mm-539
===
Subject: Re: Exact Rational Interpolation
> Cherednichenko, V.G.
> Rational interpolation: analytical solution. (Russian,
English)
> [J] Sib. Mat. Zh. 43, No.1, 188-193 (2002);
> translation in Sib. Math. J. 43, No.1, 151-155 (2002).
[ISSN 0037-4474]
>Peter, we concluded that this approach is not numerically
practical
>for interpolation with arbitrary number of points.
>After many failed attempts, I decided to give up on rational
interpolation.
what do you mean by arbitrary? No more than about 10 data
points of
course.
high degree interpolation makes no sense. if low degree (say
(7,5))
does not suf[CapitalThorn]ce, you must work piecewise.
hth
peter
===
Subject: Re: Householder QR decomposition
>I have a matrix A
>A = ( -1 -1 )
> ( 0 -a )
> ( 0 a )
>that I would like to decompose in A=QR according to
Householder method.
>I think that Q = I - 2 u transpose(u)
>but I don't know how to [CapitalThorn]nd u.
>According to what I've read, u is the vector v / norm(v)
>with v = ( 0 ) ( 0 )
> ( -a ) - ( sqrt(2)*a )
> ( a ) ( 0 )
>but it does not work here.
>Could someone explain why?
> Franck
should be
v = [0; sign(-a)*(abs(a)+sqrt(2)*abs(a)) ; a ];
or use a Givenstranformation
Q = [ 1 0 0 ; 0 -1/sqrt(2) 1/sqrt(2) ; 0 1/sqrt(2) 1/sqrt(2)
];
hth
peter
===
Subject: Re: Householder QR decomposition
> I have a matrix A
> A = ( -1 -1 )
> ( 0 -a )
> ( 0 a )
> that I would like to decompose in A=QR according to
Householder method.
> I think that Q = I - 2 u transpose(u)
> but I don't know how to [CapitalThorn]nd u.
> According to what I've read, u is the vector v / norm(v)
> with v = ( 0 ) ( 0 )
> ( -a ) - ( sqrt(2)*a )
> ( a ) ( 0 )
> but it does not work here.
> Could someone explain why?
Let's see if we can [CapitalThorn]gure out what u should 
be.
Letting ^t denote transpose, and letting e_1 denote the
vector (1 0
0)^t, we want
A = QR, so R = Q^t A = QA, since Q = Q^t.
QAe_1 is the [CapitalThorn]rst column of R, which will be a vector of the
form
(sigma 0 0)^t. Since Q is Euclidean-norm-preserving, sigma =
||Ae_1||_2 = ||(-1 0 0)^t||_2 = 1.
QAe_1 = (I - 2uu^t)Ae_1 = Ae_1 - 2u(u^t Ae_1) = (1 0 0)^t.
(u^t Ae_1) is a scalar, say alpha, so we can write
Ae_1 - 2 alpha u = (1 0 0)^t.
Thus, 2 alpha u = Ae_1 - (1 0 0)^t.
But we can determine u without knowing alpha since ||u||_2 =
1.
Therefore, normalizing Ae_1 - (1 0 0)^t gives u:
u = (Ae_1 - (1 0 0)^t) / ||Ae_1 - (1 0 0)^t||_2.
In your case, Ae_1 = (-1 0 0)^t, so Ae_1 - (1 0 0)^t = (-2 0
0)^t.
Then u = (-1 0 0)^t.
Let's check it out:
QAe_1 = [I - 2(-1 0 0)^t (-1 0 0)](-1 0 0)^t
= (-1 0 0)^t - 2(-1 0 0)^t
= (1 0 0)^t.
This is correct. Notice that column 2 of A, namely the
constant a,
does not enter into the development of u.
Dave
===
Subject: Re: Ill posed matrix-inversion
>I've a question.
>Suppose I want to solve the Eq:
>Tx = y
>This will give:
>T'Tx=T'y
>Howeever when T'T is ill-posed you will solve:
>(T'T + eI)x=T'y
>In which e is the damping.
>Now are my questions:
>1 : Can you determine the optimal-value for e when the
condition number of
>T'T is known?
optimal in which sense? cross-validation is a means , but
optimal?
if you know the level of noise in y you might try to minimize
the error bound for x .
>2: How to estimate the condition number of T'T
>Maurice
which information are you willing to provide? from the values
of T'*T
alone
you have no chance. Cholesky-decomposition ? Transformation
to tridiagoanl
form?
QR-decomposition of T?
hth
peter
===
Subject: Ill posed matrix-inversion
> I've a question.
> Suppose I want to solve the Eq:
> Tx = y
> This will give:
> T'Tx=T'y
> Howeever when T'T is ill-posed you will solve:
> (T'T + eI)x=T'y
> In which e is the damping.
> Now are my questions:
> 1 : Can you determine the optimal-value for e when the
condition
> number of
> T'T is known?
What works for lots of practical problems are the L-curve
criterion,
problems to [CapitalThorn]nd usable resources.
===
Subject: Re: Ill posed matrix-inversion
Yhanks for your answer but I'm aware of the different
techniques
availabe like L-curve, TSVD, GCV etc.
But I'm interested in the relation between the damping en 
the
ill-conditioning of the matrix.
Because f.e. in L-curve techniques you need to solve for each
step the
set of eq's for different e-values.
Furthermore I'm solving a set of eq's in the 
Fourier domain so
- They are complex
- I'm doing this for different freq's
If I can on forehand relate the value of e to the structure
of T'T this
would help me a big deal.
So do you know (or someone else) a reference or direction for
my
problem that is also understandable for no-mathematicians
(phyicist in
my case)?
Maurice
>I've a question.
>Suppose I want to solve the Eq:
>Tx = y
>This will give:
>T'Tx=T'y
>Howeever when T'T is ill-posed you will solve:
>(T'T + eI)x=T'y
>In which e is the damping.
>Now are my questions:
>1 : Can you determine the optimal-value for e when the
condition
>number of
>T'T is known?
> What works for lots of practical problems are the L-curve
criterion,
> problems to [CapitalThorn]nd usable resources.
===
Subject: Re: Ill posed matrix-inversion
> I've a question.
> Suppose I want to solve the Eq:
> Tx = y
> This will give:
> T'Tx=T'y
> Howeever when T'T is ill-posed you will solve:
> (T'T + eI)x=T'y
> In which e is the damping.
> Now are my questions:
> 1 : Can you determine the optimal-value for e when the
condition
> number of T'T is known?
> What works for lots of practical problems are the L-curve
criterion,
> problems to [CapitalThorn]nd usable resources.
As you say, these are practical rather than optimal - what I
mean
is, there is no (objective) best answer to this problem.
Similar to
SVD thresholding, where you need to de[CapitalThorn]ne a small singular
value.
===
Subject: C++ vs. JAVA
by support1.mathforum.org (8.11.6/8.11.6/The Math Forum,
$Revision:
1.9 primary) id i8DE64E12703;
I am a student of Computer Science and keenly interested in
learning
different programming languages. I have learnt C and C++ and
now
trying to learn JAVA. What I found is that JAVA almost sounds
like C++
in various ways. My question is which of these languages
offer a
better habit of programming in real-world problems and
software
development?
===
Subject: Re: C++ vs. JAVA
===
Subject: Re: C++ vs. JAVA
On a broad outlook C++ & Java look similar with same basic oo
principles. But, once you start programming, you will start
seeing the
differences and how your design very much depends on the
language you
pick. As suggested above, there is a lot to learn in mem
allocation,
garbage collection etc...Since you're just begining to learn
let me
explain the [CapitalThorn]rst simple difference (?) In java objects are
passed by
reference and in C++ you have references and pointers. This
by itself
will lead a lot of things to learn. When to use references
and when to
use pointers, which is fast, why is it fast, how is mem
allocation
done etc etc...other differences can be, Say, C++ allows
multiple
inheritance, while Java has interfaces and strictly doesnot
allow
multiple inheritance. Why? if you look from out side it seems
almost
the same, but one should be aware of every such subtle fundae
and what
it exactly mean, to become an excellent programmer or
architect.
I would highly recommend you to buy Programming with Objects A
Comparative presentation of OOP with C++ and Java by Avinash
Kak. I
would rate 5+/5 :) If you can understand & master this book
you should
be in the top20% of the lot.
--Ramoj Paruchuri
===
Subject: Re: C++ vs. JAVA
> I am a student of Computer Science and keenly interested in
learning
> different programming languages. I have learnt C and C++
and now
> trying to learn JAVA. What I found is that JAVA almost
sounds like C++
> in various ways. My question is which of these languages
offer a
> better habit of programming in real-world problems and
software
> development?
In view of the never-ending debates over languages, you
should post your
question to a language group. Prepare yourself by thoroughly
understanding
the difference between compiled and interpreted languages,
and the
subtleties of managing memory in platforms which support
dynamic memory
allocation, garbage collection, caches, etc. Then search for
information
on IEEE ßoating point math and error control.
Having done that, be prepared for the fact that many
individuals will
place familiarity with a language at higher value than other,
more
objective, performance criteria.
Good luck.
--
There are two things you must never attempt to prove: the
unprovable --
and the obvious.
--
Democracy: The triumph of popularity over principle.
--
http://www.crbond.com
===
Subject: OT ( or at least skew ;) On what group should I pose
this
question
I'm minding a hyperactive [reportedly bright ] 3rd grade boy
with
behavior problems for ~1 hour once a week ( he's bored and
disruptive in
adult meeting and (will not)/(can not) participate in age
appropriate
alternative. )
When asked what is his favorite subject, his face lights up
and replies
math!
Can some one point me to some suitable printed material that
comes in
small chunks which would get his attention. I believe he has
two basic
needs:
1. an adult taking his interests seriously
2. tackling something that gives him sense of accomplishment
===
Subject: Re: OT ( or at least skew ;) On what group should I
pose this
question
In message <10kbransfecige3@corp.supernews.com>, Richard
Owlett
>I'm minding a hyperactive [reportedly bright ] 3rd grade 
boy
with
>behavior problems for ~1 hour once a week ( he's bored and
disruptive
>in adult meeting and (will not)/(can not) participate in age
>appropriate alternative. )
How old is 3rd grade?
>When asked what is his favorite subject, his face lights up
and replies
>math!
He may well be clever then (but not necessarily at maths).
Since you
needed to ask this question it is clear that he wrong footed
you.
Try sci.math - similar questions have come up before.
>Can some one point me to some suitable printed material that
comes in
>small chunks which would get his attention. I believe he has
two basic
>needs:
> 1. an adult taking his interests seriously
> 2. tackling something that gives him sense of accomplishment
Lots of recreational mathematics puzzles have been published
in the past
from the works of Lewis Caroll through to Martin Gardeners
books. Pick a
few of the simpler practical ones and they have almost
universal appeal.
eg Hexaßexagons, Mobius strips.
Mazes, puzzles like tower of Hanoi, and strategic board games
checkers,
go and chess are other things that often appeal to budding
young
mathematics. Simple easy to learn rules with complex outcomes.
--
Martin Brown
===
Subject: Re: OT ( or at least skew ;) On what group should I
pose this
question
> In message <10kbransfecige3@corp.supernews.com>, Richard
Owlett
> I'm minding a hyperactive [reportedly bright ] 3rd grade
boy with
> behavior problems for ~1 hour once a week ( he's bored and
disruptive
> in adult meeting and (will not)/(can not) participate in
age
> appropriate alternative. )
> How old is 3rd grade?
8 or 9
> When asked what is his favorite subject, his face lights
up and
> replies math!
> He may well be clever then (but not necessarily at maths).
Since you
> needed to ask this question it is clear that he wrong
footed you.
Others had commented that they thought the boy was bright.
His comment on liking math was the [CapitalThorn]rst time we met.
I well know the difference between liking and ability [ I like
singing. Musically inclined friends prefer sitting on the far
side of
the congregation ;]
> Try sci.math - similar questions have come up before.
> Can some one point me to some suitable printed material
that comes in
> small chunks which would get his attention. I believe he
has two basic
> needs:
> 1. an adult taking his interests seriously
> 2. tackling something that gives him sense of
accomplishment
> Lots of recreational mathematics puzzles have been
published in the past
> from the works of Lewis Caroll through to Martin Gardeners
books. Pick a
> few of the simpler practical ones and they have almost
universal appeal.
> eg Hexaßexagons, Mobius strips.
> Mazes, puzzles like tower of Hanoi, and strategic board
games checkers,
> go and chess are other things that often appeal to budding
young
> mathematics. Simple easy to learn rules with complex
outcomes.
===
Subject: Re: OT ( or at least skew ;) On what group should I
pose this
question
> I'm minding a hyperactive [reportedly bright ] 3rd grade
boy with
> behavior problems for ~1 hour once a week ( he's bored and
disruptive in
> adult meeting and (will not)/(can not) participate in age
appropriate
> alternative. )
> When asked what is his favorite subject, his face lights up
and replies
> math!
> Can some one point me to some suitable printed material
that comes in
> small chunks which would get his attention. I believe he
has two basic
> needs:
> 1. an adult taking his interests seriously
> 2. tackling something that gives him sense of accomplishment
Hi
See plus.math.org for a wealth of material
As well as the many interesting, thought and experimentation
provoking,
cover exactly what you are looking for.
Good Luck
-Greg
===
Subject: Re: OT ( or at least skew ;) On what group should I
pose this
question
> I'm minding a hyperactive [reportedly bright ] 3rd grade
boy with
> behavior problems for ~1 hour once a week ( he's bored and
disruptive in
> adult meeting and (will not)/(can not) participate in age
appropriate
> alternative. )
> When asked what is his favorite subject, his face lights up
and replies
> math!
> Can some one point me to some suitable printed material
that comes in
> small chunks which would get his attention. I believe he
has two basic
> needs:
> 1. an adult taking his interests seriously
> 2. tackling something that gives him sense of accomplishment
Sorry this answer is also slightly OT, but this is a really
excellent
book I wish I had encountered when I was younger (not 3rd
grade
mind... :)
R.B.J.T. ALLENBY, RINGS, FIELDS AND GROUPS: INTRODUCTION TO
ABSTRACT
ALGEBRA
===
Subject: Re: OT ( or at least skew ;) On what group should I
pose this
question
>I'm minding a hyperactive [reportedly bright ] 3rd grade
boy with
>behavior problems for ~1 hour once a week ( he's bored and
disruptive in
>adult meeting and (will not)/(can not) participate in age
appropriate
>alternative. )
>When asked what is his favorite subject, his face lights up
and replies
>math!
>Can some one point me to some suitable printed material
that comes in
>small chunks which would get his attention. I believe he
has two basic
>needs:
> 1. an adult taking his interests seriously
> 2. tackling something that gives him sense of
accomplishment
> Sorry this answer is also slightly OT, but this is a really
excellent
> book I wish I had encountered when I was younger (not 3rd
grade
> mind... :)
> R.B.J.T. ALLENBY, RINGS, FIELDS AND GROUPS: INTRODUCTION TO
ABSTRACT
> ALGEBRA
Can you give a formal reference suitable for an inter-library
loan
request? I'm a techie living in RURAL SW MO [ they 
understand
cows,
chickens, hogs, manure, fertilizer ...] ?
===
Subject: Re: OT ( or at least skew ;) On what group should I
pose this
question
> Can you give a formal reference suitable for an
inter-library loan
> request? I'm a techie living in RURAL SW MO [ they
understand cows,
> chickens, hogs, manure, fertilizer ...] ?
Sounds like it time they learnt them some algebra ;)
Main Author: Allenby, R. B. J. T.
Title: Rings, [CapitalThorn]elds and groups : an introduction to abstract
algebra
Edition: 2nd ed.
Publisher: London : Edward Arnold, 1991 (various printings).
Physical Description: xxvi, 383 p ; 24 cm.
ISBN: 0340544406
===
Subject: Re: OT ( or at least skew ;) On what group should I
pose this
question
>Can you give a formal reference suitable for an
inter-library loan
>request? I'm a techie living in RURAL SW MO [ they
understand cows,
>chickens, hogs, manure, fertilizer ...] ?
> Sounds like it time they learnt them some algebra ;)
Request entered. They should at least see algebra in passing.
> Main Author: Allenby, R. B. J. T.
> Title: Rings, [CapitalThorn]elds and groups : an introduction to 
abstract
algebra
> Edition: 2nd ed.
> Publisher: London : Edward Arnold, 1991 (various printings).
> Physical Description: xxvi, 383 p ; 24 cm.
> ISBN: 0340544406
===
Subject: Re: OT ( or at least skew ;) On what group should I
pose this
question
> I'm minding a hyperactive [reportedly bright ] 3rd grade
boy with
> behavior problems for ~1 hour once a week ( he's bored and
disruptive in
> adult meeting and (will not)/(can not) participate in age
appropriate
> alternative. )
> When asked what is his favorite subject, his face lights up
and replies
> math!
