Re: friendly challenge 3

• To: mathgroup at smc.vnet.net
• Subject: [mg35099] Re: friendly challenge 3
• From: Samuel Kutter <sk256 at phy.cam.ac.uk>
• Date: Tue, 25 Jun 2002 03:39:39 -0400 (EDT)
• Organization: University of Cambridge, England
• References: <af6hk6\$lkc\$1@smc.vnet.net>
• Sender: owner-wri-mathgroup at wolfram.com

```Hello everybody,

vaguely remembered a test from undergraduate lectures... - I hope, it is
all right!

Sam

In[268]:=
sign2[M_?MatrixQ] :=
Count[Eigenvalues[M], _?(# > 0 &)] - Count[Eigenvalues[M], _?(# < 0 &)]

In[290]:=
<<LinearAlgebra`MatrixManipulation`

In[376]:=
SubMat[M_,p_]:=TakeMatrix[M,{1,1},{p,p}]

In[384]:=
Signew[M_]:=
Module[{count,sign},count=0; sign=1;
Do[If[sign Det[SubMat[M,i]]>0,count=count+1,count=count-1;
sign=- sign]; ,{i,1,Dimensions[M][[1]]}];count];

In[441]:=
m = Table[Random[Integer, {1, 9}], {30}, {30}]; n = m + Transpose[m];
sign2[n]//Timing
Signew[n]//Timing
%%[[1]]/%[[1]]

Out[441]=
{8.383 Second,2}    /*not EXACTLY a fast machine! ;-) */
Out[442]=
{0.66664 Second,2}
Out[443]=
12.575              /*factor 12 faster!*/

On Mon, 24 Jun 2002, Andrzej Kozlowski wrote:

> 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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