mm-124
Any idea what is the analytical solution forF(x) -
0.8F(0.2+1.1x) = 1/(x+3.0)F(innity)=0I
know,F(x)=Sum[(0.8^i)/(x_i+3.0)], x_(i+1)=0.2+1.1x_i,
i=0..innitybut, does it look familiar for anybody?More
general, what would be a good reference for discrete
equations likeF(x) - H(x)F(a+bx) = K(x)
Any idea what is
the analytical solution for>F(x) - 0.8F(0.2+1.1x) =
1/(x+3.0)>F(innity)=0The xed point of the mapping x -> 1/5
+ 11/10 x being -2, letsrst do a change of variables y = x
+ 2, G(y) = F(x):G(y) - 4/5 G(11/10 y) = 1/(y+1)The general
solution for y > 0 will allow G to be arbitrary on, say, the
interval [1, 11/10); given G(y_0) = g_0 with y_0 in this
interval,and y_n = (11/10)^n y_0, G(y_n) = g_n where g_{n+1}
= 5/4 (g_n - 1/(y_n+1))and thus g_n = (5/4)^n g_0 -
sum_{j=0}^{n-1} (5/4)^(n-j)/(y_j+1)But arranging for g_n -> 0
will be tricky. Its convenient to write z_n = 1/(y_n+1), so
z_n -> 0 as n -> innity.Even though we have an explicit
formula for z_n, well write it ascoming from a recursion. So
we have the discrete dynamical systemg_{n+1} = 5/4 (g_n -
z_n)z_{n+1} = 10/11 z_n/(1 - z_n)which has a hyperbolic xed
point at the origin. What we want is the stable manifold, a
curve through (0,0) with theproperty that if (g_0, z_0) is on
this curve, (g_n, z_n) -> (0,0) as n -> innity.Suppose [ and
I think there are theorems to back this up ]the stable
manifold is given by a function g = h(z) that is analyticat
z=0, so we have a convergent Taylor series g =
sum_{j=1}^innity h_j z^jNow we want h to satisfy the
functional equation h(10/11 z/(1-z)) = 5/4 (h(z) - z)
Expanding everything in powers of z, from the coefcient of
each power z^n we will get an equation to determine h_n in
terms of the previous ones: according to Maple, 10 /10 100
2 -- h[1] z + |-- h[1] +  h[2]| z + 11 11 121 / /10 200
1000  3 |-- h[1] +  h[2] + - h[3]| z + 11 121 1331
/ /10 10000 300 3000  4 5 |-- h[1] + -- h[4] +  h[2]
+ - h[3]| z + O(z ) = 11 14641 121 1331 / 2 3 4 (5/4
h[1] - 5/4) z + 5/4 h[2] z + 5/4 h[3] z + 5/4 h[4] z 5 + O(z
)and thus 968 713416 2739975304 h[1] = 11/3, h[2] = , h[3]
= , h[4] = - 123 21771 16064579I believe (from
looking at it numerically with 19 terms instead of 20)that
this series has radius of convergence approximately 1/10. So
thesecoefcients wouldnt be of much use near y = 1 (or z =
1/2), but theywould provide a good numerical approximation
near, say, y = 100 (or z = 1/101). I doubt that the function
h(z) can be written in closed form.>I
know,>F(x)=Sum[(0.8^i)/(x_i+3.0)], x_(i+1)=0.2+1.1x_i,
i=0..innityThats incorrect.
> > Any idea what is the
analytical solution for> > F(x) - 0.8F(0.2+1.1x) = 1/(x+3.0)>
F(innity)=0> > I know,> > F(x)=Sum[(0.8^i)/(x_i+3.0)],
x_(i+1)=0.2+1.1x_i, i=0..innity> > but, does it look
familiar for anybody?> > More general, what would be a good
reference for discrete equations like> > F(x) - H(x)F(a+bx) =
K(x)> Author: Goldberg, Samuel.Title: Introduction to
difference equations, withillustrative examples from
economics, psychology, and sociology.Publication info: New
York, Science Editions, 1961 [c1958]-- Julian V.
^^^^^^^^http://galileo.phys.virginia.edu/~jvn/ Science knows
only one commandment: contribute to science. -- Bertolt
Brecht, Galileo.