> Can some one point me to some suitable printed material
that comes in
> small chunks which would get his attention. I believe he
has two basic
> needs:
> 1. an adult taking his interests seriously
> 2. tackling something that gives him sense of accomplishment
I would start by teaching him some algebra. Teach him about
positive and
negative numbers. Give him some problems to solve that can be
solved by
algebra. You might explain the basic rule that if two sides
of an equation
are equal and you do the same thing to both sides, they
remain equal. (Of
course you don't divide by zero. Explain 
in[CapitalThorn]nity---that will
give him
some-
thing to chew on!)
If he has a decent attention span you might try him on some
simple combina-
torial problems, such as how many ways to distribute n things
in m boxes.
(Use small de[CapitalThorn]nite values for m and n, such as m=2, n=5.)
A problem that I always enjoyed, and that should keep him
busy for a while
is the twelve pearls:
1. You have twelve identical-looking pearls. 11 have the same
weight
but the 12th differs by being either heavier or lighter than
the
others.
2. Using a simple balance, and weighing groups of pearls
against
each other, can you pick out the odd pearl and determine
whether
it is heavier or lighter than the others, using only 3
weighings?
Give him examples of Pythagorean triples such as 3,4,5 and
5,12,13.
The Pythagorean Theorem is straightforward and can be
explained visually
using
the construction shown in Brownowski's The Ascent of Man
(also on the
Web).
I would eventually introduce him to functions and the ideas
of analytic
geometry by having him graph simple functions by hand on
graph paper.
I wouldn't teach him about computer graphing programs until
he has the
idea what they are for. You might also give problems
involving the
graphical solution of pairs of equations.
Once he knows algebra, ask him to devise formulas for summing
1. the integers from 1 to N.
2. the squares of the integers from 1 to N.
Explain Zeno's Paradox and pose it in such a way that he 
adds
up
the inverse powers of 2, say:
1 + 2^{-1} + 2^{-2} + ...
---then encourage him to guess the answer.
In fact it is good pedagogy for potential mathematicians to
encourage
guessing followed by veri[CapitalThorn]cation.
The only way to [CapitalThorn]nd out what your young ward can do is to try
him out
with some of these things. He may surprise you: if he has a
serious bent
for math, he should grasp the things I mentioned as if he
were remembering
them rather than learning them from scratch. I am not such a
mathematician
myself, but I have met a few.
--
Julian V. Noble
Professor Emeritus of Physics
jvn@lessspamformother.virginia.edu
^^^^^^^^^^^^^^^^^^
http://galileo.phys.virginia.edu/~jvn/
For there was never yet philosopher that could endure the
toothache patiently.
-- Wm. Shakespeare, Much Ado about Nothing. Act v. Sc. 1.
===
Subject: Re: OT ( or at least skew ;) On what group should I
pose this
question
Am 13.09.04 21:00 schrieb Richard Owlett:
> I'm minding a hyperactive [reportedly bright ] 3rd grade
boy with
> behavior problems for ~1 hour once a week ( he's bored and
disruptive in
> adult meeting and (will not)/(can not) participate in age
appropriate
> alternative. )
> When asked what is his favorite subject, his face lights up
and replies
> math!
> Can some one point me to some suitable printed material
that comes in
> small chunks which would get his attention. I believe he
has two basic
> needs:
> 1. an adult taking his interests seriously
> 2. tackling something that gives him sense of accomplishment
Gottfried Helms
===
Subject: Re: OT ( or at least skew ;) On what group should I
pose this
question
===
> Subject: On what group should I pose this question
Well, how about sci.math? They discuss this
kind of question every once in a while.
--
Quidquid latine dictum sit, altum viditur.
===
Subject: modelling thin wire in Remcom's FDTD
X-RFC2646:  Original
Good evening everyone,
I've noticed that whenever I use Remcom's FDTD 
package to
compute the input
impedance of a thin wire dipole, that it divides the courant
stability
timestep by 2. Could anyone tell me why it does this?
Alan
===
Subject: Synergetics Section 260.42 application
The last graphic in the section Finding Roots Of Equations
Numerically
With Bucky Numbers at:
http://users.adelphia.net/~cnelson9/Links/index_lnk_11.html
which is one of the sections of Synergetics Coordinates
Applications
at:
http://users.adelphia.net/~cnelson9/
is followed by a question about the speed of the convergence
of a
general root [CapitalThorn]nding method. The guesses for the third
iteration on,
could be found with the vector equation method shown in the
section
Solving Matrix Problems Using Bucky Numbers at:
http://users.adelphia.net/~cnelson9/Links/index_lnk_10.html
I don't think Newton's method can be beat, 
I've tried and
tried, but if
anyone could beat Newton it would be Bucky Fuller. How does
the speed of
convergence compare for polynomials in general? What is the
traditional
name for this kind of root [CapitalThorn]nding method? Has it been done
with the
Cartesian coordinate system?
What is the closest to Synergetics Section 260.42? at:
http://mathworld.wolfram.com/topics/Root-Finding.html
Cliff Nelson
Dry your tears, there's more fun for your ears,
Forward Into The Past 2 PM to 5 PM, Sundays,
California time, at: http://www.kspc.org/
Don't be a square or a blockhead; see:
http://users.adelphia.net/~cnelson9/
===
Subject: c1 continuity with 2 bezier curves
I am using 2 quadratic bezier curves (3 control points each)
joint
together such that the [CapitalThorn]rst control point of the second curve
is the
same as the last control point of [CapitalThorn]rst curve).
It is clear that there is no c1 continuity at the point where
the 2
curves join.
Is there a way around this problem?
Tejas
===
Subject: thermal analysis of high speed (60,000rpm)spindles
for internal
grinding
by support1.mathforum.org (8.11.6/8.11.6/The Math Forum,
$Revision:
1.9 primary) id i8ECF6T03974;
in my problem there is shaft which is rotating at 60,000rpm,
and power
rate of 4kW. the shaft is rotating between 4 ball bearings
and on
shaft motor windings are there ( rotor) ( motor and shafts are
integral parts) as the heat is generated between the stator
and rotor
1. how to cacalculate the heat generated from motor
2. heat generated due to friction in all 4 bearings
3. heat at motor windage shearing air
- using equations ( theoretically)
then whether i need to create 3D or 2D model ( axi symmetric
)in
ansys, eventhough i have taken ansys training i could not
able get
solution plz help me in getting the solution.
===
Subject: Re: polar grid to cartesian grid
by support1.mathforum.org (8.11.6/8.11.6/The Math Forum,
$Revision:
1.9 primary) id i8ECF8J04050;
Refer to Henry Stark's (1981) paper titled 
Ôdirect fourier
reconstruction in computerized tomography' and also his 
paper
titled
Ôsampling theorem in polar coordinates' .
- Vaidyanathan Ravi Shankar.
===
Subject: Re: ANSYS - 3D problems
by support1.mathforum.org (8.11.6/8.11.6/The Math Forum,
$Revision:
1.9 primary) id i8EDi8Y13012;
hai
i am srikkanth in completed my master degree in cad now i am
searching for job. i am intrested to take some problem in
ansys and do
the analysis. i am searching for projects, if u have any
problem
please send to me
bye
k.srikkanth
>I'm Timm from RWTH-Aachen, germany.
>My task is to simulate mechanical resonances of a
3-dimensional
problem
>with extended areas of sandwich-like structures. Imagine a
(folded)
>stack capacitor or somethink simular rectangular shaped.
>In the moment I'm working on a silicon graphics workstation
(R4600,
>192MB) with ANSYS 5.1, in future maybe I'll use a Gray-J90.
>Alas, my problem description exceeds any limits: like
[CapitalThorn]lesize,
>workspace, swapspace, calculation time and so on.
>(Surely, I know, it's a 3D problem! :-) )
>Do you have some ideas to shrink the problem? Or do you know
some
>interesting links or hints?
>Timm Leuerer
>leuerer@iwe.rwth-aachen.de
===
Subject: LU Factorization of a singular matrix
Hi
I understand that only non-singular matrices have unique LU
factorization and that singular matrices could have in[CapitalThorn]nitely
many LU
Factorization. How can I prove this? Also, could anyone give
me an
example (of 3x3 matrix, say) of how singular matrix can have
in[CapitalThorn]nitely many factorizations PA = LU?
Jake
===
Subject: Re: LU Factorization of a singular matrix
To be more speci[CapitalThorn]c, how can I prove that singular matrices
could have
in[CapitalThorn]nitely many LU factorization? Also, an example of a
singular
matrix that has more than one LU factorization would be
helpful.
Jake
> Hi
> I understand that only non-singular matrices have unique LU
> factorization and that singular matrices could have
in[CapitalThorn]nitely many LU
> Factorization. How can I prove this? Also, could anyone
give me an
> example (of 3x3 matrix, say) of how singular matrix can have
> in[CapitalThorn]nitely many factorizations PA = LU?
> Jake
===
Subject: Re: LU Factorization of a singular matrix
> To be more speci[CapitalThorn]c, how can I prove that singular matrices
could have
> in[CapitalThorn]nitely many LU factorization?
Pretty easy. Take any LU factorization of a nonsingular
matrix, set
L_{nn}=0, then U_{nn} can have any value you want.
V.
--
email: lastname at cs utk edu
homepage: www cs utk edu tilde lastname
===
Subject: Re: LU Factorization of a singular matrix
> Hi
> I understand that only non-singular matrices have unique LU
> factorization
False on several counts:
(1) If you allow permutations (PA=LU) then even a
non-singular matrix can
have more than one factorization (depending on the pivoting
strategy).
(2) I understand that there is a tacit assumption that one of
the matrices
L, U has unit diagonal (say L); if you relax this assumption,
you can have
in[CapitalThorn]nitely many factorizations.
(3) If you insist that P=I, then some non-singular matrices
will fail to
have LU-factorization (consider A(1,1)=0).
> and that singular matrices could have in[CapitalThorn]nitely many LU
> Factorization. How can I prove this? Also, could anyone
give me an
> example (of 3x3 matrix, say) of how singular matrix can have
> in[CapitalThorn]nitely many factorizations PA = LU?
What is closer to uniqueness is this: suppose A is an m-by-n
matrix of
rank r>0, then P*A=L*U where L is m-by-r lower triangular
with unit
diagonal and R is r-by-n upper triangular with non-singular
diagonal. For
at least one permutation matrix P, the factorization will
exist, and with
the given P, the L and U will be unique.
If your A is square (m=n) and you look for non-unique LU with
L and U
square, you pad it with something that will conform to lower
(upper)
triangular form and cancels out. Example:
A =
[2 3 2 ]
[2 3 2 ]
[2 3 2 ]
P = I
L =
[1 0 0]
[1 1 0]
[1 c 1]
U =
[2 3 2]
[0 0 0]
[0 0 0]
and c can be arbitrary.
But it's a waste: this A is actually U*V where
U =
[1 ]
[1 ]
[1 ]
V =
[2 3 2]
> Jake
===
Subject: Parallel sparse cholesky
I'm testing some new load balancing policies designed to
heterogeneous
clusters. Now, I'm seeking the appropriate parallel
applications to
serve as benchmarks to evaluate the policy.
One option would be the Sparse Cholesky Factorization. After
reading
some papers, I decided that this application would be a good
benchmark, although few papers have been published on the use
of
workstation clusters for executing this type of application.
I'd like to ask if someone could provide a message-passing
sparse
cholesky implementation. I'm using PVM on the tests, but the
implementation does not need to be implemented using PVM. It
can be in
any language, or even pseudo-language, provided that use the
message-passing paradigm.
William
===
Subject: Re: Parallel sparse cholesky
> I'd like to ask if someone could provide a message-passing
sparse
> cholesky implementation.
Petsc does not have a parallel Cholesky. Pspases is a direct
solver; if
you are looking for an incomplete factorization look at
BlockSolve95.
V.
--
email: lastname at cs utk edu
homepage: www cs utk edu tilde lastname
===
Subject: Re: Parallel sparse cholesky
>I'm testing some new load balancing policies designed to
heterogeneous
>clusters. Now, I'm seeking the appropriate parallel
applications to
>serve as benchmarks to evaluate the policy.
>One option would be the Sparse Cholesky Factorization. After
reading
>some papers, I decided that this application would be a good
>benchmark, although few papers have been published on the
use of
>workstation clusters for executing this type of application.
>I'd like to ask if someone could provide a message-passing
sparse
>cholesky implementation. I'm using PVM on the tests, but 
the
>implementation does not need to be implemented using PVM. It
can be in
>any language, or even pseudo-language, provided that use the
>message-passing paradigm.
>William
some entry says:
Re: sparse matrix computations
... PETSc is a rather comprehensive package for all things
matrix:
http://www.mcs.anl.gov/petsc
You can use it from both C and Fortran. ...
and from my annotations:
begin{citation}
PSPASES : A Scalable Parallel Direct Solver Library for
Sparse Symmetric
Positive De[CapitalThorn]nite Systems
http://www.cs.umn.edu/~mjoshi/pspases
by Mahesh Joshi, George Karypis, Vipin Kumar
Department of Computer Science, University of Minnesota,
Mineapolis.
&
Anshul Gupta
IBM Thomas J. Watson Research Center, Yorktown Heights, New
York.
We are glad to announce a new version (1.0) of PSPASES.
PSPASES is a stand-alone MPI-based parallel library for
solving linear
systems of equations involving sparse symmetric positive
de[CapitalThorn]nite matrices.
The library ef[CapitalThorn]ciently implements the scalable parallel
algorithms
developed by authors, for each of the four phases of direct
solution
method; viz. ordering, symbolic factorization, numerical
Cholesky
factorization, and solution of triangular systems.
PSPASES is implemented using standard MPI and BLAS calls,
which makes it
portable to most of today's parallel computers and networks 
of
workstations. We have tested PSPASES extensively on IBM SP,
network of
IBM RS6000 workstations, Cray T3E, SGI Origin 2000 and
PowerChallenge
architectures.
PSPASES uses ParMETIS and METIS as default libraries for
computing [CapitalThorn]ll-
reducing ordering, but it also accepts user supplied
ordering. Different
functional interfaces are provided for each of the phases of
the solver
and a simple interface is also provided for easy use. The
user can use
these interfaces to solve multiple systems with same nonzero
structures;
to solve same system for multiple right hand sides; and to
get different
statistical information such as the memory requirements of
the solver
and the quality of the ordering.
Visit the PSPASES web site at
http://www..cs.umn.edu/~mjoshi/pspases
to obtain the software, the manual, related technical papers,
usage notes,
and to see some performance benchmarking results. Please read
the terms
and conditions for use, given at the website, before
downloading and
using the PSPASES software.
If your machine does not have access to the web, then you may
obtain the
software via an anonymous ftp to ftp.cs.umn.edu. Change the
directory
to
users/kumar/mahesh and get the [CapitalThorn]les pspases-1.0.tar.gz,
README.INSTALL,
and README.USAGE (remember to use binary transfer mode).
Any comments, questions or bugs regarding PSPASES can be
directed to
Mahesh Joshi (mjoshi@cs.umn.edu).
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
hth
peter
===
Subject: unique number group between 1-1000
Hi! I'm doing a sweepstake:
http://www.standardsbigbid.com/londonapartment/php/Help.php#
works
and thought this would be the best news group to discuss the
possibilities...I know for a fact (via other news groups/
forums) that the
following amounts, so far, are not unique:
1.00, 1.01, 1.11,1.13,
1.18 1.45, 1.49, 1.75,
2.00, 2.33, 2.63,
3.00, 3.11, 3.26, 3.57,
4.00, 4.64, 5.00,
5.235.73, 6.00,
6.17,6.19, 7.00, 7.58,
7.83, 8,00, 8.58, 8.71,
9.57, 9.87, 10.00,
11.00, 11.14, 11.17,
11.63,11.71, 11.73, 12.00,
12.27, 13.00,
13.02, 13.13, 13.18, 14.00,
14.02, 14.89, 15,00, 15.56,
19.76,
20.00, 27.47, 29.09 31.91
theshak
===
Subject: Re: How to [CapitalThorn]nd the cube root of a number ( By Vedic
maths)
by support1.mathforum.org (8.11.6/8.11.6/The Math Forum,
$Revision:
1.9 primary) id i8FDPjf08900;
how do you [CapitalThorn]nd the cubed root of a number, and the 4th root
and the
5th root, etc... what does it mean do [CapitalThorn]nd the root of a
number...what
are you doing?
===
Subject: Re: How to [CapitalThorn]nd the cube root of a number ( By Vedic
maths)
>how do you [CapitalThorn]nd the cubed root of a number, and the 4th root
and the
>5th root, etc... what does it mean do [CapitalThorn]nd the root of a
number...what
>are you doing?
given number a not = 0 , [CapitalThorn]nd number x such that x^n=a, n the
given degree.
(there are always n solutions, some complex also for positive
a if n>=3)
hth
peter
===
Subject: Calling FFTW from C++? (x-posted)
I am currently working on a program using FFTW->
http://www.fftw.org .
My basic C program to compute the 2D Fourier transform of a
bunch of
data works [CapitalThorn]ne when I compile it with gcc. However, I would
like to
Unfortunately, I cannot get my program to compile with g++.
