friendly challenge 3

*To*: mathgroup at smc.vnet.net*Subject*: [mg35076] friendly challenge 3*From*: Andrzej Kozlowski <andrzej at platon.c.u-tokyo.ac.jp>*Date*: Mon, 24 Jun 2002 03:21:01 -0400 (EDT)*Sender*: owner-wri-mathgroup at wolfram.com

While the season for "friendly challenges" lasts, here is something that has just come up in my own work. Let's define the signature of a symmetric matrix as the number of positive eigenvalues - the number of negative ones. I need an efficient function to compute this. There is the obvious and rather pedestrian one: sign2[M_?MatrixQ] := Count[Eigenvalues[M], _?(# > 0 &)] - Count[Eigenvalues[M], _?(# < 0 &)] For example, let's construct a symmetric matrix of random integers (all matrices I am considering have integer entries): m = Table[Random[Integer, {1, 9}], {30}, {30}]; n = m + Transpose[m]; the above sign2 gives: In[4]:= sign2[n]//Timing Out[4]= {3.5 Second,0} My best function, sign1 gives (on 400 mghz PowerBOok G4) In[5]:= sign1[n]//Timing Out[5]= {1.44 Second,0} Nearly two and a half times as fast. Can anyone do better? Andrzej