I just pushed the resolution for this thread. I still need to tweak it to support SIMPLE since you need<div>different solvers for the parts of the Schur factorization, but I will do that soon.</div><div><br></div><div> Matt<br>
<br><div class="gmail_quote">On Tue, Jul 10, 2012 at 9:03 PM, Barry Smith <span dir="ltr"><<a href="mailto:bsmith@mcs.anl.gov" target="_blank">bsmith@mcs.anl.gov</a>></span> wrote:<br><blockquote class="gmail_quote" style="margin:0 0 0 .8ex;border-left:1px #ccc solid;padding-left:1ex">
<div class="im"><br>
On Jul 10, 2012, at 3:41 AM, Dmitry Karpeev wrote:<br>
<br>
> Evidently, I am confused: the current code (Matt's <a href="http://petsc.cs.iit.edu/petsc/petscdev/annotate/0d4ccb990bb8/src/ksp/pc/impls/fieldsplit/fieldsplit.c#l592" target="_blank">http://petsc.cs.iit.edu/petsc/petscdev/annotate/0d4ccb990bb8/src/ksp/pc/impls/fieldsplit/fieldsplit.c#l592</a>)<br>
> as well as what I had (<a href="http://petsc.cs.iit.edu/petsc/petsc-dev/annotate/a36eb42b26ee/src/ksp/pc/impls/fieldsplit/fieldsplit.c#l555" target="_blank">http://petsc.cs.iit.edu/petsc/petsc-dev/annotate/a36eb42b26ee/src/ksp/pc/impls/fieldsplit/fieldsplit.c#l555</a>)<br>
> both set the inner KSP(A00) prefix to that of the outer KSP(A00). This is because schur's inheriting the prefix from the 0 split is<br>
> equivalent to setting it with PetscSNPrintf(schurprefix, sizeof schurprefix, "%sfieldsplit_%s_", ((PetscObject)pc)->prefix ? ((PetscObject)pc)->prefix : "", jac->head->splitname);CHKERRQ(ierr);<br>
> That works for me. Matt, I thought that didn't work for SIMPLE?<br>
<br>
</div> 1) there should be a possibility of a different prefix so that the two solve approximations can be different.<br>
<br>
2) the current code seems stupid, since both solvers are the same (same options) but it creates 2 different KSPs and needs to set them both up even though they are generally the same.<br>
<br>
I think we need to change MatCreateSchurComplement() so that it takes an optional KSP (and hence does not need to create its own) or change the PCSetUp_FieldSplit() to not create a KSP(A00) and use the one from MatSchurComplementGetKSP() when they should be the same. Then provide a PCFieldSplitSchurSetUseDifferentA00KSP() option to trigger having two different KSPs, each with its own prefix.<br>
<span class="HOEnZb"><font color="#888888"><br>
<br>
Barry<br>
</font></span><div class="HOEnZb"><div class="h5"><br>
<br>
<br>
><br>
> Dmitry.<br>
><br>
> On Mon, Jul 9, 2012 at 1:56 PM, Barry Smith <<a href="mailto:bsmith@mcs.anl.gov">bsmith@mcs.anl.gov</a>> wrote:<br>
><br>
> On Jul 9, 2012, at 1:17 AM, Dmitry Karpeev wrote:<br>
><br>
> ><br>
> ><br>
> > On Sun, Jul 8, 2012 at 7:46 PM, Barry Smith <<a href="mailto:bsmith@mcs.anl.gov">bsmith@mcs.anl.gov</a>> wrote:<br>
> ><br>
> > On Jul 8, 2012, at 3:36 PM, Dmitry Karpeev wrote:<br>
> ><br>
> > > As currently designed, there are always two KSPs for A00 - one is created as part of MatSchurComplement and the other one sits in the FS split.<br>
> ><br>
> > Really? Isn't that often non-efficient? For any expensive to build and memory intensive preconditioner (like ILU(k), LU, AMR) it is crazy to create them both if they are the same.<br>
> > Actually, until recently these two KSPs had different options prefixes: the outer had -fieldsplit_0 and the inner -fieldsplit_1,<br>
> > so if they were set up from the command line, they could (and did for me) end up being different, so this reuse of the KSP would be a new thing.<br>
><br>
><br>
> Huuuh? -fieldplit_1 is most definitely NOT the prefix for the KSP inside the application of the Schur complement. It is for the KSP that is used to solve the Schur complement system.<br>
><br>
