mm-454
====
Subject:
: Re: Gauss - Kronrod ( I need help!)> I've to do
something for school about numerical integration.> But I could
really need some help understanding the method of
Gauss-Kronrod> (or at least the Gauss quadrature).> It would
be very nice if someone could explain it to me.>
BriegGauss-Legendre quadrature since it is the most efficient
(integrates agiven degree polynomial with optimal number of
points.) This pagedescribes it just about as well as
anything:http://mathworld.wolfram.com/
Legendre-GaussQuadrature.htmlAnd they also have a page on
Gauss-Kronrod quadrature (which
reusespoints)http://mathworld.wolfram.com/
Gauss-KronrodQuadrature.htmlthough not as helpful, there are
some good references there.Hope that helps,John===
===
Subject:
: Re:
Who are You Dennis ? by support1.mathforum.org
(8.11.6/8.11.6/The Math Forum, $Revision: 1.9 primary) id
i2GMCl408175;>My calculus class was challenged to determine
and prove which is the>greater of the two: e^pi or pi^e. I
have worked on this for quite>some>time, but cannot prove my
conclusion. Any suggestions?The function have the form x^y. We
want to know the direction of theinequality in pi^e ? e^pi.
Take the log on both side you now havelog(pi)/pi ? log(e)/e.
If we look at the function log(x)/x and takethe derivative we
have (log(x)/x)' = (1-log(x))/x^2. The function isdecreasing
thus pi^e < e^pi.DL===
===
Subject:
: Re: Who are You Dennis ?>My
calculus class was challenged to determine and prove which is
the>greater of the two: e^pi or pi^e. I have worked on this
for quite> some time, but cannot prove my conclusion. Any
suggestions?> The function have the form x^y. We want to know
the direction of the> inequality in pi^e ? e^pi. Take the log
on both side you now have> log(pi)/pi ? log(e)/e. If we look
at the function log(x)/x and take> the derivative we have
(log(x)/x)' = (1-log(x))/x^2. The function is> decreasing thus
pi^e < e^pi. DL===David solution is OK ! The problem ,, Which
is the greater of the two: e^ pi or pi^e is attributed to
Euler. An alternative solution is to investigate if there
exists a positive constant A such that (1) A^x >= x^A for all
positive x .By considering f:(0,infty)--->R , f(x)=ln(x)/x ,
it follows that (1) isequivalent to(1') f(A) >= f(x) for all
positive x.It follows that A must be an extremum point
(absolut maximum). One finds A=e the unique possibility.
Therefore , we have followingDefinition: Napier's constant ,,e
is the unique positive number A whichsatisfies A^x >= x^A for
all x>0 .I remember that a similar discussion was published in
The Amer.Math.Monthly in period 1950-1970.=====
===
Subject:
: Re: Who
are You Dennis ?>My calculus class was challenged to determine
and prove which is the>greater of the two: e^pi or pi^e. I
have worked on this for quite> some>time, but cannot prove
my conclusion. Any suggestions?> The function have the form
x^y. We want to know the direction of the> inequality in pi^e
? e^pi. Take the log on both side you now have> log(pi)/pi ?
log(e)/e. If we look at the function log(x)/x and take> the
derivative we have (log(x)/x)' = (1-log(x))/x^2. The function
is> decreasing thus pi^e < e^pi.You obviously meant decreasing
for log(x)>=1 i.e. x>=e. Since bothpi and e are greater than or
equal to e, the desired conclusionfollows. Indeed your argument
shows that e^x >= x^e for all x>0.> DLUse of tools
distinguishes Man from Beast. And UNIX users from WINDOZE
lusers.===
===
Subject:
: Help, if you can by support1.mathforum.org
(8.11.6/8.11.6/The Math Forum, $Revision: 1.9 primary) id
i2GNLUE15631;What do these numbers have in common - 7, 3, 10,
11, and 12 ??===
===
Subject:
: Re: Help, if you can>What do these
numbers have in common - 7, 3, 10, 11, and 12 ?? One of the
four words in Help, if you can shares the property,as does one
of the seven words in What do these numbers have in commonJohn
Adams served two terms as Vice President and one as President,
but lostreelection. Later his son became President despite
losing the popular vote.That son lost his reelection attempt
badly. Now history is repeating itself.pmontgom@cwi.nl
Microsoft Research and CWI Home: San Rafael,
California===
===
Subject:
: Re: Help, if you can> What do these
numbers have in common - 7, 3, 10, 11, and 12 ??They are all
positive integers.===
===
Subject:
: Re: Help, if you can> What do
these numbers have in common - 7, 3, 10, 11, and 12 ??> They
are all positive integers.except -7===
===
Subject:
: Re: Help, if
you can>What do these numbers have in common - 7, 3, 10, 11,
and 12 ??>They are all positive integers.>except -7 I assume
the - is a dash, not a minus. In which case theseare the
smallest nonobvious quadratic resides modulo 37: 9^2 == 7 15^2
== 3 11^2 == 10 14^2 == 11 7^2 == 12nonobvious excludes 1, 4,
9, 16. Check that 2, 5, 6, 8,13, 14, 15, 17, 18, 19, 20 are
quadratic non-residues modulo 37. Look elsewhere for another
of my answers.John Adams served two terms as Vice President
and one as President, but lostreelection. Later his son became
President despite losing the popular vote.That son lost his
reelection attempt badly. Now history is repeating
itself.pmontgom@cwi.nl Microsoft Research and CWI Home: San
Rafael, California===
===
Subject:
: Re: Modelling dynamics of linked
members <c2tg9a$gs5$1@lust.ihug.co.nzThis may not be quite what
you're looking for, but early (late 80's, early 90's)
simulations of polymers were done in a similar way, without
the external forces, usually. The method of choice was Monte
Carlo. If you techniques to minimize the energy, which should
give you the correct dynamics, even if it isn't
pretty.CheersSandra> Go to http://www3.addall.com/ type
mechanism linkage For> more hits, go to Used Books there. Then
visit your university library.> I think what makes the problem
I'm interested in rather different from > the problems treated
in robotics and mechanical machinery etc. is that > all
rotational degrees of freedom exist at the junctions.
