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Re: simpler way to get a particular banded matrix?

  • To: mathgroup at smc.vnet.net
  • Subject: [mg98205] Re: simpler way to get a particular banded matrix?
  • From: Nicholas <physnick at gmail.com>
  • Date: Thu, 2 Apr 2009 04:48:55 -0500 (EST)
  • References: <gqq3um$7ni$1@smc.vnet.net> <gqsn7a$3rn$1@smc.vnet.net>

On Mar 31, 5:19 am, rip pelletier <bitbuc... at comcast.net> wrote:
> In article <gqq3um$7n... at smc.vnet.net>,
>  Bill Rowe <readn... at sbcglobal.net> wrote:
>
>
>
> > On 3/29/09 at 2:48 AM, bitbuc... at comcast.net (rip pelletier) wrote:
>
> > >The following command constructs a matrix (typically denoted M0, it
> > >arises in filter banks in general, in Burrus, Gopinath, & Guo "intro
> > >to wavelets and wavelet transforms: a primer" in particular).
>
> > >I'm sure there's a way to do this without typing out all 5 bands.
> > >Here's what i did:
>
> > >SparseArray[Band[{1,1}]->H,{6,6}]+
> > >SparseArray[Band[{2,3}]->H,{6,6}]+
> > >SparseArray[Band[{3,5}]->H,{6,6}]+
> > >SparseArray[Band[{-2,-3},{1,1},{-1,-1}]->rH,{6,6}]+
> > >SparseArray[Band[{-3,-5},{1,1},{-1,-1}]->rH,{6,6}]
>
> > >where
>
> > >H={h0,h1,h2,h3,h4,h5}; rH=Reverse[H];
>
> > >I get what I want:
>
> > >h0  0  0  0  0  0
> > >h2 h1 h0 0  0  0
> > >h4 h3 h2 h1 h0 0
> > >0  h5 h4 h3 h2 h1
> > >0  0  0  h5 h4 h3
> > >0  0  0  0  0  h5
>
> > >Any easier ways to get this?
>
> > You can include a list of bands rather than summing separate
> > arrays with a single band specified. That is:
>
> > In[17]:= s =
> >    SparseArray[Band[{1, 1}] -> H, {6, 6}] +
> >     SparseArray[Band[{2, 3}] -> H, {6, 6}] +
> >     SparseArray[Band[{3, 5}] -> H, {6, 6}] +
> >     SparseArray[Band[{-2, -3}, {1, 1}, {-1, -1}] -> rH, {6, 6}] +
> >     SparseArray[Band[{-3, -5}, {1, 1}, {-1, -1}] -> rH, {6, 6}];
>
> > In[18]:= s ==
> >   SparseArray[{Band[{1, 1}] -> H, Band[{2, 3}] -> H[[;; 4]],
> >     Band[{3, 5}] -> H[[;; 2]], Band[{2, 1}] -> H[[3 ;;]],
> >     Band[{3, 1}] -> H[[5 ;;]]}]
>
> > Out[18]= True
>
> > Note, I've also omitted specifying the array dimensions since
> > this is fixed by the main diagonal and I have specified the off
> > main diagonal bands to use just the part of H needed which
> > eliminates the warning messages your code generates. However, I
> > still specified each of the bands. So, this may not satisfy your
> > criteria for easier.
>
> Thank you. It goes a long way toward letting me generalize this matrix,
> and you showed me how to do some things I didn't know. I appreciate
> this. Conceptually, I grok this matrix from its bands, rather than from
> its rows or columns.
>
> Vale,
>    Rip
>
> --
> NB eddress is r i p 1 AT c o m c a s t DOT n e t

What about:

M0 = Table[
  Total@Table[
    H[[k]] KroneckerDelta[2 i - 1, j + k - 1], {k, 1, 6}], {i, 1,
   6}, {j, 1, 6}]

This is a little different from the others, but it might be more
straight forward.  I don't know how it scales though.

Nick.


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