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

```

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