mm-166

Can anyone point me to some good online references or
better yet sometested code (C or FORTRAN) for calculating
slepian functions or prolatespheroidal wave functions?They
seem to be very difficult to evaluate accurately over an
arbitrarydomain. There are some functions in Matlab to
evaluate them from -1to 1 but I have my doubts about the
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ON TUESDAY HE averaged12 miles per hour more, and it took him 9
minutes less to get towork.How far does he travel to
(miles/hr)D = distance (miles)S * (45 / 60) = D(S + 12) *
((45-9)/60) = DSo,S * (45/60) = (S+12) *(36/60), which implies
S=48 miles/hr.Hence, D= 48 * (45/60) = 36 miles.You better not
be cheating on homework. If you are, you should feel
veryguilty.> On Monday, Roger drove to work in 45 minutes. ON
TUESDAY HE averaged> 12 miles per hour more, and it took him 9
minutes less to get to> work.How far does he travel to work?>
support1.mathforum.org (8.11.6/8.11.6/The Math Forum,
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only the non-zero entries of a sparse matrix using the
expression: ij=(row-1)*matrix_size +columns. Is it possibleto
reconstruct the the row and the columns of the matrix
dependingonly on the index, ij. Where matrix_size is the size
of a given squarematrix. This is becuase I can not save the
row and the columns of eachentry in the first pass of my
program.I have tried to find the row and the columns like
this:columns=ij-matrix_size*INT(ij/matrix_size)row=INT((
ij-columns)/matrix_size)+columns, where INT produce theinteger
part of the expressions above.I, however, did not quite get the
right answer ? Is there any one outthere who can comment on it
have indexed only the non-zero entries of a sparse matrix
>using the expression: ij=(row-1)*matrix_size +columns. Is it
possible >to reconstruct the the row and the columns of the
matrix depending >only on the index, ij. Where matrix_size is
the size of a given square >matrix. This is becuase I can not
save the row and the columns of each >entry in the first pass
of my program. >I have tried to find the row and the columns
like this: >columns=ij-matrix_size*INT(ij/matrix_size)
>row=INT((ij-columns)/matrix_size)+columns, where INT produce
the >integer part of the expressions above. >I, however, did
not quite get the right answer ? Is there any one out >there
who can comment on it ? >Yacob >aren't there many i and j
which make ij=k constant ? hence I cannot see how you may
reconstruct two values i and j from one k without knowing some
problem by now, but your post showed up on ournews server
again this evening, so this gives me a convenient time to
saythat I made an error in my initial reply. (I forgot to take
into accountthat your rows and columns are indexed 1 to n
rather than 0 to n-1.)This makesrow=INT(ij/matrix_size)+1 =
INT( (row-1) + column/matrix_size) + 1 = row +
INT(column/matrix_size)give the wrong answer when
columns=matrix_size (sinceINT(column/matrix_size) will not be
0 in that case).The correct solution should have beenrow
=INT((ij-1)/matrix_size) + 1column = ij -
(row-1)*matrix_sizeThat'll teach me not to post without
coffee.Rick>> I have indexed only the non-zero entries of a
sparse matrix> using the expression: ij=(row-1)*matrix_size
+columns. Is it possible> to reconstruct the the row and the
columns of the matrix depending> only on the index, ij. Where
matrix_size is the size of a given square> matrix. This is
becuase I can not save the row and the columns of each> entry
in the first pass of my program.>> I have tried to 
find the row
and the columns like this:>>
columns=ij-matrix_size*INT(ij/matrix_size)>
row=INT((ij-columns)/matrix_size)+columns, where INT produce
the> integer part of the expressions above.>> I, however, did
not quite get the right answer ? Is there any one out> there
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During my recentwork, I want to find the exact form of the sum
or multiply of I_0. orfind the maximum of the following
function:f(x)=prod_{n=-N}^{N}1/Msum_{m=0}^{M-1}(I_0(x*cos(
2pi*m/M*n))+I_0(x*sin(2pi*m/M*n)))Any good approximation
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that can do a complex 3000 point inplace fft. I don't have
Everyone,> I need the source code in C that can do a complex
3000 point in> place fft. I don't have much memory to work
with so the smaller the> AlexIf you want a comprehensable