>Any idea what is the analytical
solution for>>F(x) - 0.8F(0.2+1.1x) =
1/(x+3.0)>F(innity)=0>> The xed point of the mapping x ->
1/5 + 11/10 x being -2, lets> rst do a change of variables
y = x + 2, G(y) = F(x):>> G(y) - 4/5 G(11/10 y) = 1/(y+1)>>
The general solution for y > 0 will allow G to be arbitrary
on, say, the> interval [1, 11/10); given G(y_0) = g_0 with
y_0 in this interval,> and y_n = (11/10)^n y_0, G(y_n) = g_n
where> g_{n+1} = 5/4 (g_n - 1/(y_n+1))> and thus> g_n =
(5/4)^n g_0 - sum_{j=0}^{n-1} (5/4)^(n-j)/(y_j+1)>> But
arranging for g_n -> 0 will be tricky.>> Its convenient to
write z_n = 1/(y_n+1), so z_n -> 0 as n -> innity.> Even
though we have an explicit formula for z_n, well write it
as> coming from a recursion. So we have the discrete
dynamical system>> g_{n+1} = 5/4 (g_n - z_n)> z_{n+1} = 10/11
z_n/(1 - z_n)>> which has a hyperbolic xed point at the
origin.> What we want is the stable manifold, a curve through
(0,0) with the> property that if (g_0, z_0) is on this curve,
(g_n, z_n) -> (0,0)> as n -> innity.>> Suppose [ and I think
there are theorems to back this up ]> the stable manifold is
given by a function g = h(z) that is analytic> at z=0, so we
have a convergent Taylor series>> g = sum_{j=1}^innity h_j
z^j>> Now we want h to satisfy the functional equation>
h(10/11 z/(1-z)) = 5/4 (h(z) - z)h(10z/(11-z)) = 5/4 (h(z) -
z)( after z = 1/(y+1), h(z) = G(y) )Boundary condition now is
h(0)=0, so you proposed Taylor approximation nearzero:>
Expanding everything in powers of z, from the coefcient of
each power> z^n we will get an equation to determine h_n in
terms of the previous> ones: according to Maple,> 10 /10
100  2> -- h[1] z + |-- h[1] +  h[2]| z +> 11 11 121
/>> /10 200 1000  3> |-- h[1] +  h[2] + - h[3]| z +>
11 121 1331 />> /10 10000 300 3000  4 5> |-- h[1] + --
h[4] +  h[2] + - h[3]| z + O(z ) 11 14641 121 1331
/>> 2 3 4> (5/4 h[1] - 5/4) z + 5/4 h[2] z + 5/4 h[3] z + 5/4
h[4] z>> 5> + O(z )>> and thus> 968 713416 2739975304> h[1]
= 11/3, h[2] = , h[3] = , h[4] = -> 123
21771 16064579>> I believe (from looking at it numerically
with 19 terms instead of 20)> that this series has radius of
convergence approximately 1/10. So these> coefcients
wouldnt be of much use near y = 1 (or z = 1/2), but they>
would provide a good numerical approximation near, say, y =
100> (or z = 1/101). I doubt that the function h(z) can be
written in> closed form.>>I
know,>>F(x)=Sum[(0.8^i)/(x_i+3.0)], x_(i+1)=0.2+1.1x_i,
i=0..innity>> Thats incorrect.Other than a typo (x_0
instead of x) I thinkF(x_0)=Sum[(0.8^i)/(x_i+3.0)],
x_(i+1)=0.2+1.1x_i, i=0..innityis a correct solution. It can
be directly checked by plug-in into theinitial FE.In
Mathematica I calculate it ask[x_] = 1/(x + 3.0);g[x_] = 0.2
+ 1.1x;f[x_] := NSum[(0.8^i) k[Nest[g, x, i]], {i, 0,
[Innity]}];The only problem that it takes signicant
amount of time to calculate F(x)by that recursion. I was
looking for a closed form solution (if such exist)to increase
the computation speed.Vadym
Any idea what is the
analytical solution for >F(x) - 0.8F(0.2+1.1x) =
1/(x+3.0)>F(innity)=0> The xed point of the mapping x ->
1/5 + 11/10 x being -2, lets> rst do a change of variables
y = x + 2, G(y) = F(x):> G(y) - 4/5 G(11/10 y) = 1/(y+1)> The
