mm-969

Subject: Re: [Q] Shape functions for conforming plate bending
element
> We are developing a Þnite element code for plate bending
elements. The
> nodal degrees of freedom are u, v, w, dw/dx, dw/dy and
d2w/dx*dy.
> The interpolation functions are Hermite functions which
impose that w,
> dw/dx, dw/dy or d2w/dx*dy should be one in one particular
node, and that
> all other degrees of freedom should be zero. The
formulation of the
> stiffness matrix is based on the traditional variational
principles.
> This works Þne for rectangular plate elements, but once the
initial
> element geometry is distorted (for example to a trapezium),
the results
> are rather inaccurate. There is about 1 % error on the
displacements,
> but the error on the bending moments is much larger (10 %
to 50 %).
> I do not Þnd any documentation on how this problem is
solved in
> commercial codes, because these distorted meshes yield good
results in
> Abaqus.
> Does anybody have any suggestions ?
> Wim Van Paepegem
> E-mail : Wim.VanPaepegem@UGent.be
The 16-DOF Hermitian plate element, along with three
other variants, was derived and tested long ago by Bogner,
Fox and Schmit. Reference:
F. K. Bogner, R. L. Fox and L. A. Schmit Jr. (1966).
The generation of interelement compatible stiffness
and mass matrices by the use of interpolation formulas,
{it Proc. Conf. on Matrix Methods in Structural Mechanics},
WPAFB, Ohio, 1965, in {it AFFDL TR 66-80}, pp. 397--444.
This was the Þrst all-FEM-oriented Conference. The Proceedings
may be purchased as used book through internet search engines
like http://www3.addall.com
The 16-DOF BFS element is very accurate for rectangular
geometries.
If extended to nonrectangular shapes (there are several ways
to do
that), it fails the patch test and thus becomes useless. That
explains why this model (and variants) vanished from the FEM
scene.
===
Subject: Re: Errors incurred in one variable space when
solving a diffusion
equation in another?
I assume you mean the simple Taylor¹s theorem which is the
basis
of many numerical schemes. In my prior attempts to understand
this
problem I looked at the error terms in the variable space I
was
solving the problem in and used the variable transforms to
try and
understand how they affected the solution in the other
variable space.
This gives me some idea, but no real feel for what is going
on. One
person who had this problem described it to me as an anomalous
diffusion in the other variable space. I tried to see if I
could show
that this is indeed a good description of what the numerical
errors
are doing, but could not derive anything that looked like a
diffusion
term. This would be very convenient, then I could compare the
diffusion coefÞcient that I derived to real diffusion
coefÞcients in
that space. Can this be understood as a diffusion in the
other space?
Is there a different way to understand it? Maybe it could be
viewed
in a different way. I hope that I have done a better job of
explaining what I am looking for. I am far from an expert in
this
Þeld and while I will continue looking in other directions to
Þnd my
answer, I thought I would see if the experts had a quick
reference for
me.
Shawn
>
> I am numerically solving a diffusion equation in one
variable
>space, but I am concerned about how the errors in that
variable space
>will affect a different space where the answers are more
physically
>meaningful to many people. To illustrate my concern, if I
were to do
>a calculation in a two dimensional space (say {u,v}) for a
system that
>was only one dimensional in the variables {x,y} (say the
system only
>depended on x in this variable space). How would errors in
the {u,v}
>space show up in {x,y} space if the variables transformed as
x=x(u,v),
>y=y(u,v) instead of something like x=x(u), y=y(v)? The
errors in
>{u,v} space would likely show up as some kind of anomalous
diffusion
>in the y direction in {x,y} space an there would probably be
anomalous
>cross-diffusion as well, but how do I estimate that? I would
think
>this has been looked at before, so I am hoping someone can
point me to
>a text that discusses this?
>
>Shawn
> ?????? you have
> x=x(u,v), y=y(u,v)
> and solved a problem in terms of u,v with some errors
> what are the erros induced in x ,y
> is this your question???