fft that doesn't take a week to figureout how to 
install (not
mentioning any names here...), you could doworse than
fftn.File at
a comprehensable fft that doesn't take a week to 
figure> out how
to install (not mentioning any names here...), you could do>
worse than fftn.If you get an old version of FFTW, such as
2.x, then you can justcopy all the source files to a single
directory and compile them there.Compiling 3.0 is
Everyone, > I need the source code in C that can do a complex
3000 point in >place fft. I don't have much memory to work
with so the smaller the >Alex >what about fftw3:
http://www.fftw.orgif this is too ambitious for you, then you
Everyone,> I need the source code in C that can do a complex
3000 point in> place fft. I don't have much memory to work
with so the smaller the> AlexCheck out
support1.mathforum.org (8.11.6/8.11.6/The Math Forum,
calculate the integral of 1/(1 - x^4)^(1/2). This is a tricky
one. Anyone got any ideas? I'd really appreciate
CUNGQR was my first major problem. The second wasthat my
version of MATLAB (5.3.0.10183) is producing slightly
differentresults, numerically. I don't think 
it's just a
precision issue. Somethingappears to have changed across
know that after version 5.3, Matlab switched the core matrix
computationlibrary from LINPACK/EISPACK to LAPACK. So, my
version of Matlab isessentially using the same source code as
my Fortran LAPACK library (whichis cxml.lib under Compaq
Visual Fortran). You can read about the switch atthe Mathworks
website (www.mathworks.com) - just do a search on
LAPACK.http://www.mathworks.com/company/newsletter/
clevescorner/winter2000.cleve.shtml In any case, while there
may be small numerical differences between Matlab5.3 and the
current LAPACK 3.0, I would expect them to be _very_ small for
aQR decomposition. Another thing to remember is that Matlab
does everythingin double precision, whereas the LAPACK
routines you're using are singleprecision, so that could be
the cause of any differences you've seen. Youcan try using 
the
ZGEQRF and ZUNGRQ routines from LAPACK instead (theequivalent
of double precision complex), but make sure you declare
yourinput matrices to be double precision as well.John>> Using
CUNGRQ instead of CUNGQR was my first major problem. The 
second
was> that my version of MATLAB (5.3.0.10183) is producing
slightly different> results, numerically. I don't think 
it's
just a precision issue. Something> appears to have changed
across releases.>> What version of MATLAB are you using?>
symmetric positive> matrix with 1's on the diagonals, and 
with
eigenvalues that I can> pre-specify.>> If I do (in matlab)>
u=rand(3,3); % 3 by 3 matrix of u(0,1)> H=orth(u); %
orthonormal basis of u> D=diag([.4 0 0]) the true eigenvalues
are .4 0 0> x=H*D*H' % covariance matrix>> and then I do
svd(x), the eigenvalues of x are .4,0,0 as expected.>> Is
there some way to do a similar for the correlation matrix
(with 1> on the diagonals) and still allow me to specify the
eigenvalue?>You will find a discussion of such inGeorge
Marsaglia and Ingram Olkin, ``Generating correlation
You will find a discussion of such in> George Marsaglia and
Journ. Scient. and Statistical . Computing v 5 , 470--475,
1984,> George, - I am able to use a specific, but much more
simple solution, which I implemented in my
factor-analysis-program (just for experimental reasons those
days). Since you're surely familiar with the optimization
criterion, which is needed to compute the rotation-angles for
getting the PCA-position of a given loadings-matrix, I'll 
say,
the solution is just a small modification of that criterion. I
implemented this under the rationale to get equal
sums-of-squares-of-loadings (Equal SSqL) of the factors;
somehow opposite to that of the PC-rotation, where you
optimize for maximum SSqL in the first factor and minimum in
the second. This procedure converges as fast as PC-rotation
and can be used optimally for this job.GottfriedExampleLet D
be the desired diagonal-matrix containing the Eigenvalues:
1.60 0.00 0.00 0.00 0.00 1.20 0.00 0.00 0.00 0.00 0.80 0.00
0.00 0.00 0.00 0.40then its root can be used for the common
rotation-process (just as a degenerate loadingsmatrix)D2
(D2*D2=D) 1.26 0.00 0.00 0.00 0.00 1.10 0.00 0.00 0.00 0.00
0.89 0.00 0.00 0.00 0.00 0.63You apply a column-rotation T to
D2 for getting equal SSqL ineach column. This rotation
