Add Hadamard bound include and update determinant function#331
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LinBox computes the determinant of an integer matrix
Aby Chineseremaindering of modular determinants. In
linbox/solutions/det.h, theinteger-tag dispatch is:
SOLUTION_CRA_DETresolves to:lif_cra_det(fromlinbox/algorithms/hybrid-det.h) when LinBox isbuilt with
__LINBOX_HAVE_NTL, orcra_detotherwise.The non-NTL
cra_detpath — which is the one used in the SageMathconda environment, and is also the default in many distro packages —
uses a heuristic CRT driver:
ChineseRemainder< CRABuilderEarlySingle< Field > > cra(LINBOX_DEFAULT_EARLY_TERMINATION_THRESHOLD); cra(dd, iteration, genprime);CRABuilderEarlySinglestops as soon as the running CRT reconstructionhas not changed across
LINBOX_DEFAULT_EARLY_TERMINATION_THRESHOLDconsecutive primes. This is a probabilistic termination criterion:
if the residues for several primes happen to agree on a wrong lift
before the combined modulus exceeds
2|det(A)|,cra_detreturns thewrong determinant.
The
lif_cra_det(NTL) path is not a safe fallback either: in ourtesting it returned
0for nonsingular dense integer matrices in thisbuild configuration.