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convergence of QR-algorithm
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highegg
science forum beginner


Joined: 11 Feb 2005
Posts: 5

PostPosted: Thu Mar 24, 2005 7:28 pm    Post subject: Re: Proof of the fundamental theorem of Calculus (FTOC) Reply with quote

On 21 Nov 03 06:09:04 -0500 (EST), John Gabriel wrote:
Quote:
Your web page is interesting but I think there is a far easier way to
prove not only the FTOC but every important calculus theorem - my
average tangent theorem not only defines and proves the mean value
theorem for the first time since Isaac Newton, but it proves both
parts of the Ftoc in such simple mathematics so that even a high
school student can understand it. It also proves finite differences,
Taylor's Theorem, Quadrature/Cubature, Arc Lengths, Newton's
Approximation Formula and much more. I invite you to check out:

a
href="http://www.geocities.com/john_gabriel">http://www.geocities.com/john_gabriel</a

John Gabriel

Hi,
your ATT proof looks interesting, but I don't understand what simple
argument makes you think that you may exchange the summation and
limit, even when the summation contains increasing number of terms
(depending on n).

highegg
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highegg
science forum beginner


Joined: 11 Feb 2005
Posts: 5

PostPosted: Thu Mar 24, 2005 7:28 pm    Post subject: Re: Proof of the fundamental theorem of Calculus (FTOC) Reply with quote

On 21 Nov 03 06:09:04 -0500 (EST), John Gabriel wrote:
Quote:
Your web page is interesting but I think there is a far easier way to
prove not only the FTOC but every important calculus theorem - my
average tangent theorem not only defines and proves the mean value
theorem for the first time since Isaac Newton, but it proves both
parts of the Ftoc in such simple mathematics so that even a high
school student can understand it. It also proves finite differences,
Taylor's Theorem, Quadrature/Cubature, Arc Lengths, Newton's
Approximation Formula and much more. I invite you to check out:

a
href="http://www.geocities.com/john_gabriel">http://www.geocities.com/john_gabriel</a

John Gabriel


Hi John,

in fact your average tangent gradient theorem fails for
function f(x)=x^2 sin(1/x), f(0)=0,
which is differentiable everywhere (not continuously),
but the righthand side in your theorem doesn't converge to the secant
derivative - it oscillates baaaadly.

Jaroslav Hajek
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Jason
science forum addict


Joined: 24 Mar 2005
Posts: 72

PostPosted: Thu Mar 24, 2005 7:28 pm    Post subject: Re: Proof of the fundamental theorem of Calculus (FTOC) Reply with quote

Jaroslav,
You are incorrect! John states that it has to be continuous
everywhere in the interval.

Jason.
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Lee
science forum beginner


Joined: 10 Jun 2005
Posts: 6

PostPosted: Thu Mar 24, 2005 7:28 pm    Post subject: Re: I need help in writing a recursion formula for a series. Reply with quote

Feb 15
Hello i need the same help with this formula


On 9 Feb 05 15:34:42 -0500 (EST), monica wrote:
Quote:
On 3 Feb 05 16:54:04 -0500 (EST), Amber wrote:
I too need to know how to do this.



On 17 Oct 04 00:23:07 -0400 (EDT), alethia wrote:
a = 1
n / n
2


Please help, I do not understand how to write this formula into a
recrusion formula. Thank you

***I too need help, did you ever find the formual for this? if so,
can
you show me... thanks!
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Luis A. Afonso
science forum beginner


Joined: 24 Mar 2005
Posts: 1

PostPosted: Thu Mar 24, 2005 7:28 pm    Post subject: Re: I need help in writing a recursion formula for a series. Reply with quote

u_n= 1 / (2^n)

u_(n+1)= 1/ ( 2^(n+1))

u_(n+1)/ u_n = [1 / (2^n)] / [1/ ( 2^(n+1))]

= [2^n] / [2^(n+1)] = 1/2

u_(n+1)= u_n / 2
u_0 = 1 / (2^0)=1 / 1 = 1
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Antti Lange , lange@fkf.n
science forum beginner


Joined: 24 Mar 2005
Posts: 1

PostPosted: Thu Mar 24, 2005 7:28 pm    Post subject: Re: bordered block diagonal (bbd) matrix Reply with quote

Hi,
I found its analytic inversion already in 1975. Please see Equation
(3) of
http://personal.fimnet.fi/business/terhikki.lange/FKFformula2.gif
Yours,
Antti
www.fkf.net

On Thu, 20 Jan 2005 10:19:47 -0600, Ben Gu wrote:
Quote:
Hi,

i am doing some work on bordered block diagonal (bbd) matrix. Does
anyone know any good books talking about this subject?