Effectively > my linked chain of members is more like a string
with attached weights > than a mechanical linkage system. I
can't think of a mechanical system > with this character. I'm
guessing that it might have chaotic behaviour.> I'm looking
for suggestions on appropriate mathematical simulation
methods.> Gib===
===
Subject:
: Re: Modelling dynamics of linked
members> This may not be quite what you're looking for, but
early (late 80's, early > 90's) simulations of polymers were
done in a similar way, without the > external forces, usually.
The method of choice was Monte Carlo. If you > techniques to
minimize the energy, which should give you the correct >
dynamics, even if it isn't pretty.> Cheers> SandraBeam lattice
models for molecular crystals are a bit older than that:early
1960s. And they were done by spectral methods, not in the
timedomain.My previous post was about something totally
different: flexiblemultibodydynamics in the time domain. This
includes robotics - which isrestricted to rigid body trees or
chains - as a special case.===
===
Subject:
: Re: Help with Number
PatternNext number is: 41122314can be found at link:
http://www.research.att.com/~njas/sequences/index.html> What
is the next number in this set??> 1> 11> 21> 1112> 3112>
211213> 312213> 212223> 114213> 31121314> ????????===
===
Subject:
:
Re: Help with Number Pattern17, I'm sure of it.> What is the
next number in this set??> 1> 11> 21> 1112> 3112> 211213>
312213> 212223> 114213> 31121314> ????????===
===
Subject:
: Re: How
do i find the equation of this curveCherese
<cher_ruth@hotmail.com> schrieb im Newsbeitrag> Litres 1/4 1/2
3/4 1 1-1/4 1-1/2> Time (minutes) 9.85 7.83 7.52 7.42 7.02
6.98Try this solution:Polynomial Approximationy = c_0 + c_1*x
+ c_2*x^2 + ... + c_k*x^k + ... + c_p*x^pk = 0, 1, ... , p x y
c_k 0.250000 9.850000 15.890000 0.500000 7.830000 -35.946000
0.750000 7.520000 55.380000 1.000000 7.420000 -34.240000
1.250000 7.020000 4.800000 1.500000 6.980000 1.536000Email
account locked up due to spams===
===
Subject:
: Re: eigenvalues of
large, symmetric, matrces with many zeroes. >Actually, I need
to compute things like >sum_i (1/n) frac{lambda_i}{a -
blambda_i} >where lambda_i, i=1ldots n are the eigenvalues,
>and a and b are parameters. >Any idea ?this is
trace((inverse(a*I-b*A)*A) = sum_{i=1 to n}
(e_i'(inverse(a*I-b*A)*Ae_i)hence no eigenvalue computation is
involved here .solve (a*I-b*A)z_i = Ae_i and sum up sum_i
(e_i'z_i) hopefully you will be able to compute a sparse
decomposition of a*I-b*Aof course also iterative solvers might
be applicable. anyway this is cheaperthan solving the complete
eigenvalue problem for such a large matrix (if it is not
banded with a very small band, such that reduction to
tridiagonalform is feasible using the bandwidht reduction
algorithm of H.R.Schwarz.hthpeter >I need to compute the
eigenvalues of a symmetric, >real, nXn matrix A where: 1. each row of A is a vector of zeroes with >at most 4 ones
>2. n is very large, of order 10^5 or 10^6. >Any
suggestions are greatly appreciated. >Federico Echenique
> I assume you want only some ... (the largest , the
smallest, some in some > interval .... , otherwise you must
proceed completely different ...) >
http://www.ime.unicamp.br/~chico/arpack++ >
http://www.caam.rice.edu/software/ARPACK > hth >
peter===
===
Subject:
: Re: help! power iteration for eigenvalue
approximation >i've been looking around for answers to this
problem but i just couldn't get >a clue: >i am using power
iteration method to compute the first k dominant >eigenvalues
and eigenvectors of a symmetric matrix A. here is what i do:
>1. use power iteration to get the dominant eigenvalue k1 and
eigenvector v1 >of matrix A >2. residueA = A - k1 * v1 *
v1.transpose >3. use power iteration to get the dominant
eigenvalue k2 and eigenvector v2 >of residueA >4. ... repeat
the above until i have enough approximation terms >for some
reason i don't know, the eigenvalue produced by this process
is not >in order i want. for example, >A= >[ 0 0 0 0 1 2 3 4 >
0 0 0 1 2 3 4 3 > 0 0 1 2 3 4 3 2 > 0 1 2 3 4 3 2 1 > 1 2 3 4 3
2 1 0 > 2 3 4 3 2 1 0 0 > 3 4 3 2 1 0 0 0 > 4 3 2 1 0 0 0 0 ];
>using matlab, the first three eigenvalues are (in order of
absolute value) >14.0759, -9.4306, 4.5685 >but with the power
iteration method, i get them in order: 14.0759, >4.5685,
-9.4306 >here is the problem picked out: >the residue matrix
after first iteration >residue1=[ >-0.6592 -0.9584 -1.2071
-1.3525 -0.3525 0.7929 2.0416 >3.3408 >-0.9584 -1.3933 -1.7550
-0.9663 0.0337 1.2450 2.6067 >2.0416 >-1.2071 -1.7550 -1.2106
-0.4767 0.5233 1.7894 1.2450 >0.7929 >-1.3525 -0.9663 -0.4767
0.2251 1.2251 0.5233 > 0.0337 -0.3525 >-0.3525 0.0337 0.5233
1.2251 > 0.2251 -0.4767 -0.9663 -1.3525 > 0.7929 1.2450 1.7894
> 0.5233 -0.4767 -1.2106 -1.7550 -1.2071 > 2.0416 2.6067 1.2450
> 0.0337 -0.9663 -1.7550 -1.3933 -0.9584 > 3.3408 2.0416 >
0.7929 -0.3525 -1.3525 -1.2071 -0.9584 -0.6592]; >its
eigenvalues are -9.4306, 4.5685 ... but the power iteration
method gives >me 4.5685, which is not the dominant eigenvalue.
