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
>
>

```

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