> -> look up Taylors theorem
> hth
> peter
===
Subject: Re: Help with golf shaft þex equation
Hugh
> Answered in part below:
>>
>> Frequency is measured in numbers from 2.0 to 8.0. 5.0
being regular
>> stiffness. 6.0 is stiff, 7.0 is extra stiff (Tiger Woods
swings at
>> 7.0).
>>
>> Is there a precise deÞnition of this number (e.g. the
amount of load
>> required to produce a given deþection of the head)? If
not, then
>> there¹s no way to Þnd an analytic equation for it.
> Yes there is. In fact up until about 20 years ago they used
to use a
> board where you hung a weight on the clubhead. By how much
it þexed
is
> how you could tell how stiff it was.
> In that case, you could get a scientiÞc formula out of that
information
> if you wanted to go through the effort. If the shape of a
golf club
shaft
> is deÞned entirely by its length, you can solve the beam
bending
> differential equations to get formulas for the lowest
vibrational
> frequency as a function of length and the deþection as a
function of
> length and weight, and then solve for one in terms of the
other. If the
> shaft is simple enough (e.g. a cylinder with the same
diameter
everywhere)
> then someone will have already done the calculus for you
and you can use
> their algebraic formulas, like the vibration formula in the
other reply
> you got.
>> But people keep stealing our graphs. Is there any way to
determine
a
>> formula instead of relying on a graph?
>>
>> You can always turn the graph into an approximate formula.
Scan it,
>> then measure enough points and Þnd a curve (piecewise
smooth cubic
>> will be easy if the graph is smooth enough) that Þts it to
whatever
>> tolerance you need.
> Yeah, that¹s what I¹m looking for, a formula. The line is
just that, a
> line. Cycles is on the left side of the graph and length is
on the
> bottom. The graph is a line, or actually, many lines all
parallel to
> each other representing 2.0, 3.0 ,4.0, 5.0 etc....
> The lines are all parallel, and all straight? That¹ll Þt
with a single
> linear function, then. Try picking three points on the
graph (low
> stiffness + short length, low stiffness + long length, and
high stiffness
> + short length), and plug in the coordinates for each of
those points
into
> the equation cycles = A*length + B*stiffness + C. Three
equations
and
> three unknowns, so you can solve for A, B, and C. Then try
plugging in a
> fourth point (high stiffness + long length) and see how
close the
equation
> comes to the graph. If it Þts, you¹re done. If not, then
(presuming
you
> did the arithmetic right) the lines either weren¹t as
straight or weren¹t
> as parallel as you thought, and things get a little more
complicated.
> ---
> Roy Stogner
===
Subject: Eigenvalues of a Non-Symmetric Real Tridiagonal
Matrix
Hi!

I¹m looking for an efÞcient algorithm to compute (selected)
eigenvalues and (if desired) the corresponding eigenvectors
of a
non-symmetric real tridiagonal matrix. Is there such a
routine? I¹ve
been searching netlib.org without success, actually, all
algorithms
that I found treat tridiagonal matrix as symmetric (why?). In
this
case, how can I transform to a symmetric matrix? Under what
conditions
this transformation can be done? Any help is welcome,

yours sincerely,

Bernhard.
===
Subject: Re: Eigenvalues of a Non-Symmetric Real Tridiagonal
Matrix
> Hi!
> I¹m looking for an efÞcient algorithm to compute (selected)
> eigenvalues and (if desired) the corresponding eigenvectors
of a
> non-symmetric real tridiagonal matrix. Is there such a
routine? I¹ve
> been searching netlib.org without success, actually, all
algorithms
> that I found treat tridiagonal matrix as symmetric (why?).
In this
> case, how can I transform to a symmetric matrix? Under what
conditions
> this transformation can be done? Any help is welcome,
If the product of the off-diagonal element is positive you
can Þnd a
diagonal similarity transformation to symmetric form.
