Re: QR decomposition using Householder


In article <zYvAe.20987$bh1.7567@xxxxxxxxxxxxxxxxxxxx>,
"BemusedByQM" <groover892002@xxxxxxxxxxx> writes:
>
>"Peter Spellucci" <spellucci@xxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxx> wrote in
>message news:datip0$3k0$1@xxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxx
>>
>> In article <txLze.1660$WW5.712@xxxxxxxxxxxxxxxxxxxx>,
>> "Jeremy Watts" <jwatts1970@xxxxxxxxxxx> writes:
>> >Hello,
>> >
>> >I have been reading about the QR decomposition using Householder
>> >transformations, but it seems to use 'column pivotting' rather than the
>> >usual 'row pivotting' used with partial pivotting.
>> >
>> >Why is column pivotting used in this instance rather than row?
>> >
>> >
>> because working with an unitary matrix from the left on A keeps
>> column length fixed but not row length.
>> by having the column of max length determining the next pivot you get the
>> R-part
>> with monotonically nonincreasing diagonal. by the way often also row
>> interchanges
>> are use before the Householder reflector is applied to get the
>> (abs(a(i,i))+norm(a(i:n,i))) as large as possible which has an additional
>> stabilizing effect.
>
>so both column and row interchanges can be used in order to get a large as
>possible absolute value for the pivot element?

no. the pivot element does not change: it is the lenght of the remaining column
-norm(a(i:n,i))*sign(a(i,i)), but the normal vector u defining the reflecting
hyperplane changes: its first element
sign(a(i,i))*abs(a(i,i)+norm(a(i:n,i))) becomes as large as possible.
hth
peter




>> hth
>> peter
>
>
.

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