazyArrays.ApplyLayout{typeof(\)}})
    @ LazyArrays C:\Users\djv23\.julia\packages\LazyArrays\QwVhu\src\linalg\inv.jl:140
  simplifiable(M::ArrayLayouts.Mul{<:Any, <:ArrayLayouts.DiagonalLayout{<:ArrayLayouts.AbstractFillLayout}})      
    @ LazyArrays C:\Users\djv23\.julia\packages\LazyArrays\QwVhu\src\linalg\mul.jl:375

A well placed redirection if one is in both of those cases would be one example of a proper fix. In any case that has to happen in LazyArrays.jl. For now I would recommend the quick fix I mentioned.

DanielVandH · 2024-07-03T08:35:20Z

Thanks for looking into it @TSGut. I am still getting used to understanding all of the design across these packages :)

I'll use this temporary fix for now but I'll try and get some fixes through as well.

dlfivefifty · 2024-07-03T09:25:39Z

I would suggest in general being explicit about Lazy array constructions as it will reduce compile time

DanielVandH · 2024-07-09T08:41:08Z

Any ideas why the first testset (the Jacobi test in the test_neg1b.jl file) might have become insanely slow all of a sudden? I don't believe I did anything significant.. the slowness is coming only from the first evaluation for each (t, a, c) triple. i.e. for n=1:25 for the first x (currently 0 but this is an issue even for x=0.5 for example) every single evaluation is extremely slow. Running the loop n=1:25 again without even changing x is fast once again. Did something change with compilation?

dlfivefifty · 2024-07-09T08:47:38Z

Hmm its possible the changes in LazyArrays.jl I just made have made compilation really slow...

dlfivefifty · 2024-07-09T08:48:24Z

It might be Windows specific? I'm having trouble with these tests passing in time:

https://github.com/JuliaApproximation/ClassicalOrthogonalPolynomials.jl/actions/runs/9846940208/job/27202852824?pr=195

DanielVandH · 2024-07-09T08:50:53Z

I started seeing this issue last night but only got to confirm it just now, so I'm not sure it'd be those changes just now. If it's Windows only that'll be pretty rough to debug

dlfivefifty · 2024-07-09T08:57:34Z

Can you pin LazyArrays.jl at 2.1.4 and see if it goes away?

If it is a compile time issue there might be a type instability introduced

dlfivefifty · 2024-07-09T08:58:20Z

Also, are you on LazyArrays v2.1.5 or 2.1.6? If 2.1.5 can you try 2.1.6 just in case the issue is gone?

DanielVandH · 2024-07-09T09:04:04Z

I was on v2.1.5. The issue is still on 2.1.6. It's gone on 2.1.4

DanielVandH · 2024-07-09T09:08:50Z

Would it somehow be due to JuliaArrays/LazyArrays.jl#334? This is the only method that I think might be encountered that changed in 2.1.5 as far as I can tell (not 100% sure what JuliaArrays/LazyArrays.jl#335 does).

dlfivefifty · 2024-07-09T09:36:50Z

Can you try reverting just that commit and see if it fixes the issue?

DanielVandH · 2024-07-09T09:46:36Z

Reverting JuliaArrays/LazyArrays.jl#334 didn't change anything. I'll have to learn how to properly revert JuliaArrays/LazyArrays.jl#335 and I'll test

DanielVandH · 2024-07-09T10:00:29Z

Hopefully I did the revert properly here DanielVandH/LazyArrays.jl@6fd2bc8. This revert does fix the issues I'm seeing.

dlfivefifty · 2024-07-10T14:46:31Z

I suspect the answer is the former is inlining and removing all the simplify whilst the latter is not. It's possible that some thresholds are different between Windows and other OSes.

So I think for now we just make the fix you found.

The design of simplify definitely pushes the limits of type inference. In fact if it is called at runtime it has exponential complexity 🤦‍♂️ The compiler avoids this because the function lookup table acts as a database of whats called, but definitely not ideal. A generated function might be better but I haven't got around to trying it (and there's the risk it makes things worse).

DanielVandH · 2024-07-10T22:15:25Z

Been spending ages again trying to figure out why my expand tests were failing. Turns out deleting the

function colsupport(lay::AbstractInvLayout{<:TriangularLayout}, A, j)
    B, = arguments(lay, A)
    return colsupport(B, j)
end 

function rowsupport(lay::AbstractInvLayout{<:TriangularLayout}, A, k)
    B, = arguments(lay, A)
    return rowsupport(B, k)
end

I added makes them pass again. Is there something wrong with these definitions? Or any suggestions about what could be happening? I'm going to make a PR to revert them for the time being but it would be nice to try and see what the issue is.

Actually it seems to be a combination of this and something else I need to identify.. I'll keep trying.