> We need to start all over again at page 1 since one of us doesn't know what he is talking about.<br>
><br>
> With block Jacobi and block Gauss-Seidel (called multiplicative in fieldsplit) there is a KSP for first block and a KSP for the second block; their prefixes are -fieldsplit_0 and fieldsplit_1.<br>
><br>
> With the Schur complement preconditioner there are potentially three solves; the 0,0 block, the 0,0 block INSIDE the application of the Schur complement and the one associated with the Schur complement system.<br>
><br>
> I see someone has fucked with my perfect code.<br>
><br>
> ierr = KSPGetOptionsPrefix(jac->head->next->ksp, &Dprefix); CHKERRQ(ierr);<br>
> ierr = KSPSetOptionsPrefix(jac->kspschur, Dprefix); CHKERRQ(ierr);<br>
> /* really want setfromoptions called in PCSetFromOptions_FieldSplit(), but it is not ready yet */<br>
> /* need to call this every time, since the jac->kspschur is freshly created, otherwise its options never get set */<br>
> ierr = KSPSetFromOptions(jac->kspschur);CHKERRQ(ierr);<br>
><br>
> Not sure what this is suppose to be doing and what is in jac->head->next->ksp ? If it has a level_0 that is strange.<br>
><br>
> BTW:<br>
><br>
> static PetscErrorCode PCApply_FieldSplit_Schur(PC pc,Vec x,Vec y)<br>
> {<br>
><br>
> has<br>
><br>
> ierr = MatSchurComplementGetKSP(jac->schur,&ksp);CHKERRQ(ierr);<br>
><br>
> which means currently the KSP used for the 0,0 block is ALWAYS exactly the same as the one used inside the Schur complement?<br>
><br>
> Or has Matt changed this?<br>
><br>
><br>
> Barry<br>
><br>
><br>
><br>
><br>
> > Reusing the factors from ILU(k) or LU would be a good thing, since the same solve is done as many as 3 times sometimes. The same would be true for an ASM-type preconditioner with (I)LU on the blocks -- even more so, since submatrices are getting pulled out.<br>
> ><br>
> ><br>
> > > Are you proposing that the option mechanism we have been discussing trigger the creation of the extra KSP,<br>
> ><br>
> > Yes (maybe not just the option mechanism, but shouldn't there be one that is shared by default and only two if they are different?<br>
> > Here's what I think we could do: if there is a -fieldsplit_0_schur_ prefix in the database (this requires a new PetscOptionsHasNamePrefix()), then set this prefix on the inner KSP(A00) inside S and prefix -fieldsplit_0_ on the outer KSP(A00); otherwise pull out the inner KSP(A00) from S and reuse it as the outer KSP(A00) with prefix -fieldsplit_0_.<br>
> ><br>
> ><br>
> > Another scenario I can imagine (I'm not advocating this, though) is to have both prefixes -fieldsplit_0_ and -fieldsplit_0_schur refer<br>
> > to the inner and outer KSPs, if they are shared (by default), and to separate KSPs, if the sharing is turned off with a separate option.<br>
> > I'm not entirely sure how to implement this (it would require an aliasing mechanism in the PetscOptions database), nor whether this is actually reasonable.<br>
> ><br>
> > Dmitry.<br>
> ><br>
> > Barry<br>
> ><br>
> ><br>
> > > or do you have a different set of options in mind?<br>
> ><br>
> ><br>
> > ><br>
> > > Dmitry.<br>
> > ><br>
> > ><br>
> > ><br>
> > > On Sunday, July 8, 2012, Barry Smith <<a href="mailto:bsmith@mcs.anl.gov">bsmith@mcs.anl.gov</a>> wrote:<br>
> > > ><br>
> > > > On Jul 8, 2012, at 2:03 AM, Dmitry Karpeev wrote:<br>
> > > ><br>
> > > >><br>
> > > >><br>
> > > >> On Sat, Jul 7, 2012 at 4:01 PM, Barry Smith <<a href="mailto:bsmith@mcs.anl.gov">bsmith@mcs.anl.gov</a>> wrote:<br>
> > > >><br>
> > > >> On Jul 7, 2012, at 3:23 AM, Dmitry Karpeev wrote:<br>
> > > >><br>
> > > >> ><br>
> > > >> ><br>
> > > >> > On Fri, Jul 6, 2012 at 9:56 PM, Barry Smith <<a href="mailto:bsmith@mcs.anl.gov">bsmith@mcs.anl.gov</a>> wrote:<br>