produces this loadingsmatrix:L = D2 * TL 0.89 0.13 -0.88 0.15
-0.08 0.76 -0.11 -0.78 -0.01 0.64 0.19 0.60 0.45 -0.01 0.43
-0.12Once you have T you can create the desired
correlation-matrixR by either computing R = L'*Lor (which is
equivalent) R = T' * D * T 1.00 0.04 -0.58 0.14 0.04 1.00
-0.08 -0.19 -0.58 -0.08 1.00 0.01 0.14 -0.19 0.01
1.00(Rotation-method Equal-Sqared-Loadings-sums in my
factor-analysis- tools Inside-[R] & MatMate (both small
handmade programs for study of Matrix/factor-analysis
techniques))The rotation-criterion is the same summing of
loadingssquares likein PC-rotation, but the angle to be
rotated is then just enhancedby 90 deg, so x/y-axis are
exchanged to y/-x, and the iterationconverges to minimal
should provide a bit more accurate data:Gottfried Helms
containing the Eigenvalues: 1.600 0.000 0.000 0.000 0.000
1.200 0.000 0.000 0.000 0.000 0.800 0.000 0.000 0.000 0.000
0.400> then its root can be used for the common
rotation-process> (just as a degenerate loadingsmatrix)> D2
(D2*D2=D) 1.265 0.000 0.000 0.000 0.000 1.095 0.000 0.000
0.000 0.000 0.894 0.000 0.000 0.000 0.000 0.632> You apply a
column-rotation T to D2 for getting equal SSqL in> each column.
This rotation produces this loadingsmatrix:> L = D2 * T> L
0.865 0.233 0.893 -0.023 0.229 -0.732 -0.011 0.782 -0.088
0.636 -0.065 0.620 -0.438 -0.076 0.446 0.063> Once you have T
you can create the desired correlation-matrix> R by either
computing> R = L'*L> or (which is equivalent)> R = 
T' * D * T
1.000 0.011 0.580 0.078 0.011 1.000 0.141 -0.189 0.580 0.141
1.000 -0.041 0.078 -0.189 -0.041 1.000and L*L' = 1.600 0.000
-0.000 0.000 0.000 1.200 0.000 -0.000 -0.000 0.000 0.800 0.000
0.000 -0.000 0.000 0.400There are some differences, which are
so high, that I haveto check about the uniqueness of this
squares problem where the model functions are essentially of
the form:yi=a0*exp(a1+a2/xi) + a3*exp(a4+a5/xi) + .....(This
is the sum of a number of first order kinetics processes where
temperature (x) changes with time.)The number of exp-terms is
between 1 and 6 and the number of xi,yi is between 64 and
256.This problem is, at least in my case, very ill conditioned
in the sense that the Hessian matrix (and thus the variance
co-variance matrix) has a very small determinant and the ratio
between largest and smallest eigenvalue is very large. In
addition to that are the parameters a1 and a2, a4 and a5 and
so on, highly intercorrelated. Problems which I think are
linked.Is there a trick, e.g. changing of variables, that
least squares problem where the model functions are
>essentially of the form: >yi=a0*exp(a1+a2/xi) +
a3*exp(a4+a5/xi) + ..... >(This is the sum of a number of
first order kinetics processes where >temperature (x) changes
with time.) >The number of exp-terms is between 1 and 6 and
the number of xi,yi is >between 64 and 256. >This problem
is, at least in my case, very ill conditioned in the sense
>that the Hessian matrix (and thus the variance co-variance
matrix) has a >very small determinant and the ratio between
largest and smallest >eigenvalue is very large. In addition to
that are the parameters a1 and >a2, a4 and a5 and so on, highly
intercorrelated. Problems which I think >are linked. >Is
there a trick, e.g. changing of variables, that could improve
suggestions, >Janwillem >as said already by Brian, he
parameters a1, a4, ... are redundant. this willcause already
singularity in your Jacobian.Nevertheless, as rnold stated , a
six-term exponential fit is a quiteambitious undertaking. the
job might possibly become easier ifyou observe the partial
separability of the problem, which allows you toexpress a0,
a3, (the factors before the exp) as functions of theparameters
appearing inside the exp: given those, you can solve for the
others by a linear least squares problem. hence you may use
an appropriate software (dqed) or minimize in alternation:
guess the nonlinear parameters,solve for the linear ones,
then guess again the nonlinear ones ..there is ready to use
software for your task,
seehttp://plato.la.asu.edu/topics/problems/nlolsq.htmlhthpeter
functions are >essentially of the form:>>yi=a0*exp(a1+a2/xi) +
a3*exp(a4+a5/xi) + .....>>(This is the sum of a number of 
first
order kinetics processes where >temperature (x) changes with
time.)>>The number of exp-terms is between 1 and 6 and the
number of xi,yi is >between 64 and 256.>>This problem is, at
