# [SciPy-dev] Best interface for computing the logarithm of the determinant?

Nathaniel Smith njs@pobox....
Tue Feb 16 14:49:12 CST 2010

```So when you have a matrix whose determinant you want, it's often wise
to compute the logarithm of the determinant instead of the determinant
itself, because determinants involve lots and lots of multiplications
and the result might otherwise underflow/overflow. Therefore, in
scikits.sparse, I'd like to provide an API for doing this (and this is
well-supported by the underlying libraries).

But the problem is that for a general matrix, the determinant may be
zero or negative. Obviously we can deal with this, but what's the best
API? I'd like to use one consistently across the different
factorizations in scikits.sparse, and perhaps eventually in numpy as
well.

Some options:

1) Split off the sign into a separate return value ('sign' may be 1, -1, 0):
sign, value = logdet(A)
actual_determinant = sign * exp(value)

2) Allow complex/infinite return values, even when A is a real matrix:
logdet(eye(3)) == pi*1j
logdet(zeros((3, 3))) == -Inf

3) "Scientific notation" (This is what UMFPACK's API does): return a
mantissa and base-10 exponent:
mantissa, exponent = logdet(A)
actual_determinant = mantissa * 10 ** exponent

4) Have separate functions for computing the sign, and the log of the
absolute value (This is what GSL does, though it seems pointlessly
inefficient):
sign = sgndet(A)
value = logdet(A)
actual_determinant = sign * exp(value)

These are all kind of ugly looking, unfortunately, but that seems
unavoidable, unless someone has a clever idea.

Any preferences?

-- Nathaniel
```