Sage Reference Manual: Matrices and Spaces of Matrices - Mirrors

Sage Reference Manual: Matrices and Spaces of Matrices, Release 6.1.1
Traceback (most recent call last):
...
TypeError: base ring of the matrix needs a field of fractions, not Ring of integers modulo 6
sage: R. = PolynomialRing(QQ, ’y’)
sage: B = matrix(R, [[y+1, y^2+y], [y^2, y^3]])
sage: P, L, U = B.LU(pivot=’partial’)
Traceback (most recent call last):
...
TypeError: cannot take absolute value of matrix entries, try ’pivot=nonzero’
sage: P, L, U = B.LU(pivot=’nonzero’)
sage: P
[1 0]
[0 1]
sage: L
[ 1 0]
[y^2/(y + 1) 1]
sage: U
[ y + 1 y^2 + y]
[ 0 0]
sage: L.base_ring()
Fraction Field of Univariate Polynomial Ring in y over Rational Field
sage: B == P*L*U
True
sage: F. = FiniteField(5^2)
sage: C = matrix(F, [[a + 3, 4*a + 4, 2, 4*a + 2],
... [3, 2*a + 4, 2*a + 4, 2*a + 1],
... [3*a + 1, a + 3, 2*a + 4, 4*a + 3],
... [a, 3, 3*a + 1, a]])
sage: P, L, U = C.LU(pivot=’nonzero’)
sage: P
[1 0 0 0]
[0 1 0 0]
[0 0 1 0]
[0 0 0 1]
sage: L
[ 1 0 0 0]
[3*a + 3 1 0 0]
[ 2*a 4*a + 2 1 0]
[2*a + 3 2 2*a + 4 1]
sage: U
[ a + 3 4*a + 4 2 4*a + 2]
[ 0 a + 1 a + 3 2*a + 4]
[ 0 0 1 4*a + 2]
[ 0 0 0 0]
sage: L.base_ring()
Finite Field in a of size 5^2
sage: C == P*L*U
True
With no pivoting strategy given (i.e. pivot=None) the routine will try to use partial pivoting, but then
fall back to the nonzero strategy. For the nonsingular matrix below, we see evidence of pivoting when
viewed over the rationals, and no pivoting over the integers mod 29.
sage: entries = [3, 20, 11, 7, 16, 28, 5, 15, 21, 23, 22, 18, 8, 23, 15, 2]
sage: A = matrix(Integers(29), 4, 4, entries)
sage: perm, _ = A.LU(format=’compact’); perm
124 Chapter 7. Base class for matrices, part 2