least in my case, very ill conditioned in the sense >that the
Hessian matrix (and thus the variance co-variance matrix) has
a >very small determinant and the ratio between largest and
smallest >eigenvalue is very large. In addition to that are
the parameters a1 and >a2, a4 and a5 and so on, highly
intercorrelated. Problems which I think >are linked.>>Is there
a trick, e.g. changing of variables, that could improve the
>behaviour of the system.Ill-conditioning is a feature of the
problem that you're trying to solverather than an aspect of
the algorithm you used to solve it. In otherwords, the
ill-conditioning is telling you that the available data
doesn'tprovide much information about the values of the
parameters. The ill-conditioning will show up when you compute
confidence intervalsfor the fitted parameters (you 
will compute
confidence intervals, right?) inthe form of very broad
confidence intervals. Answers like a1=10 plus orminus 50
might not be very helpful.In this case, some of the difficulty
comes from the problem formulation.Considering the first term
only, a0*exp(a1+a2/xi)=a0*exp(a1)*exp(a2/xi)
a0*exp(a1+a2/xi)=(a0*exp(a1))*exp(a2/xi)In the a0*exp(a1)
part, there's really only one parameter! Forexample, the
solution a0=1, a1=1 is exactly equivalent to a solutionwith
a0=e and a1=0. Nothing in this nonlinear regression will
helpyou figure out whether a0 is really 1 or e. You might try
rewritingyour model as yi=c1*exp(c2/xi)+c3*exp(c4/xi)+...Your
nonlinear least squares problem is called a separable
problem becausesome of the parameters occur linearly and some
occur nonlinearly. There arespecial purpose algorithms for
such problems that can help somewhat. SeeHans Bruun 
Nielsen's
technical report at
http://www.imm.dtu.dk/documents/ftp/tr00/tr01_00.abstract.html
-- Brian Borchers borchers@nmt.eduDepartment of Mathematics
http://www.nmt.edu/~borchers/Socorro, NM 87801 FAX:
the model functions are > essentially of the form:>
yi=a0*exp(a1+a2/xi) + a3*exp(a4+a5/xi) + .....> (This is the
sum of a number of first order kinetics processes where >
temperature (x) changes with time.)> The number of exp-terms
is between 1 and 6 and the number of xi,yi is > between 64 and
256.> This problem is, at least in my case, very ill
conditioned In fact it is singular, since you cannot estimate
a0 and a1 separatelysince y depends only on the product
a0*exp(a1); similarly for theother terms. Set a1=a4=0 to
remove the redundancy, and things willbecome better. If
problems remain, try adding to your least squareobjective a
tiny multiple of a properly weighted sum of the squares ofyour
parameters.But you cannot expect getting very good results
since exponentialfitting problems are usually quite poorly
posed. Which means thateven when you have a good fit, drawing
conclusions about the Ôtrue'values of the 
parameters is
dangerous - variances are governed bythe diagonal entries of
the inverse Hessian at the solutionwhich tend to be
$199.00.> Oh sorry, I was referring to the academic
license which I believe is>> still free. > I searched on the
intel website, but I could not find a free academic> license
version for the MKL. Where would I find this, or has it been>
discontinued?> UshnishNo way! you have to pay for it. So
know a source for scheme implementations of nonlinear
source for scheme implementations of nonlinear >optimization
algorithms such as levenberg-marquardt ?? >what Ôs that?
anyone have a formula to compute the balance on a mortgage
formula to compute the balance on a mortgage loan>
amortization after n payments?After i payments on an n-payment
mortgage, there are n-i payments remaining.Hence, from the
formula for the present value of an annuity, the principal
Premaining after i payments of amount A
isP=(A/r)[1-(1+r)^(-n+i)],where r is the interest rate per
formula to compute the balance on a mortgage loan>>
amortization after n payments?>> After i payments on an
n-payment mortgage, there are n-i payments> remaining. Hence,
from the formula for the present value of an> annuity, the
principal P remaining after i payments of amount A is>
P=(A/r)[1-(1+r)^(-n+i)],> where r is the interest rate per
period (e.g., annual rate/12).Thus, the ratio R of the
principal remaining after i payments to theoriginal principal
always suggest using LAPACK (or CLAPACK if you prefer C) from www.netlib.org as opposed to anything from Numerical Recipes.
