Re: [eigen] Skyline matrix

On Thu, Nov 12, 2009 at 1:04 PM, guillaume saupin <guillaume.saupin@xxxxxx> wrote:

Gael Guennebaud wrote:

Ok, I've put it in the unsupported/ folder. To use it you just have add path/to/eigen2/unsupported to your set of include paths.

Since I guess you will certainely want to do some modifications in the future, I recommend you to create an account on bitbucket, and I will give you write access. Feel free to commit any changes you like in this module, but if your changes affect some other modules, please let's first dicuss them here !

I've just created my account on bitbucket. My id is simply guillaume.saupin.

Now I have to admit there are still a lot of work to do before having something really usable to offer to other users. In my opinion, the SkylineMatrix class should be taillored to be optimal for the two following tasks:
1 - setup/assemble the matrix
2 - build efficient high-level algorithms

Both tasks require a powerful API taking into account the specificities of the skyline storage. In particular this discard low level coeff(i,j) methods or things like that. Iterators are not very well suited here because 1) you need various kind of iterators (upper, lower), and 2) they cannot take advantage of the sequential storage to enable unrolling and vectorization.

So, for the matrix assembly, the challenge is to reduce the need for memory reallocation and memory copies. Here we first need a very low level API allowing to fill the matrix with constant time insertions.

One possibility is to let the user specify the profile for each rows and cols, and allocate the memory accordingly. Then the insertion can be done in constant time.

Then to offer more flexibility to the user, perhaps the simpler is to let the user fill a {Dynamic}SparseMatrix, and then implement a fast conversion method. The latter is a must to have feature anyway.

If we provide an easy and fast API to fill the matrix, I'm not sure that it will be very useful to create a SkylineMatrix from a SparseMatrix.

In order to easily implement efficient algorithms (products and solvers), the SkylineMatrix should provide higher level access functions than the coeff() function or the iterators. In particular I'm thinking about the following set of functions:

diagonal(); // return a vector _expression_ of the diagonal
lowerCol(j); // returns an _expression_ of the j-th column of the strictly lower triangular part
upperRow(i);
centeredBlock(i, rows, cols); // returns an _expression_ of the rows x cols submatrix at the coeff (i, i)

The first two functions would return a special _expression_ wrapping a dense VectorBlock plus a start index and a length with optimized arithmetic operators. Such an _expression_, let's call it a RangeVector could also be useful for the Band storage.

This is a good idea. What can I use as a starting point to implement those VectorBlocks ? Should it inherit from an already existing class ?

For diagonal(), it is easy. Just use a Matrix<Scalar,Size,1> to store it (instead of a raw pointer), and let it return a ref to it.

For the lowerCol(), upperRow() we have to create a new matrix type, well actually a vector type inheriting AnyMatrixBase<> with support for basics operations. However, I think it is safer to wait a bit that I've done the change we discussed on a recent thread about some refactoring of the _expression_ template mechanism. Once this change done, it should be much simpler to add new special matrices. Hopefully.

A quick question: I did not read your source files carefully, but I thought that this SkylineMatrix stores the lower triangular using column-major vectors, and the upper part using row-major vectors. Right ? Then what is the purpose of the StorageOrder option ?

Then writting a triangular solver, a LU decomposition, a Cholesky decomposition, etc. would be almost trivial. Actually, if we add the concept of matrix partitionner as I described in another thread, one could write the exact same code for both a dense or a skyline matrix !

I'm also thinking whether it could make sense to make the "bandwidth" of the matrix a template parameter. The default would be "Dynamic". This would allow to optimize the code for matrices having a small fixed size bandwidth.

This could be a good idea. What kind of optimization can be done on such SkylineMatrices ?

Mainly loop unrolling of the most inner loops. To make that possible, the "VectorRange" discussed above should be able to deal with compile time ranges.

gael.

Thanks,

guillaume

gael.

On Mon, Nov 9, 2009 at 4:30 PM, guillaume saupin <guillaume.saupin@xxxxxx <mailto:guillaume.saupin@xxxxxx>> wrote:

Hi !