Thanks a lot!
Ben
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maertens roger
science forum beginner


Joined: 24 Mar 2005
Posts: 1

PostPosted: Thu Mar 24, 2005 7:28 pm    Post subject: Re: Pascals triangle Reply with quote

Where can i find a summary of all the proven properties of the numbers
of the pascals triangle.

I need a property of the SUM OF TWO NUMBERS IN THE SAME COLUMN of the
Pascals triangle.

example: In the first column (1,1,1,1,...), the sum of two numbers
equals 2.
There is no number 2 in that column.
So, in the first column, there are no solutions

In the second column (1,2,3,4,....), the som of two
numbers is another number included in that column
So, in the second column, there are infinite solutions

In the third column (1,3,6,10,15,21,..), the som of the two
numbers 6 + 15 = 21 also is a member of that column.
But it seems that there is only one.
So, in the third column, there seems to be only one solution

In the other colums, there seems to be no solutions.

WHO CAN PROVE MATHEMATICALLY THERE ARE NO SULUTIONS FOR COLUMN FOUR
AND FURHER ON?
I need this prove to solve a practical problem.

Thanks by Roger


Maertens Roger
Koeiendreef, 13
8310 Assebroek, Brugge
Belgium
0032 (0)50 35 62 41
Who
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Dave Rusin
science forum Guru


Joined: 25 Mar 2005
Posts: 487

PostPosted: Thu Mar 24, 2005 7:28 pm    Post subject: Re: Pascals triangle Reply with quote

I have set follow-ups to sci.math since this isn't Numerical Analysis.

In article <9i6kvlu44aav@legacy>,
maertens roger <maertens.roger@skynet.be> wrote:
Quote:
Where can i find a summary of all the proven properties of the numbers
of the pascals triangle.

Too many to summarize.

Quote:
I need a property of the SUM OF TWO NUMBERS IN THE SAME COLUMN of the
Pascals triangle.

example: In the first column (1,1,1,1,...), the sum of two numbers
equals 2.
There is no number 2 in that column.
So, in the first column, there are no solutions

In the second column (1,2,3,4,....), the som of two
numbers is another number included in that column
So, in the second column, there are infinite solutions

In the third column (1,3,6,10,15,21,..), the som of the two
numbers 6 + 15 = 21 also is a member of that column.
But it seems that there is only one.
So, in the third column, there seems to be only one solution

You didn't look very far, e.g. 55=45+10. More generally, you need the
solutions to (2n-1)^2 - 1 = (2m-1)^2 - (2k-1)^2 , which is to say you need
factorizations of x^2-1 subject to some parity constraints. There are
many solutions.

Quote:
In the other colums, there seems to be no solutions.

That may be; I didn't check. I note that in some contexts the products
n(n-1)(n-2)...(n-k+1) play a role similar to the powers n^k, so
the question you are asking is in some sense similar to Fermat's Last
Theorem, except that it lacks the homogeneity that make FLT pretty.
So if there is no obvious proof that all the solutions to
C(x,k) + C(y,k) = C(z,k) are the trivial ones, then any proof valid
for every k is likely to be very hard.

But of course this is pure speculation on my part; I didn't give it
more than a moment's consideration.

dave
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bv
science forum addict


Joined: 16 May 2005
Posts: 59

PostPosted: Thu Mar 24, 2005 7:28 pm    Post subject: Re: Efficient internal values calculation of Bulirsch-Stoer type integrator Reply with quote

guy wrote:
Quote:

I found it potentially good at some of my cases, anyhow the others
include real second order kinetics such as in the system of two
equations:

dydt[0] = C1-C2*y[0]+C3*y[1]+C4*y[0]*y[1]; // C1,C2,C3,C4>0
dydt[1] = C5*y[0]-C3*y[1]-C4*y[0]*y[1]; // C5>0

Where I could not find a way of converting it into the form
d\vec{y}/dt = A*\vec{y} + \vec{u}