Matlab matrix> computations (starting in Matlab 6.0) are
performed using LAPACK. And> LAPACK is free!> John>
I know about lapack and clapack, and I know they are high
performance> libraries, but they are just so cumbersome.
That's a pain, for > people who are not numerical-analysts,
and only use these libraries> as black-boxes.> I actually
write code in c++ and I know about lapack++, and I just > saw
it was superseeded by TNT. That's nice, I thought, but then 
I
> see they have their own implementation of 1D array, complex,
etc. > Why do that, when there's already vector, valarray, 
and
complex in> C++? > I do have the freedom to choose what
routines I use, but is there> a C++ numerical library which
combines the performance of lapack or> NR, with the elegance
and convenience of C++? Also, one which does> not re-invent
the wheel and which is free.It may be useful to look for the
postings on comp.lang.c++ under thesubject Available C++
libraries by Nikki Locke.I agree with you that there does not
seem to be any perfect solutionsfor matrix computations in C++
in the same way that LAPACK exists forFORTRAN (and C). I have
developed my own class library whose primarygoal is to
interface to the core functionality of LAPACK/ATLAS/IntelMKL
while hiding the details of interfacing with these libraries.
Ifyou're interested in seeing what I've got, 
I'm willing to
CLAPACK if you prefer C) from> www.netlib.org as opposed to
anything from Numerical Recipes. Matlabmatrix> computations
(starting in Matlab 6.0) are performed using LAPACK. And>
LAPACK is free!>> John>> I know about lapack and clapack,
and I know they are high performance> libraries, but they are
just so cumbersome. That's a pain, for> people who are not
numerical-analysts, and only use these libraries> as
black-boxes.>> I actually write code in c++ and I know about
lapack++, and I just> saw it was superseeded by TNT. That's
nice, I thought, but then I> see they have their own
implementation of 1D array, complex, etc.> Why do that, when
there's already vector, valarray, and complex in> C++?>> I 
do
have the freedom to choose what routines I use, but is there>
a C++ numerical library which combines the performance of
lapack or> NR, with the elegance and convenience of C++? Also,
one which does> not re-invent the wheel and which is
free.>Silviu,I'm not sure what platform you're 
using, but if
it's Windows/Visual Studio,then at least there are 
precompiled
binaries of CLAPACK at netlib. Ifyou're on a Unix platform, 
the
install procedure is fairly straightforward(though I understand
what you mean by cumbersome). Once you've gonethrough the
procedure once it really isn't all that bad, and you
simplycannot beat the performance of CLAPACK for the price
($0.00).LAPACK++, as you've noted, is dead and has been
replaced by TNT. I havemixed feelings about TNT, particularly
the VERY slow pace of development andlack of real
functionality. The focus of TNT these days seems to be
onproviding better multi-dimensional array classes that Dr.
Pozo believeswill help with portability and maintenance
issues with C++ codes. I'm notan expert C++ programmer, so I
cannot comment intelligently on how well TNTachieves its stated
objectives.If you're digging around for C++ libraries, you
might want to browse
throughhttp://www.thefreecountry.com/sourcecode/
mathematics.shtmlThe Boost libraries seem to be popular these
what platform you're using, but if it's 
Windows/Visual
Studio,> then at least there are precompiled binaries of
CLAPACK at netlib. If> you're on a Unix platform, the 
install
procedure is fairly straightforward> (though I understand
what you mean by cumbersome). Once you've gone> through the
procedure once it really isn't all that bad, and you simply>
cannot beat the performance of CLAPACK for the price
($0.00).and installation is trivial. What I find cumbersome in
lapack isthe function prototypes and the row-column conversion
of matrices.> If you're digging around for C++ libraries, 
you
might want to browse through>
http://www.thefreecountry.com/sourcecode/mathematics.shtml>
The Boost libraries seem to be popular these days as well
(www.boost.org).> (gsl). It looks more humane, although,
being written in C, it mustimplement its own vector, complex,
etc. I don't know how performant it is though. Any 
experience
C++ libraries that are outthere.I do know that you have to be
very careful when performance claims are madewith C++. For
example, I believe Blitz++ claims _amazing_ performance,
butonly for very specific processors/compilers.Your best bet 
is