general solution for y > 0 will allow G to be arbitrary on,
say, the> interval [1, 11/10); given G(y_0) = g_0 with y_0
in this interval,> and y_n = (11/10)^n y_0, G(y_n) = g_n
where> g_{n+1} = 5/4 (g_n - 1/(y_n+1))> and thus> g_n =
(5/4)^n g_0 - sum_{j=0}^{n-1} (5/4)^(n-j)/(y_j+1)>> But
arranging for g_n -> 0 will be tricky.>> Its convenient to
write z_n = 1/(y_n+1), so z_n -> 0 as n -> innity.> Even
though we have an explicit formula for z_n, well write it
as> coming from a recursion. So we have the discrete
dynamical system>> g_{n+1} = 5/4 (g_n - z_n)> z_{n+1} = 10/11
z_n/(1 - z_n)>> which has a hyperbolic xed point at the
origin.> What we want is the stable manifold, a curve through
(0,0) with the> property that if (g_0, z_0) is on this curve,
(g_n, z_n) -> (0,0)> as n -> innity.>> Suppose [ and I think
there are theorems to back this up ]> the stable manifold is
given by a function g = h(z) that is analytic> at z=0, so we
have a convergent Taylor series>> g = sum_{j=1}^innity h_j
z^j>> Now we want h to satisfy the functional equation>
h(10/11 z/(1-z)) = 5/4 (h(z) - z)> h(10z/(11-z)) = 5/4 (h(z)
- z)> ( after z = 1/(y+1), h(z) = G(y) )Oops, yes youre
right: it should be z_n = 10 z/(11 - z). Sorry about that
slip-up. So now the Taylor series coefcients start 88 5896
2058584 h[1] = 11/3, h[2] = , h[3] = --, h[4] =
--, 123 21771 16064579 90906635864 h[5] = -
1302050192529and moreover it looks like the radius of
convergence is now close to 1.For example, to calculate F(0)
= G(2) = h(1/3), the power series (for 19 terms) gives
1.3136893661154333 approximately. Since this is only
approximate and the recursion is unstable, we would expect
that starting with this value the terms will eventually
diverge, but (using 17 digit arithmetic) I found the g_n
decreasing in absolute value until n = 91.So this is quite a
good approximation.>I know,> >F(x)=Sum[(0.8^i)/(x_i+3.0)],
x_(i+1)=0.2+1.1x_i, i=0..innity>> Thats incorrect.> > Other
than a typo (x_0 instead of x) I think> >
F(x_0)=Sum[(0.8^i)/(x_i+3.0)], x_(i+1)=0.2+1.1x_i,
i=0..innity> > is a correct solution. It can be directly
checked by plug-in into the> initial FE.Oops again. I take
that back. Yes, youre right.I still dont think theres a
closed-form solution, but there is a transformation of the
series that may be useful. If a = .8 and b = 1.1, we can
write your series as G(y_0) = sum_{n=0}^innity a^n/(b^n y_0
+ 1)Expand the summand in a series in powers of 1/(b^n y_0),
and you getG(y_0) = sum_{n=0}^innity sum_{m=1}^innity a^n
(-1)^(m+1)/(b^n y_0)^m = sum_{m=1}^innity sum_{n=0}^innity
a^n (-1)^(m+1)/(b^(mn) y_0^m) = sum_{m=1}^innity
(-1)^(m+1)/(1-a/b^m) y_0^m) which will converge more rapidly
than the original series if y_0 is large.Robert Israel
israel@math.ubc.caDepartment of Mathematics
http://www.math.ubc.ca/~israel University of British Columbia
Vancouver, BC, Canada V6T 1Z2
=I would like to know whether
Maple V (or later) includes proceduresthat support Boolean
function denition (in terms of and, or, and notoperators)and
Boolean function manipulation (eg. expansion and evaluation
ofBooleanexpressions). What version of Maple do I need and
what syntax do I usetocarry out such algebraic manipulations?
