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Date: Wed, 25 Jan 2023 01:50:56 -0800 (PST)
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Subject: Re: Real_Arrays on heap with overloaded operators and clean syntax
From: Jim Paloander
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Dear Gautier,
thank you for the draft. Indeed something like that would be great possibly=
without polynomials or rational numbers etc. You just need doubles and/or =
floats and complex numbers. Sparse matrices are another animal, and once ad=
opted they are another dimension of complexity. This is because apart from =
the triplet format which is easily expandable, one may also need to convert=
the triplet to CSR and there multiple entries with same row and column ind=
ex need first to be summed so we have a single entry (i, j, value) for each=
(i, j) before converting to CSR. So I mean, one would like to build a libr=
ary then and I am not so sure how easy would be to hardcode this library so=
that operations on matrices/vectors are as efficient as in Fortran. For De=
nse matrices it may still be possible to support mathematical expressions w=
ithout introducing temporary objects. But if you also bring sparse into the=
game then?
(alpha,beta,gamma scalars, while x,y,z,w vectors, A, B_sparse matrices)
does this have a chance to be executed without temporaries?
y :=3D alpha*x + beta*A*x + gamma*B_sparse*(z-w)
this has a chance to be executed without temporaries Fortran already does i=
t.
y :=3D alpha*x + beta*A*x
I would create a SparseMatrix library and break the statement above in 2
y1 :=3D B_sparse*(z-w)
y :=3D alpha*x + beta*A*x + gamma*y1;
I could survive with that. The problem dear Gautier is the temporary object=
s in such libraries. Very correctly J.P.Rosen suggested BLAS. The problem i=
s that for simple vector operations (BLAS1) there is no speedup at all and =
the operations of BLAS1 cannot be executed for the whole expression, just i=
n an `axpy (a*x+y)` manner one by one. This would require transversal of t=
he arrays many times in a long expression involving vectors as (y :=3D x1 +=
x2*alpha + beta*(x3-gamma*x4) + alpha*beta*(x1-x2)). For BLAS2 operations =
(matrix times vector) you may see a performance gain using BLAS but for lar=
ge matrices, very large. Where you definitely see such a performance gain i=
s for BLAS3 and there we definitely need something like that.=20
This discussion started about techniques to avoid generation of temporaries=
. Even when you manipulate complex numbers you introduce temporaries. I nee=
d to come back to you guys with some number to understand the problem bette=
r. I will be back with these numbers. But until then, please let me know, d=
o you feel that there is a need of a performance uplifting and mathematics =
enabling ADA library which is not Real_Arrays or Containers, but a real Mat=
h library allowing Real and Complex math manipulation and integrating possi=
bly some Lapack routines as Real_Arrays do at the moment?