DanielVandH · 2024-07-11T06:55:05Z

Ignore the above. It has nothing to do with those methods. Something else between [email protected] and [email protected] is breaking my tests, but it's not that. I don't know what it is yet though.. none of the other tests fail, its only my b=-1. Back to the debugger

dlfivefifty · 2024-07-11T07:01:57Z

There were a few other similar changes I made replacing AbstractPaddedLayout, try changing those back?

DanielVandH · 2024-07-11T13:38:54Z

Ignore the above. It has nothing to do with those methods.

I don't know what went wrong where I determined that I was wrong initially. Turns out that reverting it so fix my problems. I'm going to make a PR reverting it, but I don't fully understand the issue. It would be nice to eventually get that method back in.

dlfivefifty · 2024-07-11T13:53:22Z

Is it a compile time thing or run time? Try throwing an error?

DanielVandH · 2024-07-11T13:56:53Z

It's runtime. The coefficients from expand were just not giving the correct values (e.g. expanding 5 + (1 - x) was giving something like coefficients = [3.4, -2] instead of [5, 1]).

…oefficients

…e parameters take that long) to ~1m

DanielVandH · 2024-07-13T18:38:10Z

Pretty much all done now except for figuring out weighted expansions and fixing families. Have to special case it since the existing code assumes there is a normalisation constant. Not sure how to do expansions that involve the $b=-1$ weight yet. Weighted expansions and derivatives done (including $b=-1$ weight not allowed). Looking into fixing issues with efficiently constructing families now, that's all that's left I believe.

Do we need derivatives of fully weighted $b=-1$ polynomials? I don't know how to make sense of it when the $b$ weight is included since we'd want $\mathcal D[(1-x)^{-1}P^{t,(a,-1,c)}] =(1-x)^{-2}P^{t,(a+1,-2,c+1)}D$. Not differentiable at $x=1$ anyway but the issue is decrementing past $-1$.

CI will also fail until JuliaLinearAlgebra/InfiniteLinearAlgebra.jl#186 gets merged and registered. Might also want JuliaArrays/LazyArrays.jl#341 done before completing this

DanielVandH · 2024-07-14T09:49:32Z

Everything should be good now. Tests all pass locally. Not sure what CI will do until

CI will also fail until JuliaLinearAlgebra/InfiniteLinearAlgebra.jl#186 gets merged and registered. Might also want JuliaArrays/LazyArrays.jl#341 done before completing this

is resolved.

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DanielVandH and others added 4 commits June 15, 2024 10:54

temp commit

ec13034

stash

3dbe4d2

Merge branch 'JuliaApproximation:master' into neg1b

4e01fb8

Put current changes in

a4b74ad

DanielVandH changed the title ~~WIP: Implement $\boldsymbol P^{t, (u, -1, w)}$~~ WIP: Implement $b=-1$ Jun 19, 2024

DanielVandH changed the title ~~WIP: Implement $b=-1$~~ WIP: Implement b=-1 Jun 19, 2024

DanielVandH and others added 3 commits July 1, 2024 18:23

Merge branch 'master' into neg1b

f74b9ae

Update SemiclassicalOrthogonalPolynomials.jl

58739e3

other changes

8a67de0

current state

1bdaa2b

Fix expansions

a1f57d6

DanielVandH mentioned this pull request Jul 10, 2024

Change signature of materialize! to fix regression on Windows JuliaArrays/LazyArrays.jl#337

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DanielVandH mentioned this pull request Jul 11, 2024

Revert "Add (row/col)support for inv(Triangular)" JuliaArrays/LazyArrays.jl#338

Merged

DanielVandH and others added 6 commits July 11, 2024 20:45

Finally fix all the tests. Fixed issue with expansions with integer c…

acad7fb

…oefficients

Add comment to address later [no ci]

40e5eb8

Merge branch 'JuliaApproximation:master' into neg1b

39d964e

Fix connections

94d427f

Testing is a bit excessive. Trims the time down from 14m (why do thre…

c4b1d71

…e parameters take that long) to ~1m

Properly test differentiation [no ci]

42ad289

DanielVandH added 5 commits July 13, 2024 23:12

Fix weighted expansions

eef5a45

Weighted derivatives come for free.

77b2cd5

Remove old comments and show [no ci]

ce92ea0

Fix expansions

4dd15ca

Add a comment explaining the expansion code

7c090d8

DanielVandH marked this pull request as ready for review July 14, 2024 09:48

DanielVandH changed the title ~~WIP: Implement b=-1~~ Implement b=-1 Jul 14, 2024

DanielVandH closed this Jul 14, 2024

DanielVandH reopened this Jul 14, 2024

dlfivefifty merged commit a1e4c3e into JuliaApproximation:master Jul 14, 2024
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DanielVandH deleted the neg1b branch July 14, 2024 13:18

DanielVandH mentioned this pull request Nov 9, 2024

Use a better method for the derivative of the $b = -1$ polynomials #122

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