> > > >> ><br>
> > > >> > On Jul 6, 2012, at 9:35 PM, Matthew Knepley wrote:<br>
> > > >> ><br>
> > > >> > > On Fri, Jul 6, 2012 at 8:18 PM, Barry Smith <<a href="mailto:bsmith@mcs.anl.gov">bsmith@mcs.anl.gov</a>> wrote:<br>
> > > >> > ><br>
> > > >> > > I don't understand this thread but I see nothing wrong with options like<br>
> > > >> > ><br>
> > > >> > > -fieldsplit_0_fieldsplit_0_fieldsplit_0....<br>
> > > >> > ><br>
> > > >> > > when using three nested levels of fieldsplit in the same way that three levels of block Jacobi (or ASM) gives -sub_sub_sub....<br>
> > > >> > ><br>
> > > >> > > The recursive nature of the prefixes should be completely natural and not require any special code ....<br>
> > > >> > ><br>
> > > >> > > Providing a prefix in the command line for other options seems terrible to me. Using the word inner also seems terrible; when you have _sub_sub that is clearly inner<br>
> > > >> > ><br>
> > > >> > > Could someone explain to me what prefixes are being generated that are not the normal recursive process and why?<br>
> > > >> > ><br>
> > > >> > > Sure, that is how it works in general. The point here is the distinction between A^{-1} in the (0,0) block and<br>
> > > >> > > A^{-1} embedded in S, which I will call A^{-1}_S. Right now we have<br>
> > > >> > ><br>
> > > >> > > a) A^{-1}_S and S have the same prefix<br>
> > > >> > ><br>
> > > >> > > which Dmitry does not want (perfectly reasonable).<br>
> > > >> > Before we had<br>
> > > >> > ><br>
> > > >> > > b) A^{-1} and A^{-1}_S had the same prefix<br>
> > > >> > ><br>
> > > >> > > which I do not want since it makes things like SIMPLE hard. I wanted<br>
> > > >> > ><br>
> > > >> > > c) A^{-1}_S has prefix <prefix of S>_sub<br>
> > > >> > ><br>
> > > >> > > but Dmitry said this was a hassle for normal setups and suggested that<br>
> > > >> > > we have some option that allows A^{-1} and A^{-1}_S to have different<br>
> > > >> > > prefixes.<br>
> > > >> > ><br>
> > > >> ><br>
> > > >> > Thanks, this makes things much clearer.<br>
> > > >> ><br>
> > > >> > 1) I don't like your prefix _sub (what the heck does _sub mean in this case) but I agree with you that having a different prefix there is good<br>
> > > >> ><br>
> > > >> > 2) I don't like Dmitry's solution. It introduces an entirely new paradigm that we don't have anywhere else in PETSc.<br>
> > > >> ><br>
> > > >> > My thoughts ----------<br>
> > > >> ><br>
> > > >> > For PCMG we have prefixes for the levels mg_levels_%d_ if the user uses -mg_levels_ksp_type it applied to ALL the levels but if the user does -mg_levels_3_ksp_type it is applied only to the 3rd level. This is done by having the special treatment of _%d_ integers n the prefix that the options database can handle. It would be nice if we could use this same basic paradigm to handle this new case that supports both what Dmitry and you want but not in a hacky ugly special case way. For example (not so good) just use<br>
> > > >> ><br>
> > > >> > -fieldsplit_0_ksp_type sets the same for both<br>
> > > >> > -fieldsplit_0_0_ksp_type for the 0,0 block -fieldsplit_0_1_ksp_type for the solve inside the application of S.<br>
> > > >> ><br>
> > > >> > this is not good because it uses 0 and 1 for the two solves (and 0 and 1 have no particular meaning here) but the advantage is that it reuses current paradigms.<br>
> > > >> ><br>
> > > >> > Going further we could have<br>
> > > >> ><br>
> > > >> > -fieldsplit_0_ksp_type sets the same for both<br>
> > > >> > -fieldsplit_0_outter_ksp_type for the 0,0 block and -fieldsplit_0_inner_ksp_type for the one inside the S<br>
> > > >> ><br>