I compile it as follows:
g++ ft2d.c -o ft2d -I/usr/local/include -L /usr/local/lib
-lfftw3 -lm
The errors I get are:
ft2d.c: In function `int main()':
ft2d.c:13: invalid conversion from `void*' to `double 
(*)[2]'
ft2d.c:14: invalid conversion from `void*' to `double 
(*)[2]'
Here is my code:
#include <fftw3.h>
#include <math.h>
int main(void)
{
fftw_complex *in, *out;
fftw_plan p;
int nx = 5;
int ny = 5;
int i;
/* Allocate the input and output arrays, and create the plan
*/
in = fftw_malloc(sizeof(fftw_complex) * nx * ny);
out = fftw_malloc(sizeof(fftw_complex) * nx * ny);
p = fftw_plan_dft_2d(nx, ny, in, out, FFTW_FORWARD,
FFTW_ESTIMATE);
/* Now [CapitalThorn]ll the input array with the input data */
/* We'll simply make a sine wave */
for (i = 0; i < (nx * ny); i++) {
in[i][0] = sin(i / 10.0 * M_PI); /* Real part */
in[i][1] = 0.0; /* Imaginary part */
}
/* Actually execute the plan on the input data */
fftw_execute(p);
/* Print out the results */
for (i = 0; i < (nx * ny); i++)
printf(Coef[CapitalThorn]cient %d: %f%+f*in, i, out[i][0], out[i][1]);
/* Clean up after ourselves */
fftw_destroy_plan(p);
fftw_free(in); fftw_free(out);
return 0;
}
Eventually I will do something more complex than a sine wave,
but you
get the idea.
If anyone knows what the problem is, I would greatly
appreciate any
help. This is all that fftw.org has about calling it from C++:
http://fftw.org/faq/section2.html#cplusplus
--Erica
===
Subject: Re: Calling FFTW from C++? (x-posted)
> My basic C program to compute the 2D Fourier transform of a
bunch of
> data works [CapitalThorn]ne when I compile it with gcc. However, I would
like to
> Unfortunately, I cannot get my program to compile with g++.
> I compile it as follows:
> g++ ft2d.c -o ft2d -I/usr/local/include -L /usr/local/lib
-lfftw3 -lm
> The errors I get are:
> ft2d.c: In function `int main()':
> ft2d.c:13: invalid conversion from `void*' to `double
(*)[2]'
> ft2d.c:14: invalid conversion from `void*' to `double
(*)[2]'
IMHO if you want to mix C and C++ in a project, you should
leave your C [CapitalThorn]les as .C [CapitalThorn]les, (and compile them 
with gcc not
g++).
If you include the corresponding .H [CapitalThorn]le from a C++ source 
[CapitalThorn]le
do this:
extern C {
#include ft2d.h
};
Alternatively you could write a wrapper [CapitalThorn]le that uses this,
that would avoid having to do this in every other unit that
has
to include the C header.
There's a chapter in the c.l.c++ FAQ about mixing C and C++ 
in
projects, have a read of it and see if there is any more
advice.
===
Subject: Re: Calling FFTW from C++? (x-posted)
> I am currently working on a program using FFTW->
http://www.fftw.org .
> My basic C program to compute the 2D Fourier transform of a
bunch of
> data works [CapitalThorn]ne when I compile it with gcc. However, I would
like to
> Unfortunately, I cannot get my program to compile with g++.
> I compile it as follows:
> g++ ft2d.c -o ft2d -I/usr/local/include -L /usr/local/lib
-lfftw3 -lm
> The errors I get are:
> ft2d.c: In function `int main()':
> ft2d.c:13: invalid conversion from `void*' to `double
(*)[2]'
> ft2d.c:14: invalid conversion from `void*' to `double
(*)[2]'
You don't note in your code what lines 13 and 14 are, but if
I'm counting
correctly, the errors are on the fftw_malloc lines, right?
You probably
need to cast the results of those function calls to
(fftw_complex *). It
appears that the fftw_malloc call returns a void*, and the
compiler is
complaining about that type not matching your in and out
variables'
declared
type.
-Howard
> Here is my code:
> #include <fftw3.h>
> #include <math.h>
> int main(void)
> fftw_complex *in, *out;
> fftw_plan p;
> int nx = 5;
> int ny = 5;
> int i;
> /* Allocate the input and output arrays, and create the
plan */
> in = fftw_malloc(sizeof(fftw_complex) * nx * ny);
> out = fftw_malloc(sizeof(fftw_complex) * nx * ny);
===
Subject: Re: Calling FFTW from C++? (x-posted)
> I am currently working on a program using FFTW->
http://www.fftw.org .
> My basic C program to compute the 2D Fourier transform of
a bunch of
> data works [CapitalThorn]ne when I compile it with gcc. However, I
would like to
> Unfortunately, I cannot get my program to compile with
g++.
> I compile it as follows:
> g++ ft2d.c -o ft2d -I/usr/local/include -L /usr/local/lib
-lfftw3 -lm
> The errors I get are:
> ft2d.c: In function `int main()':
> ft2d.c:13: invalid conversion from `void*' to `double
(*)[2]'
> ft2d.c:14: invalid conversion from `void*' to `double
(*)[2]'
> You don't note in your code what lines 13 and 14 are, but
if I'm counting
> correctly, the errors are on the fftw_malloc lines, right?
You probably
> need to cast the results of those function calls to
(fftw_complex *). It
> appears that the fftw_malloc call returns a void*, and the
compiler is
> complaining about that type not matching your in and out
variables'
declared
> type.
> -Howard
> Here is my code:
> #include <fftw3.h>
> #include <math.h>
> int main(void)
> {
> fftw_complex *in, *out;
> fftw_plan p;
> int nx = 5;
> int ny = 5;
> int i;
> /* Allocate the input and output arrays, and create the
plan */
> in = fftw_malloc(sizeof(fftw_complex) * nx * ny);
> out = fftw_malloc(sizeof(fftw_complex) * nx * ny);
===
Subject: Re: Calling FFTW from C++? (x-posted)
X-RFC2646:  Original
> I am currently working on a program using FFTW->
http://www.fftw.org .
> My basic C program to compute the 2D Fourier transform of a
bunch of
> data works [CapitalThorn]ne when I compile it with gcc. However, I would
like to
> Unfortunately, I cannot get my program to compile with g++.
> I compile it as follows:
> g++ ft2d.c -o ft2d -I/usr/local/include -L /usr/local/lib
-lfftw3 -lm
> The errors I get are:
> ft2d.c: In function `int main()':
> ft2d.c:13: invalid conversion from `void*' to `double
(*)[2]'
> ft2d.c:14: invalid conversion from `void*' to `double
(*)[2]'
This is one of the differences between C++ and C. In C void*
automatically
converts to any other pointer, in C++ you must cast.
> Here is my code:
> #include <fftw3.h>
> #include <math.h>
> int main(void)
> fftw_complex *in, *out;
> fftw_plan p;
> int nx = 5;
> int ny = 5;
> int i;
> /* Allocate the input and output arrays, and create the
plan */
> in = fftw_malloc(sizeof(fftw_complex) * nx * ny);
in = (fftw_complex*)fftw_malloc(sizeof(fftw_complex) * nx *
ny);
> out = fftw_malloc(sizeof(fftw_complex) * nx * ny);
out = (fftw_complex*)fftw_malloc(sizeof(fftw_complex) * nx *
ny);
That should do it.
john
===
Subject: Simple oscillator numerical integration
I need to integrate (numericaly) the well know simple
oscillator
equation
x''+ 2 csi w x' + w^2 x = f(t)
where:
x is the displacement
x'the velocity
x'' the acceleration
w is the frequency
csi is the oscillator dumping
f(t) is the external applied force
The problem arise in the case whe add a dumping force in the
form:
fd(t)=C x'^a
where:
C is a constant
a is a exponent usualy in the range 0.1 to 1
In my situation the dumpers work like a friction dumper force
a=0.1
(similar to a diriclet force depending of the sign of x').
The full equation become:
x''+ 2 csi w x' + w^2 x = 
f(t)+C x'^a
I try some tecnique:
* standards integrator (newmark o theta wilson) delaing the
dumping
force in the next step.
* Runge Kutta decoupling the system in a system of two [CapitalThorn]rst
order
ODE..
but, in any case, the result is bad because:
* the response of the dumper can be very strong respect to
the applied
force and the system become unstable. Also is outside my
aspectative
that the force in the dumper is great than the applied force.
* at the end of f(t) application the system oscillate for a
unde[CapitalThorn]ned
time around a con[CapitalThorn]guration and dont show the classical
oscillation
decay of csi dumped system (probably for the dif[CapitalThorn]culty tho
capture
the correct sign of velocity...)
Any one as a suggestion? Some relaxing technique?
Pepito
===
Subject: Re: Simple oscillator numerical integration
> I need to integrate (numericaly) the well know simple
oscillator
> equation
> x''+ 2 csi w x' + w^2 x = 
f(t)
< ... >
> Any one as a suggestion? Some relaxing technique?
> Pepito
For highly-nonlinearly damped systems, it is a good idea to
try an
explicit integrator [CapitalThorn]rst, since you dont need to worry about
iterating a corrector (which is largely meaningless in path
dependent problems).
But the best choice depends on the application. I had
good results with CD for dry friction damping, in which there
is a
signi[CapitalThorn]cant hysteretic cycle.
In any case, the key step is: REDUCE TO FIRST ORDER. Methods
designed to work directly with SO systems often fail
miserably for
nonlinear damping.
===
Subject: Re: Simple oscillator numerical integration
>I need to integrate (numericaly) the well know simple
oscillator
>equation
>x''+ 2 csi w x' + w^2 x = 
f(t)
>where:
>x is the displacement
>x'the velocity
>x'' the acceleration
>w is the frequency
>csi is the oscillator dumping
>f(t) is the external applied force
>The problem arise in the case whe add a dumping force in the
form:
>fd(t)=C x'^a
>where:
>C is a constant
>a is a exponent usualy in the range 0.1 to 1
>In my situation the dumpers work like a friction dumper
force a=0.1
>(similar to a diriclet force depending of the sign of x').
>The full equation become:
>x''+ 2 csi w x' + w^2 x = 
f(t)+C x'^a
>I try some tecnique:
>* standards integrator (newmark o theta wilson) delaing the
dumping
>force in the next step.
>* Runge Kutta decoupling the system in a system of two [CapitalThorn]rst
order
>ODE..
>but, in any case, the result is bad because:
>* the response of the dumper can be very strong respect to
the applied
>force and the system become unstable. Also is outside my
aspectative
>that the force in the dumper is great than the applied force.
>* at the end of f(t) application the system oscillate for a
unde[CapitalThorn]ned
>time around a con[CapitalThorn]guration and dont show the classical
oscillation
>decay of csi dumped system (probably for the dif[CapitalThorn]culty tho
capture
>the correct sign of velocity...)
>Any one as a suggestion? Some relaxing technique?
>Pepito
there are three points:
a in [0.1,1] you are sure? if you write this in the standard
form you get
x'' 
+r(x,x',t)*x' + w^2 * x = f(t)
r(x,x',t)=(2*csi*w -C*abs(x')^{a-1})
r is the damping factor
this means if abs(x') becomes small, then there is no 
damping
but a
strong acceleration? I had thougth of something like C<0 and
a>1 ?
secondly, if strong damping takes place, then the stepsize
must be reduced
drastically in order to remain in the stable regime of the
method.
thirdly, the integrators you mention assume a high order
differentiable
x which is not the case if x' changes sign. hence if this
occurs you must
carefully detect time where this takes place and restart the
integrator
there,
otherwise again there will be an additional oscillating
effect in the
numerical solution.
hth
peter
===
Subject: Re: maximum entropy method
by support1.mathforum.org (8.11.6/8.11.6/The Math Forum,
$Revision:
1.9 primary) id i8GC6PO32543;
I want souce code for maximum entropy method for signal
restoration/estimation
Is any body has material on that
===
Subject: best software environment for numerical analysis
I am about to embark on a software project with intensive
numerical
analysis and i have been out of it(ie software) for a while.
below is my project but i wanted to know what the best
language to use
is. My best guess is to use c++ and use kdevelop 3.1 on Suse
Linux
9.1, and that i will be able to compile other people's C or
C++ code
from the internet. (other linux distro's
15 years ago i did a fortran simulation utilised in
diffraction
co-ef[CapitalThorn]cients for antenna desgn. now i assume all the fortran
stuff is
now essentially available in C or C++(or at least the stuff i
need).
I assume for numerical analysis, from what i have recently
read on the
internet, delphi and Python are NOT the way to go.
Project Description: I am writing a blackjack simulation for
card
counting anaylis. will be calculations of 0% advantage,
con[CapitalThorn]dence
intervals variance etc for numerous rule variations of a game
that is
well suited to object orientated design. Comparisons of actual
recorded results with expected results etc.
So my questions are: any info appreciated. :) :)
1. Delphi/python and other high level langauges are NOT the
way to go?
2. C++ is roughly on the right track?
3. kdevelop will give me reasonable CASE tools etc/ software
engineering environment? (i want to spend the least amount of
time
doing the software)
4. If i wanted to get the best tools available does that mean
spending
lots of $$$$ on tools like Rational or other. is this right?
just how
bad is kdevelop compared to these other tools?
===
Subject: Re: best software environment for numerical analysis
>I am about to embark on a software project with intensive
numerical
>analysis and i have been out of it(ie software) for a while.
>I assume for numerical analysis, from what i have recently
read on the
>internet, delphi and Python are NOT the way to go.
It appears that you intend to write software which is really
for a one
off calculation albeit for a number of scenarios.
Modern personal computers are incredibly fast. Apparently the
new Mac
G5 is on par with a Cray supercomputer from 1985.
Given how fast computers are and how slow programmers are for
a one
off calculation you should be looking for a language which
will be
easy to program in and which is not prone to the introduction
of
errors.
If you are already familiar with Pascal then you should
seriously
consider using Delphi. I have been playing around with some
computation in Delphi. I was concerned with performance so I
replaced
the innermost loop (doing DAXPY) with inline assembler and
got only a
small improvement even though that loop consumed a majority
of the
total time for the whole program.
I do not think you will suffer much or any performance
penalty using
Delphi over C or C++.
===
Subject: Re: best software environment for numerical analysis
>I am about to embark on a software project with intensive
numerical
>analysis and i have been out of it(ie software) for a while.
>I assume for numerical analysis, from what i have recently
read on the
>internet, delphi and Python are NOT the way to go.
>It appears that you intend to write software which is really
for a one
>off calculation albeit for a number of scenarios.
>Modern personal computers are incredibly fast. Apparently
the new Mac
>G5 is on par with a Cray supercomputer from 1985.
This might be, but they are far from ef[CapitalThorn]cient for quite
a few types of problems which were much better handled
before the desire for speed was used to throw out good
mathematical instructions.
>Given how fast computers are and how slow programmers are
for a one
>off calculation you should be looking for a language which
will be
>easy to program in and which is not prone to the
introduction of
>errors.
This is assuming that the operations wanted are even
present in the language. Low-accuracy computations of
the type envisioned by the current generation of hardware
and software designers (the present double precision)
is really a misnomer), no matter how fast, do not do a
reasonable job of handling different or more accurate
computations.
--
This address is for information only. I do not claim that
these views
are those of the Statistics Department or of Purdue
University.
Herman Rubin, Department of Statistics, Purdue University
hrubin@stat.purdue.edu Phone: (765)494-6054 FAX: (765)494-0558
===
Subject: Re: best software environment for numerical analysis
> I assume for numerical analysis, from what i have recently
read on
> the internet, delphi and Python are NOT the way to go.
before you discount Delphi, you might want to take a look at
some of the things that folks have done with it in the [CapitalThorn]elds
of science/engineering/mathematics...
see http://www-rab.larc.nasa.gov/nmp/fNMPapps.htm
--
Mark Vaughan
____________
Visit the Numerical Methods in Pascal web page at
http://www-rab.larc.nasa.gov/nmp/fNMPhome.htm
===
Subject: Re: best software environment for numerical analysis
>I am about to embark on a software project with intensive
numerical
>analysis and i have been out of it(ie software) for a while.
>below is my project but i wanted to know what the best
language to use
>is. My best guess is to use c++ and use kdevelop 3.1 on Suse
Linux
>9.1, and that i will be able to compile other people's C or
C++ code
>from the internet. (other linux distro's
>15 years ago i did a fortran simulation utilised in
diffraction
>co-ef[CapitalThorn]cients for antenna desgn. now i assume all the fortran
stuff is
>now essentially available in C or C++(or at least the stuff
i need).
Fortran is still OK. You can get it free from Salford Software
http://www.salfordsoftware.co.uk/compilers/support/
downloads.html
You can get both 95 and 77 free.
A.L.
===
Subject: Re: best software environment for numerical analysis
>I am about to embark on a software project with intensive
numerical
>analysis and i have been out of it(ie software) for a while.
