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C ( g ) = Q g ( 1 ) c g ( 2 ) c g ( 3 ) + r g ( 1 ) Q g ( 2 ) c g ( 3 ) + r g ( 1 ) r g ( 2 ) Q g ( 3 )

In general

C ( g ) = i = 1 n j = 1 i - 1 r g ( j ) Q g ( i ) j = i + 1 n c g ( j ) .

Therefore, the most efficient factorization of i A i is described by the permutation g ( · ) that minimizes C .

It turns out that for the Kronecker product of more than two matrices, the ordering of operations that describesthe most efficient factorization of i A i also depends only on the ratios ( r i - c i ) / Q i . To show that this is the case, suppose u ( · ) is the permutation that minimizes C , then u ( · ) has the property that

r u ( k ) - c u ( k ) Q u ( k ) r u ( k + 1 ) - c u ( k + 1 ) Q u ( k + 1 )

for k = 1 , , n - 1 . To support this, note that since u ( · ) is the permutation that minimizes C , we have in particular

C ( u ) C ( v )

where v ( · ) is the permutation defined by the following:

v ( i ) = { u ( i ) i < k , i > k + 1 u ( k + 1 ) i = k u ( k ) i = k + 1 .

Because only two terms in [link] are different, we have from [link]

i = k k + 1 j = 1 i - 1 r u ( j ) Q u ( i ) j = i + 1 n c u ( j ) i = k k + 1 j = 1 i - 1 r v ( j ) Q v ( i ) j = i + 1 n c v ( j )

which, after canceling common terms from each side, gives

Q u ( k ) c u ( k + 1 ) + r u ( k ) Q u ( k + 1 ) Q v ( k ) c v ( k + 1 ) + r v ( k ) Q v ( k + 1 ) .

Since v ( k ) = u ( k + 1 ) and v ( k + 1 ) = u ( k ) this becomes

Q u ( k ) c u ( k + 1 ) + r u ( k ) Q u ( k + 1 ) Q u ( k + 1 ) c u ( k ) + r u ( k + 1 ) Q u ( k )

which is equivalent to [link] . Therefore, to find the best factorization of i A i it is necessary only to compute the ratios ( r i - c i ) / Q i and to order them in an non-decreasing order. The operation A i whose index appears first in this list is applied to the data vector x first, and so on

As above, if r u ( k + 1 ) > c u ( k + 1 ) and r u ( k ) < c u ( k ) then [link] is always true. Therefore, in the most computationally efficientfactorization of i A i , all matrices with fewer rows than columns are always applied to the data vector x before any matrices with more rows than columns.If some matrices are square, then their ordering does not affect the computational efficiency as longas they are applied after all matrices with fewer rows than columns and before all matrices with more rows than columns.

Once the permutation g ( · ) that minimizes C is determined by ordering the ratios ( r i - c i ) / Q i , i A i can be written as

i = 1 n A i = i = n 1 I a ( i ) A g ( i ) I b ( i )

where

a ( i ) = k = 1 g ( i ) - 1 γ ( i , k )
b ( i ) = k = g ( i ) + 1 n γ ( i , k )

and where γ ( · ) is defined by

γ ( i , k ) = { r k if g ( i ) > g ( k ) c k if g ( i ) < g ( k ) .

Some matlab code

A Matlab program that computes the permutation that describes the computationally most efficientfactorization of i = 1 n A i is cgc() . It also gives the resulting computational cost.It requires the computational cost of each of the matrices A i and the number of rows and columns of each.

function [g,C] = cgc(Q,r,c,n)% [g,C] = cgc(Q,r,c,n);% Compute g and C % g : permutation that minimizes C% C : computational cost of Kronecker product of A(1),...,A(n) % Q : computation cost of A(i)% r : rows of A(i) % c : columns of A(i)% n : number of terms f = find(Q==0);Q(f) = eps * ones(size(Q(f))); Q = Q(:);r = r(:); c = c(:);[s,g] = sort((r-c)./Q);C = 0; for i = 1:nC = C + prod(r(g(1:i-1)))*Q(g(i))*prod(c(g(i+1:n))); endC = round(C);

The Matlab program kpi() implements the Kronecker product i = 1 n A i .

function y = kpi(d,g,r,c,n,x) % y = kpi(d,g,r,c,n,x);% Kronecker Product : A(d(1)) kron ... kron A(d(n)) % g : permutation of 1,...,n% r : [r(1),...,r(n)] % c : [c(1),..,c(n)]% r(i) : rows of A(d(i)) % c(i) : columns of A(d(i))% n : number of terms for i = 1:na = 1; for k = 1:(g(i)-1)if i>find(g==k) a = a * r(k);else a = a * c(k);end endb = 1; for k = (g(i)+1):nif i>find(g==k) b = b * r(k);else b = b * c(k);end end% y = (I(a) kron A(d(g(i))) kron I(b)) * x; y = IAI(d(g(i)),a,b,x);end

where the last line of code calls a function that implements ( I a A d ( g ( i ) ) I b ) x . That is, the program IAI(i,a,b,x) implements ( I a A ( i ) I b ) x .