If not, you can use the LR algorithm with shifts, but it may
be
numerically unstable; that¹s why it is not in general use. See
A. Dax and S. Kaniel,
The ELR method for computing the eigenvalues of a general
matrix,
Arnold Neumaier
===
Subject: What keywords that I should search for references
for this kind of
problems - vector comparison? Set partitions?
I have two vectors of 1 by 3 dimension and each cell can only
take value
of either 0 or 1. For example, vector A can be (0,1,1),
(1,0,1) or
(1,1,0) and vector B can be (0,1,1), (1,0,1) or (1,1,0).
My question is that how many ways (permutations?) that vector
B is
greater than vector A?
I need the deÞnition of comparing vectors. In my example
above, (1,1,0)
> (0,1,1) and (1,0,1) > (0,1,1). Therefore, there are two
situations that
vector B > vector A.
Couple you please tell me what keywords (vector comparison,
set
partition, combinatorial, etc) that I should use to search
for the
references of this kind of problem?
John
===
Subject: Re: A very traditional question
> I¹d like to know about methods for Þnding a few (or a few
hundred) of
> the smallest real eigenvalues and associated eigenvectors
of a large (say
> 60k*60k) sparse (say seven entries per row) matrix. The
matrix has no
> particular structure beyond the sparsity, though I suspect
its bandwidth
> is of the order of the square root of its size.
What you want is an implementation of the Lanczsos algorithm.
It is
described in
many books; e.g. Golub & van Loan.
But if you want to know for sure that you got the k smallest
eigenvalues,
which can be done (only) if your matrix is Hermitian,
you need a spectrum slicing technique, described, e.g. in
Parlett¹s book.
But this means that you need to do many factorizations, and
your matrix
may well be too large for that.
Arnold Neumaier
===
Subject: Re: A very traditional question
> What you want is an implementation of the Lanczsos
algorithm. It is
described in
> many books; e.g. Golub & van Loan.
Sorry, wrong spelling - should be ŒLanczos algorithm¹
AN.
===
Subject: Re: A very traditional question
> I¹d like to know about methods for Þnding a few (or a few
hundred) of
> the smallest real eigenvalues and associated eigenvectors
of a large (say
> 60k*60k) sparse (say seven entries per row) matrix. The
matrix has no
> particular structure beyond the sparsity, though I suspect
its bandwidth
> is of the order of the square root of its size.
Do you know anything in advance about how many eigenvalues
correspond to
each eigenvector? The simplest way to get the largest
eigenvalue of a
matrix that I remember is to take an arbitrary starting
vector and
repeatedly multiply by the matrix and normalize the result.
That will
converge (how rapidly depends on the ratio of the largest two
eigenvalues)
to an eigenvector whose eigenvalue (the length you end up
dividing by) is
the largest the matrix has.
You can get other eigenvalues by changing which matrix you
multiply by.
If you want the smallest eigenvalue of A, you can get it as 1
divided by
the largest eigenvalue of A^-1. If you want the second largest
eigenvalue of A after you¹ve already found the largest
eigenvalue (call
it a), you can get it as the largest eigenvalue of A-a*I.
> Google isn¹t very helpful, it gives me the manuals for a
great number
> of packages that can compute these things, but I¹m at the
dichotomy of
> either wanting a import this python package and say
> matrix.SmallSparseEigenvalues(50), or wanting a description
whereby
> I could actually implement the algorithm.
If you want a description, try adding terms like raleigh
quotient,
power method and QR factorization to your search, and read
whatever
that gives you instead of relying on my fuzzy memory above.
> The context is http://tom.womack.net/projects/lagrange.html
- I¹d like
> to produce better images of the mode functions, and that¹s
going to
> need eigenvectors for sparse matrices. I¹d also appreciate
comments
> on what I¹ve got up there already, if people don¹t mind too
much.
You say you want an automatic way to nicely distribute
points, and you¹re
familiar with Voronoi cells; have you heard of centroidal
Voronoi
tesselations? All you do is iteratively move vertices toward
the
centroids of the Voronoi cells they generate, and you get a
neatly spaced
set of vertices which will give you a nice triangulation.