>below is my project but i wanted to know what the best
language to use
>is. My best guess is to use c++ and use kdevelop 3.1 on
Suse Linux
>9.1, and that i will be able to compile other people's C or
C++ code
>from the internet. (other linux distro's
>15 years ago i did a fortran simulation utilised in
diffraction
>co-ef[CapitalThorn]cients for antenna desgn. now i assume all the
fortran stuff is
>now essentially available in C or C++(or at least the stuff
i need).
>Fortran is still OK. You can get it free from Salford
Software
>http://www.salfordsoftware.co.uk/compilers/support/
downloads.html
>You can get both 95 and 77 free.
The Salford Fortran compiler is free for personal use, but it
is a Windows
product, and the OP intends to use Linux. The free Fortran 95
compilers for
Linux are
Intel (for non-commercial use)
http://www.intel.com/software/products/compilers/ßin/
G95
http://www.g95.org
gfortran
http://www.gfortran.org/
G95 and gfortran are not complete but are under active
development. So far,
G95 runs all my F95 standard-conforming codes.
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World! >100,000
Newsgroups
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===
Subject: Re: best software environment for numerical analysis
>I am about to embark on a software project with intensive
numerical
>analysis and i have been out of it(ie software) for a while.
>below is my project but i wanted to know what the best
language to use
>is. My best guess is to use c++ and use kdevelop 3.1 on Suse
Linux
>9.1, and that i will be able to compile other people's C or
C++ code
>from the internet. (other linux distro's
>15 years ago i did a fortran simulation utilised in
diffraction
>co-ef[CapitalThorn]cients for antenna desgn. now i assume all the fortran
stuff is
>now essentially available in C or C++(or at least the stuff
i need).
You were probably using Fortran 77. Fortran 90 and 95 have
greatly improved
the language. For code using arrays, especially
multidimensional arrays,
it easier to work in Fortran 95 than C++ IMO. Fortran 90 is a
strict
superset
of F77 and Fortran 95 is a superset for practical purposes
(F95 comopilers
still support the few F77 features of[CapitalThorn]cially deleted), so
your F77
knowledge
is still good.
There is a free Fortran 95 compiler for Linux, called G95, at
http://www.g95.org
. I have used it on SuSE Linux 9.0 .
>I assume for numerical analysis, from what i have recently
read on the
>internet, delphi and Python are NOT the way to go.
Unless speed is very important, I think Python is [CapitalThorn]ne,
especially in
conjunction
with the Numeric or Numarray modules and SciPy. There are
many math
libraries
callable from Python, and it is possible to call C and
Fortran libraries.
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===
Subject: Re: best software environment for numerical analysis
>You were probably using Fortran 77.
yes it was....
>Fortran 90 and 95 have greatly improved
>the language. For code using arrays, especially
multidimensional arrays,
>it easier to work in Fortran 95 than C++ IMO. Fortran 90 is
a strict
superset
>of F77 and Fortran 95 is a superset for practical purposes
(F95 comopilers
>still support the few F77 features of[CapitalThorn]cially deleted), so
your F77
knowledge
>is still good.
>There is a free Fortran 95 compiler for Linux, called G95, at
http://www.g95.org
> I have used it on SuSE Linux 9.0 .
i have found using arrays within pascal to be messy at the
time.
I have become jack of all trades and master of none. all a
long time
ago. i have done a Ôbit' of: assembly language 
in DOS,
fortran77 on
solaris, some embedded ADA, DOS pascal, EXCEL Visual Basic. I
want to
try industry standard that will be here in a while.
A fair amount of tricky code, and therefor at risk of error,
is in the
play of the hands. many different rules at different casinos,
slightly
different. hence object orientation techniques - data
abstraction,
infomation hiding etc are very important. eg shoe.shufße, or
shoe.ßip_next_card, shoe.number_of_decks=6.
I know ADA and pascal can pass functions as variables to
procedures.
this type of stuff was useful last time i tried. i assume you
can do
this in C++ as well.
>Unless speed is very important, I think Python is [CapitalThorn]ne,
especially in
conjunction
>with the Numeric or Numarray modules and SciPy. There are
many math
libraries
>callable from Python, and it is possible to call C and
Fortran libraries.
speed is important do to the high variance of the game and
hence
number of simulations required to determine the long run. but
there
is more than one way to skin a cat. maybe i can scrouge CPU
usage from
somewhere else.
The [CapitalThorn]rst part of my analysis will be to do probabilties as
the count
varies for different hands. the statistical aspects of
variance etc
will be later.
I get the impression Python can be useful as a glue language
to call
routines written in other languages. this in itself seems
useful even
if most of the code is implented in other languages.
can python call ADA stuff as well?
I have been doing some searching on the internet to compare
languages
but i am using internet cafes at the moment so internet
access is
limited. my usenet server has a 24hour'ish delay on updating
posted
messages.(astraweb.com) . so there is a delay to read your
guys tips.
>http://www.newsfeed.com The #1 Newsgroup Service in the
World! >100,000
Newsgroups
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===
Subject: Re: best software environment for numerical analysis
> [...]
> speed is important do to the high variance of the game and
hence
> number of simulations required to determine the long run.
but there
> is more than one way to skin a cat. maybe i can scrouge CPU
usage from
> somewhere else.
> [...]
> I get the impression Python can be useful as a glue
language to call
> routines written in other languages. this in itself seems
useful even
> if most of the code is implented in other languages.
I once used Python + C to do some analysis on large images.
It was
disappointing: I ended up re-writing essentially all of the
code in C.
At [CapitalThorn]rst, I prototyped in Python, but by the time it took the
prototype to [CapitalThorn]nish its test run on real data I usually had
the C
implementation ready (which ran 100-1000x faster). I also
didn't like
that I had to do the memory management manually when
interfacing C to
Python.
I also tried the pynumerics libraries. But as my code didn't
vectorize well results were not satisfying. Also, I found the
pointer
semantics of pynumeric's vectors and matrices annoying:
Octave/
Matlab's value semantics seems more natural to me, and
side-effects
are easier to control.
Today I use Python for scripting non-time-critical stuff only.
===
Subject: Re: best software environment for numerical analysis
> I get the impression Python can be useful as a glue
language to call
> routines written in other languages. this in itself seems
useful even
> if most of the code is implented in other languages.
Yes, I am familar with doing this with C (a little).
> can python call ADA stuff as well?
Never tried it.
-- Lou Pecora (my views are my own)
===
Subject: Re: best software environment for numerical analysis
by support1.mathforum.org (8.11.6/8.11.6/The Math Forum,
$Revision:
1.9 primary) id i8HBajY21887;
>There is a free Fortran 95 compiler for Linux, called G95,
at <a
href=http://www.g95.org>http://www.g95.org</a>
GNU gcc-4.0 will include gfortran instead of g77. This is
available,
as a beta release, either from the gcc snapshots or as
Linux/Cygwin
binaries at http://f77.linksysnet.com.
G95 and gfortran forked nearly two years ago - don't ask me
why
because I wasn't there! It does seem like a waste of scarce
effort,
though.
>Unless speed is very important, I think Python is [CapitalThorn]ne,
especially in
conjunction
>with the Numeric or Numarray modules and SciPy. There are
many math
libraries
>callable from Python, and it is possible to call C and
Fortran
libraries.
This choice is highly subjective and dependent on what you
are trying
to do.
(i) For high performance computing and large codes, fortran
or C are
the way to go. Both have extensive libraries, automatic
vectorizers
etc., etc..
(ii) For day to day, medium scale calculations/simulations,
the
various interpreted languages, mentioned in this thread, are
ideal.
(iii) For syntactic simplicity in numerical calculations, with
rasonable performance, matlab/octave/Scilab/FreeMat/... are
ideal.
(iv) Python with SciPy, R and all the rest were designed for
other
things and, in their present manifestations, can only be
described
(IMO) as being clumsy for numerical work - particularly if
you are
coming from a fortran world.
(v) Frankly, matlab is the best; both in terms of performance
and
libraries. However, you have to have a VERY, VERY deep pocket.
(vi) For compatibility and performance, octave is next best
and is
FREE. It does lack the fancy graphics and the GUI but, as has
already
been said on this thread, that can be dealt with by other
means and
will change in the future. Above all octave has good
libraries and
code written for Matlab runs on octave, pretty much as it
comes. I
use Matlab at work and octave at home. Code transfers between
the two
with no problem.
(vii) Scilab is also good, particularly in version 3, but has
diverged
substantially from matlab. As David Bateman said, the license
has a
commercial restriction: .... Any commercial use or
circulation of the
DERIVED( or COMPOSITE) SOFTWARE shall have been previously
authorized
by INRIA and ENPC..... .
Paul T
===
Subject: Re: best software environment for numerical analysis
>There is a free Fortran 95 compiler for Linux, called G95,
at <a
>href=http://www.g95.org>http://www.g95.org</a>
>GNU gcc-4.0 will include gfortran instead of g77. This is
available,
>as a beta release, either from the gcc snapshots or as
Linux/Cygwin
>binaries at http://f77.linksysnet.com.
>G95 and gfortran forked nearly two years ago - don't ask me
why
>because I wasn't there! It does seem like a waste of scarce
effort,
>though.
>Unless speed is very important, I think Python is [CapitalThorn]ne,
especially in
>conjunction
>with the Numeric or Numarray modules and SciPy. There are
many math
>libraries
>callable from Python, and it is possible to call C and
Fortran
>libraries.
>This choice is highly subjective and dependent on what you
are trying
>to do.
>(i) For high performance computing and large codes, fortran
or C are
>the way to go. Both have extensive libraries, automatic
vectorizers
>etc., etc..
>(ii) For day to day, medium scale calculations/simulations,
the
>various interpreted languages, mentioned in this thread, are
ideal.
>(iii) For syntactic simplicity in numerical calculations,
with
>rasonable performance, matlab/octave/Scilab/FreeMat/... are
ideal.
Fortran 95 has array functionality comparable to Matlab, and
I [CapitalThorn]nd its
syntax
to pretty simple. The question is whether you prefer a
statically typed
language
like Fortran, with explicit declarations -- I do.
>(iv) Python with SciPy, R and all the rest were designed for
other
>things and, in their present manifestations, can only be
described
>(IMO) as being clumsy for numerical work - particularly if
you are
>coming from a fortran world.
I came to Python from Fortran (which I still use) and have
not found it
clumsy,
just slower for number-crunching. The lower speed is usually
not a problem.
>(v) Frankly, matlab is the best; both in terms of
performance and
>libraries. However, you have to have a VERY, VERY deep
pocket.
Deep pockets to use it on one machine, and even deeper
pockets to create
a stand-alone executable with the Matlab compiler, which
costs $20,000 for
a U.S. commercial license -- a colleague just asked about
this.
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Subject: Re: best software environment for numerical analysis
>(v) Frankly, matlab is the best; both in terms of
performance and
>libraries. However, you have to have a VERY, VERY deep
pocket.
> Deep pockets to use it on one machine, and even deeper
pockets to create
> a stand-alone executable with the Matlab compiler, which
costs $20,000
for
> a U.S. commercial license
And fork out another ten grand or so if you'd like to run it
in
real-time. Not bad pricing (gouging) strategy for a *[CapitalThorn]x* to
make Matlab
usable for real, non homework, calculations.
===
Subject: Re: best software environment for numerical analysis
by support1.mathforum.org (8.11.6/8.11.6/The Math Forum,
$Revision:
1.9 primary) id i8GFdee20816;
Phil:
Forever philosophical be your name for it proves the
existence of odds
and probability!
I am myself not only a blackjack player, but also a gamblers
gambler.
I have authored a huge web site dedicated to gambling science,
including blackjack. The main blackjack page is:
http://www.saliu.com/blackjack.html
(Dont be disappointed by the cold mathematical truth on card
counting!)
I have done also a lot of programming in gambling, lottery,
mathematics, statistics, etc. I use Visual Basic sometimes.
But my
tool of choice is the 32-bit Basic compiler from PowerBasic.
It really
is powerful. My FTP download site has dozens of free programs
compiled
with PowerBasic Console Compiler (PBCC).
I also offer free source code in PBCC to generate unique
random
numbers (http://www.saliu.com/random-numbers.html). That
might be of
interest to you, since I assume you want to generate and
analyze
of combinatorics: PermuteCombine.EXE. It calculates and
generates all
possible sets of numbers and words: exponents, permutations,
arrangements, and combinations. There is also software to
calculate
the probability and odds for many gambling situations. My
freeware is
available from this entry point:
http://www.saliu.com/infodown.html
The programs have meaningful descriptions, plus links to more
detailed
information.
If you want perfect accuracy in odds calculation, only
generation in
lexicographical order will do. But todays computers are
incapable of
working with trillions of trillions of gazillions of googols.
For now,
we can only settle for random simulation in gambling,
especially
blackjack (4 or more decks).
Hope to see your program soon. Hopefully, youll make it
freeware!
Laius Usail,
Doctor in Occult Science of Gambling
http://www.saliu.com/occult-science-gambling.html
>I am about to embark on a software project with intensive
numerical
>analysis and i have been out of it(ie software) for a while.
>below is my project but i wanted to know what the best
language to
use
>is. My best guess is to use c++ and use kdevelop 3.1 on Suse
Linux
>9.1, and that i will be able to compile other people's C or
C++ code
>from the internet. (other linux distro's
>15 years ago i did a fortran simulation utilised in
diffraction
>co-ef[CapitalThorn]cients for antenna desgn. now i assume all the fortran
stuff
is
>now essentially available in C or C++(or at least the stuff
i need).
>I assume for numerical analysis, from what i have recently
read on
the
>internet, delphi and Python are NOT the way to go.
>Project Description: I am writing a blackjack simulation for
card
>counting anaylis. will be calculations of 0% advantage,
con[CapitalThorn]dence
>intervals variance etc for numerous rule variations of a
game that is
>well suited to object orientated design. Comparisons of
actual
>recorded results with expected results etc.
>So my questions are: any info appreciated. :) :)
>1. Delphi/python and other high level langauges are NOT the
way to
go?
>2. C++ is roughly on the right track?
>3. kdevelop will give me reasonable CASE tools etc/ software
>engineering environment? (i want to spend the least amount
of time
>doing the software)
>4. If i wanted to get the best tools available does that mean
spending
>lots of $$$$ on tools like Rational or other. is this right?
just how
>bad is kdevelop compared to these other tools?
===
Subject: Re: best software environment for numerical analysis
>I have done also a lot of programming in gambling, lottery,
>mathematics, statistics, etc. I use Visual Basic sometimes.
But my
>tool of choice is the 32-bit Basic compiler from PowerBasic.
It really
>is powerful. My FTP download site has dozens of free
programs compiled
>with PowerBasic Console Compiler (PBCC).
I did once start writing a pascal implementation of
blackjack. problem
was the language compiler at the time, borland pascal DOS
IDE, did not
support objects. this made it tricky going(ie lot of extra
work). Same
problem here with BASIC, also type checking may not be up to
scratch
on BAsic.
>I also offer free source code in PBCC to generate unique
random
>numbers (http://www.saliu.com/random-numbers.html). That
might be of
>interest to you, since I assume you want to generate and
analyze
>of combinatorics: PermuteCombine.EXE. It calculates and
generates all
>possible sets of numbers and words: exponents, permutations,
>arrangements, and combinations. There is also software to
calculate
>the probability and odds for many gambling situations. My
freeware is
>available from this entry point:
needs to be open source language compatable....I am trying to
refresh
>If you want perfect accuracy in odds calculation, only
generation in
>lexicographical order will do. But todays computers are
incapable
of
>working with trillions of trillions of gazillions of
googols. For now,
>we can only settle for random simulation in gambling,
especially
>blackjack (4 or more decks).
lexicographical order. i assume this considers all
combinations and
permutations of all cards remaining. interesting, had not
thought of
this.
curiosity only...not planning an implentation. do you know
how many
calculations for one deck, assuming your at the 1st hand of
shoe. 2
decks etc.
===
Subject: Re: best software environment for numerical analysis
> 1. Delphi/python and other high level langauges are NOT the
way to go?
Well, actually Python CAN be the way to go. There are plenty
of people
who use Python to do numerical calculations, especially for
rapid tests
of algorithms and ideas. VERY good for that. One of the nicest
languages you'll ever use. However, for blazing speed when
writing your
own (especially if it has loops) you can write C or C++
extensions that
can be called from Python. There are lots of packages for
exending
Python out there already. Well known and used is the numeric
(or
numarray) linear algebra package (built on LAPAC IIRC). I
would
certainly check it out. A high price option that's 
Ôsimilar'
in that it
is easy to program (though not as clean as Python) and has
very good
plotting routines is MatLAB. If you can afford it, that might
be a good
RAD/Test tool. Oh, yeah. Python is free.
> 2. C++ is roughly on the right track?
I guess if you're develping a app for commercial release
(this wasn't
clear) this is eventually where you want to go. Eventually.
That
development part can still include the above.
Don't know answers to 3. adn 4. Sorry.
-- Lou Pecora (my views are my own)
===
Subject: Re: best software environment for numerical analysis
> 1. Delphi/python and other high level langauges are NOT
the way to go?