Is such functionality alsoavailablein other packages such as
Matlab or Mathematica?Neil
=Maple used to have a symbolic
logic package, but some time ago this was removed; it
doesnt appear to have been reinstated into Maple 9. A logic
package was available through the Maple Applications Center
(http://www.mapleapps.com), but I just checked the site to
discover that it is no longer available. I dont know why
Maple doesnt support symbolic logic - Ifind it very
annoying. MuPAD does however. I dont know about
Mathematica.You can do some Boolean work in Maple with
evalb.- Alasdair> > I would like to know whether Maple V
(or later) includes procedures> that support Boolean function
denition (in terms of and, or, and not> operators)> and
Boolean function manipulation (eg. expansion and evaluation
of> Boolean> expressions). What version of Maple do I need
and what syntax do I use> to> carry out such algebraic
manipulations? Is such functionality also> available> in
other packages such as Matlab or Mathematica?> > > Neil--
=computer. I looked for both ./congure or Makele in both
linuxdirectory and shared directory which I have got after
untar the lesthat I downloaded from mupad website. However,
I could notfind anyles like that. Could you tell me how I
can install this software inmy computer?
Could you tell me how I can install this software in> my
computer? Just follow the README: Either (preferred form) use
an rpm or untarboth share_252.tgz and bin_linux_252.tgz in a
directory supposed tohold the installation. Then, add the
share/bin directory to your$PATH.-- +--+ +--+| |+-|+
Christopher Creutzig (ccr@mupad.de)
=spherical coordinates
or cylindrical coordinates. I tried tofind thefunction using:
info(),but could notfind it. Could you help me?
Hey all,I
am having some problems importing the procedures on one
module to be usedunderstanding, I should be using
with(module), but Im not sure where itshould be located in
the module. Should I write a proc to be executed uponloading
that calls with(module)? is there a better way? if anyone
needs anymore details about this problem, just let me know. -
Chris
=|>I am having some problems importing the procedures
on one module to be used|>in another, and I was hoping
someone out there could give me a hand. > moduleB:= module()
export pB; pB:= proc(x) moduleA:-pA(x) + 1 end proc end
module:or > moduleC:= module() export pC; pC:= proc(x) use
moduleB in pB(x)+1 end use end proc end module:You could also
say > moduleD:= module() export pD; use moduleC in pD:=
proc(x) pC(x)+1 end proc end use end module:but that will
only work if moduleC is dened before moduleD is.Robert
Israel israel@math.ubc.caDepartment of Mathematics
http://www.math.ubc.ca/~israel University of British Columbia
Vancouver, BC, Canada V6T 1Z2
=Im not a mathematician,
hopefully you can help me with my question:I was looking at
some equations in a text book that were written insummation
notation and felt that they would be clearer if rewritten
asmatrix operations. In the course of rewriting, it struck me
that thereare actually many families of summation notation
equations that can bereasonably translated into matrix/vector
operations. The reverse is alsotrue; many basic matrix
operations can be easily translated into summationstyle
symbology. This especially comes to my mind in the case of
tryingto write computer procedures for linear algebra; you
could (although itsnot always the most efcient way) code
many matrix operations as a seriesof control loops iterating
over matrix subscripts, which seems to meclosely related to
standard summation notation.A simple example of what Im
getting at is that sum(i=1)(i=n)(Ai*Bi) canbe easily
rewritten as A dot B provided you make the
intuitivesubstitution that vector A is the n-dimensional
vector [A1 A2 ... An] andB likewise. This becomes matrix
operation if you bother to keep track ofrows versus
columns.Conversely, I could convert a simple dot product to
the more cumbersomesum notation, something which it seems I
am implicitly doing if I try tocode a dot product as a
for(i=0, i<n, i++) kind of control loop.What I want to know
is: has this relationship and the interconversion ofthese
families of equations been formalized in any signicant way?
Isthere any symbolic formalism for taking an equation in
summation notationand rewriting it as standard matrix/vector
operations (or vice versa,although this way is easier to
gure out intuitively at least for me)?Like I said, Im not a
mathematician, so if this is a nonsensical questionand you can
explain specically why it is nonsensical that would
alsointerest me.Saur
= > Im not a mathematician,
hopefully you can help me with my question: > > I was looking
at some equations in a text book that were written in >
summation notation and felt that they would be clearer if
rewritten > as matrix operations. In the course of rewriting,
it struck me that > there are actually many families of
summation notation equations that > can be reasonably
translated into matrix/vector operations. The > reverse is
also true; many basic matrix operations can be easily >
translated into summation style symbology. This especially
comes to > my mind in the case of trying to write computer
procedures for linear > algebra; you could (although its not
always the most efcient way) > code many matrix operations as
a series of control loops iterating > over matrix subscripts,
which seems to me closely related to standard > summation
notation. > > A simple example of what Im getting at is that
sum(i=1)(i=n)(Ai*Bi) > can be easily rewritten as A dot B
provided you make the intuitive > substitution that vector A
is the n-dimensional vector [A1 A2 ... > An] and B likewise.