Here comes the cleaned code for the skyline matrix. If you could
integrate it inside Eigen, it would be perfect.

thanks,

guillaume

guillaume saupin wrote:

Gael Guennebaud wrote:

On Thu, Nov 5, 2009 at 2:40 PM, guillaume saupin
<guillaume.saupin@xxxxxx <mailto:guillaume.saupin@xxxxxx>
<mailto:guillaume.saupin@xxxxxx
<mailto:guillaume.saupin@xxxxxx>>> wrote:

Gael Guennebaud wrote:

Hi,

thanks for the patch !

I'd be ok to have it in Eigen. In a first step I would
probably put it in the unsupported/ set of modules
because it
sems quite preliminary for now.

No problem. This is perfect if we can access this
Matrix from
eigen. I've written some unit tests with boost. Where
should I put
them ?

However before we can apply the patch some cleaning is
required: at the very least the copyrights have to
be updated.
It would also be nice to remove all the commented
stuff.

Then we can discuss about the usefulness of having a
SkylineMatrixBase. Indeed, such a base class can be
needed for
two reasons:

The truth is that I have not really thought about that.
I just
copy/paste the SparseMatrix code ;)

1 - To implement the _expression_ template paradigm
for basics
operations (+, -, scaling, coefficients wise
operations like
abs, sin, cos etc.). However, here I'm not sure
that makes a
lot of sense. It seems to me that a such a
specialized matrix
is only usefull for solving problems, so supporting
such
operators is not really needed, right ?

Our main use for SkylineMatrix is for solving systems.
Nevertheless we will soon use at least a Skyline *
Dense product,
and probably a Dense * Skyline product. So template
expressions
may be required to do these products without
temporaries and with
a simple syntax. But it is maybe possible to do that
without a
Base class ?

Yes and no, because actually matrix products are special
in the sense that they must be directly evaluated. On the
other hand ET might still be useful to allow for more
complex products involving transpose, negate and scaling ops.

I will need this kind of complex products, especially those
involving transpose, in my ColumnMatrix. Do you think that it
must inherit MatrixBase to benefit from ET ?

The problem is that our current design does not allow to
reuse our Transpose/CwiseUnaryOp/etc. _expression_ classes
for something else than dense matrices. So I'm more and
more convinced that our ET mechanism should be updated to
be more flexible. But that's another topic !

2 - Another reason would be to have various
SkylineMatrix
storage variants which would be sufficiently
different to
require different classes, but close enough to
share the same
(solver) algorithms. Again, I'm not sure that's the
case here.

Right. I don't think too that we'll introduce another
storage.

Nevertheless, it's still fine to keep it for now,
and wait a
bit for more use cases to see what's the best strategy.

Thanks anyway for your advises. I'll clean the code and
send it
back to you.

ok, cool.

gael.

gael.

On Wed, Nov 4, 2009 at 3:24 PM, guillaume saupin
<guillaume.saupin@xxxxxx
<mailto:guillaume.saupin@xxxxxx>
<mailto:guillaume.saupin@xxxxxx
<mailto:guillaume.saupin@xxxxxx>>
<mailto:guillaume.saupin@xxxxxx
<mailto:guillaume.saupin@xxxxxx>
<mailto:guillaume.saupin@xxxxxx
<mailto:guillaume.saupin@xxxxxx>>>> wrote:

You'll find enclosed the code as a patch.

guillaume saupin wrote:

Hello !