>Well, actually Python CAN be the way to go. There are plenty
of people
>who use Python to do numerical calculations, especially for
rapid tests
>of algorithms and ideas. VERY good for that. One of the
nicest
>languages you'll ever use.
this sounds good. rapid tests of ideas.I had another idea i
wanted to
do that required fast analysis. use python to test the
hypothesis.
What is of major importance is the quality. i read a lot of
stuff
about type checking done at compile time versus run time and
the
checking of the actual values during run time.(otherwise
called static
or dynamic type checking). Obviously the software will be
unit tested
as i can but any extra Ôway' of being more sure 
is useful. a
LOT of my
money will be bet using the analysis - so bug free code is
critical. I
do have other sources of analysis but not the aspects i want
to
investigate.
For me it doesn't really matter if the languages does checks
at
compile time or run time as long as the checks are done.
since it is a
simulation, ie start it, then wait [CapitalThorn]ve hours, then look at
text
output. i understand python does do the checks.
but python does not have the object orientated data hiding
forced on
the programmer.
>However, for blazing speed when writing your
>own (especially if it has loops) you can write C or C++
extensions that
>can be called from Python. There are lots of packages for
exending
>Python out there already. Well known and used is the numeric
(or
>numarray) linear algebra package (built on LAPAC IIRC). I
would
>certainly check it out.
could always do it [CapitalThorn]rst in python and then convert parts or
all of
it to C++ as required.
>Oh, yeah. Python is free.
An important aspect of the language is using what everyone
else is
using to gain access to libraries/code that is already there
and
hopefully tested. support too, also maybe everybody is using
this for
a reason, ie it is good. free is good too.
>A high price option that's 
Ôsimilar' in that it
>is easy to program (though not as clean as Python) and has
very good
>plotting routines is MatLAB. If you can afford it, that
might be a good
>RAD/Test tool.
I know people use Matlab for numerical analysis but i don't
know much
about it as i have never used it. i imagine it is not so much
object
orientated but is for heavy duty numerical analysis.
At [CapitalThorn]rst my program will be mostly the game
simulation/optimised play
which are well modelled with objects and probability analysis
from
results.
The next part that will include variance analysis, possily
error
analysis that will be very numerical as well as possible
shufße
analysis.
> 2. C++ is roughly on the right track?
>I guess if you're develping a app for commercial release
(this wasn't
>clear) this is eventually where you want to go. Eventually.
That
>development part can still include the above.
App is for my own use and possibly a handful of associates
only.
why do you say c++ is eventually where you want to go.
because python
would need to be on the target system as well or users may
not like
slow response or for robustness?
===
Subject: Re: best software environment for numerical analysis
>I guess if you're develping a app for commercial release
(this wasn't
>clear) this is eventually where you want to go.
Eventually. That
>development part can still include the above.
> App is for my own use and possibly a handful of associates
only.
> why do you say c++ is eventually where you want to go.
because python
> would need to be on the target system as well or users may
not like
> slow response or for robustness?
I was thinking more in terms of an optimized, binary app. But
I guess
Python could still be comercial. I've never really done
commercial so I
was guessing here.
I use Python and C++ for my own research which means rarely
does anyone
else use my apps. The combination of languages is very nice.
I would
check out S- or R-Plus. I know people who have used those
packages and
like them. I've never used them. Finally, for MatLAB I would
guess you
will put out several thousand $, but most engineering people
swear by
it. It's speed is roughly equivalent to Python. That means 
if
you want
to optimize/speed things up you are looking at some compiled
language in
addition, too.
-- Lou Pecora (my views are my own)
===
Subject: Re: best software environment for numerical analysis
[...snip]
A high price option that's 
Ôsimilar' in that it
> is easy to program (though not as clean as Python) and has
very good
> plotting routines is MatLAB. If you can afford it, that
might be a good
> RAD/Test tool. Oh, yeah. Python is free.
Also Python has a very nice plotting library:
http://matplotlib.sourceforge.net/
And for a Matlab replacement I recomend Scilab
(www.scilab.org) also if it
hasn't for example a PDEtool... :-(
hi,
nelson
===
Subject: Re: best software environment for numerical analysis
>Also Python has a very nice plotting library:
>http://matplotlib.sourceforge.net/
excellent, i had been thinking about xplot.
>And for a Matlab replacement I recomend Scilab
(www.scilab.org) also if it
>hasn't for example a PDEtool... :-(
>hi,
> nelson
===
Subject: Re: best software environment for numerical analysis
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> certainly check it out. A high price option that's
Ôsimilar' in that it
> is easy to program (though not as clean as Python) and has
very good
> plotting routines is MatLAB. If you can afford it, that
might be a good
> RAD/Test tool. Oh, yeah. Python is free.
You could also try Scilab and Octave as free alternatives to
Matlab.
Russell.
===
Subject: Re: best software environment for numerical analysis
> certainly check it out. A high price option that's
Ôsimilar' in that
> it is easy to program (though not as clean as Python) and
has very
> good plotting routines is MatLAB. If you can afford it,
that might be
> a good RAD/Test tool. Oh, yeah. Python is free.
> You could also try Scilab and Octave as free alternatives
to Matlab.
> Russell.
The license of Scilab restricts its use in commericial
environments.
Octave is GPLed, so you can use it in commerical
environments, but if
you distribute changes you made to Octave or its functions
you must
distribute the source code... Your own functions are yours of
course :-)
D.
===
Subject: Re: best software environment for numerical analysis
I prefer R, which is the successor of S. It's free (beer and
freedom), has much better graphics than Octave, has a good
high-level
control language, and has a huge array of software available.
It was
designed with statistics in mind, but is extremely useful for
a wide
range of numerical tasks. (IMO, the only time Octave is
really a good
choice is if you have to run existing Matlab, and even then,
it rarely
works, as Octave is missing many of Matlab's features.)
rif
===
Subject: Re: best software environment for numerical analysis
> I prefer R, which is the successor of S. It's free (beer 
and
> freedom), has much better graphics than Octave, has a good
high-level
> control language, and has a huge array of software
available. It was
> designed with statistics in mind, but is extremely useful
for a wide
> range of numerical tasks. (IMO, the only time Octave is
really a good
> choice is if you have to run existing Matlab, and even
then, it rarely
> works, as Octave is missing many of Matlab's features.)
First of all let me tell you that I am in no way an
experienced R-user,
so I cannot talk from that point of view, but I do use Matlab
and Octave a
lot for my research. I try to write compatible code, and from
my
experience and in my opinion, there is indeed still a
difference and
certainly not all of your Matlab code will run under Octave,
but please
note that there is already quite some compatibility (as David
Bateman
already mentioned in this thread) and the Octave-developers
are really
putting their efforts in keeping Octave compatible with
Matlab. This
used to be different in the past...
Please understand that Ôkeeping up 
compatability' is not
always easy,
especially when all the work is done by volunteers working in
their
spare free time...
It's also a good idea to always try to keep up with the
latest releases
of Octave to pro[CapitalThorn]te from all the best stuff that is out 
there.
In my opinion, the only big thing Octave still is missing is
a nice
graphical user interface like Matlab has it. But let's all
face it: if
we can type our M-[CapitalThorn]les in an editor, run our scripts and
functions at
the Octave prompt, and look at our graphics within GNU-plot,
then is it
really necessary to have other bells and whistles around?
Bart
--
Share what you know. Learn what you don't.
===
Subject: Re: best software environment for numerical analysis
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> In my opinion, the only big thing Octave still is missing
is a nice
> graphical user interface like Matlab has it. But let's all
face it: if
> we can type our M-[CapitalThorn]les in an editor, run our scripts and
functions at
> the Octave prompt, and look at our graphics within
GNU-plot, then is it
> really necessary to have other bells and whistles around?
Unfortunately Gnuplot is the main reason I don't use Octave.
I primarily
use Matlab to analyse data and Gnuplot simply doesn't offer 
a
good means
of doing that (it might be okay once you have [CapitalThorn]gured out all
its
idiosyncracies).
Russell.
===
Subject: Re: best software environment for numerical analysis
> I prefer R, which is the successor of S. It's free (beer 
and
> freedom), has much better graphics than Octave, has a good
high-level
> control language, and has a huge array of software
available. It was
> designed with statistics in mind, but is extremely useful
for a wide
> range of numerical tasks. (IMO, the only time Octave is
really a good
> choice is if you have to run existing Matlab, and even
then, it rarely
> works, as Octave is missing many of Matlab's features.)
> rif
Rif,
As some one who uses heavily and contributes to Octave, I [CapitalThorn]nd
your
critism laughable. Though there is not much point responding
as you
don't say what it is that you [CapitalThorn]nd lacking in 
Octave. Recent
versions of
Octave have NDArrays, int/uint and math on these. Octave even
has
features ahead of matlab (eg. +=, and || though I think
matlab just
introduced these) and features from the very latest version
of matlab
such as anonymous function handles (eg @ (<function-args>)
<function-body>). Want a toolbox that isn't in the base
distribution,
you need look no further than octave.sourceforge.net.
As for graphics, yes there are some issues with octave, that
are being
worked on now and I'm certain that future versions of octave
will have
comparable graphics capabilities to matlab.
Even without fancy graphics, the fact that Octave is now so
higher
compatiable with Matlab, make it a perfect candidate for
simulation in a
cluster where you'd need as many Matlab license as nodes in
the cluster.
If you care to identify what it is exactly you [CapitalThorn]nd lacking
with Octave,
I'm sure I could make a better response to your complaint.
D.
===
Subject: Re: best software environment for numerical analysis
I must have misspoke. The thrust of my message should have
been I
really like R, why not take a look at it? rather than Octave
is not
a good program, avoid it. I did not mean to badmouth Octave. I
haven't used it in a year or so, since I started using R; 
it's
possible it has improved greatly in that time frame, and I
have no
basis for making a judgement about its current capabilities.
Also,
please understand that my comment was made in the context of
not
liking Matlab much either --- I strongly prefer R to Matlab,
and
that's why I'd personally only use Octave when 
I wanted to
run Matlab
code.
Similarly, I retract my statement that R is a successor to S.
rif
===
Subject: Re: best software environment for numerical analysis
> I must have misspoke. The thrust of my message should have
been I
> really like R, why not take a look at it? rather than
Octave is not
> a good program, avoid it. I did not mean to badmouth
Octave. I
> haven't used it in a year or so, since I started using R;
it's
> possible it has improved greatly in that time frame, and I
have no
> basis for making a judgement about its current
capabilities. Also,
> please understand that my comment was made in the context
of not
> liking Matlab much either --- I strongly prefer R to
Matlab, and
> that's why I'd personally only use Octave 
when I wanted to
run Matlab
> code.
Just out of curiosity and because I don't know R myself and
might want
to learn it's capabilities(*): what kind of things make R so
superior to
Matlab/Octave? In what is R better than Octave/Matlab?
Bart
(*) the right tool for the right job, you know ;-)
--
Share what you know. Learn what you don't.
===
Subject: Re: best software environment for numerical analysis
> I must have misspoke. The thrust of my message should have
been I
> really like R, why not take a look at it? rather than
Octave is not
> a good program, avoid it. I did not mean to badmouth
Octave. I
> haven't used it in a year or so, since I started using R;
it's
> possible it has improved greatly in that time frame, and I
have no
> basis for making a judgement about its current
capabilities. Also,
> please understand that my comment was made in the context
of not
> liking Matlab much either --- I strongly prefer R to
Matlab, and
> that's why I'd personally only use Octave 
when I wanted to
run Matlab
> code.
>Just out of curiosity and because I don't know R myself and
might want
>to learn it's capabilities(*): what kind of things make R 
so
superior to
>Matlab/Octave? In what is R better than Octave/Matlab?
R and S-Plus are designed for and used by statisticians, so
there is
statistical
software for those programs not available in Matlab -- you'd
have to write
your own code. See http://cran.r-project.org for a list of R
packages.
Usually,
R can run S-Plus code.
http://www.newsfeed.com The #1 Newsgroup Service in the
World! >100,000
Newsgroups
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===
Subject: Re: best software environment for numerical analysis
>R and S-Plus are designed for and used by statisticians, so
there is
statistical
>software for those programs not available in Matlab -- 
you'd
have to write
>your own code. See http://cran.r-project.org for a list of R
packages.
Usually,
>R can run S-Plus code.
I have another problem that is purely statistical, not a
simulation.
A randomizing machine has a bias, and from a sample of spins
i want
to calculate the bias. Obviously as the sample size increases
the
con[CapitalThorn]dence of the bias calculation will narrow. There are four
randomizing devices in the machine and for each device the
next
outcome of a spin is a function of the previous outcome, but
you don't
know which of the four previous outcomes was the starting
point for
each of the four devices.
Now if there was only one device this would be straight
forward(although some work for me since i am not
mathematician) to
reverse backwards and calculate actual probabilities with a
certain
con[CapitalThorn]dence level for a given sample size.
So i need to do the analysis of a sample, or analyze the data
as it is
collected to determine a bias with a useful con[CapitalThorn]dence
interval.
Will this R and S help me best do this?
===
Subject: Re: best software environment for numerical analysis
There's a powerpoint presentation at
http://www.bioconductor.org/workshops/Heidelberg02/
r-intro.ppt that
does a good job explaining R's features. Basically, I 
[CapitalThorn]nd the
language itself more full-featured than Matlab, closer to
functional
programming, with lots of built-in data types that seem to
mesh well
for numerical work, and a nice infrastructure or automagically
downloading software from the central R repository.
rif
===
Subject: Re: best software environment for numerical analysis
>I prefer R, which is the successor of S. It's free (beer 
and
>freedom), has much better graphics than Octave, has a good
high-level
>control language, and has a huge array of software
available. It was
>designed with statistics in mind, but is extremely useful
for a wide
>range of numerical tasks. (IMO, the only time Octave is
really a good
>choice is if you have to run existing Matlab, and even then,
it rarely
>works, as Octave is missing many of Matlab's features.)
R is worth considering, but calling it a successor (implying
better or
more modern) of S is a bit misleading IMO. S-Plus is a
commercial product
that is still under development, and R is a viable
open-source ALTERNATIVE
in many cases.
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World! >100,000
Newsgroups
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===
Subject: Re: Science and Religion
by support1.mathforum.org (8.11.6/8.11.6/The Math Forum,
$Revision:
1.9 primary) id i8GF9aV18070;
Art:
Forever axiomatic be your name for it proves the necessity of
the
quest for knowledge!
Science and religion in agreement? They can only agree to
disagree
at best! At worst, which is also equivalent to most of the
time,
religion orders an armada of inquisitors to war. That is, war
against
the nerds who can only [CapitalThorn]ght with pencils and defend
themselves with
paper shields!
Science uses reason as a tool in searching Truth, while
religion
attempts to establish Order in randomness. What religion and
science
have in common is they are fundamental components of
consciousness.
You may want to read these bibles:
http://www.saliu.com/civilization.html
(Civilization is the permutation of the fundamental elements
of
Consciousness: Art, Religion, Science, Philosophy)
and
http://www.saliu.com/almighty_number.html
(Almighty Number, Randomness, Universe, God, Order, History,
Repetition).
Laius Usail,
Doctor in Occult Science of Randomness
>Science and Religion
>If you're interested in science and in religion, and wish
>the two were more in agreement, you're invited to a free
>download of the book Science Without Bounds:
>A Synthesis of Science, Religion, and Mysticism
>Free electronic copies available for personal and
educational use
>at <a
href=http://www.adamford.com/swb>http://www.adamford.com/swb<
/a>
>Here's what some people have said about Science Without
Bounds.
>One of the best on the Internet . . . for quality, accuracy
of
>content, presentation and usability. - Britannica Presents
Internet
>Guide Awards
>Style is tight, clean and refreshingly free from scienti[CapitalThorn]c,
mystical,
>philosophical and theological jargon. . . . It's the kind 
of
book
>where you'll need to stop and do 15 minutes of deep 
thinking
and
>pondering every few pages . . . Outstanding book,
engrossing, a
>classic - Wendy Christensen, Inscriptions, the weekly e-zine
for
>professional writers, (InscriptionsMagazine.com)
>Selected by The McKinley Group's professional editorial 
team
as a
>3-Star site . . . a special mark of achievement in Magellan,
>McKinley's comprehensive Internet directory of over 1.5
million sites
>and 40,000 reviews.
>Found it very intriguing. A lot of your ideas had me nodding
my head
>in agreement. - Karl B.
>I was astonished to read your book non-stop over 7 hours
last night.
>It was the best synthesis of the collective understanding I
have read
>in my 48 years. - S. R.
>I'm impressed by the breadth of your understanding and the
amount of
>work you've obviously put into it. - R.W.K.
>Wow! This is about all I can say at the moment. - Ella B.
>Science and Religion
>If you're interested in science and in religion, and wish
>the two were more in agreement, you're invited to a free
>download of the book Science Without Bounds:
>A Synthesis of Science, Religion, and Mysticism
>Free electronic copies available for personal and
educational use
>at <a
href=http://www.adamford.com/swb>http://www.adamford.com/swb<
/a>
>Here's what some people have said about Science Without
Bounds.