>so i am confused what the power iteration is really doing,
because in some >cases it finds the eigenvalue with maximum
ABSOLUTE VALUE, while in others >it finds the one with maximum
VALUE. did i do anything wrong? please help >me!!! >rui the
only explanation I have that you might have an unlucky choice
of the first eigenvector guess for the second eigenvalue and a
weak terminationcriterion. often you might have the effect that
seemingly, with a badly chosen initial vector, the iteration
does not converge to the dominant eigenvalue, but if the
termination criterion is very sharp, then finallyaccumulation
of roundoff drives the iteration into the right direction.a
side remark:your problem should not be solved using this
simple scheme plus deflation but the simultaneous vector
iteration which finds the desired eigenspace simultaneously.
see golub & van loan for a description.code is in
http://www.netlib.org/svdpackhthpeter===
===
Subject:
: Spherical
Harmonics Kit by support1.mathforum.org (8.11.6/8.11.6/The
Math Forum, $Revision: 1.9 primary) id i2HDlY705838;HI alldo
anyone know about a spherical harmonics kitthat works on
windows platform ? specialy Visual C++.i'll be glad to have
some hintsThanx in advanceTarik===
===
Subject:
: Re: C code for
finding the roots of a polynomial by support1.mathforum.org
(8.11.6/8.11.6/The Math Forum, $Revision: 1.9 primary) id
i2HDmCJ06132;Hi everybodydoes any one know where can I find
the C programe to find the roots ofthe polynomial and the
product of an expression of the
form(s-2)(s-1)(s.^2+2ac-6s).............>Hi all,> Does anyone
know where I can find a C function for finding>the roots of a
polynomial with real coefficients? I've found>lots of fortran
routines, but the folks I'm working with have >all moved to C.
>Bud G.>-->Dr. William K Glunt | Are you ABNORMAL?>APSU Dept of
Math and CS| Then you are probably better than
mostpeople!>Clarksville TN | Are alien space monsters bringing
aSTARTLING NEW>home phone 615 645 8938 | WORLD? from _The book
of the SubGenius_===
===
Subject:
: Re: C code for finding the roots
of a polynomial> Hi everybody> does any one know where can I
find the C programe to find the roots of> the polynomial and
the product of an expression of the form>
(s-2)(s-1)(s.^2+2ac-6s).............Try looking at my web site
www.hvks.com/numerical.htm click on portsand I have posted 3
general algorithm for finding roots on polynomialswith real
coeeficients. The three methods are:1) Newton (By Madsen)2)
Jenkins-Traub famous roots finderand finally3) Renormalized
Graeffe Iteration by Malajovich.Henrik===
===
Subject:
: searching
algo by support1.mathforum.org (8.11.6/8.11.6/The Math Forum,
$Revision: 1.9 primary) id i2HDlW005746;>Dear Friend:> I use
the package of ARPACK, for our project, we have to write
our>own subprogram to calculate w=A*v.> I read the matrix in
the Harwell-Boeing format. Then I can>calculate the w=A*v
faster ( because we can use CSR format
includedin>Harwell-Boeing format for this calculation).> For
the serial version of ARPACK, I get my resolution easily.