This becomes matrix operation if you bother to > keep track
of rows versus columns. > > Conversely, I could convert a
simple dot product to the more > cumbersome sum notation,
something which it seems I am implicitly > doing if I try to
code a dot product as a for(i=0, i<n, i++) kind of > control
loop. > > What I want to know is: has this relationship and
the interconversion > of these families of equations been
formalized in any signicant > way? Is there any symbolic
formalism for taking an equation in > summation notation and
rewriting it as standard matrix/vector > operations (or vice
versa, although this way is easier to gure out > intuitively
at least for me)? > > Like I said, Im not a mathematician, so
if this is a nonsensical > question and you can explain
specically why it is nonsensical that > would also interest
me. >There are programming languages in which the
n-dimensional vector is abasic data type, with all the right
operations on it.APL was the rst such language. Its modern
version is J:http://www.jsoftware.com/J is a highly developed
well thought out formalism for what you areasking for, and
much more.You can download a J interpreter from the J
Software web site, and alsopapers and books showing
applications of J in various areas of mathematics.Nemo
>
What I want to know is: has this relationship and the
interconversion> > of these families of equations been
formalized in any signicant> > way?> There are programming
languages in which the n-dimensional vector is a> basic data
type, with all the right operations on it.>> APL was the rst
such language. Its modern version is J:>
http://www.jsoftware.com/>> J is a highly developed well
thought out formalism for what you are> asking for, and much
more.>> You can download a J interpreter from the J Software
web site, and also> papers and books showing applications of
J in various areas ofmathematics.For purely mathematical
purposes, most of the elementary notions of linear(or
multilinear) algebra are denable without using bases and
matrices atall; ee.g. linear mappings, kernels, traces,
determinants, and transposes.But for the real world, I
cant improve on Nemos answer.(I was pretty amazed by APL
when I played with it at the University ofToronto in the
early 70s. Source code for an APL interpreter was in one
ofthe popular programming magazines 2 or 3 years ago, but I
dont recall thedetails.)Larry
= Does anybody know whether
these software packages are still availablefor purchase in
forms that will operate under Windows XP or other
Windowsoperating system? J. Ogilvie
Does anybody know
whether these software packages are still available>for
purchase in forms that will operate under Windows XP or other
Windows>operating system?> J. Ogilvie>John -Yes, you can still
purchase it, although I am unsure of the address to contact.
If you post to the maxima news group, someone can point you
to the source from which to buy commercial macsyma and
pdease. I have no problem running it under xp. If you have
trouble getting an answer, please contact me. I am sure I can
track down the necessary address.Dick Fell
=John -Dick> Does
anybody know whether these software packages are still
available>for purchase in forms that will operate under
Windows XP or other Windows>operating system?> J.
Ogilvie>
=Im using Maple v. 7, and am stumped trying to
gure out how to use subs (orsomething equivalent) with
matrices.Here is an example of the problem Im having:>
with(linalg):> mat:=array([[a,b],[c,d]]); [a b] mat := [ ] [c
d]> subs(a=0.05,mat); matIn other words, substituting a=0.05
into the relevant element(s) of the matrixmat isnt
working.How can I do this? Seems like it should be
straightforward, but Ill be dangedif I can gure it
out.
Im using Maple v. 7, and am stumped trying to
gure out how to use subs(or> something equivalent) with
matrices.>> Here is an example of the problem Im having:>>
with(linalg):> mat:=array([[a,b],[c,d]]);>> [a b]> mat := [
]> [c d]>> subs(a=0.05,mat);>> mat> In other words,
substituting a=0.05 into the relevant element(s) of
thematrix> mat isnt working.>Use:
subs(a=0.05,evalm(mat));
> In other words, substituting
a=0.05 into the relevant element(s) of the>matrix>> mat isnt
working.>Use: subs(a=0.05,evalm(mat));>>Makes sense, now
that you mention it.
Use: subs(a=0.05,evalm(mat)); By
the way: What is the point in having evalm in the rst
place?In each and every example I have seen so far, evalm was
justnecessary to make Maple do anything at all with an
expression whereit was completely obvious (and easy to deduce
for a computer) thatthe computation should be done on
matrices.-- +--+ +--+| |+-|+ Christopher Creutzig
(ccr@mupad.de)
=does anybody know if there is an interface
from SciLab to MuPAD? iwant to adress MuPAD from SciLab.