>One of the best on the Internet . . . for quality, accuracy
of
>content, presentation and usability. - Britannica Presents
Internet
>Guide Awards
>Style is tight, clean and refreshingly free from scienti[CapitalThorn]c,
mystical,
>philosophical and theological jargon. . . . It's the kind 
of
book
>where you'll need to stop and do 15 minutes of deep 
thinking
and
>pondering every few pages . . . Outstanding book,
engrossing, a
>classic - Wendy Christensen, Inscriptions, the weekly e-zine
for
>professional writers, (InscriptionsMagazine.com)
>Selected by The McKinley Group's professional editorial 
team
as a
>3-Star site . . . a special mark of achievement in Magellan,
>McKinley's comprehensive Internet directory of over 1.5
million sites
>and 40,000 reviews.
>Found it very intriguing. A lot of your ideas had me nodding
my head
>in agreement. - Karl B.
>I was astonished to read your book non-stop over 7 hours
last night.
>It was the best synthesis of the collective understanding I
have read
>in my 48 years. - S. R.
>I'm impressed by the breadth of your understanding and the
amount of
>work you've obviously put into it. - R.W.K.
>Wow! This is about all I can say at the moment. - Ella B.
===
Subject: Re: Help to solve an integral
by support1.mathforum.org (8.11.6/8.11.6/The Math Forum,
$Revision:
1.9 primary) id i8GF9af18066;
>I am looking for help to solve an integral.
>The integral is a de[CapitalThorn]nite integral for x varying from 0
until pi of
>the following function:
>Exp[-Sqrt[-1]*a(x+b*Sqrt[1+c^2-2*c*Cos[x]])]
>b and c are real values, and a is a value taking Integer
numbers.
>Mathematica does not help.
>Thibaut
Bonjour,
I don't know if its useful but you can integrate
approximately if you
replace the cos by a square signal :
cos[x_]:=If[x<Pi/2,1,-1];
Integrate[Exp[-I*a(x+b*Sqrt[1+c^2-2*c*cos[x]])],{x,0,Pi}]//
FullSimplify[#,
Element[b,Reals]&&Element[c,Reals]&&Element[a,Integers]]&
((-I)*(-1 + I^a)*(E^(I*a*b*Abs[-1 + c]) + I^a*E^(I*a*b*Abs[1 +
c])))/(a*E^(I*a*(Pi + b*(Abs[-1 + c] + Abs[1 + c]))))
Salutations,
V.Astanoff
===
Subject: How do I [CapitalThorn]t a line through three points in space?
Hi all!
How do I go about [CapitalThorn]tting a straight line (vector) through
three
points in space? Each of the three points is given by three
coordinates (x,y,z) that are measured with some uncertainty.
The textbooks I have all discuss using the method of least
squares for
[CapitalThorn]tting a straight line through a number of (x,y) points in a
plane
but I need a method that works in three dimensions.
Although this is probably a trivial problem for you
mathematicians I
would appreciate any help you can offer.
/Red B.
===
Subject: Re: How do I [CapitalThorn]t a line through three points in 
space?
Am 17.09.04 11:03 schrieb Red B.:
> Hi all!
> How do I go about [CapitalThorn]tting a straight line (vector) through
three
> points in space? Each of the three points is given by three
> coordinates (x,y,z) that are measured with some uncertainty.
> The textbooks I have all discuss using the method of least
squares for
> [CapitalThorn]tting a straight line through a number of (x,y) points in
a plane
> but I need a method that works in three dimensions.
> Although this is probably a trivial problem for you
mathematicians I
> would appreciate any help you can offer.
> /Red B.
With two coordinates it's just linear regression, with more
coordinates
is's multiple regression or pca (depending on the
distance-measure).
Gottfried Helms
===
Subject: Re: How do I [CapitalThorn]t a line through three points in 
space?
> Hi all!
> How do I go about [CapitalThorn]tting a straight line (vector) through
three
> points in space? Each of the three points is given by three
> coordinates (x,y,z) that are measured with some uncertainty.
> The textbooks I have all discuss using the method of least
squares for
> [CapitalThorn]tting a straight line through a number of (x,y) points in
a plane
> but I need a method that works in three dimensions.
> Although this is probably a trivial problem for you
mathematicians I
> would appreciate any help you can offer.
> /Red B.
This is a way to [CapitalThorn]nd an acceptable solution :
Let the line pass thru the center of gravity and
then compare the 6 directions de[CapitalThorn]ned by the cg
and one point or parallel with two points.
A model in mathematica :
lineFit[OM1_,OM2_,OM3_]:=
Module[{},
norm[vec_]:=Sqrt[vec.vec];
OG=(OM1+OM2+OM3)/3;
v[1]=OM1-OG;
v[2]=OM2-OG;
v[3]=OM3-OG;
v[4]=OM2-OM1;
v[5]=OM3-OM2;
v[6]=OM1-OM3;
d[i_]:=norm[Cross[v[i],(OM1-OG)]]/norm[v[i]]
+norm[Cross[v[i],(OM2-OG)]]/norm[v[i]]
+norm[Cross[v[i],(OM3-OG)]]/norm[v[i]] ;
res={v[#],d[#]}&/@{1,2,3,4,5,6} ;
Sort[res,Last[#1]<Last[#2]&]
];
example :
lineFit[{1.,1.,1.},{5.,1.,2.},{3.,5.,3.}]
{{{0., 2.66666, 1.}, 4.10812}, {{-2., -1., -1.}, 4.49407},
{{-2., -4., -2.}, 4.77648}, {{2., -1.33333, 0.}, 4.86747},
{{-2., 4., 1.}, 5.10627}, {{4., 0., 1.}, 5.6753}}
points aligned :
lineFit[{1.,1.,1.},{2.,2.,2.},{4.,4.,4.}]
{{{-3., -3., -3.}, 0.}, {{2., 2., 2.}, 0.}, {{1., 1., 1.},
0.},
{{ 1.66666, 1.66666, 1.66666}, 0.},
{{-0.33333, -0.33333, -0.33333}, 0.},
{{-1.33333, -1.33333, -1.33333}, 0.}}
hth
V.Astanoff
===
Subject: Re: How do I [CapitalThorn]t a line through three points in 
space?
Two follow-on questions:
1. Will the best [CapitalThorn]t line run through the centroid of the
triangle?
2. Will the best [CapitalThorn]t line reside in the plane of this 
triangle?
Will this extend to points in R4?
Randall
===
Subject: Re: How do I [CapitalThorn]t a line through three points in 
space?
Am 18.09.04 18:03 schrieb anonymous:
> Two follow-on questions:
> 1. Will the best [CapitalThorn]t line run through the centroid of the
triangle?
> 2. Will the best [CapitalThorn]t line reside in the plane of this
triangle?
> Will this extend to points in R4?
> Randall
Yes to all questions.
Gottfried Helms
Example:
[9] a = {{1,2,3}, _
{4,1,5}, _
{7,1,3}} // three arbitrary points
a:
x-coord y-coord z-coord
|------------------------------|
p1 | 1.00 2.00 3.00 |
p2 | 4.00 1.00 5.00 |
p3 | 7.00 1.00 3.00 |
[10] center = meansp(a) // [CapitalThorn]nding center
center :
x-coord y-coord z-coord
+------------------------------+
c | 4.00 1.33 3.67 |
[11] a1 = a-center //translate centering to origin
a1 :
x-coord y-coord z-coord
+------------------------------+
p1' | -3.00 0.67 -0.67 |
p2' | . -0.33 1.33 |
p3' | 3.00 -0.33 -0.67 |
[12] t = gettrans(a1,pca)
[13] pca = a1*t // [CapitalThorn]nding best line in [CapitalThorn]rst axis 
(column 1 of
matrix)
pca :
x-coord y-coord z-coord
+------------------------------+
p1 | -3.07 0.66 . |
p2 | 0.06 -1.37 . |
p3 | 3.01 0.71 . |
The three points are in a plane. The best approximate line is
the
[CapitalThorn]rst axis (x-axis)
[14] pca_plus = {pca,{1,0,0},{2,0,0}} // add two points
marking the main
axis
pca_plus :
x-coord y-coord z-coord
+------------------------------+
p1 | -3.07 0.66 . |
p2 | 0.06 -1.37 . |
p3 | 3.01 0.71 . |
+------------------------------+
r1 | 1.00 . . |
r2 | 2.00 . . |
Two points on the main axis are added to the system.
Now rotate back into original position:
[15] a1_plus = pca_plus*t' // rotate back
a1_plus :
x-coord y-coord z-coord
+------------------------------+
p1' | -3.00 0.67 -0.67 |
p2' | . -0.33 1.33 |
p3' | 3.00 -0.33 -0.67 |
+------------------------------+
r1' | 0.99 -0.17 0.01 |
r2' | 1.97 -0.33 0.01 |
... and translate back to old center:
[16] a_plus = a1_plus + center // translate back
a_plus :
x-coord y-coord z-coord
+------------------------------+
p1 | 1.00 2.00 3.00 |
p2 | 4.00 1.00 5.00 |
p3 | 7.00 1.00 3.00 |
+------------------------------+
r1 | 4.99 1.17 3.67 |
r2 | 5.97 1.00 3.68 |
Note:
r2 could also be the center-point, since the translation
was origin to center, and the origin lies on the [CapitalThorn]rst
principal axis.
Example four points:
[17] a = {{1,2,3,4},{4,1,5,7},{7,1,3,2},{5,9,2,1}} // four
arbitrary points
a :
x-coord y-coord z-coord w-coord
+----------------------------------------+
| 1.00 2.00 3.00 4.00 |
| 4.00 1.00 5.00 7.00 |
| 7.00 1.00 3.00 2.00 |
| 5.00 9.00 2.00 1.00 |
[18] center = meansp(a) // [CapitalThorn]nding center
center :
x-coord y-coord z-coord w-coord
+----------------------------------------+
| 4.25 3.25 3.25 3.50 |
[19] a1 = a-center //translate centering to origin
a1 :
x-coord y-coord z-coord w-coord
+----------------------------------------+
| -3.25 -1.25 -0.25 0.50 |
| -0.25 -2.25 1.75 3.50 |
| 2.75 -2.25 -0.25 -1.50 |
| 0.75 5.75 -1.25 -2.50 |
[20] t = gettrans(a1,pca)
[21] pca = a1*t // [CapitalThorn]nding best line in [CapitalThorn]rst axis 
(column 1 of
matrix)
pca :
x-coord y-coord z-coord w-coord
+----------------------------------------+
| -1.74 2.47 -1.82 . |
| -4.04 0.57 1.94 . |
| -0.58 -3.74 -0.78 . |
| 6.36 0.70 0.66 . |
The four points are spanning a 3-d-space
Best [CapitalThorn]tting line is the [CapitalThorn]rst axis.
[22] pca_plus = {pca,{1,0,0,0},{2,0,0,0}} // add two points
marking the
main axis
pca_plus :
x-coord y-coord z-coord w-coord
+----------------------------------------+
| -1.74 2.47 -1.82 . |
| -4.04 0.57 1.94 . |
| -0.58 -3.74 -0.78 . |
| 6.36 0.70 0.66 . |
+----------------------------------------+
| 1.00 . . . |
| 2.00 . . . |
[23] a1_plus = pca_plus*t' // rotate back
a1_plus :
x-coord y-coord z-coord w-coord
+----------------------------------------+
| -3.25 -1.25 -0.25 0.50 |
| -0.25 -2.25 1.75 3.50 |
| 2.75 -2.25 -0.25 -1.50 |
| 0.75 5.75 -1.25 -2.50 |
+----------------------------------------+
| 0.16 0.82 -0.24 -0.50 |
| 0.33 1.63 -0.48 -1.00 |
[24] a_plus = a1_plus + center // translate back
a_plus :
x-coord y-coord z-coord w-coord
+----------------------------------------+
| 1.00 2.00 3.00 4.00 |
| 4.00 1.00 5.00 7.00 |
| 7.00 1.00 3.00 2.00 |
| 5.00 9.00 2.00 1.00 |
+----------------------------------------+
| 4.41 4.07 3.01 3.00 |
| 4.58 4.88 2.77 2.50 |
===
Subject: Re: How do I [CapitalThorn]t a line through three points in 
space?
Randall
===
Subject: Re: How do I [CapitalThorn]t a line through three points in 
space?
>Hi all!
>How do I go about [CapitalThorn]tting a straight line (vector) through
three
>points in space? Each of the three points is given by three
>coordinates (x,y,z) that are measured with some uncertainty.
>The textbooks I have all discuss using the method of least
squares for
>[CapitalThorn]tting a straight line through a number of (x,y) points in a
plane
>but I need a method that works in three dimensions.
>Although this is probably a trivial problem for you
mathematicians I
>would appreciate any help you can offer.
>/Red B.
you can do least squares of course also here, but it is
somewhat
tricky to do a line [CapitalThorn]t because:
how do you want to represent the line?
how will you measure the error?
for example you can represent the line as the intersection of
two planes.
since this is possible in many ways and optimization codes
dislike
solution continua we must make it unique in some way: make
the planes
orthogonal to each other. represent the plane in Hesse form:
a*x+b*y+c*z+d=0 (*) side condition a^2+b^2+c^2=1 a>=0
e*x+f*y+g*z+h=0 (*) side condition e^2+f^2+g^2=1 b>=0
side condition a*e+b*f+c*g=0
inserting x,y,z for the three points gives six residuals whose
sum of squares you may want to minimize subject to the 5
constraints.
a nonlinear optimization code will do that.
this seems like overkill, since from the three points you
could also
conclude which of the coef[CapitalThorn]ciients a,b,c,d,e,f cannot be zero
and [CapitalThorn]x
that to one, remaining with six residuals, six unknowns and
the
orthogonality
constraint. but this already weights the errrors differently
and in an
unpredictable way, hence I discourage this..
or you use the parametric form
x(t)=x0+d*t x0 , d vectors, t the line parameter
d=(d1,d2,d3) with d1*x0_1+d2*x0_2+d3*x0_3=0 (orthogonality)
and d1^2+d2^2+d3^2=1, d1>=0 . (normalization)
you could go a step further and represent d by a vector in
spherical
coordinates
d=(cos(phi)*cos(psi),sin(phi)*cos(psi),sin(psi)), now with
phi and psi
as unknowns, but the orthogonality constraint remains to make
the solution
unique (with phi in [-pi/2,pi/2])
then the orthogonal distance of the point
P(i)=(x(i),y(i),z(i))
from the line is
euclidean length of P(i)-x0-(d'*P(i))*d
and squaring and adding this you get the error functional
which is to be
minimized subject to the constraints
for codes which can do nonlinear least squares with
constraints see
http://plato.la.asu.edu/topics/problems/nlolsq.html
I recommend ENLSIP (just at the bottom of this page)
hth
peter
===
Subject: Re: How do I [CapitalThorn]t a line through three points in 
space?
>How do I go about [CapitalThorn]tting a straight line (vector) through
three
>points in space? Each of the three points is given by three
>coordinates (x,y,z) that are measured with some
uncertainty.
>you can do least squares of course also here, but it is
somewhat
>tricky to do a line [CapitalThorn]t because:
>how do you want to represent the line?
>how will you measure the error?
>for example you can represent the line as the intersection
of two planes.
Is there really a context (consistent with the OP's
presentation of his/her
problem) in which it would make sense to select a line which
is not
contained in the line speci[CapitalThorn]ed by the three points? This
reduces the
problem to [CapitalThorn]tting a line to three points in a single plane.
(Of course one
still has to decide the criterion by which that line will be
considered
optimal.)
dave
===
Subject: How to solve this PDE? (Pleaaaaaaaaaaaaase!!)
Hi every one.
I would like to [CapitalThorn]nd function A(x,t) by solving this PDE:
dA/dt=cosh(ax).dA/dx + b.cosh(ax).t
a and b are constants. Does anyone has any idea of how I
should
deal with it?
Nas
===
Subject: Re: How to solve this PDE? (Pleaaaaaaaaaaaaase!!)
> Hi every one.
> I would like to [CapitalThorn]nd function A(x,t) by solving this PDE:
> dA/dt=cosh(ax).dA/dx + b.cosh(ax).t
> a and b are constants. Does anyone has any idea of how I
should
> deal with it?
> Nas
Maple solves diff(A(x,t),t) = cosh(c*x)*(diff(A(x,t),x)+b*t)
readily as :
(A(x,t) =
2*b/c*int(arctan(exp(c*x)),x)-b*t*x-2*b/c*arctan(exp(c*x))*x+
arbitrary_funct
ion((t*c+2*arctan(exp(c*x)))/c));
or equivalently :
(A(x,t) =
-I*b/c^2*dilog(1-I*exp(c*x))+I*b/c^2*dilog(1+I*exp(c*x))-b*t*
x-2*b/c*arctan(
exp(c*x))*x+arbitrary_function((t*c+2*arctan(exp(c*x)))/c));
Chris
===
Subject: IMS 2005: First announcement and call for submissions
With many apologies for multiple postings.
