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One might get the impression that there is a strict dichotomy that divides cache-aware and cache-oblivious algorithms, but the two arenot mutually exclusive in practice. Given an implementation of a cache-oblivious strategy, one can further optimize it for the cachecharacteristics of a particular machine in order to improve the constant factors. For example, one can tune the radices used, thetransition point between the radix- n algorithm and the bounded-radix algorithm, or other algorithmic choices as describedin "Memory strategies in FFTW" . The advantage of starting cache-aware tuning with a cache-oblivious approach is that the starting point already exploitsall levels of the cache to some extent, and one has reason to hope that good performance on one machine will be more portable to otherarchitectures than for a purely cache-aware “blocking” approach. In practice, we have found this combination to be very successful withFFTW.

Memory strategies in fftw

The recursive cache-oblivious strategies described above form a useful starting point, but FFTW supplements them with a number ofadditional tricks, and also exploits cache-obliviousness in less-obvious forms.

We currently find that the general radix- n algorithm is beneficial only when n becomes very large, on the order of 2 20 10 6 . In practice, this means that we use at most a single step of radix- n (two steps would only be used for n 2 40 ). The reason for this is that the implementation of radix n is less efficient than for a bounded radix: the latter has the advantage that an entire radix butterfly can beperformed in hard-coded loop-free code within local variables/registers, including the necessary permutations and twiddlefactors.

Thus, for more moderate n , FFTW uses depth-first recursion with a bounded radix, similar in spirit to the algorithm of [link] but with much larger radices (radix 32 is common) and base cases (size 32 or 64 iscommon) as produced by the code generator of "Generating Small FFT Kernels" . The self-optimization described in "Adaptive Composition of FFT Algorithms" allows the choice of radix and the transition to the radix- n algorithm to be tuned in a cache-aware (but entirely automatic) fashion.

For small n (including the radix butterflies and the base cases of the recursion), hard-coded FFTs (FFTW's codelets ) are employed. However, this gives rise to an interesting problem: acodelet for (e.g.) n = 64 is 2000 lines long, with hundreds of variables and over 1000 arithmetic operations that can be executed inmany orders, so what order should be chosen? The key problem here is the efficient use of the CPU registers, which essentially form anearly ideal, fully associative cache. Normally, one relies on the compiler for all code scheduling and register allocation, but but thecompiler needs help with such long blocks of code (indeed, the general register-allocation problem is NP-complete). In particular, FFTW'sgenerator knows more about the code than the compiler—the generator knows it is an FFT, and therefore it can use an optimalcache-oblivious schedule (analogous to the radix- n algorithm) to order the code independent of the number ofregisters [link] . The compiler is then used only for local “cache-aware” tuning (both for register allocation and the CPUpipeline). One practical difficulty is that some “optimizing” compilers will tend to greatly re-order the code,destroying FFTW's optimal schedule. With GNU gcc, we circumvent this problem by using compiler flags that explicitly disable certain stages of theoptimizer. As a practical matter, one consequence of this scheduler is that FFTW's machine-independent codelets are no slower thanmachine-specific codelets generated by an automated search and optimization over many possible codelet implementations, as performedby the SPIRAL project [link] .

Questions & Answers

Do somebody tell me a best nano engineering book for beginners?
s. Reply
what is fullerene does it is used to make bukky balls
Devang Reply
are you nano engineer ?
what is the Synthesis, properties,and applications of carbon nano chemistry
Abhijith Reply
so some one know about replacing silicon atom with phosphorous in semiconductors device?
s. Reply
Yeah, it is a pain to say the least. You basically have to heat the substarte up to around 1000 degrees celcius then pass phosphene gas over top of it, which is explosive and toxic by the way, under very low pressure.
how to fabricate graphene ink ?
for screen printed electrodes ?
What is lattice structure?
s. Reply
of graphene you mean?
or in general
in general
Graphene has a hexagonal structure
On having this app for quite a bit time, Haven't realised there's a chat room in it.
what is biological synthesis of nanoparticles
Sanket Reply
what's the easiest and fastest way to the synthesize AgNP?
Damian Reply
types of nano material
abeetha Reply
I start with an easy one. carbon nanotubes woven into a long filament like a string
many many of nanotubes
what is the k.e before it land
what is the function of carbon nanotubes?
I'm interested in nanotube
what is nanomaterials​ and their applications of sensors.
Ramkumar Reply
what is nano technology
Sravani Reply
what is system testing?
preparation of nanomaterial
Victor Reply
Yes, Nanotechnology has a very fast field of applications and their is always something new to do with it...
Himanshu Reply
good afternoon madam
what is system testing
what is the application of nanotechnology?
In this morden time nanotechnology used in many field . 1-Electronics-manufacturad IC ,RAM,MRAM,solar panel etc 2-Helth and Medical-Nanomedicine,Drug Dilivery for cancer treatment etc 3- Atomobile -MEMS, Coating on car etc. and may other field for details you can check at Google
anybody can imagine what will be happen after 100 years from now in nano tech world
after 100 year this will be not nanotechnology maybe this technology name will be change . maybe aftet 100 year . we work on electron lable practically about its properties and behaviour by the different instruments
name doesn't matter , whatever it will be change... I'm taking about effect on circumstances of the microscopic world
how hard could it be to apply nanotechnology against viral infections such HIV or Ebola?
silver nanoparticles could handle the job?
not now but maybe in future only AgNP maybe any other nanomaterials
I'm interested in Nanotube
this technology will not going on for the long time , so I'm thinking about femtotechnology 10^-15
can nanotechnology change the direction of the face of the world
Prasenjit Reply
At high concentrations (>0.01 M), the relation between absorptivity coefficient and absorbance is no longer linear. This is due to the electrostatic interactions between the quantum dots in close proximity. If the concentration of the solution is high, another effect that is seen is the scattering of light from the large number of quantum dots. This assumption only works at low concentrations of the analyte. Presence of stray light.
Ali Reply
how did you get the value of 2000N.What calculations are needed to arrive at it
Smarajit Reply
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Source:  OpenStax, Fast fourier transforms. OpenStax CNX. Nov 18, 2012 Download for free at http://cnx.org/content/col10550/1.22
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