MATLAB Mathematics - SERC - Index of

Sparse Matrix Operations
selects P via numerical partial pivoting, but does not pivot to improve sparsity
in the LU factors. On the other hand, the four-output syntax
[L,U,P,Q]=lu(S)
selects P via threshold partial pivoting, and selects P and Q to improve sparsity
in the LU factors.
You can control pivoting in sparse matrices using
lu(S,thresh)
where thresh is a pivot threshold in [0,1]. Pivoting occurs when the diagonal
entry in a column has magnitude less than thresh times the magnitude of any
sub-diagonal entry in that column. thresh = 0 forces diagonal pivoting.
thresh = 1 is the default. (The default for thresh is 0.1 for the four-output
syntax).
When you call lu with three or less outputs, MATLAB automatically allocates
the memory necessary to hold the sparse L and U factors during the
factorization. Except for the four-output syntax, MATLAB does not use any
symbolic LU prefactorization to determine the memory requirements and set
up the data structures in advance.
Reordering and Factorization. If you obtain a good column permutation p that
reduces fill-in, perhaps from symrcm or colamd, then computing lu(S(:,p))
takes less time and storage than computing lu(S). Two permutations are the
symmetric reverse Cuthill-McKee ordering and the symmetric approximate
minimum degree ordering.
r = symrcm(B);
m = symamd(B);
The three spy plots produced by the lines below show the three adjacency
matrices of the Bucky Ball graph with these three different numberings. The
local, pentagon-based structure of the original numbering is not evident in the
other three.
spy(B)
spy(B(r,r))
spy(B(m,m))
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