-------------------------------------------------------------
------------
7TH INTERNATIONAL MATHEMATICA SYMPOSIUM 2005
The University of Western Australia
Perth, Western Australia
AUSTRALIA
5-8 August 2005
THE SYMPOSIUM
The International Mathematica Symposium is an
interdisciplinary
conference for users of Mathematica in mathematics, natural
and life
sciences, social sciences and law, engineering, graphics and
design,
arts and music, education, industry, and commerce. Held six
times since
its inception in 1995,
IMS 1995 (Southampton, United Kingdom),
IMS 1997 (Rovaniemi, Finland),
IMS 1999 (Linz, Austria),
IMS 2001 (Tokyo, Japan),
this is the [CapitalThorn]rst time it will be held in the southern
hemisphere.
The Symposium is refereed, and a published Proceedings (book
or CD with
ISBN) is produced for each conference.
If you use Mathematica in research or teaching, or if you
have developed
or are developing products based on Mathematica, then the
Symposium is
an opportunity to share your results with like-minded
colleagues. IMS
has also built up a deserved reputation as an exceptionally
convivial
and friendly gathering.
In 2005 it will be held at the School of Physics of The
University of
Western Australia. The Symposium will run from 5-8 August,
including
optional Mathematica training sessions.
Since the conference is being held during our second
semester, College
accommodation is not available. However special rates with
local hotels
and serviced apartments have been negotiated to make this
conference
affordable.
CALL FOR PAPERS
The closing date for abstracts is February 28, 2005. Please
submit your
proposals by email to Paul Abbott
<mailto:paul@physics.uwa.edu.au>
CALL FOR CONTRIBUTIONS
The IMS 2005 Organizing Committee also seeks proposals for
* Discussion Panels and Workshops,
* Art Exhibitions & Installations,
* Poster Presentations,
* Software Demonstrations, and
* Displays / Exhibition Booths by vendors and sponsors.
The closing date for such submissions is April 30 2005.
CONFERENCE WEBSITE
The conference website is under construction and its URL will
be
announced, along with a list of plenary speakers and details
of the
paper format and submission procedure, in the Second
announcement.
--
Paul Abbott Phone: +61 8 6488 2734
School of Physics, M013 Fax: +61 8 6488 1014
The University of Western Australia (CRICOS Provider No
00126G)
35 Stirling Highway
Crawley WA 6009 mailto:paul@physics.uwa.edu.au
AUSTRALIA http://physics.uwa.edu.au/~paul
===
Subject: 3D Fourier transform of a radial function
I would like to compute the 3D Fourier transform F(k) of a
radial
function f(r)=f(|r|) :
F(k) = int_{0}^{infty} f(r) 4 pi sin(k r)/(k r) dr
where the angular integrations have been performed.
Is there an ef[CapitalThorn]cient algorithm (and available routine) for
computing
F(k) like for FFT? How to handle the oscillatory behavior of
the
integrand in a numerical integration?
Julien Toulouse
===
Subject: Re: 3D Fourier transform of a radial function
Sorry, there was a mistake in the formula. The correct one is
of course
F(k) = int_{0}^{infty} f(r) 4 pi k sin(k r)/r dr
Julien Toulouse
===
Subject: Re: 3D Fourier transform of a radial function
I don't suppose the polar fourier transform is what 
you're
looking for?
===
Subject: Matrix inversion by Gauss-Jordan elimination
Hi. I'm trying to write a computer program to invert a 
matrix.
Unfortunately I don't have much experience or knowledge of
linear
algebra.
I have successfully implemented a matrix inverse function
which uses
determinants and the adjoint matrix. However, just as I
[CapitalThorn]nished that
code, I found a web page describing this method as slow.
Also, my code
allocates and deallocates a lot of memory, which is bad as
well. Most
of the matrices I will have to invert will be of the range
20x20 to
100x100. However, later on I'm likely to need to invert
matrices of
sizes similar to 1000x1000.
Any recommendations of other methods (that I'm likely to be
able to
understand) suitable for computer programs would be
appreciated.
What I'm not con[CapitalThorn]dent is that the method by 
which I can
successfully
invert small matrices by hand is the correct method for use
in a
computer program. I.e. will it succeed in [CapitalThorn]nding the inverse
for all
invertible matrices?
I'll list here a worked example done by my current (correct?
incorrect?) version of my program. If anyone would be so kind
as to
read this and criticse my method as correct or incorrect, it
would be
greatly appreciated. What I'm not sure about is choosing the
proper
pivots (I believe they are called).
I was going to post an example done by hand, but eventually
found it
easier to write the program for the method I *think* might be
right,
rather than do all the working with no mistakes :-(. It does
calculate
an inverse (checked by multiplying against the original
matrix), but
is it guaranteed to always get an inverse where possible?
First, take a matrix:
0 5 6
1 2 3
6 2 3
Append the identity matrix.
0 5 6 1 0 0
1 2 3 0 1 0
6 2 3 0 0 1
The next step is looped for column 1 to 3. I'll start with
column 1.
Find the [CapitalThorn]rst row which has a number other than 0 in column
1. I'm
assuming that I can choose ANY such row (this assumption is
the one
I'm most worried about, please advise. Can I just choose ANY
such row,
or is the real situation more complicated?). In the above
case, I can
swap rows 1 and 2.
1 2 3 0 1 0
0 5 6 1 0 0
6 2 3 0 0 1
Now, we need a 1 in the [CapitalThorn]rst column of the [CapitalThorn]rst 
row, and I
have one.
So onto the next step. I need to get rid of the 6 in the [CapitalThorn]rst
column
of the last row. So, I minus 6 x the [CapitalThorn]rst row from the last
one.
1 2 3 0 1 0
0 5 6 1 0 0
0 -10 -15 0 -6 1
Now move onto the second column. We [CapitalThorn]nd any row (other than
the [CapitalThorn]rst
one) that has a number other than 0 in the second column. The
second
row will do. This has a 5 rather than a 1, so we'll divide
the entire
row by 5. This gives:
1 2 3 0 1 0
0 1 1.2 0.2 0 0
0 -10 -15 0 -6 1
Now we need to remove the 2 in the [CapitalThorn]rst row. We minus 2 * row
2 from
the [CapitalThorn]rst row.
1 0 0.6 -0.4 1 0
0 1 1.2 0.2 0 0
0 -10 -15 0 -6 1
And, now we remove -1/10 * the second row from the last row.
1 0 0.6 -0.4 1 0
0 1 1.2 0.2 0 0
0 0 -3 2 -6 1
Finally we get to the last row. We have -3 instead of 1 in
the third
column. So, we divide row 3 by -3.
1 0 0.6 -0.4 1 0
0 1 1.2 0.2 0 0
0 0 1 -0.666667 2 -0.333333
Now we need to remove the numbers in the last column from
rows 1 and
2. For the [CapitalThorn]rst row, we remove 0.6 * row3. I.e.
1 0 0 0 -0.2 0.2
0 1 1.2 0.2 0 0
0 0 1 -0.666667 2 -0.333333
And [CapitalThorn]nally, we remove 1.2 * row3 from row2.
1 0 0 0 -0.2 0.2
0 1 0 1 -2.4 0.4
0 0 1 -0.666667 2 -0.333333
Then, the matrix on the right hand side is the inverse of our
original
matrix.
0 -0.2 0.2
1 -2.4 0.4
-0.666667 2 -0.333333
Then, I multiply my original matrix with the inverse.
1 0 0
0 1 0
0 0 1
Note: I did round the number -5.17978e-17 to zero in the above
working, but otherwise it's the same as my program output.
Can anyone criticise my interpretation of the Gauss-Jordan
elimination
method? Please?!?!?
Ross-c
===
Subject: Re: Matrix inversion by Gauss-Jordan elimination
Since you asked this question, it IS guaranteed to give you
an inverse
whenever one exists. And here's why:
You're essentially solving
A*x = e_n
where e_n is a column vector of zeros with a 1 in the nth
place.
You are then using the de[CapitalThorn]nition of the inverse as being
x = inv(A) * e_n
But inv(A) * e_n is by de[CapitalThorn]nition the nth column of inv(A). So
when you
have x, you have the nth column of your inverse.
The method you're using has a shortcut in it, where instead
of doing the
process one e_n vector at a time, it uses an identity matrix.
This
doesn't change the process itself, it just saves some 
tedious
recalculations.
Hope that helps... I haven't done enough matrix programming
to know
shortcuts for you.
clemenr@wmin.ac.uk says...
(snip)
> I was going to post an example done by hand, but eventually
found it
> easier to write the program for the method I *think* might
be right,
> rather than do all the working with no mistakes :-(. It
does calculate
> an inverse (checked by multiplying against the original
matrix), but
> is it guaranteed to always get an inverse where possible?
(snip)
===
Subject: Re: Matrix inversion by Gauss-Jordan elimination
> Can anyone criticise my interpretation of the Gauss-Jordan
elimination
> method? Please?!?!?
No, nothing to critizise.
You'll only encounter some problems when you are dealing 
with
general
matrices:
- accumulation of rounding-off errors
- division of a big number by a very small number
- zero at any pivot-position.
The exchange of rows does not affect the result, except you
should
keep track of this exchanges to reorder your result later
approriately.
I have implemented that the way, that I only access the
matrix-rows
by an index-array and can reorder the result easily at the
end.
If it is useful in your case, you could consider to not to
divide at
each step. You could always do:
a1 a2 a3 | 1 0 0
b1 b2 b3 | 0 1 0 -> 0 b2' b3' | 
y1' y2' y3'
c1 c2 c3 | 0 0 1 0 c2' c3' | 
z1' z2' z3' -> 0 0 c3 | z1
z2 z3
where b2' = a1 b2 - a2 b1 instead of the common 
b2' = b2-
a2b1/a1
b3' = a1 b3 - a3 b1
c2' = a1 c3 - a2 c1 and so on.
For matrix-inversion of that type nothing changes (only if
you would
compute the determinant you have to keep track of the factors
of
omitted divisions for later use)
Then you need only to divide at *the last step* z1/c3 and so
on.
Also you can perform that process inplace, you don't need
temporary
matrices - just compute the new values in the current cells.
The second problem is inevitable. If any matrix *has* a big
ratio
between its biggest and smallest eigenvalue, then you will
have
to deal with very big relative numerical errors.
---
With symmetric matrices the eigenvalue/eigenvector
decomposition
is (relatively) simple; in this case you could then simply
compute the reciprocal of the eigenvalues:
(let A be the given matirx, t an orthogonal rotation matrix
with t*t'=I)
A = t' * D * t where D is diagonal
then simply all diagonal-elements of D have to be inverted and
inv(A) = t * inv(D) * t'
This process of [CapitalThorn]nding a rotation-matrix t involves cos/sin-
computations, which seems to be costly, but it is known, that
the computational-process is numerically very stable against
round-off errors.
---
matrix-inversion-procedures, even for large matrices like that
of size 1000*1000, which discuss rounding errors in detail as
well
problems were discussed in a very instructive manner is
19 dubious ways to compute the matrix exponential (search
HTH -
Gottfried Helms
===
Subject: How to [CapitalThorn]t an equation
by support1.mathforum.org (8.11.6/8.11.6/The Math Forum,
$Revision:
1.9 primary) id i8IFma129265;
Hi all,
I have a scenario with three variables, X,Y,Z and Result This
is what
I know about them:
X could go be from 1  100 say meters
Y could be 0  31 say ranking
Z could be 1- 100 say meters
Result is in meters (1-100)
For values of X between say 1-20 meters, and 80-100 meters it
impacts
the result very much i.e low values of Result i.e. like a
bell shaped
curve.
If Y increases then the Result increases for the most part of
course
there could be some exceptions.
IF Z increases then the Result increases again with an
exception that
if Z becomes like 50 meters then the probability of achieving
that
high Result (meters) is very low.
How do I put together an equation that consists of X, Y, Z to
achieve
the Result. Should I assume a correlation coef[CapitalThorn]cient and then
try to
[CapitalThorn]t an equation. I have no clue about this.
Any help is greatly appreciated.
===
Subject: Re: Golf--shooting your age
by support1.mathforum.org (8.11.6/8.11.6/The Math Forum,
$Revision:
1.9 primary) id i8IFmbi29334;
Are ther records for golfers who shoot their age on their
birthday? I
campton, nh.
===
Subject: Re: zeros of a polynomial of third order
by support1.mathforum.org (8.11.6/8.11.6/The Math Forum,
$Revision:
1.9 primary) id i8IFmcV29356;
>Given the following monic polynomial:
>P=xi^3 - (mu+1)*(mu+2)/2*rho*xi^2 + mu*(mu+2)*rho*xi -
mu*(mu+1)/2*rho=0
>with mu a real number and 0<mu<1. rho is a complex number
with
>|rho|<1.
>Suppose that mu and rho are such that the largest root (the
root
which
>has the largest modulus) of P, has a modulus equal to one.
>Then we want to show that, if mu increases while rho is kept
constant,
>the modulus of the largest root of P also increases.
>It seems a simple problem (and it probably is), but I can't
[CapitalThorn]nd a
proof
>for it. Does anyone has some idea?
>Christophe
Try using Cardan's formula to express the roots in terms of
p, mu, and
rho. The required variations can then be studied.
Otolorin
===
Subject: Re: zeros of a polynomial of third order
by support1.mathforum.org (8.11.6/8.11.6/The Math Forum,
$Revision:
1.9 primary) id i8HHJMH21332;
>Given the following monic polynomial:
>P=xi^3 - (mu+1)*(mu+2)/2*rho*xi^2 + mu*(mu+2)*rho*xi -
mu*(mu+1)/2*rho=0
>with mu a real number and 0<mu<1. rho is a complex number
with
>|rho|<1.
>Suppose that mu and rho are such that the largest root (the
root
which
>has the largest modulus) of P, has a modulus equal to one.
>Then we want to show that, if mu increases while rho is kept
constant,
>the modulus of the largest root of P also increases.
>It seems a simple problem (and it probably is), but I can't
[CapitalThorn]nd a
proof
>for it. Does anyone has some idea?
>Christophe
Try express the roots of the cubic equation using Cardan's
formular.
The variations of the parameters can then be studied.
Otolorin
===
Subject: Re: zeros of a polynomial of third order
>Given the following monic polynomial:
>P=xi^3 - (mu+1)*(mu+2)/2*rho*xi^2 + mu*(mu+2)*rho*xi -
mu*(mu+1)/2*rho=0
>with mu a real number and 0<mu<1. rho is a complex number
with |rho|<1.
>Suppose that mu and rho are such that the largest root (the
root which
>has the largest modulus) of P, has a modulus equal to one.
>Then we want to show that, if mu increases while rho is kept
constant,
>the modulus of the largest root of P also increases.
You were probably right to ask this in a numerical analysis
forum,
but I can only think of how to do with in a symbolic-algebra
way.
Sorry, I'm a guy of limited talents.
I can show that the largest root of P increases for _small_
changes in
mu.
Write rho = r1 + r2 i and the roots as xi = r(c+si).
Expanding out
into real and imaginary parts gives two equations, which,
together with
c^2+s^2=1, enable us to solve for r, c, s in terms of mu, r1,
r2.
(That's no surprise, of course: You're still 
just solving for
xi in
terms of mu and rho of course, but you're doing so in a
different way.)
Since you are only interested in (the largest value of) r ,
not c and s,
you can eliminate c and s and obtain a single polynomial Q
whose
roots are the values of r . It's not pretty: 
it's of degree
18 in r.
I can try to write it a little more compactly: let nu = mu +
1 and write
rr for |rho|^2 and R for r^2; then we can eliminate mu, r2,
and r.
I get Q =
(nu-1)^6*nu^6*rr^3-4*(nu-1)^6*(nu+1)^2*nu^4*rr^3*R+2*(nu-1)^4
*(-4+3*nu^2)
*(nu+1)^2*nu^4*rr^3*R^2-4*(nu-1)^4*rr^2*nu^2*(-8*r1+3*nu^2+rr
*nu^6+4*rr*nu^5
+6*rr*nu^4+4*rr*nu^3+rr*nu^2+2*r1*nu^4+4*r1*nu^3-6*r1*nu^2-16
*r1*nu)*R^3
+(nu-1)^2*rr^2*(nu+1)^2*nu^2*(-16+32*r1+32*nu-16*nu^2+rr*nu^6
+2*rr*nu^5
+rr*nu^4+16*r1*nu^4-48*r1*nu^2)*R^4-4*(nu-1)^2*rr^2*(nu+1)^2*
(16-nu^4
-16*nu^2+2*r1*nu^6+4*r1*nu^5+2*r1*nu^4)*R^5+8*(nu-1)*rr*nu^2*
(-6+12*r1
+2*r1^2*nu^5+6*r1^2*nu^4+6*r1^2*nu^3+2*r1^2*nu^2+6*nu-2*rr+rr
*nu^5+3*rr*nu^4
+rr*nu^3-5*rr*nu^2-6*rr*nu-12*r1*nu^3-12*r1*nu^2+12*r1*nu)*R^
6-32*(nu-1)*rr
*(nu+1)^2*(r1*nu^3+r1*nu^2-4*nu+4)*R^7+16*(nu+1)^2*nu^2*rr*R^
8-64*R^9
You can decide whether R is increasing as mu increases by
computing
dR/d(mu) = - (dQ/d(nu)) / (dQ/dR).
