Re: [eigen] notes on Householder and Givens |

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*To*: eigen@xxxxxxxxxxxxxxxxxxx
*Subject*: Re: [eigen] notes on Householder and Givens
*From*: Benoit Jacob <jacob.benoit.1@xxxxxxxxx>
*Date*: Wed, 4 Feb 2009 18:48:53 +0100
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2009/2/4 Keir Mierle <mierle@xxxxxxxxx>:
> On Wed, Feb 4, 2009 at 7:04 AM, Benoit Jacob <jacob.benoit.1@xxxxxxxxx>
>> Here is a good formula giving such a vector v:
>>
>> v = x + ( x_0 / abs(x_0) ) norm(x) e_0
>
> So if x is close to -alpha*e_0, won't catastrophic cancellation occur? Isn't
> this exactly the case that we'll be producing by doing the pivoting below?
No, look, there's no problem!
Let alpha be any nonzero complex number.
Take x = alpha * e_0
Now we have:
x_0 = alpha
abs(x_0) = abs(alpha)
norm(x) = abs(alpha)
so v = alpha * e_0 + alpha / abs(alpha) * abs(alpha) * e_0
simplifying v, we get:
v = 2 * alpha * e_0
No problem at all!
Cheers,
Benoit