> > > >> > to implement this we would need to add support for %s in options prefixes. Maybe _<%s>_ so the options processing accepts a match with the string inside the <> or if that is not in the options database it accepts an option without the entire _<%s>_ This would require some small additions to PetscOptionsFind_private() like the<br>
> > > >> ><br>
> > > >> > if (!*flg) {<br>
> > > >> > PetscInt j,cnt = 0,locs[16],loce[16];<br>
> > > >> > size_t n;<br>
> > > >> > ierr = PetscStrlen(tmp,&n);CHKERRQ(ierr);<br>
> > > >> > /* determine the location and number of all _%d_ in the key */<br>
> > > >> > for (i=0; i< (PetscInt)n; i++) {<br>
> > > >> > if (tmp[i] == '_') {<br>
> > > >> > for (j=i+1; j< (PetscInt)n; j++) {<br>
> > > >> > if (tmp[j] >= '0' && tmp[j] <= '9') continue;<br>
> > > >> > if (tmp[j] == '_' && j > i+1) { /* found a number */<br>
> > > >> > locs[cnt] = i+1;<br>
> > > >> > loce[cnt++] = j+1;<br>
> > > >> > }<br>
> > > >> > break;<br>
> > > >> > }<br>
> > > >> > }<br>
> > > >> > }<br>
> > > >> ><br>
> > > >> ><br>
> > > >> > What does everyone think?<br>
> > > >> > I'm fine with this, except for the small detail that "outer" and "inner" infixes might be obscure to the user.<br>
> > > >> > I would advocate -fieldsplit_0_ksp_type by itself setting up both the inner and outer A^{-1}, and -fieldsplit_0_schur_ksp_type<br>
> > > >> > overriding the inner solver settings -- I think "Schur" is more descriptive then "inner" and "outer".<br>
> > > >><br>
> > > >> So you want -fieldsplit_0_ksp_type to apply to both or to the 0,0 block and -fieldsplit_0_schur_ksp_type to refer to just the one inside the Schur but no special one just for the 0,0 block, that will always be determined by -fieldsplit_0_ksp_type ?<br>
> > > >><br>
> > > >> How do you suggest we implement this in a clean way?<br>
> > > >><br>
> > > >> Right, the subtlety is that we want to apply the _schur prefix to the inner object only if it exists in the options database.<br>
> > > >> I think this can be solved fairly simply by providing PetscOptionsHasNamePrefix(const char pre[], const char name[], const char partial[], PetscBool *match) that returns true when partial is a prefix in pre##name. Maybe there is a better name for this routine to avoid confusion of "Prefix" with pre.<br>
> > > >><br>
> > > >> Is that clean enough? Any other ideas?<br>
> > > >><br>
> > > > Are there always TWO ksps (one for the 0,0 solve block and one for the solve inside the Schur complement? There really shouldn't always be two since usually they are the same, right? So really we need an option to indicate one wants different solvers in the the two locations and when the one in Schur is created it gets the special prefix. Much like up and down smoothing in PCMG is usually the same KSP but need not be.<br>
> > > ><br>
> > > > Barry<br>
> > > ><br>
> > > >> Note another issue: the inner and outer A00 KSP share the DM object. That is probably okay, though,<br>
> > > >> because the DM is supposed to describe the (sub)problem, which is the same for both solvers.<br>
> > > >><br>
> > > >> Dmitry.<br>
> > > >><br>
> > > >><br>
> > > >> Barry<br>
> > > >><br>
> > > >> ><br>
> > > >> > Dmitry.<br>
> > > >> ><br>
> > > >> ><br>
> > > >> > Barry<br>
> > > >> ><br>
> > > >> ><br>
> > > >> ><br>
> > > >> ><br>
> > > >> ><br>
> > > >> > > Matt<br>
> > > >> > ><br>
> > > >> > ><br>
> > > >> > > Barry<br>
> > > >> > ><br>
> > > >> > > On Jul 6, 2012, at 8:39 AM, Matthew Knepley wrote:<br>
> > > >> > ><br>
> > > >> > > > On Fri, Jul 6, 2012 at 7:28 AM, Dmitry Karpeev <<a href="mailto:karpeev@mcs.anl.gov">karpeev@mcs.anl.gov</a>> wrote:<br>
> > > >> > > ><br>
> > > >> > > > On Fri, Jul 6, 2012 at 8:17 AM, Matthew Knepley <<a href="mailto:knepley@gmail.com">knepley@gmail.com</a>> wrote:<br>
