Sage Reference Manual: Matrices and Spaces of Matrices - Mirrors

Sage Reference Manual: Matrices and Spaces of Matrices, Release 6.1.1
OUTPUT:
Suppose that A is an m × n matrix, then an LU decomposition is a lower-triangular m × m matrix L with
every diagonal element equal to 1, and an upper-triangular m × n matrix, U such that the product LU,
after a permutation of the rows, is then equal to A. For the ‘plu’ format the permutation is returned as an
m × m permutation matrix P such that
A = P LU
It is more common to place the permutation matrix just to the left of A. If you desire this version, then use
the inverse of P which is computed most efficiently as its transpose.
If the ‘partial’ pivoting strategy is used, then the non-diagonal entries of L will be less than or equal to 1 in
absolute value. The ‘nonzero’ pivot strategy may be faster, but the growth of data structures for elements
of the decomposition might counteract the advantage.
By necessity, returned matrices have a base ring equal to the fraction field of the base ring of the original
matrix.
In the ‘compact’ format, the first returned value is a tuple that is a permutation of the rows of LU that
yields A. See the doctest for how you might employ this permutation. Then the matrices L and U are
merged into one matrix – remove the diagonal of ones in L and the remaining nonzero entries can replace
the entries of U beneath the diagonal.
The results are cached, only in the compact format, separately for each pivot strategy called. Repeated
requests for the ‘plu’ format will require just a small amount of overhead in each call to bust out the
compact format to the three matrices. Since only the compact format is cached, the components of the
compact format are immutable, while the components of the ‘plu’ format are regenerated, and hence are
mutable.
Notice that while U is similar to row-echelon form and the rows of U span the row space of A, the rows of
U are not generally linearly independent. Nor are the pivot columns (or rank) immediately obvious. However
for rings without specialized echelon form routines, this method is about twice as fast as the generic
echelon form routine since it only acts “below the diagonal”, as would be predicted from a theoretical
analysis of the algorithms.
Note: This is an exact computation, so limited to exact rings. If you need numerical results, convert the
base ring to the field of real double numbers, RDF or the field of complex double numbers, CDF, which
will use a faster routine that is careful about numerical subtleties.
ALGORITHM:
“Gaussian Elimination with Partial Pivoting,” Algorithm 21.1 of [TREFETHEN-BAU].
EXAMPLES:
Notice the difference in the L matrix as a result of different pivoting strategies. With partial pivoting, every
entry of L has absolute value 1 or less.
sage: A = matrix(QQ, [[1, -1, 0, 2, 4, 7, -1],
... [2, -1, 0, 6, 4, 8, -2],
... [2, 0, 1, 4, 2, 6, 0],
... [1, 0, -1, 8, -1, -1, -3],
... [1, 1, 2, -2, -1, 1, 3]])
sage: P, L, U = A.LU(pivot=’partial’)
sage: P
[0 0 0 0 1]
[1 0 0 0 0]
[0 0 0 1 0]
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