# 0.10 Implementing ffts in practice  (Page 15/21)

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## Plans for higher vector ranks

These plans extract a vector loop to reduce a DFT problem to a problem of lower vector rank, which is then solved recursively. Anyof the vector loops of $\mathbf{V}$ could be extracted in this way, leading to a number of possible plans corresponding to different looporderings.

Formally, to solve $dft\left(\mathbf{N},\mathbf{V},\mathbf{I},\mathbf{O}\right)$ , where $\mathbf{V}=\left\{\left(,n,,,\iota ,,,o,\right)\right\}\cup {\mathbf{V}}_{1}$ , FFTW generates a loop that, for all $k$ such that $0\le k , invokes a plan for $dft\left(\mathbf{N},{\mathbf{V}}_{1},\mathbf{I}+k·\iota ,\mathbf{O}+k·o\right)$ .

## Indirect plans

Indirect plans transform a DFT problem that requires some data shuffling (or discontiguous operation) into a problem that requires noshuffling plus a rank-0 problem that performs the shuffling.

Formally, to solve $dft\left(\mathbf{N},\mathbf{V},\mathbf{I},\mathbf{O}\right)$ where $\left|\mathbf{N}\right|>0$ , FFTW generates a plan that first solves $dft\left(\left\{\right\},\mathbf{N}\cup \mathbf{V},\mathbf{I},\mathbf{O}\right)$ , and then solves $dft\left(copy-o\left(\mathbf{N}\right),copy-o\left(\mathbf{V}\right),\mathbf{O},\mathbf{O}\right)$ . Here we define $copy-o\left(t\right)$ to be the I/O tensor $\left\{\left(,n,,,o,,,o,\right),\mid ,\left(,n,,,\iota ,,,o,\right),\in ,t\right\}$ : that is, it replaces the input strides with the output strides. Thus, an indirect plan firstrearranges/copies the data to the output, then solves the problem in place.

## Plans for prime sizes

As discussed in "Goals and Background of the FFTW Project" , it turns out to be surprisingly useful to be able to handle large prime $n$ (or large prime factors). Rader plans implement the algorithm from [link] to compute one-dimensional DFTs of prime size in $\Theta \left(nlogn\right)$ time. Bluestein plans implement Bluestein's “chirp-z” algorithm, which can also handle prime $n$ in $\Theta \left(nlogn\right)$ time [link] , [link] , [link] . Generic plans implement a naive $\Theta \left({n}^{2}\right)$ algorithm (useful for $n\lesssim 100$ ).

## Discussion

Although it may not be immediately apparent, the combination of the recursive rules in "The space of plans in FFTW" can produce a number of useful algorithms. To illustrate these compositions, we discuss three particular issues: depth- vs. breadth-first, loop reordering,and in-place transforms.

As discussed previously in sections "Review of the Cooley-Tukey FFT" and "Understanding FFTs with an ideal cache" , the same Cooley-Tukey decomposition can be executed in either traditionalbreadth-first order or in recursive depth-first order, where the latter has some theoretical cache advantages. FFTW is explicitlyrecursive, and thus it can naturally employ a depth-first order. Because its sub-problems contain a vector loop that can be executed ina variety of orders, however, FFTW can also employ breadth-first traversal. In particular, a 1d algorithm resembling thetraditional breadth-first Cooley-Tukey would result from applying "Cooley-Tukey plans" to completely factorize the problem size before applying the loop rule "Plans for higher vector ranks" to reduce the vector ranks, whereas depth-first traversal would result fromapplying the loop rule before factorizing each subtransform. These two possibilities are illustrated by an example in [link] .

where we get a research paper on Nano chemistry....?
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
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
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