---
Roy Stogner
===
Subject: Re: A very traditional question
Originator: twomack@chiark.greenend.org.uk ([193.201.200.170])
>> I¹d like to know about methods for Þnding a few (or a few
hundred) of
>> the smallest real eigenvalues and associated eigenvectors
of a large
(say
>> 60k*60k) sparse (say seven entries per row) matrix. The
matrix has no
>> particular structure beyond the sparsity, though I suspect
its bandwidth
>> is of the order of the square root of its size.
> Do you know anything in advance about how many eigenvalues
> correspond to each eigenvector?
I presume you mean that the other way round, how many
eigenvectors per
eigenvalue; in which case I¹m not quite certain, but I know
the answer
is not 1 and I suspect it¹s not very much more than 2 - you
see on the
web page some examples of ŒconÞguration¹ and ŒconÞguration
rotated
45 degrees¹, which I believe had extremely close eigenvalues,
probably
would have been equal had I done the calculations exactly
rather than
to double-precision.
> The simplest way to get the largest eigenvalue of a
> matrix that I remember is to take an arbitrary starting
vector and
> repeatedly multiply by the matrix and normalize the result.
That will
> converge (how rapidly depends on the ratio of the largest
two
eigenvalues)
> to an eigenvector whose eigenvalue (the length you end up
dividing by) is
> the largest the matrix has.
I¹d spent a while on my walk home from work thinking about
that, but
it does unavoidably give the _largest_ eigenvalue, and I
can¹t see why
it wouldn¹t fail fairly horribly if the matrix has
nearly-equal
eigenvalues.
>You can get other eigenvalues by changing which matrix you
multiply by.
>If you want the smallest eigenvalue of A, you can get it as
1 divided by
>the largest eigenvalue of A^-1.
But I don¹t think computing A^-1 explicitly is practical in
this
situation, since computing Av for a vector v is only really
practical
because A is very sparse, and A^-1 is not remotely sparse.
I suppose there are techniques for solving sparse linear
equations
efÞciently, so instead of computing A^-1 and then calculating
A^-1 v,
I could compute w : Aw = v directly. What are the relevant
terms to
look up for that kind of thing?
> If you want the second largest
> eigenvalue of A after you¹ve already found the largest
eigenvalue (call
> it a), you can get it as the largest eigenvalue of A-a*I.
I don¹t think largest eigenvalue of A-a*I works; what I was
thinking
might
work would be the iteration
temp = A * v [ or temp = sparse_solve(A,v), depending ...]
temp = temp - (temp dot evec1)*evec1 - (temp dot evec2)*evec2
...
where I take the image of temp in the orthogonal complement
of the set of
known eigenvectors. Does that sound reasonable; does it have
a name?
> You say you want an automatic way to nicely distribute
points, and you¹re
> familiar with Voronoi cells; have you heard of centroidal
Voronoi
> tesselations? All you do is iteratively move vertices
toward the
> centroids of the Voronoi cells they generate, and you get a
neatly spaced
> set of vertices which will give you a nice triangulation.
Ooh, that¹s a really nice argument, now all I need is the
Voronoi
code. There¹s an awful lot of wondrous stuff out there, the
awkward
bit is just writing the interfaces to plug it into python ...
Tom
===
Subject: Re: A very traditional question
>> Do you know anything in advance about how many eigenvalues
>> correspond to each eigenvector?
> I presume you mean that the other way round, how many
eigenvectors per
> eigenvalue;
Right; sorry for expressing it poorly.
> I¹d spent a while on my walk home from work thinking about
that, but
> it does unavoidably give the _largest_ eigenvalue, and I
can¹t see why
> it wouldn¹t fail fairly horribly if the matrix has
nearly-equal
> eigenvalues.
Yup, that¹s pretty much worst-case. If you have exactly equal
largest eigenvalues, your guess vector should be projected
toward the
space spanned by the eigenvectors, but if you have just
nearly equal
eigenvalues, then you¹ll get to the span of their
eigenvectors quickly but
converge toward the one true eigenvector horribly slowly.