I've finally (after some performances tests)
implemented a
basic skyline matrix using the following
storage :

- an array stores the diagonal elements
- an array stores the upper elements in a column
oriented way
- an array stores the lower elements in a
row oriented way

I've implemented a LU decomposition for this
format. I
use the
following algorithm (with minor modification
to handle the
sparseness of the matrix):

for each row
-- compute the lower part
for each col < row
LU(row, col) = (LU(row, col) - sum(k
= 0 ; k < col)
[LU(row, k) * LU(k, col)] ) / LU(row, row)
-- elements are
access contiguously / Can be easily vectorize
-- compute the upper part
for each rrow < row
LU(rrow, row) -= sum(k = 0 ; k < rrow)
[LU(rrow, k)
* LU(k,
row)] -- elements are access contiguously /
Can be easily
vectorize
-- compute the diag part
LU(row, row) -= sum(k = 0 ; k < row)
[LU(rrow, k) * LU(k,
row)] -- elements are access contiguously /
Can be easily
vectorize

This algorithm is well suited for our
storage format.
After a
few tests, it seems even more efficient to
store the
the upper
and lower elements in the same array, with
the elements
from
one row followed by the elements of the cols
of same index,
and so on ...

I've also implemented a product for the
Skyline * Dense
operation.

Do you think that it would be possible to
insert this code
into eigen ?

guillaume saupin wrote:

Gael Guennebaud a écrit :

On Tue, Oct 27, 2009 at 10:40 AM,
guillaume saupin
<guillaume.saupin@xxxxxx
<mailto:guillaume.saupin@xxxxxx>
<mailto:guillaume.saupin@xxxxxx
<mailto:guillaume.saupin@xxxxxx>>
<mailto:guillaume.saupin@xxxxxx
<mailto:guillaume.saupin@xxxxxx>
<mailto:guillaume.saupin@xxxxxx
<mailto:guillaume.saupin@xxxxxx>>>
<mailto:guillaume.saupin@xxxxxx
<mailto:guillaume.saupin@xxxxxx>
<mailto:guillaume.saupin@xxxxxx
<mailto:guillaume.saupin@xxxxxx>>
<mailto:guillaume.saupin@xxxxxx
<mailto:guillaume.saupin@xxxxxx>
<mailto:guillaume.saupin@xxxxxx
<mailto:guillaume.saupin@xxxxxx>>>>> wrote:

Gael Guennebaud a écrit :

this is not an easy question
because we are
still unsure how
to manage efficiently and
with minimal
code all
these kind of
"special" matrices.

Of course an easy solution
would be to
make it
inherit
SparseMatrixBase and let the
generic
algorithms
based on
sparse iterators do the job.
However, this
approach won't be
optimal because the iterators
won't take
advantage of the
specificity of a
SkylineMatrix which, e.g.,
should allow
vectorization.

Actually skyline matrices are
very
similar to
banded matrices,
and in particular it seems to
me that
they are
similar enough
to share the same algorithms.
So for
instance
these two kind
of matrices could inherit the
same
matrix base
and the
algorithms could be
implemented in a generic
way via the
concept of "range inner
vector" which is a
column (or a row)
vector with an associated
start index... We
also need
efficient basic operators
dealing with such
vector. E.g. the
addition of two such vector
can be
efficiently
done by first
copying the non overlapping
part (start and
end), and then sum
the overlapping part using
Eigen's core
module
feature.

The way we implemented skyline
matrix is not
really
similar to
what is usually done for banded
matrix, as
we store
the diagonal,
the upper, and the lower elements
in separate
arrays. This way, we
can achieve quite efficient LU
decomposition
(and
solving) due to
coherent memory accesses.

ok I see, then it is not really what
people call
skyline storage
(http://www.netlib.org/linalg/html_templates/node96.html),
but it is more like an extension of
the tridiagonal
storage.

I think that our storage is actually
quite similar
to the
one proposed there for non symetric skyline
(referred as
NSK format by Saad) matrix. As in these
format, we
store
the upper element in a column-oriented
way, and
the lower
element in a row-oriented way.
The only difference is that we store the
diagonal
elements
in a separate array instead of keeping
an index
array to
these elements. This allows contiguous
access to
diagonal
elements.

However, I don't see how your
storage can yield
to a
more efficient implementation of the LU
decomposition
than the standard skyline storage.
For instance
here
is a basic algorithm without
pivoting for the
standard
skyline storage:

for(int k = 0; k+1 < rows; ++k)
{
int rrows = rows-k-1;
int rsize = size-k-1;

// line 1
lu.col(k).end(rrows) /= lu.coeff(k,k);

// line 2
lu.corner(BottomRight,rrows,rsize).noalias() -=
lu.col(k).end(rrows) *
lu.row(k).end(rsize);
}

where lu is the working matrix,
lu.col(k) is
assumed
to return a "range vector" as I
described in my
previous email.

