On 27 February 2024 17:25:51 GMT, 'Animesh Shree' via sage-devel
<sage-devel@googlegroups.com> wrote:
>This works good if input is square and I also checked on your idea of
>padding zeros for non square matrices.
>I am currently concerned about the permutation matrix and L, U in case of
>padded 0s. Because if we pad then how will they affect the outputs, so that
>we can extract p,l,u for unpadded matrix.
please read details I wrote on how to deal with the non-square case. There is
no padding needed.
>
>On Tuesday, February 27, 2024 at 10:03:25 PM UTC+5:30 Dima Pasechnik wrote:
>
>>
>>
>> On 27 February 2024 15:34:20 GMT, 'Animesh Shree' via sage-devel <
>> sage-...@googlegroups.com> wrote:
>> >I tried scipy which uses superLU. We get the result but there is little
>> bit
>> >of issue.
>> >
>> >
>> >--For Dense--
>> >The dense matrix factorization gives this output using permutation matrix
>> >sage: a = Matrix(RDF, [[1, 0],[2, 1]], sparse=True)
>> >sage: a
>> >[1.0 0.0]
>> >[2.0 1.0]
>> >sage: p,l,u = a.dense_matrix().LU()
>> >sage: p
>> >[0.0 1.0]
>> >[1.0 0.0]
>> >sage: l
>> >[1.0 0.0]
>> >[0.5 1.0]
>> >sage: u
>> >[ 2.0 1.0]
>> >[ 0.0 -0.5]
>> >
>>
>> you'd probably want to convert the permutation matrix into a permutation.
>>
>>
>> >--For Sparse--
>> >But the scipy LU decomposition uses permutations which involves taking
>> >transpose, also the output permutations are represented as array.
>>
>> It is very normal to represent permutations as arrays.
>> One can reconstruct the permutation matrix from such an array trivially
>> (IIRC, Sage even has a function for it)
>>
>> I am not sure what you mean by "taking transpose".
>>
>> >sage: p,l,u = a.LU(force=True)
>> >sage: p
>> >{'perm_r': [1, 0], 'perm_c': [1, 0]}
>> >sage: l
>> >[1.0 0.0]
>> >[0.0 1.0]
>> >sage: u
>> >[1.0 2.0]
>> >[0.0 1.0]
>> >
>> >
>> >Shall I continue with this?
>>
>> sure, you are quite close to getting it all done it seems.
>>
>>
>> >On Tuesday, February 6, 2024 at 11:29:07 PM UTC+5:30 Dima Pasechnik wrote:
>> >
>> >> Non-square case for LU is in fact easy. Note that if you have A=LU as
>> >> a block matrix
>> >> A11 A12
>> >> A21 A22
>> >>
>> >> then its LU-factors L and U are
>> >> L11 0 and U11 U12
>> >> L21 L22 0 U22
>> >>
>> >> and A11=L11 U11, A12=L11 U12, A21=L21 U11, A22=L21 U12+L22 U22
>> >>
>> >> Assume that A11 is square and full rank (else one may apply
>> >> permutations of rows and columns in the usual way). while A21=0 and
>> >> A22=0. Then one can take L21=0, L22=U22=0, while A12=L11 U12
>> >> implies U12=L11^-1 A12.
>> >> That is, we can first compute LU-decomposition of a square matrix A11,
>> >> and then compute U12 from it and A.
>> >>
>> >> Similarly, if instead A12=0 and A22=0, then we can take U12=0,
>> >> L22=U22=0, and A21=L21 U11,
>> >> i.e. L21=A21 U11^-1, and again we compute LU-decomposition of A11, and
>> >> then L21=A21 U11^-1.
>> >>
>> >> ----------------
>> >>
>> >> Note that in some cases one cannot get LU, but instead must go for an
>> >> PLU,with P a permutation matrix.
>> >> For non-square matrices this seems a bit more complicated, but, well,
>> >> still doable.
>> >>
>> >> HTH
>> >> Dima
>> >>
>> >>
>> >>
>> >>
>> >> On Mon, Feb 5, 2024 at 6:00 PM Nils Bruin <nbr...@sfu.ca> wrote:
>> >> >
>> >> > On Monday 5 February 2024 at 02:31:04 UTC-8 Dima Pasechnik wrote:
>> >> >
>> >> >
>> >> > it is the matter of adding extra zero rows or columns to the matrix
>> you
>> >> want to decompose. This could be a quick fix.
>> >> >
>> >> > (in reference to computing LU decompositions of non-square matrices)
>> --
>> >> in a numerical setting, adding extra zero rows/columns may not be such
>> an
>> >> attractive option: if previously you know you had a maximal rank
>> matrix,
>> >> you have now ruined it by the padding. It's worth checking the
>> >> documentation and literature if padding is appropriate/desirable for
>> the
>> >> target algorithm/implementation.
>> >> >
>> >> > --
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>> >> .
>> >>
>> >
>>
>
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