That circular drum problem is pretty awful in this regard
too, isn¹t it?
Every eigenfunction that isn¹t axially symmetric should give
you an
inÞnite number of other eigenfunctions with the same
eigenvalue, just by
rotating around the axis. It¹s not just conÞguration rotated
45
degrees which will give you problems, it¹s conÞguration
rotated 0.1
degrees, conÞguration rotated 0.01 degrees, etc. right up to
whatever
your mesh can hold.
>>You can get other eigenvalues by changing which matrix you
multiply by.
>>If you want the smallest eigenvalue of A, you can get it as
1 divided by
>>the largest eigenvalue of A^-1.
> But I don¹t think computing A^-1 explicitly is practical in
this
> situation, since computing Av for a vector v is only really
practical
> because A is very sparse, and A^-1 is not remotely sparse.
Right.
> I suppose there are techniques for solving sparse linear
equations
> efÞciently, so instead of computing A^-1 and then
calculating A^-1 v, I
> could compute w : Aw = v directly. What are the relevant
terms to look
> up for that kind of thing?
Probably linear solver. This is what you¹d want to do, and it
shouldn¹t
be too hard.
Are you getting symmetric matrices out of your Þnite
difference approach,
by the way? If not you might want to try Þnite elements
instead; the
code won¹t be any harder than the least squares Þts you¹re
doing now, and
for Laplace¹s operator it¹ll give you a symmetric matrix and
make more
linear solver options (conjugate gradient is easy to
implement) practical.
> what I was thinking might work would be the iteration
> temp = A * v [ or temp = sparse_solve(A,v), depending ...]
temp = temp
> - (temp dot evec1)*evec1 - (temp dot evec2)*evec2 ...
> where I take the image of temp in the orthogonal complement
of the set
> of known eigenvectors. Does that sound reasonable; does it
have a name?
It sounds familiar, which probably means it¹s something I
learned once and
forgot the name to. Sorry. ;-) It¹s similar to the deþation
approach
Ron Shepard suggested, though.
> Ooh, that¹s a really nice argument, now all I need is the
Voronoi code.
> There¹s an awful lot of wondrous stuff out there, the
awkward bit is
> just writing the interfaces to plug it into python ...
Have you ever played with SWIG? ( http://www.swig.org/ ) I¹ve
heard
recommendations for it, it may be simpler than Python¹s
native facility
for wrapping C/C++/Fortran code (it¹s certainly simpler than
Perl¹s...),
and you¹ll probably Þnd wrapping somebody else¹s numerical
code to be no
harder than debugging your own.
---
Roy Stogner
===
Subject: Re: A very traditional question
> If you want the second largest
> eigenvalue of A after you¹ve already found the largest
eigenvalue (call
> it a), you can get it as the largest eigenvalue of A-a*I.
In case anyone else hasn¹t spotted the error yet, I¹m clearly
being
confused here - A-a*I may take the eigenvalue a out of the
picture as far
as power method iteration is concerned, but it will also
shift around
every other eigenvalue and so won¹t give you the second
largest eigenvalue
of A. I think I was misremembering shifted inverse iteration,
where you
use (A-a*I)^-1 to Þnd the eigenvalue closest to a.
Sorry,
---
Roy Stogner
===
Subject: Re: A very traditional question
> If you want the second largest
> eigenvalue of A after you¹ve already found the largest
eigenvalue (call
> it a), you can get it as the largest eigenvalue of A-a*I.
> In case anyone else hasn¹t spotted the error yet, I¹m
clearly being
> confused here - A-a*I may take the eigenvalue a out of the
picture as far
> as power method iteration is concerned, but it will also
shift around
> every other eigenvalue and so won¹t give you the second
largest
eigenvalue
> of A. I think I was misremembering shifted inverse
iteration, where you
> use (A-a*I)^-1 to Þnd the eigenvalue closest to a.