Now, you are stating that 1 < nu < 2 and that 0 < rr < 1 and
that one
of the roots of this polynomial is R = 1. Under those
conditions there are
numbers t and u with nu = (1+t^2)/(2t) and rr =
(1+u)^2/(2(1+u^2));
the fact that R=1 then becomes an equation in t, u, and r1
which we
can solve for r1; for one of the choices of u we will have r1
=
(-1+2*t-2*u-11*t^2-19*t^4+7*t^6-14*t^5+4*t^7-u^2-7*t^6*u-3*t^
6*u^2+45*t^4*u-
45
*t^4*u^2+t^8*u+2*t^8*u^2+18*t^5*u^2+6*t^7*u+7*t^2*u^2-18*t^3*
u-4*t^3*u^2+4*t
^5
*u+3*t^2*u-6*t*u^2-7*u^3*t^2+19*u^3*t^4+11*u^3*t^6+u^3*t^8-4*
u^3*t+14*u^3*t^
3
-2*u^3*t^7)/(t+1)^3/(1+t^2)^2/(t-1)/(1+u^2)/(1+u)
Then we can compute the derivative above at R = 1 ; it's
miraculously
simple:
-32*(u-1)^2*(2*t^4-t^3-t+2)*t^3/(-3*u^2+16*t*u^2-30*t^2*u^2+t
^4*u^2-2*u+16*t
*u
-12*t^2*u+16*t^3*u-2*t^4*u+1-30*t^2+16*t^3-3*t^4)/(1+t^2)/(t-
1)^2/(t+1)^2
Many of the factors are square, or clearly positive. (Note
that we may
always take t between 1 and 2+sqrt(3) because of the
inequalities
on mu.) The only complicated factor is a quadratic in u, whose
discriminant never vanishes for t in (1 , 2+sqrt(3) ), and
which is
positive at u=0 (at least, it is for those t's). So the
quadratic is
always positive for these t, and hence the derivative
dR/d(mu) > 0
under the conditions given in the problem.
By the way, not every u and t (in this range) give rise to a
mu and rho which satisfy the premises of the problem. For one
thing,
the value of r2 will have to be one of the square roots of rr
- r1^2
which puts the constraint that this expression is indeed a
square;
it's actually a bunch of square factors times
-2*(-1-2*t^2+t^4-u+2*t^2*u+t^4*u)*(u-1)
so that's what must be positive. For our values of t this
means we
must have u in the interval (-1 + 4t^2/(t^4+2t^2-1) , 1 ) .
The other problem is that I have parameterized the set of all
mu and rho
which have _a_ root of Q equal to 1; that includes all the
mu's and
rho's
whose largest root of P has magnitude 1 but it may include
others with
(say) their smallest root of this magnitude.
dave
===
Subject: Re: tackling singular matrix
by support1.mathforum.org (8.11.6/8.11.6/The Math Forum,
$Revision:
1.9 primary) id i8IJ1ao13750;
why dont you ask Dr. Chiranjeeb Bhattacharya?
>HI GROUP
>IVE TRIED LINEAR LEAST SQUARE FITTING TO A SET OF DATA AND
SOLVED
THE
>NORMAL EQUATIONS BY GAUSSIAN ELIMINATION (NORMALISATION
FOLLOWED BY
>PARTIAL PIVOTING STRATEGY). BUT THE RESULTANT MATRIX PRIOR TO
>BACKSUBSTITUTION
>I KNOW THERE ARE MANY COMPLEX STRATEGIES LIKE SVD AND SO ON,
BUT I
>WANT TO IMPLEMENT A RELATIVELY SIMPLER ONE. ALSO IT WOULD BE
OF
>IMMENSE HELP TO ME IF YOU CAN GIVE A BRIEF DESCRIPTION OF THE
>ALGORITHM TO USE. FOR YOUR CONVENIENCE IM GIVING BELOW THE
SINGULAR
>MARTRIX THAT RESULTED:
>2.9E-5 2.9E-5 3.9E-3 6.1E-2 1
>-8.4E-22 1.5E-4 3.01E-3 5.5E-2 1
>6.3E-22 -2.7E-21 1.4E-3 4.3E-2 1
>-4.8E-22 3.6E-21 -2.11E-21 2.5E-2 1
>6.8E-22 5.9E-21 7.1E-20 -7E-19 1.7E-2
>NB: I TRIED TO FIT A POLYNOMIAL OF DEGREE 4.
>
> ....DHRUBOJYOTI SINHA
===
Subject: Re: tackling singular matrix
>HI GROUP
>I?VE TRIED LINEAR LEAST SQUARE FITTING TO A SET OF DATA
AND SOLVED
> THE
>NORMAL EQUATIONS BY GAUSSIAN ELIMINATION (NORMALISATION
FOLLOWED BY
>PARTIAL PIVOTING STRATEGY). BUT THE RESULTANT MATRIX PRIOR
TO
>BACKSUBSTITUTION
>I KNOW THERE ARE MANY COMPLEX STRATEGIES LIKE SVD AND SO
ON, BUT I
>WANT TO IMPLEMENT A RELATIVELY SIMPLER ONE. ALSO IT WOULD
BE OF
>IMMENSE HELP TO ME IF YOU CAN GIVE A BRIEF DESCRIPTION OF
THE
>ALGORITHM TO USE. FOR YOUR CONVENIENCE I?M GIVING BELOW
THE SINGULAR
>MARTRIX THAT RESULTED:
>2.9E-5 2.9E-5 3.9E-3 6.1E-2 1
>-8.4E-22 1.5E-4 3.01E-3 5.5E-2 1
>6.3E-22 -2.7E-21 1.4E-3 4.3E-2 1
>-4.8E-22 3.6E-21 -2.11E-21 2.5E-2 1
>6.8E-22 5.9E-21 7.1E-20 -7E-19 1.7E-2
>
>NB: I TRIED TO FIT A POLYNOMIAL OF DEGREE 4.
>
> ....DHRUBOJYOTI SINHA
Hei,
if the matrix is the one you posted, you can still use a
trick to
invert it. That's because it's not really 
singular, but only
badly
scaled. A simple way is to use the same scale for columns and
rows: If
you apply
A^-1 = D * (D*A*D)^-1 * D
with
D = diag(1.0/sqrt(abs(diag(A))))
The matrix D*A*D has now 1.0 or -1.0 on the diagonal elements.
In your example, the determinant of D*A*D is virtually 1.0,
so no
problem to invert, while the non-scaled matrix has det A =
2.5883e-15.
In the Gaussian elimination, you might be confronted with tiny
pivot-elements, even though the matrix isn't really singular
(your
example), but only badly scaled. Your algroithm can't handle
that, so
just test the selected pivot element for its absolute value
and drop
out, if that is too low (e.g. < 1e15).
Volker
===
Subject: expand the right-hand side on the basis of the
eigenvectors of the
matrix
Hello!
I have a problem: Ax=b, where
A = [ 1001 1000;
1000 1001]
x = [x1 ;
x2]
b = [ 2001;
2001]
a small change of b db = [1;
-1]
Could somebody explain why the small variation in the right
hand side
produces large variations in the solution?
Here is a hint: expand the right-hand side on the basis of the
eigenvectors of the matrix.
SO. I found eigenvectos: x01 = [1; x02 = [1;
1] -1]
It forms an orthogonal basis, because the matrix A is
symmetric.
Then, when we expand the right hand side in the basis of the
eigenvectors, then we have:
b = c1 * x01 + c2 * x02, Hence c1 = 2001 and c2 = 0
But, when we use (b + db) = c1 * x01 + c2 * x02, then c1 =
2001.5 and
c2 = 0.5
So, in the [CapitalThorn]rst case c2 = 0 and in the second c2 = 0.5 .
I believe that the answer to the question: why the small
variation in
the right hand side produces large variations in the
solution? should
be somewhere here...
But what is a name of a theorem or a method which will help
me to
anser such question.
Sinserely,
Taglit
===
Subject: Re: expand the right-hand side on the basis of the
eigenvectors of
the matrix
X-RFC2646:  Original
> I have a problem: Ax=b, where
> A = [ 1001 1000;
> 1000 1001]
> b = [ 2001;
> 2001]
> a small change of b db = [1;
> -1]
> Could somebody explain why the small variation in the right
hand side
> produces large variations in the solution?
> Here is a hint: expand the right-hand side on the basis of
the
> eigenvectors of the matrix.
A is badly conditioned. Hint: plot the two equations.
===
Subject: Re: expand the right-hand side on the basis of the
eigenvectors of
the matrix
> I have a problem: Ax=b, where
> A = [ 1001 1000;
> 1000 1001]
> b = [ 2001;
> 2001]
> a small change of b db = [1;
> -1]
> Could somebody explain why the small variation in the
right hand side
> produces large variations in the solution?
> Here is a hint: expand the right-hand side on the basis
of the
> eigenvectors of the matrix.
> A is badly conditioned. Hint: plot the two equations.
It's not enough, I have to expand the right-hand side on the
basis of the
eigenvectors of the matrix to [CapitalThorn]nd the answer....
===
Subject: Re: Noise in linearized exponential decay [CapitalThorn]tting
Just some random thoughts... since the main problem I see
with the
linearized model is at the end of the decay where the signal
is small
then the log values can spray around (wildly). Say my decay
starts at
maximum amplitude of 1 and the later points are close to zero
(10^-2
or so). Then with the addition of noise the later points end
up
varying from -3 to -12 or so and they have an undue inßuence
on the
linear [CapitalThorn]t.
So, since it is the small values which appear to be
problematic, what
if I simply weight the [CapitalThorn]t using the signal value at that
point? It
seems to work OK, but gives different results from both the
simple
linearized and full non-lin-LS [CapitalThorn]tting. Is this a previously
studied
method of [CapitalThorn]tting?
===
Subject: Re: Noise in linearized exponential decay [CapitalThorn]tting
>Just some random thoughts... since the main problem I see
with the
>linearized model is at the end of the decay where the signal
is small
>then the log values can spray around (wildly). Say my decay
starts at
>maximum amplitude of 1 and the later points are close to
zero (10^-2
>or so). Then with the addition of noise the later points end
up
>varying from -3 to -12 or so and they have an undue inßuence
on the
>linear [CapitalThorn]t.
>So, since it is the small values which appear to be
problematic, what
>if I simply weight the [CapitalThorn]t using the signal value at that
point? It
>seems to work OK, but gives different results from both the
simple
>linearized and full non-lin-LS [CapitalThorn]tting. Is this a previously
studied
>method of [CapitalThorn]tting?
you mean ???
y(i)= a*exp(b*x(i)) + noise =>
log(y(i)) = log(a*exp(b*x(i)) +noise) =
= log(a) + b*x(i) + log(1+noise/y(i)) (assuming small
residuals
such that y(i) approx
=a*exp(b*x(i))
and by weighting the data by y(i) to make the variances in
the new model
of
equal size? this is only approximately true
yes, the problem has been often studied and concerns the
introduction of
bias in transformaing model equations. there is much
literature for
solving
nonlinear regression models bias free. (-> applied statistics)
hth
peter
===
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===
7.67816092 sub 7.6 = 0.07816092
0.078160919 * 88 = 6.87816092(1)
6.87816092 sub 6.8 = 0.07816092
0.078160919 * 88 = 6.878160918(2)
6.878160918 sub 6.8 = 0.078160918
0.078160918 * 88 = 6.878160763(3)
6.878160763 sub 6.8 = 0.078160763
0.078160763 * 88 = 6.878147133(4)
6.878147133 sub 6.8 = 0.078147133
0.078147133 * 88 = 6.876947743(5)
6.876947743 sub 6.8 = 0.076947742
0.076947742 * 88 = 6.771401296
6.8 * 5 = 34
sqr3 34 = 3.239611801
3.239611801 * pi = 10.17754064
10.17754064 * 6.771401296 = 68.91621188
88 constellations, 68th constellation Crux
===
Subject: Complexity optimisation for a block matrix inversion
I know, I shouldn't do this, but after posting my problem in
comp.theory without success, I found that it's maybe better
to ask
here, too.
I described my problem behind this link:
In the mean time, I found the Dijkstra-algorithm to have a
complexity
of n^2*2^(2*n), which is exponential and therefore not
suitable for
me.
Is there heuristics, how to pivot block matrices in order to
minimise
numerical effort?
Volker
===
Subject: 64 - is this the only number that is both a sq and a
cube?
Hi - I am trying to [CapitalThorn]n out if this is the only number that is
both a
square
and a cube, and if any one is working on the signi[CapitalThorn]cance of
this.
===
Subject: Re: 64 - is this the only number that is both a sq
and a cube?
16^3=64^2=4096
256^3=4096^2=16777216
....
> Hi - I am trying to [CapitalThorn]n out if this is the only number that
is both a
square
> and a cube, and if any one is working on the signi[CapitalThorn]cance of
this.
===
Subject: Re: 64 - is this the only number that is both a sq
and a cube?
>16^3=64^2=4096
>256^3=4096^2=16777216
more generally, any N^p, where p is divisible by 6, is both a
square and a
cube.
>.....
> Hi - I am trying to [CapitalThorn]n out if this is the only number that
is both a
>square
> and a cube, and if any one is working on the signi[CapitalThorn]cance
of this.
===
Subject: Re: 64 - is this the only number that is both a sq
and a cube?
X-RFC2646:  Original
thank you, that's great - all makes sense now.
>16^3=64^2=4096
>256^3=4096^2=16777216
> more generally, any N^p, where p is divisible by 6, is both
a square and a
> cube.
>.....
> Hi - I am trying to [CapitalThorn]n out if this is the only number
that is both a
>square
> and a cube, and if any one is working on the signi[CapitalThorn]cance
of this.
===
Subject: Proof of Cholesky decomposition
Does someone know how is proven, by recurrence, the existence
of Cholesky
decomposition for a symmetric positive de[CapitalThorn]nite matrix?
Franck
===
Subject: Re: Proof of Cholesky decomposition
>Does someone know how is proven, by recurrence, the
existence of Cholesky
>decomposition for a symmetric positive de[CapitalThorn]nite matrix?
> Franck
cholesky=> positive de[CapitalThorn]nite: obvious.
positive de[CapitalThorn]nite=> all leading minors positive. use 
[CapitalThorn]rst
LU-decomp ;
show that product of pivots=minor. make LU-> LDL' using the
fact that
R=DL' (no pivoting , symmetry)
done
(other possibilities exist )
hth
peter
===
Subject: CumulativeInquiry BB (re DAGs/perms/dim2 posets) now
functional
USENET posters and perusers who are interested in certain
problems
arising from the interplay of theory on DAGs, perms, and dim2
posets
may be interested in a new BB available at:
http://www.CumulativeInquiry.com/Forums
Visitors can peruse WITHOUT registering or logging in; posting
requires both.
Background to the issues being thrashed out on this BB can be
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at:
http://www.CumulativeInquiry.com/Problems
Visitors should also feel free to request membership in the CI
(Cumulative Inquiry) list-server by sending a message WITH NO
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and the following single line in the message:
subscribe ci_discuss_001
You will receive a reply message containing an auth command
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key.
Then send a second message back to the CI major domo with NO
SUBJECT
line and the entire auth command as the [CapitalThorn]rst and only line in
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Although some members of this list-server also read the BB,
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to receive and send messages is via the list server. All
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this
list server are archived to the BB in chronological order.
===
Subject: number of intersection points
Suppose k is a positive integer and mu is a real number in
[0,1]. Then
consider the complex number
c=( (exp(i*theta))2/(exp(i*theta)*(1+mu)-mu) )^(1/k)
(i is the complex unit (sqrt(-1))).
If we plot these complex values in the complex plane (for
theta=0..Pi),
we become a (continuous) curve C in the complex plane. If
theta=0 for
example, then c=1+0i, and this is one point of this curve.
Then de[CapitalThorn]ne the points A=exp(i*2*Pi/k) and B=1.
Now I would like to proove that the curve C has at most 2
intersections
with the line between the complex points A and B, with theta
in ]0,Pi]
(thus max 2 intersections next to the trivial case c=1).
Christophe
===
Subject: Re: Bessel Function - J0(x)
charset=iso-8859-1
: Perhaps you should offer a version suitable for
benchmarking the accuracy
: and speed.
This version can serve as a benchmark.
D. Baruth
===
Subject: Re: Bessel Function - J0(x)
> : Perhaps you should offer a version suitable for
benchmarking the
accuracy
> : and speed.
> This version can serve as a benchmark.
> D. Baruth
Sigh ... nobody (here?) wants only your programm
but if you want to discuss your solutions you may
provide more. Either do it or not. Anyway there
are some real goodies on C R Bond's URL & they are
open ...
--
use mail for mail not nonail