Check bands for zeros in `lu!` and `qr!`

[ctkelley]

Hello again, it seems that qr! thinks it has a method for BandedMatrices,
but the results are incorrect. lu! knows it has no such method and issues
the usual complaint. Example below.

I am trying to figure out how to do a PR for the Float32 issue. This may
take me awhile because I'm still not sure where the problem lives in
your code and have never issued a PR for a project that I'm not seriously
involved with.

-- Tim

julia> B=brand(Float64,10,10,2,2);
julia> b=rand(10,);
julia> c=B\b;
julia> P=qr(B);
julia> d=P\b; norm(d-c)
1.00688e-15
julia> C=copy(B); Q=qr!(C); e=Q\b; norm(e-c)
3.58602e+00

[ctkelley]

Sorry, lu! does think it has a method, but that is wrong too.

julia> Z=lu!(B)
julia> f=Z\b;
julia> norm(f-c)
8.02474e+01

[dlfivefifty]

qr! assumes that you've pre-allocated the matrices with extra bands, as otherwise in-place qr is not possible. Would you have a better design in mind? We could check that the extra bands are zero.

[ctkelley]

Thanks, That makes sense. lu! for dense matrices allocates the extra storage for the pivots but I can see that as a real pain for the banded case. Was that documented somewhere? I missed it. This is also a very good case for using qr instead of lu. You need to double the bandwidths if I recall correctly. Is that right? — Tim

…

On Tue, Aug 18, 2020 at 10:24 AM Sheehan Olver ***@***.***> wrote: qr! assumes that you've pre-allocated the matrices with extra bands, as otherwise in-place qr is not possible. Would you have a better design in mind? We could check that the extra bands are zero. — You are receiving this because you authored the thread. Reply to this email directly, view it on GitHub <#197 (comment)>, or unsubscribe <https://github.com/notifications/unsubscribe-auth/ACOEX62UNJNCM2NE4I44GXTSBKFIPANCNFSM4QDN27QQ> .

-- C. T. Kelley Department of Mathematics, Box 8205 SAS Hall 2311 Stinson Drive North Carolina State University Raleigh, NC 27695-8205 (919) 515-7163, (919) 513-7336 (FAX) Tim_Kelley@ncsu.edu https://ctk.math.ncsu.edu

[dlfivefifty]

If A has bandwidths (l,u) then qr(A).factors has bandwidth (l,u+l)

[ctkelley]

So lu! needs 2l and 2u?

…

On Tue, Aug 18, 2020 at 10:50 AM Sheehan Olver ***@***.***> wrote: If A has bandwidths (l,u) then qr(A).factors has bandwidth (l,u+l) — You are receiving this because you authored the thread. Reply to this email directly, view it on GitHub <#197 (comment)>, or unsubscribe <https://github.com/notifications/unsubscribe-auth/ACOEX67QBCSS2VYTXZFTOXLSBKIM5ANCNFSM4QDN27QQ> .

-- C. T. Kelley Department of Mathematics, Box 8205 SAS Hall 2311 Stinson Drive North Carolina State University Raleigh, NC 27695-8205 (919) 515-7163, (919) 513-7336 (FAX) Tim_Kelley@ncsu.edu https://ctk.math.ncsu.edu

[dlfivefifty]

I didn't develop the lu! code so I'm not sure I can answer. The LAPACK documentation should provide details: http://www.netlib.org/lapack/explore-html/d2/d2d/group__double_g_bcomputational_gad1efab86e6d869915e059286ecf1bcb1.html

[ctkelley]

No problem. The LAPACK manual is less than a meeter away from me.

…

On Tue, Aug 18, 2020 at 10:58 AM Sheehan Olver ***@***.***> wrote: I didn't develop the lu! code so I'm not sure I can answer. The LAPACK documentation should provide details: http://www.netlib.org/lapack/explore-html/d2/d2d/group__double_g_bcomputational_gad1efab86e6d869915e059286ecf1bcb1.html — You are receiving this because you authored the thread. Reply to this email directly, view it on GitHub <#197 (comment)>, or unsubscribe <https://github.com/notifications/unsubscribe-auth/ACOEX6462A2MH5VFYFAT2JTSBKJI3ANCNFSM4QDN27QQ> .

-- C. T. Kelley Department of Mathematics, Box 8205 SAS Hall 2311 Stinson Drive North Carolina State University Raleigh, NC 27695-8205 (919) 515-7163, (919) 513-7336 (FAX) Tim_Kelley@ncsu.edu https://ctk.math.ncsu.edu

[ctkelley]

You need to embed your matrix into one with bandwidths ll, and lu+2 with zeros on the unused bands. Then it works fine. So does lu!

You should look at the LAPACK manual, as @dlfivefifty suggests.

[ctkelley]

Take this example, please.

julia> using BandedMatrices
julia> B=BandedMatrix{Float64}(Zeros(20,20),(2,6));
julia> D=rand(20,); D1=rand(19,); D2=rand(18,); D3=rand(17,); D4=rand(16,);
julia> Dm1=rand(19,); Dm2=rand(18,);
julia> B[band(0)].=D; B[band(1)].=D1; B[band(2)].=D2; B[band(3)].=D3;
julia> B[band(4)].=D4; B[band(-1)].=Dm1; B[band(-2)].=Dm2;
julia> b=rand(20,); C=copy(B); x1=B\b;
julia> QRB=qr!(C);
julia> x2=QRB\b;
julia> norm(x1-x2)
8.86313e-14

[dlfivefifty]

A simple for loop checking that the extra bands are zero, and throwing an error if not, should fix this. The cost is negligible to the actual qr!, though we can bypass the check in qr(A) where the bands are initialised correctly.

[MikaelSlevinsky]

There's an argument here for not naming the current code qr!(A) since it doesn't work for all banded matrices, just those with the extra data assumption. Certain assumptions on A already result in an InexactError according to the base docs.

New in 1.5 is lu! for sparse matrices. According to the docs, due to type conversions, the signature takes the form

lu!(F::UmfpackLU, A::SparseMatrixCSC) uses the UMFPACK library that is part of SuiteSparse.
As this library only supports sparse matrices with Float64 or ComplexF64 elements,
lu! converts A into a copy that is of type SparseMatrixCSC{Float64} or SparseMatrixCSC{ComplexF64} as appropriate.

Actually calling this according to the example only changes F not A even though A is the "input" data.

One option would be to use qr!(F::BT, A::BandedMatrix), where BT is either a banded QR factorization type or another banded matrix whose superdiagonals are extended.

[ctkelley]

No, please no. qr! does exactly what it does in LAPACK and LINPACK. You have to pad the matrix with a couple zero bands, but then it works correctly and is the same as the calls in C or Fortran. lu! does the same thing on BandedMatrices.

The general sparse case is very different. I am not clear on what the right thing to do is there.

[dlfivefifty]

Wait, there's a banded qr! in LAPACK??