Also, the power method gives you the vector corresponding to
the
eigenvalue of largest magnitude, which may not be the largest
eigenvalue. For example, if the largest eigenvalue happens to
be
zero, then the power method will never converge to it, even
if the
initial vector is only eps away from the exact eigenvector.
Another possibly useful approach is to use the modiÞed matrix
A-a*v*v¹ where v is the eigenvector that corresponds to the
largest eigenvalue a. This deþation approach leaves all the
other eigenvalues and vectors unchanged, but it shifts the
eigenvalue that was a to zero. Once the second eigenvalue has
been found, then the matrix can be deþated again for the third
eigenvalue, and so on. There can be numerical problems with
this
approach if v is poorly converged.
As far as methods to Þnd selected eigenpairs for sparse
matrices, I
would suggest iterative subspace methods. These methods would
be
ideal for symmetric or hermitian problems, but you sometimes
have to
be careful with nonsymmetric cases, even those that are known
to
have real eigenvalues. Look up Jacobi-Davidson and see if you
can
Þnd something that Þts your problem. If the matrix is
diagonally
dominant, then straight Davidson with a simple diagonal
preconditoner may be a good approach.
$.02 -Ron Shepard
===
Subject: Helmholtz equation, sign problem
Resolving 1-dimensional Helmholtz equation for TE and TM
modes in
wavequide planar structure, using Finite Element Method, some
of the
eigenfunctions (modes) has a wrong sign, but the absolute
values are
properly computed. I have tried many diffrent steps
(basically the
step is equal to lambda divided by 20, where lambda is the
wavelenght
of light). The wavequide structure is divided into three
regions
(substrat, wavequide and cover layer). The central difference
scheme
were being used. To compute the eigenvalues and
eigenfunctions the
Matlab functuion eig were used. The refractive index is
distributed
along the structure. I tried both Dirichlet and Neumann
boundary
conditions. Would you tell me why some eigenfunctions have
the wrong
sign?
d^2E/dx^2+k^2*(n^2-nef^2)*E=0
E - the electric Þeld intensity ( eigenfunction E(x) )
k - the wave number (constant value), related to wavelenght
of light
x - the spatial varbial
n - refractive index distributed along the structure
nef - effective refractive index (eigenvalue)
Author-Supplied-Address: bds <AT> ipp <DOT> mpg <DOT> de
===
Subject: Re: Helmholtz equation, sign problem
Mail-To-News-Contact: abuse@dizum.com
|> Resolving 1-dimensional Helmholtz equation for TE and TM
modes in
|> wavequide planar structure, using Finite Element Method,
some of the
|> eigenfunctions (modes) has a wrong sign, but the absolute
values are
|> properly computed.
[...]
Aren¹t the eigenfunctions themselves undetermined up to a
constant?
Maybe the signs depend on how the matrix of coefÞcients is
ordered...
The equation you give is indeed linear...
--
cu,
Bruce
drift wave turbulence: http://www.rzg.mpg.de/~bds/
===
Subject: Re: Helmholtz equation, sign problem
> |> Resolving 1-dimensional Helmholtz equation for TE and TM
modes in
> |> wavequide planar structure, using Finite Element Method,
some of the
> |> eigenfunctions (modes) has a wrong sign, but the
absolute values are
> |> properly computed.
> [...]
> Aren¹t the eigenfunctions themselves undetermined up to a
constant?
> Maybe the signs depend on how the matrix of coefÞcients is
ordered...
> The equation you give is indeed linear...
An eigenvector has an arbitrary scale factor (with an
arbitrary sign)
unless it is normalized to some criteria.
There are various normalization criteria.
The Helmholtz equation also applies to certain acoustic and
vibration
problems. In this case, the eigenvectors may be normalized so
that:
X^T M X = I
where
M = mass matrix
I = identity matrix
X = eigenvector matrix (column format)
X^T is the transpose of the eigenvector matrix
Tom Irvine
www.vibrationdata.com
===
Subject: Re: The 0 is the number of evil !