(exchange parameter and have MuPADcalculate thing for
SciLab).can i somehow adress MuPAD from outside ?i do know
of the MuPAD - SciLab link but i want to include MuPAD in
SciLab and not vice versa.thanx bent
system,> which runs
in any computer with Java. You can play it online.>>
www.SymbMath.com
> >Having said all that to show that I
need to prove metric invariance ->my real problem is that
I have a stereographic projection and I want to>be sure that
when I say points A and B are closer than A and C in
the>complex plane that this information is preserved through
the>stereographic projection.> > Its not true. If it were,
the stereographic projection f (from the> sphere to the
complex plane) would map a circle centred at A to a> circle
centred at f(A). The stereographic projection maps circles
to> circles (and straight lines in the case of circles
through the north> pole), but does not preserve the centres.>
Intuitively though I think the relative positions of the
centres arepreserved. So if A is closer to the origin than B
in the complex plane,this is still true if you consider the
distance around the sphere fromthe south pole (f(0)) to the
projected points f(A) and f(B). Perhaps my relating this
concept to the metric was incorrect?I think what I was
thinking goes something like this:1. The metric measures the
shortest distance between two points in aspace [this
assumption may or may not be correct; appreciate help hereif
Im wrong]. As a result using the metric you are able to say
thedistance between A and B is shorter than distance between
A and C.2. If I have a mapping function that maps between two
spaces with twodifferent metrics (which is denitely the case
in the stereographicprojection) it might be possible that as
a result of the mapping thestatement the distance between A
and B is shorter than distance betweenA and C is no longer
true for distances f(A)f(B) and f(A)f(C).I think my initial
example showed that statement (2) is a valid concern,although
in the example the spaces had the same metric.Does this help
explain my train of thought?Russell.
=is it possible to
work with two-sided polynomials in MuPAD? By two-sided
polynomials I mean something like:>> a := 3*z + 2 + 4*z^-1 +
2*z^-2; Namely, I would like to get coefcients of >> a*x; 2 3
/ x1 x2 x3 x4 x5 x6  (a0 + z a1 + z a2 + z a3) | x0 + -- +
-- + -- + -- + -- + -- | | z 2 3 4 5 6 |  z z z z z /that
are associated with nonpositive powers of z. The COEFF
command does not work, is there any alternative way?Best
ragards,Zdenek
=I took me some time (I am a MuPAD newbie)
but nally I succeeded. Here is my solution: >> a :=
_plus(a.i*z^i $ i=0..3); 2 3 a0 + z a1 + z a2 + z a3>> x :=
_plus(x.i*z^-i $ i=0..10); x1 x2 x3 x4 x5 x6 x7 x8 x9 x10 x0
+ -- + -- + -- + -- + -- + -- + -- + -- + -- +  z 2 3 4 5
6 7 8 9 10 z z z z z z z z z>> y :=
revert([coeff(poly(numer(a*x),[z]),i) $ i = 2..10]);[a0 x0 +
a1 x1 + a2 x2 + a3 x3, a0 x1 + a1 x2 + a2 x3 + a3 x4, a0 x2 +
a1 x3 + a2 x4 + a3 x5, a0 x3 + a1 x4 + a2 x5 + a3 x6, a0 x4 +
a1 x5 + a2 x6 + a3 x7, a0 x5 + a1 x6 + a2 x7 + a3 x8, a0 x6 +
a1 x7 + a2 x8 + a3 x9, a0 x7 + a1 x8 + a2 x9 + a3 x10, a0 x8 +
a1 x9 + a2 x10](well, I omitted the last two terms)Zdenek
Hurak> > is it possible to work with two-sided polynomials
in MuPAD? By two-sided> polynomials I mean something like: a := 3*z + 2 + 4*z^-1 + 2*z^-2;> > Namely, I would like
to get coefcients of> > a*x;> > 2 3 / x1 x2 x3 x4 x5 x6
> (a0 + z a1 + z a2 + z a3) | x0 + -- + -- + -- + -- + -- +
-- |> | z 2 3 4 5 6 |>  z z z z z /> > that are associated
with nonpositive powers of z. The COEFF command does> not
work, is there any alternative way?> > Best ragards,> Zdenek