# 0.10 Implementing ffts in practice  (Page 5/21)

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Finally, we should mention that there are many FFTs entirely distinct from Cooley-Tukey. Three notable such algorithms are the prime-factor algorithm for $gcd\left({n}_{1},{n}_{2}\right)=1$ [link] , along with Rader's [link] and Bluestein's [link] , [link] , [link] algorithms for prime $n$ . FFTW implements the first two in its codelet generator for hard-coded $n$ "Generating Small FFT Kernels" and the latter two for general prime $n$ (sections "Plans for prime sizes" and "Goals and Background of the FFTW Project" ). There is also the Winograd FFT [link] , [link] , [link] , [link] , which minimizes the number of multiplications at the expense of a largenumber of additions; this trade-off is not beneficial on current processors that have specialized hardware multipliers.

## Goals and background of the fftw project

The FFTW project, begun in 1997 as a side project of the authors Frigo and Johnson as graduate students at MIT, has gone through several major revisions, and as of 2008 consists of more than 40,000 lines of code. It is difficult to measure the popularity of a free-software package, but (as of 2008) FFTW has been cited in over 500 academic papers, is used in hundreds of shipping free and proprietary software packages, and the authors have received over 10,000 emails from users of the software. Most of this chapter focuses on performance of FFT implementations, but FFTW would probably not be where it is today if that were the only consideration in its design. One of the key factors in FFTW's success seems to have been its flexibility in addition to its performance. In fact, FFTW is probably the most flexible DFT library available:

• FFTW is written in portable C and runs well on many architectures and operating systems.
• FFTW computes DFTs in $O\left(nlogn\right)$ time for any length $n$ . (Most other DFT implementations are either restricted to a subset of sizes or they become $\Theta \left({n}^{2}\right)$ for certain values of $n$ , for example when $n$ is prime.)
• FFTW imposes no restrictions on the rank (dimensionality) of multi-dimensional transforms. (Most other implementations arelimited to one-dimensional, or at most two- and three-dimensional data.)
• FFTW supports multiple and/or strided DFTs; for example, to transform a 3-component vector field or a portion of amulti-dimensional array. (Most implementations support only a single DFT of contiguous data.)
• FFTW supports DFTs of real data, as well as of real symmetric/anti-symmetric data (also called discrete cosine/sinetransforms).

Our design philosophy has been to first define the most general reasonable functionality, and then to obtain the highest possibleperformance without sacrificing this generality. In this section, we offer a few thoughts about why such flexibility has proved important,and how it came about that FFTW was designed in this way.

FFTW's generality is partly a consequence of the fact the FFTW project was started in response to the needs of a real application forone of the authors (a spectral solver for Maxwell's equations [link] ), which from the beginning had to run on heterogeneous hardware. Our initial application requiredmulti-dimensional DFTs of three-component vector fields (magnetic fields in electromagnetism), and so right away this meant: (i)multi-dimensional FFTs; (ii) user-accessible loops of FFTs of discontiguous data; (iii) efficient support for non-power-of-two sizes(the factor of eight difference between $n×n×n$ and $2n×2n×2n$ was too much to tolerate); and (iv) saving a factor of two for the common real-input case was desirable. That is,the initial requirements already encompassed most of the features above, and nothing about this application is particularly unusual.

where we get a research paper on Nano chemistry....?
nanopartical of organic/inorganic / physical chemistry , pdf / thesis / review
Ali
what are the products of Nano chemistry?
There are lots of products of nano chemistry... Like nano coatings.....carbon fiber.. And lots of others..
learn
Even nanotechnology is pretty much all about chemistry... Its the chemistry on quantum or atomic level
learn
da
no nanotechnology is also a part of physics and maths it requires angle formulas and some pressure regarding concepts
Bhagvanji
hey
Giriraj
Preparation and Applications of Nanomaterial for Drug Delivery
revolt
da
Application of nanotechnology in medicine
what is variations in raman spectra for nanomaterials
ya I also want to know the raman spectra
Bhagvanji
I only see partial conversation and what's the question here!
what about nanotechnology for water purification
please someone correct me if I'm wrong but I think one can use nanoparticles, specially silver nanoparticles for water treatment.
Damian
yes that's correct
Professor
I think
Professor
Nasa has use it in the 60's, copper as water purification in the moon travel.
Alexandre
nanocopper obvius
Alexandre
what is the stm
is there industrial application of fullrenes. What is the method to prepare fullrene on large scale.?
Rafiq
industrial application...? mmm I think on the medical side as drug carrier, but you should go deeper on your research, I may be wrong
Damian
How we are making nano material?
what is a peer
What is meant by 'nano scale'?
What is STMs full form?
LITNING
scanning tunneling microscope
Sahil
how nano science is used for hydrophobicity
Santosh
Do u think that Graphene and Fullrene fiber can be used to make Air Plane body structure the lightest and strongest. Rafiq
Rafiq
what is differents between GO and RGO?
Mahi
what is simplest way to understand the applications of nano robots used to detect the cancer affected cell of human body.? How this robot is carried to required site of body cell.? what will be the carrier material and how can be detected that correct delivery of drug is done Rafiq
Rafiq
if virus is killing to make ARTIFICIAL DNA OF GRAPHENE FOR KILLED THE VIRUS .THIS IS OUR ASSUMPTION
Anam
analytical skills graphene is prepared to kill any type viruses .
Anam
Any one who tell me about Preparation and application of Nanomaterial for drug Delivery
Hafiz
what is Nano technology ?
write examples of Nano molecule?
Bob
The nanotechnology is as new science, to scale nanometric
brayan
nanotechnology is the study, desing, synthesis, manipulation and application of materials and functional systems through control of matter at nanoscale
Damian
Is there any normative that regulates the use of silver nanoparticles?
what king of growth are you checking .?
Renato
What fields keep nano created devices from performing or assimulating ? Magnetic fields ? Are do they assimilate ?
why we need to study biomolecules, molecular biology in nanotechnology?
?
Kyle
yes I'm doing my masters in nanotechnology, we are being studying all these domains as well..
why?
what school?
Kyle
biomolecules are e building blocks of every organics and inorganic materials.
Joe
how did you get the value of 2000N.What calculations are needed to arrive at it
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