Hello,
Currently, determinant(A) calculates the determinant of 'A' by factorizing
A=LU and computing prod(diag(U)) [or the logarithm of the absolute value].
The factorization is done by LAPACK routine DGETRF, which gives a status
code INFO, documented [1] as follows:
*> INFO is INTEGER
*> = 0: successful exit
*> < 0: if INFO = -i, the i-th argument had an illegal value
*> > 0: if INFO = i, U(i,i) is exactly zero. The factorization
*>has been completed, but the factor U is exactly
*>singular, and division by zero will occur if it is used
*>to solve a system of equations.
Accordingly, when INF0>0, determinant(A) behaves as det(A)=0, _not_ computing
prod(diag(U)). The problem here is that DGETRF can _also_ give positive
INFO for matrices containing NaN, which may very well not be singular for some
finite value of NaN.
I claim that, when INFO>0, determinant(A) should _not_ behave as det(A)=0
unconditionally, but rather sometimes (depending on some test) give NaN.
Here is one case where 0 is really "wrong":
> (A <- diag(c(NaN, 1)))
[,1] [,2]
[1,] NaN0
[2,]01
> det(A)
[1] 0
R isn't consistent, either:
> (B <- diag(c(1, NaN)))
[,1] [,2]
[1,]10
[2,]0 NaN
> det(B)
[1] NaN
Here, DGETRF _does_ succeed, because it does not "see" the trailing NaN in 'B'.
So: Should R change to better handle the INFO>0 case? If so, how?
Ideally (I think), the proposed change would give NaN for 'A' and 'B'
above and 0 for 'C' and 'D' below (both of which really _are_ singular):
> (C <- matrix(c(NaN, NaN, 0, 0), 2L, 2L))
[,1] [,2]
[1,] NaN0
[2,] NaN0
> det(C)
[1] NaN
> (D <- t(C))
[,1] [,2]
[1,] NaN NaN
[2,]00
> det(D)
[1] 0
Furthermore, the proposed change should _not_ decrease the performance
of determinant(A) for nonsingular 'A' ...
For those looking, the relevant C-level function is det_ge_real(),
defined in R-devel/src/modules/lapack/Lapack.c (at line 1260 in r83320).
Mikael
[1] https://github.com/Reference-LAPACK/lapack/blob/master/SRC/dgetrf.f
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