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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

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
anyone know any internet site where one can find nanotechnology papers?
Damian Reply
research.net
kanaga
sciencedirect big data base
Ernesto
Introduction about quantum dots in nanotechnology
Praveena Reply
what does nano mean?
Anassong Reply
nano basically means 10^(-9). nanometer is a unit to measure length.
Bharti
do you think it's worthwhile in the long term to study the effects and possibilities of nanotechnology on viral treatment?
Damian Reply
absolutely yes
Daniel
how to know photocatalytic properties of tio2 nanoparticles...what to do now
Akash Reply
it is a goid question and i want to know the answer as well
Maciej
characteristics of micro business
Abigail
for teaching engĺish at school how nano technology help us
Anassong
Do somebody tell me a best nano engineering book for beginners?
s. Reply
there is no specific books for beginners but there is book called principle of nanotechnology
NANO
what is fullerene does it is used to make bukky balls
Devang Reply
are you nano engineer ?
s.
fullerene is a bucky ball aka Carbon 60 molecule. It was name by the architect Fuller. He design the geodesic dome. it resembles a soccer ball.
Tarell
what is the actual application of fullerenes nowadays?
Damian
That is a great question Damian. best way to answer that question is to Google it. there are hundreds of applications for buck minister fullerenes, from medical to aerospace. you can also find plenty of research papers that will give you great detail on the potential applications of fullerenes.
Tarell
what is the Synthesis, properties,and applications of carbon nano chemistry
Abhijith Reply
Mostly, they use nano carbon for electronics and for materials to be strengthened.
Virgil
is Bucky paper clear?
CYNTHIA
carbon nanotubes has various application in fuel cells membrane, current research on cancer drug,and in electronics MEMS and NEMS etc
NANO
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.
Harper
Do you know which machine is used to that process?
s.
how to fabricate graphene ink ?
SUYASH Reply
for screen printed electrodes ?
SUYASH
What is lattice structure?
s. Reply
of graphene you mean?
Ebrahim
or in general
Ebrahim
in general
s.
Graphene has a hexagonal structure
tahir
On having this app for quite a bit time, Haven't realised there's a chat room in it.
Cied
what is biological synthesis of nanoparticles
Sanket 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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