[petsc-users] SLEPc: non-real singular vectors from SVD of real matrix

[Peder Jørgensgaard Olesen]

Thank you for your advice, Jose.


I tried using a single column of the data matrix as a basis for left singular vector space, and a row for the right one - and lo and behold, all singular vectors become real.


Somehow it did not occur to me that I was I was looking at was simply a complex phase on U and V which would cancel upon computing UΣV*. In other words, rather than an incorrect result it was a slightly inconvenient representation of a correct result.


Best regards

Peder

________________________________
Fra: Jose E. Roman <jroman at dsic.upv.es>
Sendt: 29. april 2021 13:00:03
Til: Peder Jørgensgaard Olesen
Cc: petsc-users at mcs.anl.gov
Emne: Re: [petsc-users] SLEPc: non-real singular vectors from SVD of real matrix

In complex scalars there is no way of knowing if the user-provided matrix is real or not. If there was an option of MatSetOption() we could use it.

If you set a real initial vector with SVDSetInitialSpaces(), then most probably the computed singular vectors will be real. But still rounding errors could introduce a nonzero imaginary part.

In any case, you could normalize the computed singular vectors, as is done for instance in FixSign() in this example https://slepc.upv.es/documentation/current/src/nep/tutorials/ex20.c.html

Jose


> El 29 abr 2021, a las 12:42, Peder Jørgensgaard Olesen via petsc-users <petsc-users at mcs.anl.gov> escribió:
>
> Hello
>
> I've noticed that doing a singular value decomposition of a real matrix appears to result in non-real singular vectors. This should not be the case - singular vectors of a real matrix must be real-valued.
>
> In the example attached I read a matrix from a binary file (also attached), perform the SVD, and write singular vectors to an HDF5 file which I subsequently inspect using h5dump, revealing  non-zero imaginary parts of vector elements as highlighted below:
> [pjool at svol mwes]$ h5dump -d U0_sample_svd -c "5,2" sample_svecs.h5
> HDF5 "sample_svecs.h5" {
> DATASET "U0_sample_svd" {
>    DATATYPE  H5T_IEEE_F64LE
>    DATASPACE  SIMPLE { ( 882, 2 ) / ( 882, 2 ) }
>    SUBSET {
>       START ( 0, 0 );
>       STRIDE ( 1, 1 );
>       COUNT ( 5, 2 );
>       BLOCK ( 1, 1 );
>       DATA {
>       (0,0): 0.0226108, 0.0299595,
>       (1,0): 0.035414, 0.0469237,
>       (2,0): 0.0276317, 0.0366122,
>       (3,0): 0.0145344, 0.0192581,
>       (4,0): 0.0110376, 0.0146249
>       }
>    }
>    ATTRIBUTE "complex" {
>       DATATYPE  H5T_STD_I32LE
>       DATASPACE  SCALAR
>       DATA {
>       (0): 1
>       }
>    }
> }
> }
> I also extract the imaginary part of the input matrix and print its norm to ensure that the matrix is indeed real.
>
> I'm running v3.14, but I don't believe that alone should cause the issue, since it what I'm trying to do appears like a rather common and basic task.
>
> What might be the reason behind this behavior, and what can be done to resolve it?
>
>
> Med venlig hilsen / Best Regards
>
> Peder Jørgensgaard Olesen
> PhD Student, Turbulence Research Lab
> Dept. of Mechanical Engineering
> Technical University of Denmark
> Niels Koppels Allé
> Bygning 403, Rum 105
> DK-2800 Kgs. Lyngby
> <read_bin.c><sample_data>

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