But>now I want to use a supercomputer IBM SP3 to resolve my
problem. I>used the PARPACK. This key problem is that the
examples in the>examples are hard to understand. Can you give
me a example ofnormally>use (read a matrix file ,then
calculate the w=A*v)?> Have someone had some exemples of
parallel version to calculate>w=A*v using CSR format? SPARKIT
has no such subprograms.> Thank you very much!> Best wishes to
you!!===
===
Subject:
: Ability In MathematicsTO THE
SCI.MATH.NUM-ANALYSIS NEWSGROUPI have an enquiry which people
in this Mathematics newsgroupmay wish to address if they so
desire:Firstly,I have to say that,personally,I have always
found Mathematicsdifficult.It has hindered my understandings
of Astronomy and I failedthe British 11+ examination BECAUSE
of it and so, consequently,went to a British Secondary Modern
School as opposed to themore academic British Grammar
School.Even though I hadunderstanding and competence in every
other subject I studied atschool and,later, college.So, in at
least two crucial ways, Mathematicshas been a very significant
subject in my life.So,given all this, have you any ideas as to
why,for some people,Mathematics ability comes so naturally
whilst other people find it sodifficult?Clearly some people
see patterns in Mathematics whilstothers,NO MATTER HOW MUCH
THEY TRY,cannot.Why is this so? Thank you. Brian
Devonald===
===
Subject:
: Re: Ability In Mathematics> So,given all
this, have you any ideas as to why,for some people,>
Mathematics ability comes so naturally whilst other people
find it so> difficult?Clearly some people see patterns in
Mathematics whilst> others,NO MATTER HOW MUCH THEY
TRY,cannot.Short answer -- different learning styles.I saw two
examples of this when I was in university {engineering
student}.My first year introductory calculus course was taught
in sections of ~20 students by graduate degree candidates. All
students (several hundred) took the same exams [all university
students requiring *ANY* math took this course. ]. My section
was primarily engineering students. The instructor was a pure
mathematician whose teaching style was theory oriented with
strong formal rigor. On the first course-wide exam, our
section had dubious distinction of coming in dead last (
evidently by a significant margin). Instructor received an
invitation to visit Math department chair.The following class
went something like:Instructor: What is the problem?Class: We
want examples.Instructor: You need theory.Instructor (
recognizing lost cause ;): ok HOWEVER, when the next exam
proves you wrong, we will go back to doing it the right wayThe
next exam came.I can not claim we were top section. However we
were soundly above 50th percentile. I think instructor said
something like Engineers are STRANGE. I will admit I did not
really grasp much of what we were taught until the second
semester Physics course demonstrated some of what we learned
in first semester math ;/My second example comes from my
sophomore year.A set of 3rd and 4th semester math courses were
required of and taken ONLY by math majors and engineering
majors.The course format of the 3rd semester was three
lectures and one recitation (small group) per week. The course
was large enough to require two lectures. The Math department
provided one. The Engineering college provided one. Student
assignment to lecture section was approximately random. There
was a common final exam. Once again learning/teaching styles
were evident.I had the misfortune to be in the mathematician
lecture. Not only was he a pure mathematician, but he was also
an algebraist. Our lecture hall had 2 30 ft long chalk boards
which could be raised/lowered. He covered both at least TWICE
per lecture. There was a math grad student in the same rooming
house as I. He could not understand why I hated having his hero
as a lecturer. He also could not understand why such a great
man was being wasted on undergraduates when there were grad
students fighting to take courses from him.The end result
;?Well, we gave new meaning to grading on the curve.For the
students lectured by the engineering prof, a nice conventional
Bell curve. [Class median] ~= [math major median] ~=
[engineering median] ~= [historical median]My section you ask?
;{Shall, we say, we gave administrators fits ;)The math majors
scored *SIGNIFICANTLY* higher than any group you might compare
them to.The engineers? Well create formula to arithmetically
add x points to ALL grades of engineers in that section to
force the median grade to be passing. All engineering students
of that lecturer session who received a passing grade were to
be placed in a special ( can we say 'remedial'?) section of
the following course.===
===
Subject:
: Re: Ability In
Mathematics>So,given all this, have you any ideas as to
why,for some people,>Mathematics ability comes so naturally
whilst other people find it so>difficult?Clearly some people
see patterns in Mathematics whilst>others,NO MATTER HOW MUCH
THEY TRY,cannot.>Why is this so?Some can, some can't. Are you
going to ask the same question in themusic-performance
newsgroups? In the competitive-sports newsgroups?This has
nothing to do with numerical analysis; Followup-To
set.dave===
===
Subject:
: Re: Ability In MathematicsCognitive
scientists have studied this to a degree. Mathematical ability
is a complex of abilities, which can differ among good
mathematicians. The human brain uses pointers in short term
memory to reference collections of ideas, clumps. People who
do mathematics well can do this more easily than others; and a
substantial part of this ability is learned through practice.
Einstein, although not a mathematician, is reputed to have
used internal bodily sensations to help. Most mathematicians
are strongly visual, but there are exceptions: I know at least
one reasonably good mathematician who has very poor visual
memory.I've been waiting for an opportunity to cite some wise
words from my advisor, Jimmy Savage, who, although he slighted
his ability, was a passable mathematician.In the first place,
it cannot be too strongly emphasized that a long mathematical
argument can be fully understood on first reading only when it
is very elementary indeed, relative to the reader's
mathematical knowledge. If one wants only the gist of it, he
may read such material once only; but otherwise he must expect
to read it at least once again. Serious reading of mathematics
is best done sitting bolt upright on a hard chair at a desk.
Pencil and paper are nearly indispensable; for there are
always figures to be sketched and steps in the argument to be
verified by calculation.> TO THE SCI.MATH.NUM-ANALYSIS
NEWSGROUP> I have an enquiry which people in this Mathematics
newsgroup> may wish to address if they so desire:> Firstly,I
have to say that,personally,I have always found Mathematics>
difficult.It has hindered my understandings of Astronomy and I
failed> the British 11+ examination BECAUSE of it and so,
consequently,> went to a British Secondary Modern School as
opposed to the> more academic British Grammar School.Even
though I had> understanding and competence in every other
subject I studied at> school and,later, college.So, in at
least two crucial ways, Mathematics> has been a very
significant subject in my life.> So,given all this, have you
any ideas as to why,for some people,> Mathematics ability
comes so naturally whilst other people find it so>
difficult?Clearly some people see patterns in Mathematics
whilst> others,NO MATTER HOW MUCH THEY TRY,cannot.> Why is
this so?> Thank you.> Brian DevonaldBob Wheeler ---
http://www.bobwheeler.com/ ECHIP, Inc. ---Randomness comes in
bunches.===
===
Subject:
: Network Reliability by
support1.mathforum.org (8.11.6/8.11.6/The Math Forum,
$Revision: 1.9 primary) id i2HGW2S28832;Does anyone know of a
source for source code incorporating currentstate of the are
techniques in computing network system reliability. Irealize
that these are NP-hard problems but am looking for code
thatimplements the current algorithmic state of the art.Al
Myers===
===
Subject:
: ill-conditioned system of linear equations by
support1.mathforum.org (8.11.6/8.11.6/The Math Forum,
$Revision: 1.9 primary) id i2HGW2E28838;I have a big problem
concerning the solution of an ill-conditionedsystem of linear
equations. It is of the form A.x=b, where A can besquare or
rectangular depending on the application. The problem isthat
really small errors in the experimentally obtained values of
Aand x result in absurd values of the solution x. I solve the
system bytaking the inverse of the matrix A (if a is square)
and thenmultiplying b by this inverse. I think this isn't the
problem, sincewhen I do the calculation (A.A^-1)-(unity
matrix) the resulting matrixhas elements smaller then e.g.
