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Theorem 1 [link] Given two polynomials, x ( s ) and h ( s ) , of degree N and M respectively, each with indeterminate coefficients that are elements of a field H , N + M + 1 multiplications are necessary to compute the coefficients of the product polynomial x ( s ) h ( s ) . Multiplication by elements of the field G (the field of constants), which is contained in H , are not counted and G contains at least N + M distinct elements.

The upper bound in this theorem can be realized by choosing an arbitrary modulus polynomial P ( s ) of degree N + M + 1 composed of N + M + 1 distinct linear polynomial factors with coefficients in G which, since its degree is greater than the product x ( s ) h ( s ) , has no effect on the product, and by reducing x ( s ) and h ( s ) to N + M + 1 residues modulo the N + M + 1 factors of P ( s ). These residues are multiplied by each other, requiring N + M + 1 multiplications, and the results recombined using the Chinese remainder theorem (CRT). The operations required in thereduction and recombination are not counted, while the residue multiplications are. Since the modulus P ( s ) is arbitrary, its factors are chosen to be simple so as to make the reduction and CRTsimple. Factors of zero, plus and minus unity, and infinity are the simplest. Plus and minus two and other factors complicate the actualcalculations considerably, but the theorem does not take that into account. This algorithm is a form of the Toom-Cook algorithm and ofLagrange interpolation [link] , [link] , [link] , [link] . For our applications, H is the field of reals and G the field of rationals.

Theorem 2 [link] If an algorithm exists which computes x ( s ) h ( s ) in N + M + 1 multiplications, all but one of its multiplication steps must necessarily be of the form

m k = ( g k ' + x ( g k ) ) ( g k " + h ( g k ) ) for k = 0 , 1 , . . . , N + M

where g k are distinct elements of G ; and g k ' and g k " are arbitrary elements of G

This theorem states that the structure of an optimal algorithm is essentially unique although the factors of P ( s ) may be chosen arbitrarily.

Theorem 3 [link] Let P ( s ) be a polynomial of degree N and be of the form P ( s ) = Q ( s ) k , where Q ( s ) is an irreducible polynomial with coefficients in G and k is a positive integer. Let x ( s ) and h ( s ) be two polynomials of degree at least N - 1 with coefficients from H , then 2 N - 1 multiplications are required to compute the product x ( s ) h ( s ) modulo P ( s ) .

This theorem is similar to Theorem 1 with the operations of the reduction of the product modulo P ( s ) not being counted.

Theorem 4 [link] Any algorithm that computes the product x ( s ) h ( s ) modulo P ( s ) according to the conditions stated in Theorem 3 and requires 2 N - 1 multiplications will necessarily be of one of three structures, each of whichhas the form of Theorem 2 internally.

As in Theorem 2 , this theorem states that only a limited number of possible structures exist foroptimal algorithms.

Theorem 5 [link] If the modulus polynomial P ( s ) has degree N and is not irreducible, it can be written in aunique factored form P ( s ) = P 1 m 1 ( s ) P 2 m 2 ( s ) . . . P k m k ( s ) where each of the P i ( s ) are irreducible over the allowed coefficient field G . 2 N - k multiplications are necessary to compute the product x ( s ) h ( s ) modulo P ( s ) where x ( s ) and h ( s ) have coefficients in H and are of degree at least N - 1 . All algorithms that calculate this product in 2 N - k multiplications must be of a form where each of the k residue polynomials of x ( s ) and h ( s ) are separately multiplied modulo the factors of P ( s ) via the CRT.

Questions & Answers

where we get a research paper on Nano chemistry....?
Maira Reply
what are the products of Nano chemistry?
Maira Reply
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
Google
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
Hafiz Reply
revolt
da
Application of nanotechnology in medicine
what is variations in raman spectra for nanomaterials
Jyoti Reply
I only see partial conversation and what's the question here!
Crow Reply
what about nanotechnology for water purification
RAW Reply
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
Brian Reply
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?
LITNING Reply
what is a peer
LITNING Reply
What is meant by 'nano scale'?
LITNING Reply
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 ?
Bob Reply
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?
Damian Reply
what king of growth are you checking .?
Renato
What fields keep nano created devices from performing or assimulating ? Magnetic fields ? Are do they assimilate ?
Stoney Reply
why we need to study biomolecules, molecular biology in nanotechnology?
Adin Reply
?
Kyle
yes I'm doing my masters in nanotechnology, we are being studying all these domains as well..
Adin
why?
Adin
what school?
Kyle
biomolecules are e building blocks of every organics and inorganic materials.
Joe
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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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