to check out several and find what works best for 
you.Actually,
one of my primary gripes with C++ is the lack of built
infunctionality to work with arrays (vectors/matrices/linear
algebra). I'verun into exactly the same problems that you 
are
encountering when it comesto finding good numerical linear
algebra libraries, includingvector/matrix classes that can be
made to work efficiently.IMHO, Fortran 90/95 is far better if 
a
majority of your work is focused onthe field of numerical 
linear
algebra. There is a nice Fortran 95 interfacefor LAPACK that
simplifies the calling syntax considerably (provides ageneric
interface for all LAPACK driver routines regardless of type
andautomatically allocates temporary work arrays). I use it
all the time andcouldn't be happier.>> I'm not 
sure what
platform you're using, but if it's 
Windows/VisualStudio,>
then at least there are precompiled binaries of CLAPACK at
netlib. If> you're on a Unix platform, the install procedure
is fairlystraightforward> (though I understand what you mean
by cumbersome). Once you've gone> through the procedure
once it really isn't all that bad, and you simply> cannot
beat the performance of CLAPACK for the price ($0.00).>> and
installation is trivial. What I find cumbersome in lapack is>
the function prototypes and the row-column conversion of
matrices.> If you're digging around for C++ libraries,
you might want to browsethrough>
http://www.thefreecountry.com/sourcecode/mathematics.shtml> The Boost libraries seem to be popular these days as
well(www.boost.org).> (gsl). It looks more humane,
although, being written in C, it must> implement its own
vector, complex, etc. I don't know how performant> it is
answers on my question. You are of course right with the a0,
a1 correlation in:yi=a0*exp(a1+a2/xi) + a3*exp(a4+a5/xi) +
.....This is, however due to me over simplifying the problem.
My apologies for posing a seemingly trivial problem.It should
have been:There are between 1 and 6, say k=1..m differential
equations:dy1(x)/dx=-yk(x)*exp(a1-a2/x) with initial value
x=x0 and y1(x0)=a0dy2(x)/dx=-yk(x)*exp(a4-a5/x) with initial
value x=x0 and y2(x0)=a3.... etc (x is actually (a function
of) temperature in time during heating of a sample)The model
then is in quasi Mathematica
notationYi=Sum[Integrate[dyk(x)/dx,{x,x0,xi}],{k,1,m}]for
i=1..n, n>>mEach diffeqn results in a skew bell-shaped
function resulting in a sum that has a bumpy bell shape.In
practice the a1 is of the same size as a2/x and the final x is
about 2*x0. This means that increasing both a1 and a2 makes no
dramatic change and can easily be compensated by small changes
in the parameters of an other of the diffeqns.So I thought
that the solution could be in transforming a0,a1,a2 etc into
other variables b0,b1,b2.... more or less along the lines of
orthogonalisation in improving the stability of linear
problems.Unfortunately, however, although I passed my exams
numerical analysis 30 years ago, this is too complex for me to
find out for my self. That's why I asked for 
help on the news
group.The suggestion of adding a fraction of the sum of
squares of the parameters as a kind of penalty to the ssq
clearly improves the system. In addition it is even physically
relevant because in particular the a1, a4 .. should not be too
large to have a physical meaning.Janwillem van DijkNRG
sample values taken from a bandlimited function.I assume that
sampling has been done properly (nyquist criteria).I like to
calculate the first derivative of this function by usingthe
Derivative Theorem of the Fourier transform: df[n]/dt <=>
-jnF[n]- First I do a Fourier transform. As I only have real
values, the resulting function is hermitian: f(x) = f*(-x) f*
means conjugate complex- Then the multiplication with -j
(-sqrt(-1)) is a rotation of 90deg around the time axis. Easy
saying: re_new = im_old im_new = -re_old- than I multiply the
result values with their index n for example size 8: index: 0
1 2 3 4 5 6 7 factor 0 1 2 3 k -3 -2 -1 and there is the
problem! what to do at index 4? in order to keep the hermitian
property f(4)=f*(4) I would have to force the imaginary
component to 0, but that would alter the data and therefore
lead to a less precise results. The reason why I want to keep
the hermitian property is, that I want a real function after
the inverse Fourier transform. Fortunately this is only a
problem when I have an even number of samples, so I can always
zero pad to a odd number. But that is not very satisfying, and
I would appreciate a cleaner solution.Maybe someone can
help me out here!Martin