On 8/18/12, W. Edwin Clark <wclark@mail.usf.edu> wrote:
On Sat, Aug 18, 2012 at 12:43 PM, Warren Smith <warren.wds@gmail.com> wrote:
What is the maximum |determinant| of an NxN circulant matrix whose defining first row consists entirely of +1's and -1's? (Related question: each entry lies between -1 and +1 inclusive?)
By changing slightly my (brute force) Maple program for http://oeis.org/A144926 for your question I get the following:
--I confirm your findings. My own program is in C and is rather stupid in the sense it uses a general purpose determinant routine, not specialized for circulants. Anyway, it finds N det example matrix 1 1.00 + 2 0.00 +- 3 4.00 -++ 4 -16.00 +--- 5 48.00 -++++ 6 -128.00 +----- 7 512.00 --+-+++ 8 2304.00 ++-+---- 9 6912.00 -+-++-+++ 10 22528.00 ++-+---+-- 11 273408.00 -+-+--+++++ 12 2097152.00 ++++--+-+++- 13 14929920.00 --+-+++++-+++ 14 -50331648.00 ++--+-+------- 15 390905856.00 +---++-+-+++++- 16 -1644167168.00 +---+-+++--+---- 17 12279939072.00 ++-+++--++-+---++ 18 69660573696.00 +++++-++--+-+-++-- 19 865782202368.00 +++--+++-+----++-++ 20 5566277615616.00 +-+--++-++++++--+++- 21 41248865910784.02 ---+-+++-++-++++-++++ 22 -215055782117376.19 +-+----+--+--++---+++- 23 2385859554836482.50 +-+---+-++-++++-+++--++ 24 25783171861708820.00 ++++-++-+-+---+--+++++-- 25 146322302697472192.00 +-+----++-++++--+++-++-++ where I have stopped it here since clearly the limits of double-precision IEEE reals have been reached (the true answer is always an integer), and probably exceeded. If anybody writes an actually-good program they should be able to reach N in the 40-50 range... and compute the exact-integer answers.