Here line 1 would be trivially
optimized (i.e.,
vectorized) since
lu.col(k).end(rrows) is just
a small
dense vector.

Line 2 is an outer product which
again is
trivially/automatically vectorized
sicne it is
impelmented as a sequence of:
"col_range_vector_i -=
scalar * col_range_vector".

Here the locality is pretty good
because the vector
"lu.col(k).end(rrows)" which is
reused multiple
times
is sequentially stored in memory.

But perhaps there exists a special
algorithm which
perfectly fit your storage ? Is it
possible to see
your code somewhere ? Finally, if it
is really more
efficient then that would make sense
to have it
in Eigen.

The benefit of using the NSK, i.e. of
storing the lower
matrix in a column-oriented way, and the
upper
matrix in a
row-oriented way is that in line 1 we
access contigous
elements (the column k of the lower
matrix), and in
line 2
we also access contiguous element (the
row elements
of the
upper matrix). I don't know if it's
efficient for
vectorization, but without, it is quite
efficient,
as the
locality is very good.

gael.

So I don't think that our
algorithms share
common
features with
band matrix algorithms. They seem
closer to what
can be done with
diagonal matrices.

Anyway, I've used the
SparseMatrix code as a
base
for my
SkylineMatrix code. Currently it only
supports very
basic
operations. I'll implement the LU
decomposition
soon. I will also
try to use vectorization.

guillaume

These are just some initial
thoughts and the
discussion is
very open!

gael

On Mon, Oct 19, 2009 at 12:55 PM,
guillaume saupin
<guillaume.saupin@xxxxxx
<mailto:guillaume.saupin@xxxxxx>
<mailto:guillaume.saupin@xxxxxx
<mailto:guillaume.saupin@xxxxxx>>
<mailto:guillaume.saupin@xxxxxx
<mailto:guillaume.saupin@xxxxxx>
<mailto:guillaume.saupin@xxxxxx
<mailto:guillaume.saupin@xxxxxx>>>
<mailto:guillaume.saupin@xxxxxx
<mailto:guillaume.saupin@xxxxxx>
<mailto:guillaume.saupin@xxxxxx
<mailto:guillaume.saupin@xxxxxx>>
<mailto:guillaume.saupin@xxxxxx
<mailto:guillaume.saupin@xxxxxx>
<mailto:guillaume.saupin@xxxxxx
<mailto:guillaume.saupin@xxxxxx>>>>
<mailto:guillaume.saupin@xxxxxx
<mailto:guillaume.saupin@xxxxxx>
<mailto:guillaume.saupin@xxxxxx
<mailto:guillaume.saupin@xxxxxx>>
<mailto:guillaume.saupin@xxxxxx
<mailto:guillaume.saupin@xxxxxx>
<mailto:guillaume.saupin@xxxxxx
<mailto:guillaume.saupin@xxxxxx>>>
<mailto:guillaume.saupin@xxxxxx
<mailto:guillaume.saupin@xxxxxx>
<mailto:guillaume.saupin@xxxxxx
<mailto:guillaume.saupin@xxxxxx>>
<mailto:guillaume.saupin@xxxxxx
<mailto:guillaume.saupin@xxxxxx>
<mailto:guillaume.saupin@xxxxxx
<mailto:guillaume.saupin@xxxxxx>>>>>> wrote:

Hello,

We are planning to use
your library
in our
projects, but we
need a
skyline matrix. Therefore
I'd like to
implement one for
eigen, but
I don't now where to start.

Is there a specific class
that can be
a good
starting point /
skeleton to start with ? The
SparseMatrixBase might be a
good choice.
Should this SkylineMatrix
inherit from
SparseMatrixBase, or
be a
separate class ?

Thanks,

guillaume