It may not be obvious at a first glance, however DEs often need a bit of
"coaxing" into a right shape. In your case, either

- solve as linearized DEs: d/dt y = J*y, where J is system Jacobian
- solve as linear DEs with nonlinearities as input: d/dt y = Ay + u(y)

puts "expokit" squarely back in business.
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Paul Fackler
science forum beginner


Joined: 24 Mar 2005
Posts: 2

PostPosted: Thu Mar 24, 2005 7:28 pm    Post subject: Re: matrix cookbook - new version Reply with quote

Kaare Brandt Petersen wrote:
Quote:
Dear Colleagues

(Apollogies for multiple postings)

A new and updated version of The Matrix Cookbook is available for download

http://www.imm.dtu.dk/pubdb/views/edoc_download.php/3274/pdf/imm3274.pdf

The Matrix Cookbook is a desktop reference on formulas regarding matrices
such as the derivatives of determinants and traces, identities involving
inverses, statistical moments and more.

Apart from smaller additions and corrections, this version of The Matrix
Cookbook has been updated with material on derivatives of complex matrices
and matrix norm inequalities. This is largely due to the work of Michael
Syskind Pedersen who has joined the project as author.

Comments and corrections are most welcome.

Best regards, Kaare
--
Kaare Brandt Petersen * http://2302.dk



Hi Karre
A nice document. You may find some notes of mine of interest

http://www4.ncsu.edu/~pfackler/MatCalc.pdf

I take a somewhat different approach than you do in defining derivatives
but one that it makes it easy to solve fairly complicated matrix
derivatives.

Cheers
Paul
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Gottfried Helms
science forum Guru


Joined: 24 Mar 2005
Posts: 301

PostPosted: Thu Mar 24, 2005 7:28 pm    Post subject: Re: matrix cookbook - new version Reply with quote

Am 24.02.05 08:38 schrieb Paul Fackler:
Quote:

Hi Karre
A nice document. You may find some notes of mine of interest

http://www4.ncsu.edu/~pfackler/MatCalc.pdf

I take a somewhat different approach than you do in defining derivatives
but one that it makes it easy to solve fairly complicated matrix
derivatives.

Hi Paul -


just took a short look into your text.
It reminds me to some operations, that I have done recently, which
look like a lengthy notation for a 3-D-matrix concept, which could
then be more concise, if elaborated.

For instance, computing the eigenvectors of a matrix having the
eigenvalues, (applying a vandermonde-like eigenvalue-matrix, which
I derived recently) seems to be expressible in 3-d-matrix-notation
much more concisely. The vec- and the kronecker-operator seem to
map the common 2-d-matrices into a 3-d-(notation-) space.

Have you seen a notation for 3-D-matrices anywhere?

Gottfried Helms
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Guest






PostPosted: Thu Mar 24, 2005 7:28 pm    Post subject: Re: C++ lib for Complex Matrix Algebra Reply with quote

help involving complex libraries.
i have installed lapack++2.1.2 ,the latest distribution as available on
sourceforge.net.
but i dont know how to turn on complex types.
it says " Note: To switch on the support for complex-valued matrices,
you need to define the macro LA_COMPLEX_SUPPORT in your application "

the non-trivial question is about a suitable library for my
requirements.
*calculating pseudoinverses for complex matrices
*calculating eigenvalues and their corresponding LEFT eigenvectors
for a GENERALISED eigenvalue problem involving complex matrices.
i require the final code to be compiled in gcc on a linux platform.
it would be nice if you can tell me as to which of the following
packages would suit my requirements.
1. lapack++
2. MTL
3. PETSc and SLEPc


Mumit Khan wrote:
Quote:
In article <46bots$s2@harbinger.cc.monash.edu.au>,
Kai O'Yang <oyang@fcit.monash.edu.au> wrote:
pecora@zoltar.nrl.navy.mil (Lou Pecora) writes:

There is a LaPack++, not sure of source. Check out:

Netlib: http://netlib@att.com
Math lib: http://math.jpl.nasa.gov, /start/html

Just a note, you can't use LaPack++ 1.0 with gcc/g++ 2.7.0. The
libg++ that
comes with gcc 2.7.0 changed the format of complex numbers to
templates
and LaPack++ didn't like it at all. Have to use 2.6.3 for LaPack++ .
(The authors know about it and suggested me to do so.)

