Re: friendly challenge 3

*To*: mathgroup at smc.vnet.net*Subject*: [mg35115] Re: friendly challenge 3*From*: "Carl K. Woll" <carlw at u.washington.edu>*Date*: Tue, 25 Jun 2002 03:41:51 -0400 (EDT)*Organization*: University of Washington*References*: <af6hk6$lkc$1@smc.vnet.net>*Sender*: owner-wri-mathgroup at wolfram.com

Andrzej, Is there some reason you don't want to compute the signature by first applying N and then computing the eigenvalues? Computing the eigenvalues of a matrix with integer entries ought to be much slower than computing the eigenvalues of a matrix with real entries. Carl Woll Physics Dept U of Washington "Andrzej Kozlowski" <andrzej at platon.c.u-tokyo.ac.jp> wrote in message news:af6hk6$lkc$1 at smc.vnet.net... > 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 > >