10^-14. So I conlude that the problemis not the inversion. If
A is rectangular I use the pseudo inverse inorder to derive
the least square solution of the system.Also, when I do the
calculation A.x, I find b again!Does anyone has an idea how I
can solve my problem? Does apre-conditioning of the matrix A
can help me? Does this relax theaccuracy-needs for A and x?
Can I make my calculation more robust by apllying a certain
scaling ofmy rows?Please give me a suggestion, I'm quite
desperate!Sam.===
===
Subject:
: Re: ill-conditioned system of linear
equations> I have a big problem concerning the solution of an
ill-conditioned> system of linear equations. It is of the form
A.x=b, where A can be> square or rectangular depending on the
application. The problem is> that really small errors in the
experimentally obtained values of A> and x result in absurd
values of the solution x. I solve the system by> taking the
inverse of the matrix A (if a is square) and then> multiplying
b by this inverse. I think this isn't the problem, since> when
I do the calculation (A.A^-1)-(unity matrix) the resulting
matrix> has elements smaller then e.g. 10^-14. So I conlude
that the problem> is not the inversion. If A is rectangular I
use the pseudo inverse in> order to derive the least square
solution of the system.> Also, when I do the calculation A.x,
I find b again!> Does anyone has an idea how I can solve my
problem? Does a> pre-conditioning of the matrix A can help me?
Does this relax the> accuracy-needs for A and x? > Can I make
my calculation more robust by apllying a certain scaling of>
my rows?> Please give me a suggestion, I'm quite desperate!You
need to regularize. Try solving A^T A x = A^T b but multiplythe
diagonal of A^T A by 1.0001 or a similar quantity.For more
background and more sophisticated recipes if this fails,see,
e.g.,http://www.mat.univie.ac.at/~neum/papers.html#
regtutorialArnold Neumaier===
===
Subject:
: Re: ill-conditioned
system of linear equations >I have a big problem concerning
the solution of an ill-conditioned >system of linear
equations. It is of the form A.x=b, where A can be >square or
rectangular depending on the application. The problem is >that
really small errors in the experimentally obtained values of A
>and x result in absurd values of the solution x. I solve the
system by ^ you mean b here? >taking the inverse of the matrix
A (if a is square) and then >multiplying b by this inverse. I
think this isn't the problem, since >when I do the calculation
(A.A^-1)-(unity matrix) the resulting matrix >has elements
smaller then e.g. 10^-14. So I conlude that the problem did
you also try A^-1.A-I ? this may be quite different >is not
the inversion. If A is rectangular I use the pseudo inverse in
>order to derive the least square solution of the system.
>Also, when I do the calculation A.x, I find b again! >Does
anyone has an idea how I can solve my problem? Does a
>pre-conditioning of the matrix A can help me? Does this relax
the >accuracy-needs for A and x? >Can I make my calculation
more robust by apllying a certain scaling of >my rows? >Please
give me a suggestion, I'm quite desperate! > Sam. since A and b
contain empirical data, it is hard to say whether scaling
mighthelp.bad scaling is in most cases the result of bad
modelling (e.g. representing a polynomial on an interval far
from zero (say [1000,2000]) by the standard form McLaurin
series a0=a1*x+a2*x^2+...a rule of thumb for scaling is 1.