Kai


A slight nit-pick: libg++-2.7.0 did not "change" the format of the
complex numbers, rather it tries to adhere to the draft C++ standard
and uses a templatized version of complex (eg., complex<double> which
is what most of the previous libraries used as complex).

The changes to lapack++ to make it work with gcc-2.7.0 and
libg++-2.7.0a
is trivial: simple write a script that changes all the complex to
complex<double> and you're almost there. Can't remember if I had to
also
fix any of the new for-init-statement scope rule, but that's also
trivial.
I do remember that I didn't spend more than 20 minutes on it. Anybody
who
wants a copy of the shell script, please drop me a line; it's quite
simple,
but works. I actually have my code such that it works with
cfront-based
compilers that come with the older complex numbers as well as the
newer
ones such as libg++, and I do it by typedef'ing COMPLEX to be either
complex<double> (draft standard) or Complex (deprecated cfront-based
and
most other compilers probably).

regards,
mumit -- khan@xraylith.wisc.edu
http://www.xraylith.wisc.edu/~khan/
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Wendy E. McCaughrin
science forum beginner


Joined: 18 Jul 2005
Posts: 4

PostPosted: Thu Mar 24, 2005 7:28 pm    Post subject: Re: I need help in writing a recursion formula for a series. Reply with quote

The OP did not indicate "^".
For even n, the recurrence is: a(0) = 0, a(n) = a(n-2) + 1 for n>0.

"Luis A. Afonso" <licas_@hotmail.com> wrote:


: u_n= 1 / (2^n)

: u_(n+1)= 1/ ( 2^(n+1))
:
: u_(n+1)/ u_n = [1 / (2^n)] / [1/ ( 2^(n+1))]
:
: = [2^n] / [2^(n+1)] = 1/2

: u_(n+1)= u_n / 2
: u_0 = 1 / (2^0)=1 / 1 = 1
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bv
science forum addict


Joined: 16 May 2005
Posts: 59

PostPosted: Thu Mar 24, 2005 7:28 pm    Post subject: Re: Is there any: Simulation based parameter estimation method? Reply with quote

Neon wrote:
Quote:

How do you develop parameter estimation for complex system with some
output observations, wich can only be simulated other than modeled with
close form formula?

It's done routinely since few, if any, real world problems come with a
close form solution, except, homework problems stupefied for Matlab type
sw usage. It's a subject of active research, and is also (hopefully) at
the very core of modern cae/cad toolsets. Search for "inverse problems"
to learn more.
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Jason Wells
science forum beginner


Joined: 24 Mar 2005
Posts: 1

PostPosted: Thu Mar 24, 2005 7:28 pm    Post subject: New Proof Of The Fundamental Theorem Of Calculus Reply with quote

John,
If indeed your Average Tangent Theorem is true, then it would be a
far more efficient way of proving both the Mean Value Theorem as well
as the FTOC (fundamental theorem of calculus). And yes it wouold
provide a direct link between the derivative and integral.

I can't see any flaws with it but some of your opponents, eg. Graeme
McRae and others don't seem to think so. Perhaps your most vehement
arguments were those on Cut-the-knot wher you refuted the
mathematician Vladimir. You have a very abrupt style of responding to
objections - if I may, you may try posing your responses as questions
rather than 'attacks' as has been interpreted by most.
I think McRae is not necessarily an opponent either because he plainly
admits he cannot understand your theorem.

I think you are on to something big here - at least as far as
understanding is concerned. Personally I have failed to see the direct
link even though I've always known there is one. An Archimedes may
have indeed known of differentiation should your theorem be proved.

Jason Wells.

On 21 Nov 03 06:09:04 -0500 (EST), John Gabriel wrote:
Quote:
Your web page is interesting but I think there is a far easier way to
prove not only the FTOC but every important calculus theorem - my
average tangent theorem not only defines and proves the mean value
theorem for the first time since Isaac Newton, but it proves both
parts of the Ftoc in such simple mathematics so that even a high
school student can understand it. It also proves finite differences,
Taylor's Theorem, Quadrature/Cubature, Arc Lengths, Newton's
Approximation Formula and much more. I invite you to check out:

a
href="http://www.geocities.com/john_gabriel">http://www.geocities.com/john_gabriel</a

John Gabriel
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