LU may work on singular matrices even when it should not. The same is
true of the getrf family from LAPACK. This is to be expected, since a
small perturbation of a singular matrix is nonsingular. A perturbed
nonsingular matrix remains nonsingular.
In the code below, LUcheck is from the LU decomposition essay. The matrix
A is singular. The matrix B is the transpose of A, and so also singular.
The way both routines do the calculation, A is figured out to be
nonsingular while B is correctly identified as singular.
require '~addons/math/lapack/lapack.ijs'
require '~addons/math/lapack/getrf.ijs'
getrf_z_=:getrf_jlapack_
]A=:>:i. 3 3
1 2 3
4 5 6
7 8 9
B=:|: A
LUcheck A
+-----+-----+-----------------+
|0 1 2|1 0 0|1 2 3|
| |4 1 0|0 _3 _6|
| |7 2 1|0 0 _3.55271e_15|
+-----+-----+-----------------+
getrf A
+--------------+-----------------------+-----+
| 1 0 0|7 8 9|3 3 3|
|0.142857 1 0|0 0.857143 1.71429| |
|0.571429 0.5 1|0 0 _1.58619e_16| |
+--------------+-----------------------+-----+
LUcheck B
|index error: LU
| p;(=1);p {"1 A[p=.C.(n-1);~.0,(0~:,A)i.1
getrf B
info result: 3
|attention interrupt: error
| error''
Best wishes,
John
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