> The 0 is really evil.
> You can«t do x/0 , because the 0 is evil !
> The 0 is the devil , burn it !
After 30 years of scientiÞc research, now he tells me!
--
Lou Pecora
- My views are my own.
===
Subject: Re: Need help optimizing....
> I have a routine that evaluates a polynomial equation that
have 3
variables
> x,y,z of orders 1,2,3 the coefÞcients of the polynomial are
in an array.
> This routine is quite slow and I¹d like to optimize it.
> Any suggestions?
[...]
> pts[pn].val = c[cn++] * pow(pts[pn].x, i) *
zyexp;
The problem is that you are using the pow() function inside
of a set
of nested loops, so you need to avoid that. There are two
things to
try.
The Þrst is to factor the polonomial using Horner¹s rule. This
results in a single þoating point mulitply and add
combination in
the inner-most loop.
The second approach is to precompute a table of powers of the
three
variables, and replace the pow() function with the
appropriate table
lookup in the innermost loop.
Depending on the integer powers involved, and on whether the
coefÞcients are dense or sparse, either of these two
approaches
might be optimal.
$.02 -Ron Shepard
===
Subject: Re: Need help optimizing....
> I have a routine that evaluates a polynomial equation that
have 3
variables
> x,y,z of orders 1,2,3 the coefÞcients of the polynomial are
in an
array.
> This routine is quite slow and I¹d like to optimize it.
> Any suggestions?
> [...]
> pts[pn].val = c[cn++] * pow(pts[pn].x, i) *
zyexp;
> The problem is that you are using the pow() function inside
of a set
> of nested loops, so you need to avoid that. There are two
things to
> try.
> The Þrst is to factor the polonomial using Horner¹s rule.
This
> results in a single þoating point mulitply and add
combination in
> the inner-most loop.
Yes I found out about horner¹s rule today, and while I
understand it
for polynomials with one variable I am wondering if it can be
applied to
polynomials of more than one variable.
> The second approach is to precompute a table of powers of
the three
> variables, and replace the pow() function with the
appropriate table
> lookup in the innermost loop.
> Depending on the integer powers involved, and on whether the
> coefÞcients are dense or sparse, either of these two
approaches
> might be optimal.
> $.02 -Ron Shepard
===
Subject: Re: What is gist of proof that series of reciprocal
primes
diverges.
> [[ This message was both posted and mailed: see
> the To, Cc, and Newsgroups headers for details. ]]
> I have seen proof that the harmonic series, 1+1/2+1/3+1/4+
....
diverges.
> I have also seen statements that a subset of that series,
i.e, the
> reciprocals of the prime numbers
> 1+1/2+1/3+1/5+1/7+ ....
> also diverges.
> Can anyone describe the gist of a proof?
> Because of the following observations:
> If p is prime, then
> 1
> ----- = 1 + 1/p + 1/p^2 + 1/p^3 + ...
> 1-1/p
> Now take the product over all primes p, and note that on
the right-hand
> side you get the sum of all 1/n for positive integers n,
Perhaps mention that this comes from the factorization theorem
for integers.
> which is
> inÞnite. It follows that
> prod_{p prime} (1-1/p) = 0.
> A necessary and sufÞcient condition for this is that
> sum_{p prime} 1/p = infty.
> --Ron Bruck
--
Julian V. Noble
Professor Emeritus of Physics
jvn@lessspamformother.virginia.edu
^^^^^^^^^^^^^^^^^^
http://galileo.phys.virginia.edu/~jvn/
Science knows only one commandment: contribute to science.
-- Bertolt Brecht, Galileo.
===
Subject: Loading octave on a computer
I am a novice at this stuff. I am trying to Þnd out if there
is a
.exe Þle for octave that I can run on my computer. If yes, we
can i
get it. Any help will be greatly appreciated. I am working on
a
project and adding functions to octave is one of my options.
I am
trying to familiarize myself with the program.