Description
The output matrices are mutable exactly when the input matrix is, but the matrix
A is not modified
If
Q is the
m by
m permutation matrix such that
Q_(P_i,i) = 1, and all other entries are zero, then
A = QLU.
There are several restrictions. The first is that there are only a limited number of rings for which this function is implemented. Second, if
A is a mutable matrix defined over
RR or
CC, then
A must be densely encoded. This restriction is not present if
A is
a matrix.
i1 : kk = ZZ/101
o1 = kk
o1 : QuotientRing
|
i2 : A = matrix"1,2,3,4;1,3,6,10;19,7,11,13" ** kk
o2 = | 1 2 3 4 |
| 1 3 6 10 |
| 19 7 11 13 |
3 4
o2 : Matrix kk <-- kk
|
i3 : (P,L,U) = LUdecomposition A
o3 = ({0, 1, 2}, | 1 0 0 |, | 1 2 3 4 |)
| 1 1 0 | | 0 1 3 6 |
| 19 -31 1 | | 0 0 47 22 |
o3 : Sequence
|
i4 : Q = id_(kk^3) _ P
o4 = | 1 0 0 |
| 0 1 0 |
| 0 0 1 |
3 3
o4 : Matrix kk <-- kk
|
i5 : Q * L * U == matrix A
o5 = true
|
For matrices over
RR or
CC, this function calls the lapack routines, which are restricted to 53 bits of precision.
i6 : A = matrix"1,2,3,4,5,6;1,3,6,12,13,16;19,7,11,47,48,21" ** RR
o6 = | 1 2 3 4 5 6 |
| 1 3 6 12 13 16 |
| 19 7 11 47 48 21 |
3 6
o6 : Matrix RR <-- RR
53 53
|
i7 : (P,L,U) = LUdecomposition A
o7 = ({2, 1, 0}, | 1 0 0 |, | 19 7 11 47 48
| .0526316 1 0 | | 0 2.63158 5.42105 9.52632 10.4737
| .0526316 .62 1 | | 0 0 -.94 -4.38 -4.02
------------------------------------------------------------------------
21 |)
14.8947 |
-4.34 |
o7 : Sequence
|
i8 : Q = id_ (RR^3) _ P
o8 = | 0 0 1 |
| 0 1 0 |
| 1 0 0 |
3 3
o8 : Matrix RR <-- RR
53 53
|
i9 : Q * L * U - A
o9 = | 0 -2.22045e-16 0 0 0 0 |
| 0 0 0 0 0 0 |
| 0 0 0 0 0 0 |
3 6
o9 : Matrix RR <-- RR
53 53
|
i10 : clean(1e-15,oo)
o10 = | 0 0 0 0 0 0 |
| 0 0 0 0 0 0 |
| 0 0 0 0 0 0 |
3 6
o10 : Matrix RR <-- RR
53 53
|
Mutable matrices can sometimes be useful for speed, and/or space. If
A is a mutable matrix, it must be densely encoded (which is the default).
i11 : A = mutableMatrix(CC,5,10, Dense=>true)
o11 = | 0 0 0 0 0 0 0 0 0 0 |
| 0 0 0 0 0 0 0 0 0 0 |
| 0 0 0 0 0 0 0 0 0 0 |
| 0 0 0 0 0 0 0 0 0 0 |
| 0 0 0 0 0 0 0 0 0 0 |
o11 : MutableMatrix
|
i12 : printingPrecision = 2
o12 = 2
|
i13 : setRandomSeed 0
o13 = 0
|
i14 : fillMatrix A
o14 = | .89+.67ii .91+.31ii .35+.56ii .19+.4ii .28+.61ii .034+.51ii .44+.64ii .47+.4ii .48+.82ii .5+.15ii |
| .29+.63ii .074+.81ii .25+.15ii .62+.015ii .97+.68ii .15+.66ii .19+.52ii .16+.71ii .98+.06ii .47+.77ii |
| .026+.71ii .36+.71ii .83+.54ii .22+.39ii .91+.89ii .17+.63ii .99+.57ii .91+.57ii .065+.28ii .31+.54ii |
| .89+.23ii .13+.25ii .87+.42ii .56+.87ii .17+.97ii .35+.38ii .18+.37ii .31+.73ii .98+.21ii .21+.28ii |
| .46+.78ii .74+.11ii .61+.85ii .7+.68ii .065+.88ii .24+.12ii .34+.062ii .56+.63ii .59+.83ii .096+.61ii |
o14 : MutableMatrix
|
i15 : (P,L,U) = LUdecomposition A;
|
i16 : Q = id_(CC^5) _ P
o16 = | 1 0 0 0 0 |
| 0 0 0 0 1 |
| 0 0 1 0 0 |
| 0 1 0 0 0 |
| 0 0 0 1 0 |
5 5
o16 : Matrix CC <-- CC
53 53
|
i17 : matrix Q * matrix L * matrix U - matrix A
o17 = | 0 0 0 0
| -5.6e-17 0 -5.6e-17 1.1e-16
| -5.6e-17-1.1e-16ii 0 0 0
| -1.1e-16 0 0 0
| -1.1e-16-1.1e-16ii -5.6e-17ii 1.1e-16+1.1e-16ii 0
-----------------------------------------------------------------------
0 0 0 0 0 0 |
1.1e-16ii -2.8e-17 5.6e-17 0 -1.1e-16 0 |
-1.1e-16ii 0 0 0 2.8e-17 0 |
0 0 0 0 0 0 |
-2.8e-17-1.1e-16ii 5.6e-17ii 0 0 0 0 |
5 10
o17 : Matrix CC <-- CC
53 53
|
i18 : clean(1e-15,oo)
o18 = | 0 0 0 0 0 0 0 0 0 0 |
| 0 0 0 0 0 0 0 0 0 0 |
| 0 0 0 0 0 0 0 0 0 0 |
| 0 0 0 0 0 0 0 0 0 0 |
| 0 0 0 0 0 0 0 0 0 0 |
5 10
o18 : Matrix CC <-- CC
53 53
|