# 0.10 Implementing ffts in practice  (Page 2/21)

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And yet there is a vast gap between this basic mathematical theory and the actual practice—highly optimized FFT packages are often anorder of magnitude faster than the textbook subroutines, and the internal structure to achieve this performance is radically differentfrom the typical textbook presentation of the “same” Cooley-Tukey algorithm. For example, [link] plots the ratio of benchmark speeds between a highly optimized FFT [link] , [link] and a typical textbook radix-2 implementation [link] , and the former is faster by a factor of 5–40 (with a larger ratio as $n$ grows). Here, we will consider some of the reasons for this discrepancy, and some techniques that can be used to address thedifficulties faced by a practical high-performance FFT implementation. We won't address the question of parallelization on multi-processor machines, which adds even greaterdifficulty to FFT implementation—although multi-processors are increasingly important, achieving good serial performance is a basicprerequisite for optimized parallel code, and is already hard enough!

In particular, in this chapter we will discuss some of the lessons learned and the strategies adopted in the FFTW library. FFTW [link] , [link] is a widely used free-software library that computes the discrete Fourier transform (DFT) and its various special cases.Its performance is competitive even with manufacturer-optimized programs [link] , and this performance is portable thanks the structure of the algorithms employed, self-optimization techniques, and highlyoptimized kernels (FFTW's codelets ) generated by a special-purpose compiler.

This chapter is structured as follows. First "Review of the Cooley-Tukey FFT" , we briefly review the basic ideas behind the Cooley-Tukey algorithm anddefine some common terminology, especially focusing on the many degrees of freedom that the abstract algorithm allows toimplementations. Next, in "Goals and Background of the FFTW Project" , we provide some context for FFTW's development and stress that performance, while it receives the most publicity, is not necessarily the most important consideration in the implementation of a library of this sort. Third, in "FFTs and the Memory Hierarchy" , we consider a basic theoretical model of the computer memory hierarchy and its impact onFFT algorithm choices: quite general considerations push implementations towards large radices and explicitly recursivestructure. Unfortunately, general considerations are not sufficient in themselves, so we will explain in "Adaptive Composition of FFT Algorithms" how FFTW self-optimizes for particular machines by selecting its algorithm atruntime from a composition of simple algorithmic steps. Furthermore, "Generating Small FFT Kernels" describes the utility and the principles of automatic code generation used to produce the highly optimized building blocksof this composition, FFTW's codelets. Finally, we will briefly consider an important non-performance issue, in "Numerical Accuracy in FFTs" .

## Review of the cooley-tukey fft

The (forward, one-dimensional) discrete Fourier transform (DFT) of an array $\mathbf{X}$ of $n$ complex numbers is the array $\mathbf{Y}$ given by

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
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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