> > > >> > > > On Fri, Jul 6, 2012 at 5:06 AM, Dmitry Karpeev <<a href="mailto:karpeev@mcs.anl.gov">karpeev@mcs.anl.gov</a>> wrote:<br>
> > > >> > > > Here's the line in question (also see the immediately preceding code):<br>
> > > >> > > > <a href="http://petsc.cs.iit.edu/petsc/petsc-dev/rev/0d4ccb990bb8#l1.127" target="_blank">http://petsc.cs.iit.edu/petsc/petsc-dev/rev/0d4ccb990bb8#l1.127</a><br>
> > > >> > > ><br>
> > > >> > > > As long as we are fixing this, I would rather not repeat the prefix, since we will likely want to<br>
> > > >> > > > configure this differently than the block 0 solve. Is any thing wrong with<br>
> > > >> > > ><br>
> > > >> > > > schurprefix+"_sub"<br>
> > > >> > > > If the inner and outer KSP prefixes are different, it will force one to repeat all of the configuration options for the inner and outer A00 solvers, even when it is desirable to keep them identical.<br>
> > > >> > > > This becomes tedious, if the A00 solvers configuration is involved (e.g., a nested fieldsplit with separate options for the splits etc.).<br>
> > > >> > > > I would advocate making the inner solver use the same prefix as the outer solver by default, and allowing the user to specify<br>
> > > >> > > > a separate prefix for the inner solver, if it is to be configured differently. For example:<br>
> > > >> > > > -fieldsplit_0_schur_prefix fieldsplit_0_inner -fieldsplit_0_ksp_type gmres -fieldsplit_0_inner_ksp_type preonly etc.<br>
> > > >> > > ><br>
> > > >> > > > As long as there is a way to do it.<br>
> > > >> > > ><br>
> > > >> > > > Matt<br>
> > > >> > > ><br>
> > > >> > > > Dmitry.<br>
> > > >> > > ><br>
> > > >> > > ><br>
> > > >> > > ><br>
> > > >> > > > Matt<br>
> > > >> > > ><br>
> > > >> > > > Dmitry.<br>
> > > >> > > ><br>
> > > >> > > ><br>
> > > >> > > > ---------- Forwarded message ----------<br>
> > > >> > > > From: Dmitry Karpeev <<a href="mailto:karpeev@mcs.anl.gov">karpeev@mcs.anl.gov</a>><br>
> > > >> > > > Date: Fri, Jul 6, 2012 at 6:04 AM<br>
> > > >> > > > Subject: Re: [petsc-dev] Problematic Merge of FieldSplit<br>
> > > >> > > > To: For users of the development version of PETSc <<a href="mailto:petsc-dev@mcs.anl.gov">petsc-dev@mcs.anl.gov</a>><br>
> > > >> > > ><br>
> > > >> > > ><br>
> > > >> > > > I have the following problem with the prefix choice for the MatSchurComplement KSP introduced in this changeset (<a href="http://petsc.cs.iit.edu/petsc/petsc-dev/rev/0d4ccb990bb8" target="_blank">http://petsc.cs.iit.edu/petsc/petsc-dev/rev/0d4ccb990bb8</a>).<br>
> > > >> > > > I'm talking about the "inner" KSP for A00, effecting inv(A00) in the definition S = A11 - A10 inv(A00) A01.<br>
> > > >> > > > We also have the "outer" inv(A00) KSP, which gets prefix "0". I recently set the "inner" inv(A00) KSP<br>
> > > >> > > > prefix to "0", simply by inheriting it from the "outer" solver. Now, it is completely reasonable<br>
> > > >> > > > to expect the inner and outer A00 KSPs to have different prefixes so that they can be configured differently.<br>
> > > >> > > > In fact, there was a recent petsc-users request related to this (<a href="http://lists.mcs.anl.gov/pipermail/petsc-users/2012-June/014005.html" target="_blank">http://lists.mcs.anl.gov/pipermail/petsc-users/2012-June/014005.html</a>).<br>
> > > >> > > > However, currently the inner A00 KSP inherits the prefix from the A11 KSP corresponding to the "1" field. With this prefix choice<br>
> > > >> > > > I end up configuring inv(A00) and inv(S) identically, which isn't what I want at all.<br>
> > > >> > > > I'm not sure what the right approach is, but the current one doesn't work for me.<br>
> > > >> > > ><br>