scale the variables acording to their importancein the model,
then 2. scale every row to be of equal length. but, since you
already are acquainted with the pseudoinverse:why not using
the svd, setting small singular values to zero and using
thepseudoinverse thereafter? (also in the square case).this is
one of the most successful techniques for regularizing
illconditionedsystems of linear equations where nothing else
helpshthpeter===
===
Subject:
: Numerical integration of Laguerre
polynomialsfor a while I've been stuck with the following
problem and I washoping that someone here could give me some
advise. It looks as if I'mhitting the limits of double
precision accuracy.I need to do this triple integral
numerically (physical background:matrix element between
harmonic oscillator states):int_0^infty du sqrt(u) exp(-u)
L_{kappa}^{1/2}(u) timesint_0^infty dv sqrt(v) exp(-v)
L_{k}^{1/2}(v) timesint_{-1}^{+1} dx
L_{kappa'}^{1/2}(f(u,v,x)) L_{k'}^{1/2}(g(u,v,x))where
L_{k}^{1/2} is an associated Laguerre polynomial. f and g
aresimple functions. They are linear in x but not in u and v.I
use Gauss-Laguerre quadrature for the u- and v-integrals,
andGauss-Legendre for x. This works very well for small
indeces kappa,kappa', k, k'.However, already for kappa=0,
kappa'=0 and k=30, k'=30, say, theresult of the triple
quadrature becomes unreliable since the productof the Laguerre
polynomials in the integrand becomes very large(1.D+300)
whereas the quadrature weights beome very small (1.D-300)and
over/underflow occurs.Neither could I write down an asymptotic
formula for the integralabove. Is there a better suited way to
carry out the integral (I needto it for many indeces...) in
reasonable computational time?Martin Stoll===
===
Subject:
: Re:
Numerical integration of Laguerre polynomials >for a while
I've been stuck with the following problem and I was >hoping
that someone here could give me some advise. It looks as if
I'm >hitting the limits of double precision accuracy. >I need
to do this triple integral numerically (physical background:
>matrix element between harmonic oscillator states):
>int_0^infty du sqrt(u) exp(-u) L_{kappa}^{1/2}(u) times
>int_0^infty dv sqrt(v) exp(-v) L_{k}^{1/2}(v) times
>int_{-1}^{+1} dx L_{kappa'}^{1/2}(f(u,v,x))
L_{k'}^{1/2}(g(u,v,x)) >where L_{k}^{1/2} is an associated
Laguerre polynomial. f and g are >simple functions. They are
linear in x but not in u and v. >I use Gauss-Laguerre
quadrature for the u- and v-integrals, and >Gauss-Legendre for
x. This works very well for small indeces kappa, >kappa', k,
k'. >However, already for kappa=0, kappa'=0 and k=30, k'=30,
say, the >result of the triple quadrature becomes unreliable
since the product >of the Laguerre polynomials in the
integrand becomes very large >(1.D+300) whereas the quadrature
weights beome very small (1.D-300) >and over/underflow occurs.
>Neither could I write down an asymptotic formula for the
integral >above. >Is there a better suited way to carry out
the integral (I need >to it for many indeces...) in reasonable
computational time? >Martin Stollwhat if you transform the
integrals from [0,infy[ to ]0,1] and use high order
gauss-kronrod like quadpack does?hthpeter===
===
Subject:
: Re:
wavelet-based spectrogram - help needed by
support1.mathforum.org (8.11.6/8.11.6/The Math Forum,
$Revision: 1.9 primary) id i2HGjPH30432;dear alli am
implementing wavelet based spectrogram approach forisolated
word recognition can any one send me the matlab code for
wavelet based spectrogramsthanks in
anticipationnaresh===
===
Subject:
: Re: Random generation of
covariance matrices> This doesn't look like a natural
distribution of covariance> matrices. It is more likely that
your application requires> that you generate them according to
a Wishart distribution.Though I am not familar with Wishart
distributions I cannot imaginehow it could help.My application
is as follows:The required covariance matrix represents the
dependency structure ofnormal returns of divisions of a
company. Dependent on such arandomly drawn covariance matrix I
want to calculate variousallocations of value at risk limits
for each division and analyse someproperties. Since I cannot
determine a realistic distribution forthese matrices, I want
to assume uniform variances and correlations.===
===
Subject:
: Re:
Random generation of covariance matrices>This doesn't look
like a natural distribution of covariance>matrices. It is more
likely that your application requires>that you generate them
according to a Wishart distribution.> Though I am not familar
with Wishart distributions I cannot imagine> how it could
help.> My application is as follows:> The required covariance
matrix represents the dependency structure of> normal returns
of divisions of a company. Dependent on such arandomly drawn
covariance matrix I want to calculate various> allocations of
value at risk limits for each division and analyse some>
properties. Since I cannot determine a realistic distribution
for> these matrices, I want to assume uniform variances and
correlations.The Wishart distribution is for covariance
matrices what the Gaussiandistribution is for real numbers.
Since you don't know anything aboutyour covariance matrices, I
recommend you use a Wishart distributionwhose covariance matrix
is a multiple of the identity matrix.This is a much more
realistic default assumption than what you hadin mind.Arnold
Neumaier===
===
Subject:
: Re: Random generation of covariance
matrices>This doesn't look like a natural distribution of
covariance>matrices. It is more likely that your application
requires>that you generate them according to a Wishart
distribution.>Though I am not familar with Wishart
distributions I cannot imagine>how it could help.>My
application is as follows:>The required covariance matrix
represents the dependency structure of>normal returns of
divisions of a company. Dependent on such arandomly drawn
covariance matrix I want to calculate various>allocations of
value at risk limits for each division and analyse
some>properties. Since I cannot determine a realistic
distribution for>these matrices, I want to assume uniform
variances and correlations.If one assumes that the X_j are n
independent normalvectors of size p with mean 0 and covariance
matrixC = A*A', then the sum of X_j*X_j' has the
Wishartdistribution with parameters n and C (p is the sizeof
C). For simplicity, let us assume that n >= p;there are some
easily seen complications otherwise.Then a random covariance
matrix from this can berepresented as follows:Let b_ii have a
chi distribution with n-i+1 degreesof freedom, and let b_ij,
j>i, be normal (0,1), andlet all of these be independent. For
i>j, b_ij=0.Then A*B*B'*A' is an efficiently generated
matrixwith the Wishart distribution.If you want something
else, I still suggest that yougenerate is as Q*Q', as this
will guarantee that itis a covariance matrix. The correlations
cannot beuniformly distributed.This address is for information
only. I do not claim that these viewsare those of the
Statistics Department or of Purdue University.Herman Rubin,
Department of Statistics, Purdue
Universityhrubin@stat.purdue.edu Phone: (765)494-6054 FAX:
(765)494-0558===
===
Subject:
: Re: Fast eigenvalue/eigenvector
solutions>I need a fast module for determination of the
eigenvalues and>eigenvectors of a 4x4 real symmetric matrix.
