Re: Orthogonazlie with Method->"Householder"

• To: mathgroup at smc.vnet.net
• Subject: [mg97791] Re: Orthogonazlie with Method->"Householder"
• From: ADL <alberto.dilullo at tiscali.it>
• Date: Sun, 22 Mar 2009 05:44:58 -0500 (EST)
• References: <gpvh97\$grj\$1@smc.vnet.net>

```Maris, I did the following computations also using the Rationalize

In[1]:= ClearAll[M, oM, MR, oMR];

In[2]:= M = {...}; (* deleted from the post *)

In[3]:= MR = Rationalize[M, 10^(-16)];
oMR = Orthogonalize[MR];

In[5]:= methods = {"GramSchmidt", "ModifiedGramSchmidt",
"Householder", "Reorthogonalization"};

In[6]:= (oM[#] = Orthogonalize[M, Method -> #]) & /@ methods;

In[7]:= MatrixRank[oMR]

Out[7]= 6

In[8]:= MatrixRank[oM[#]] & /@ methods

Out[8]= {6,6,12,6}

In[9]:= Chop@Max@Abs@(oM[#] - oMR) & /@ methods

Out[9]= {0,0,1.23961912632296,0}

In[10]:= \$Version

Out[10]= 7.0 for Microsoft Windows (32-bit) (February 18, 2009)

>From this, I would say that the issue is not accuracy but the fact
that "Householder" is not working properly. If it is a feature, it is
a discount feature: you get 12 at the price of 6.

What do you think about starting the famous "bugs and caveats" site
with this post?

Maris Ozols ha scritto:

> I have a large numeric matrix M (see below), whose rank is 6. Clearly,
> the rank of Orthogonalize[M] must be the same (in particular, it
> should not increase). However, if I execute the following code on my
> matrix M:
>
> MatrixRank@M
> MatrixRank@Orthogonalize[M]
> MatrixRank@Orthogonalize[M,Method->"Householder"]
>
> I get 6, 6, and 12. So it appears to me that Householder's method has
> doubled the rank of M. I am using Mathematica 7.0 in Linux.
>
> Is this a bug or a feature?
>
> ~Maris Ozols~

```

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