> > > >> > > > Note also that if A00 is treated with a recursive split, there may be numerous options for the A00 KSP.<br>
> > > >> > > > Do we want to repeat them for the inner and outer KSPs, if we want to configure them identically?<br>
> > > >> > > > It's automatic, if the two A00 KSPs share a prefix. Again, this takes away some flexibility, so maybe it's not the best solution,<br>
> > > >> > > > but I think retaining a simple option for using identical configurations is also highly desirable.<br>
> > > >> > > ><br>
> > > >> > > > Any ideas on how to handle this?<br>
> > > >> > > > Dmitry.<br>
> > > >> > > ><br>
> > > >> > > > On Tue, Jun 26, 2012 at 6:13 AM, Matthew Knepley <<a href="mailto:knepley@gmail.com">knepley@gmail.com</a>> wrote:<br>
> > > >> > > > It turns out that 'hg rollback' during an 'hg rebase' does not do what I thought it did. I think<br>
> > > >> > > > everything is cleaned up with this push, but if you made FS changes in the past month, please<br>
> > > >> > > > check that it is doing what you want with prefixes, etc.<br>
> > > >> > > ><br>
> > > >> > > > Now, nested fieldsplits from the command line work, ala<br>
> > > >> > > ><br>
> > > >> > > > -ksp_type fgmres<br>
> > > >> > > > -pc_type fieldsplit -pc_fieldsplit_type additive<br>
> > > >> > > > -pc_fieldsplit_0_fields 0,1<br>
> > > >> > > > -fieldsplit_0_pc_type fieldsplit<br>
> > > >> > > > -fieldsplit_0_pc_fieldsplit_type schur -fieldsplit_0_pc_fieldsplitschur_factorization_type full<br>
> > > >> > > > -fieldsplit_0_fieldsplit_velocity_ksp_type preonly<br>
> > > >> > > > -fieldsplit_0_fieldsplit_velocity_pc_type lu<br>
> > > >> > > > -fieldsplit_0_fieldsplit_pressure_ksp_rtol 1e-10<br>
> > > >> > > > -fieldsplit_0_fieldsplit_pressure_pc_type jacobi<br>
> > > >> > > > -pc_fieldsplit_1_fields 2<br>
> > > >> > > > -fieldsplit_temperature_ksp_type preonly<br>
> > > >> > > > -fieldsplit_temperature_pc_type lu<br>
> > > >> > > ><br>
> > > >> > > > A split with only one field gets the field name, and otherwise a split number.<br>
> > > >> > > ><br>
> > > >> > > > Matt<br>
> > > >> > > ><br>
> > > >> > > > --<br>
> > > >> > > > What most experimenters take for granted before they begin their experiments is infinitely more interesting than any results to which their experiments lead.<br>
> > > >> > > > -- Norbert Wiener<br>
> > > >> > > ><br>
> > > >> > > ><br>
> > > >> > > ><br>
> > > >> > > ><br>
> > > >> > > ><br>
> > > >> > > > --<br>
> > > >> > > > What most experimenters take for granted before they begin their experiments is infinitely more interesting than any results to which their experiments lead.<br>
> > > >> > > > -- Norbert Wiener<br>
> > > >> > > ><br>
> > > >> > > ><br>
> > > >> > > ><br>
> > > >> > > ><br>
> > > >> > > > --<br>
> > > >> > > > What most experimenters take for granted before they begin their experiments is infinitely more interesting than any results to which their experiments lead.<br>
> > > >> > > > -- Norbert Wiener<br>
> > > >> > ><br>
> > > >> > ><br>
> > > >> > ><br>
> > > >> > ><br>
> > > >> > > --<br>
> > > >> > > What most experimenters take for granted before they begin their experiments is infinitely more interesting than any results to which their experiments lead.<br>
> > > >> > > -- Norbert Wiener<br>
> > > >> ><br>
> > > >> ><br>
> > > >><br>
> > > >><br>
> > > ><br>
> > > ><br>
> ><br>
> ><br>
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</div></div></blockquote></div><br><br clear="all"><div><br></div>-- <br>What most experimenters take for granted before they begin their experiments is infinitely more interesting than any results to which their experiments lead.<br>
-- Norbert Wiener<br>
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