(It needs to be fast>because it will be running many times on
a slow machine.)> One can rather quickly reduce the matrix to
tridiagonal> form, using two Householder transformations (one
square> root for each). Then the characteristic equation can>
be obtained, and the quartic solved. Finding the > vectors
from the roots is easy, but if the roots wanted> are close to
each other, or to unwanted roots, there > can be considerable
roundoff error. Transforming back> is also quick. > BTW, when
there is bad roundoff error here, any other> method will also
have it. Not necessarily. A quartic polynomial with very
closezeros gives very poor accuracy for the roots, while
thesymmetric eigenvalue problem is perfectly conditioned andQR
gives very accurate results.Try the matrixa bb a b b a b b
awith a=1/3 and b=eps.Arnold Neumaier===
===
Subject:
: Keeping
quaternion normalizedwhat's the recommended method of keeping
the unit quaternionrepresenting orientation in a rigid body
model normalized if the stateequation is integrated by some
numerical method?Zipfel, Modeling and Simulation of Aerospace
Vehicle Dynamics,recommends adding a term k ( 1 - || q || )
qto the RHS of the state equation, which effectifly uses the
ODE solverto move q back to the surface of the unit sphere as
soon as it'sdeviating. But what are the reasons against the
simple method of justsetting q <- 1 / sqrt( || q || ) qafter
each integration time step?Kind regards-GerhardPS: Return
address given is a spam trap, please see my homepage for
myreal address!Gerhard Wesp o o Tel.: +41 (0) 43
5347636Bachtobelstrasse 56 |
http://www.cosy.sbg.ac.at/~gwesp/CH-8045 Zuerich _/===
===
Subject:
:
Re: Keeping quaternion normalized> what's the recommended
method of keeping the unit quaternion> representing
orientation in a rigid body model normalized if the state>
equation is integrated by some numerical method?> Zipfel,
Modeling and Simulation of Aerospace Vehicle Dynamics,>
recommends adding a term> k ( 1 - || q || ) q> to the RHS of
the state equation, which effectifly uses the ODE solver> to
move q back to the surface of the unit sphere as soon as it's>
deviating. But what are the reasons against the simple method
of just> setting> q <- 1 / sqrt( || q || ) q> after each
integration time step?> Kind regards> -Gerhard> PS: Return
address given is a spam trap, please see my homepage for my>
real address!> -- > Gerhard Wesp o o Tel.: +41 (0) 43 5347636>
Bachtobelstrasse 56 | http://www.cosy.sbg.ac.at/~gwesp/>
CH-8045 Zuerich _/You're thinging q <- q/||q||, right?Adding a
term k(1 - sum(q_i ^ 2))q has the advantage that you don't have
totake a square root -- doing the four multiplies is bad, but
taking a squareroot is really going to add to your execution
time.===
===
Subject:
: Re: solution of non-linear dynamical system?
by support1.mathforum.org (8.11.6/8.11.6/The Math Forum,
$Revision: 1.9 primary) id i2HIZdO12018;cute... really cute...
;-)but doesn't help a bit! :(cheers,pedro>greetings!>i'm a
biologist interested in the dynamical
systemu[t]=1-exp(u[t-1])>starting from u[0]=1, in case it
matters. >Yes it matters! What would've happened if you'd
started with u[0]=0?>Nothing? (Couldn't resist!)>-- >Surendar
Jeyadev jeyadev@wrc.xerox.bounceback.com> Remove
'bounceback' for email address===
===
Subject:
: Re: solution of
non-linear dynamical system?> dave,> thanx for your reply :-)>
i feel really stupid, though :-( after reading your reply, i
realized> that i left a minus sign out. the real equation is>
u[t]=1-exp(-u[t-1]) , so u is always positive, and decreasing.
would> that make things any easier?Mathematica's power series
calculation, with a lot of additional work, suggestsu[t] = 2/n
- 2 log(n)/(3 n^2) + (2 log(n)^2 - 2 log(n) + 1)/(9 n^3) - (10
log(n)^3-25 log(n)^2+25 log(n)-7)/(135 n^4) +
O(n^(-5+eps)),where n = t+C, for some C. Does this
help?===
===
Subject:
: Re: solution of non-linear dynamical system?
by support1.mathforum.org (8.11.6/8.11.6/The Math Forum,
$Revision: 1.9 primary) id i2I0Etf19326;arthur,this helps an
awful lot! thanx very much! :-)did you by any chance save your
work? is there any way you couldemail me your mathematica
notebook, so i could try and learn how youdid it, please? oh,
and... any chance of bounds on C, or is that asking for too
much?;-)cheers, and thanx again,pedro>the real equation
is>u[t]=1-exp(-u[t-1]) , so u is always positive, and
decreasing. would>that make things any easier?>Mathematica's
power series calculation, with a lot of >additional work,
suggests>u[t] = 2/n - 2 log(n)/(3 n^2) + (2 log(n)^2 - 2
log(n) + 1)/(9 n^3) -> (10 log(n)^3-25 log(n)^2+25
log(n)-7)/(135 n^4) +O(n^(-5+eps)),>where n = t+C, for some C.
