# Solving A{k} * x + b = 0 for large numbers of A{k} with same structure/filling.

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Dear all,

A{k} are square, non-symmetrical, large ( NxN with N in 1E5-5E5) sparse matrices (typical density <1E-4), column strictly diagonally dominant. They all have the same (block, see EDIT 1) structure/filling :

and b is Nx1 with nnz(b) typically in 1-1E4.

---- Digression start ----

I discussed the case A*x+B=0 already in another context, for B NxM with M large, where I can take advantage of a single LU factorization of a single A and solve in parallel:

[L, U, P, Q, R] = lu( A ) ;

parfor ...

X{k} = Q * (U \ (L \ (P * (R \ B_cell{k}))))

end

---- Digression end ----

My question in the current context is: is there a transformation/factorization that I could use, that exploits the fact that all A{k} have strictly the same filling/structure (i.e. non-zero elements are always at the same place), that I would apply e.g. to A{1} and then re-use to accelerate the solving of A{k}*x+b=0 for all other A{k} (e.g. by providing MLDIVIDE (or LINSOLVE with smaller, dense blocks) with matrices that shortcut part of the factorization)?

Also, could the block-structure and the partial filling of b be used in any manner?

EDIT 1, 2017/12/19 @ 17:32 UTC - On this line, if I perform a LU decomposition as shown in the digression above, for example, I get matrices P (row permutation) and Q (column reordering) that will (or should) be the same for all A{k}. Noting that:

[L, U, P, Q, R] = lu( A{k} ) ;

x{k} = Q * (U \ (L \ (P * (R \ b)))) ;

is always more efficient than:

x{k} = A{k} \ b ;

because we avoid the analysis of A, is there a way to exploit the fact that P and Q are known after the first iteration (i.e. P and Q computed for k=1 will remain the same for all ks)?

Cheers,

Cedric

EDIT 1, 2017/12/14 @ 19:39 UTC

Here is another sparse matrix where I delineated the block structure. It is associated with another simulation, hence the different size, but in this simulation all A{k} are 99149x99149 with the same structure:

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### Accepted Answer

Christine Tobler
on 19 Dec 2017

In Tim Davis' C++ library UMFPACK, there are two phases to computing the decomposition of a matrix: The first one only uses the sparsity structure, while the second one also uses the numbers. Unfortunately, such separate phases are not accessible through MATLAB directly.

Another approach, if the numbers in A{k} do not change too strongly, would be to use the LU factorization of A{1} as a preconditioner for an iterative solver (for example gmres, pcg).

##### 2 Comments

Steven Lord
on 19 Dec 2017

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