Well, I have no doubt that in most cases QR-Cholesky works fine and is
pretty efficient, however Cholesky involves squaring thus reducing
numerical accuracy. So I'm tempted to favor a full orthogonalization
method. This is also the approach followed by Lapack (dgelsy).
Nevertheless your approach has the advantage of simplicity and being
already implemented! So if you're ok I propose in a first step to add
it in unsupported/, I'm sure it might be useful to many users.
Gael.
On Tue, Jun 12, 2012 at 8:49 AM, Helmut Jarausch
<jarausch@xxxxxxxxxxxxxxxxxxx> wrote:
> On 06/11/2012 06:31:22 PM, Gael Guennebaud wrote:
>>
>> On Mon, Jun 11, 2012 at 5:56 PM, Helmut Jarausch
>> <jarausch@xxxxxxxxxxxxxxxxxxx> wrote:
>> > I've attached the corrected version of QR_Cholesky.
>> > Is it impudent to suggest adding this to Eigen?
>> >
>> > I think it's the standard way of solving the minimum length least
>> > squares
>> > problem.
>>
>> I think it would be better to perform a full orthogonalization of
the
>> form:
>>
>> A * P = Q * U * Z
>>
>> where U of the form:
>>
>> | U11 0 |
>> | 0 0 |
>>
>> This can be done by applying householder reflexions on the non zero
>> upper trapezoidal part of the matrix R.
>>
>
> I have been using that myself - more than 20 years ago.
> Recent references, like Peter Deuflhard's book on
> Numerical Methods of Nonlinear Problems, Springer 2004,
> recommendsthe use of QR-Cholesky (section 4.4.1).
> Since Peter Deuflhard and his group have a lot of experience in
> real life problems I am sure this suggestion has proven itself in
practise.
>
> But if someone has an implementation of full orthogonalization we
could do
> some benchmarking.
>
> Helmut.
>
>
>