Does this help?===
===
Subject:
: Re: solution of non-linear dynamical
system?I never do it the same way twice, and I didn't save the
notebook.Outline of method:1. Guess form of solution as
Sum(i=1..infinity n^-i, a[i][Log[n]])2. Write Onn =
(Series[n,{n,Infinity,1}]-n)/nComment: You can't just write
O[1/n], even though it displays that way.3. Write un (for u_n)
= Sum[a[i][Ln]n^-i,{i,5}] + Onn^6Comment: You need one more
term in the expressions than you can evaluate in the final
result.4. Evaluate the corresponding series for u(n+1) as:un1
= (Normal[un] /. {n->n+1,Ln->Ln+Log[1+1/n]}) + Onn^65. eqn =
un1 == 1-Exp[-un];6. LogicalExpand[eqn]7. Manually solve the 4
differential equations for a[1],...,a[4],noting that any
solution that grows exponentially is invalid.8. After
substituting the solutions of the equations into un, replace
Ln by Log[n].No idea about bounds on C. If the series
converges, or ifyou could approximate the sum by an
integral....> arthur,> this helps an awful lot! thanx very
much! :-)> did you by any chance save your work? is there any
way you could> email me your mathematica notebook, so i could
try and learn how you> did it, please?> oh, and... any chance
of bounds on C, or is that asking for too much?> ;-)> cheers,
and thanx again,> pedro> the real equation is>
u[t]=1-exp(-u[t-1]) , so u is always positive, and
decreasing.> would> that make things any
easier?>Mathematica's power series calculation, with a lot
of>additional work, suggests>u[t] = 2/n - 2 log(n)/(3 n^2) +
(2 log(n)^2 - 2 log(n) + 1)/(9 n^3) -> (10 log(n)^3-25
log(n)^2+25 log(n)-7)/(135 n^4) +> O(n^(-5+eps)),>where n =
t+C, for some C. Does this help?===
===
Subject:
: Re: To prove a
connection between 26 & 666 by support1.mathforum.org
(8.11.6/8.11.6/The Math Forum, $Revision: 1.9 primary) id
i2HK6iC22917;Because Google cherishes all her Forum and
discussion links (andlikewise this oldie!) it seems reasonable
to add my (final?)evaluation of the question, for which I
created some web
pages:http://geocities.com/delfiden/text/sss6.htmhttp://
geocities.com/freezotic/deo/s2s108s.htmlhttp://geocities.com/
freezotic/deo/s109s1000s.htmlhttp://geocities.com/freezotic/
deo/s1001s12000s.htmlhttp://geocities.com/freezotic/deo/s666.
htmlGreetingsFreezotic===
===
Subject:
: Numerical integration of
Laguerre polynomialsI'm stuck with the following problem which
arose during a physicsproject: Three-dimensional integration of
a product of four associatedLaguerre polynomials of the
formint_0^inf du sqrt(u) exp(-u) LaguerreL(kappa,1/2,u)
timesint_0^inf dv sqrt(v) exp(-v) LaguerreL(k,1/2,v)
timesint_{-1}^{+1} dx LaguerreL(kappa',1/2,f(u,v,x))
LaguerreL(k',1/2,g(u,v,x)) (sorry for the mixed LaTeX/Maple
notation). The functions f and g inthe innermost integrand are
linear in x but, unfortunately, not in uand v.I used nested
Gauss-Laguerre integrations for u and v, andGauss-Legendre for
x. This works very well for small values of theindeces kappa,
kappa', k, k' <= 30.For larger indices, kappa, kappa', k, k'
>= 30, however, I run into thetrouble that the integrand
becomes extremely large (1.D+300) whereasthe Gauss integration
weights become extremely small (1.D-299),leading to a terrible
loss in accuracy when summing them up.Since I'm unexperienced
with this kind of problem I was hoping thatsomeone could give
me a good hint.Martin Stoll===
===
Subject:
: Numerical integration
of Laguerre polynomialsHello (I'm re-sending this since my
first message seems to have got lost),I've been stuck for a
while with the following problem: I need tocalculate a matrix
element between 3D harmonic oscillator states. Itlooks as
follows:int_{0}^{infty} du u^{1/2} exp(-u) L_{kappa}^{1/2}(u)
timesint_{0}^{infty} dv v^{1/2} exp(-v) L_{k}^{1/2}(v)
timesint_{-1}^{+1} dx L_{kappa'}^{1/2}(f(u,v,x))
L_{k'}^{1/2}(g(u,v,x))where L_{k}^{1/2) is the associated
Laguerre polynomial of order kwith upper index 1/2 and f and g
are simple functions of theirvariables (but not linear). For
low excitation indeces (kappa,kappa',k,k' <= 30) I can
useGauss-Laguerre integration on the u- and v-integrals,
andGauss-Legendre on the x-integral. This works very well. For
larger indeces, however, the Gauss-Laguerre weights at large
meshpoints u, v become extremely small (~1.E-300) and even
underflowwhereas the integrand itself becomes very large
(~1.E+299), althoughtheir product is, in principle, still of
the order unity . Whensumming up the Gauss-Laguerre
integration formula this leads to a hugenumerical inaccuracy.
On the other hand I don't know of anotherintegration scheme to
do the triple integral (and I need to do many ofthem) in
reasonable computer time.Perhaps a rewrite of the integrand as
exp(-u + ln L{kappa}^{1/2})etc. would help, but I don't know of
any formulas for the logarithm ofthe Laguerre polynomials...As
I'm unexperienced with this kind of problem I was hoping
thatsomeone could give me some better ideas :) Martin Stoll
Martin Stoll Institut f.9fr Theoretische Physik, Uni
G.9attingen f: +49-551-39-7687 Tammannstrae 1, 37077
G.9attingen, Germany