The Matlab program IAI implements y = ( I m A I n ) x

function y = IAI(A,r,c,m,n,x) % y = (I(m) kron A kron I(n))x% r : number of rows of A % c : number of columns of Av = 0:n:n*(r-1); u = 0:n:n*(c-1);for i = 0:m-1 for j = 0:n-1y(v+i*r*n+j+1) = A * x(u+i*c*n+j+1); endend

It simply uses two loops to implement the m n copies of A . Each copy of A is applied to a different subset of the elements of x .

Vector/parallel interpretation

The command I A I where is the Kronecker (or Tensor) product can be interpreted as avector/parallel command [link] , [link] . In these references, the implementation of thesecommands is discussed in detail and they have found that the Tensor product is“an extremely useful tool for matching algorithms to computer architectures [link] .”

The expression I A can easily be seen to represent a parallel command:

I A = A A A .

Each block along the diagonal acts on non-overlapping sections of the data vector - so that each sectioncan be performed in parallel. Since each section represents exactly the same operation, this form isamenable to implementation on a computer with aparallel architectural configuration. The expression A I can be similarly seen to represent a vector command, see [link] .

It should also be noted that by employing `stride' permutations, the command ( I A I ) x can be replaced by either ( I A ) x or ( A I ) x [link] , [link] . It is only necessary to permute the input and output.It is also the case that these stride permutations are natural loading and storing commands for some architectures.

In the programs we have written in conjunction with this paper we implement the commands y = ( I A I ) x with loops in a set of subroutines. The circular convolution and prime length FFT programswe present, however, explicitly use the form I A I to make clear the structure of the algorithm, to make themmore modular and simpler, and to make them amenable to implementation on special architectures.In fact, in [link] it is suggested that it might be practical to develop tensor product compilers.The FFT programs we have generated will be well suited for such compilers.

Questions & Answers

find the 15th term of the geometric sequince whose first is 18 and last term of 387
Jerwin Reply
I know this work
salma
The given of f(x=x-2. then what is the value of this f(3) 5f(x+1)
virgelyn Reply
hmm well what is the answer
Abhi
how do they get the third part x = (32)5/4
kinnecy Reply
can someone help me with some logarithmic and exponential equations.
Jeffrey Reply
sure. what is your question?
ninjadapaul
20/(×-6^2)
Salomon
okay, so you have 6 raised to the power of 2. what is that part of your answer
ninjadapaul
I don't understand what the A with approx sign and the boxed x mean
ninjadapaul
it think it's written 20/(X-6)^2 so it's 20 divided by X-6 squared
Salomon
I'm not sure why it wrote it the other way
Salomon
I got X =-6
Salomon
ok. so take the square root of both sides, now you have plus or minus the square root of 20= x-6
ninjadapaul
oops. ignore that.
ninjadapaul
so you not have an equal sign anywhere in the original equation?
ninjadapaul
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Abhi
is it a question of log
Abhi
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But this is not salma it's Faiza live in lousvile Ky I garbage this so I am going collage with JCTC that the of the collage thank you my friends
salma
Commplementary angles
Idrissa Reply
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Sherica
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Uday
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what is a good calculator for all algebra; would a Casio fx 260 work with all algebra equations? please name the cheapest, thanks.
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a perfect square v²+2v+_
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algebra 2 Inequalities:If equation 2 = 0 it is an open set?
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or infinite solutions?
Kim
The answer is neither. The function, 2 = 0 cannot exist. Hence, the function is undefined.
Al
y=10×
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Differences Between Laspeyres and Paasche Indices
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No. 7x -4y is simplified from 4x + (3y + 3x) -7y
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Need to simplify the expresin. 3/7 (x+y)-1/7 (x-1)=
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. After 3 months on a diet, Lisa had lost 12% of her original weight. She lost 21 pounds. What was Lisa's original weight?
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China
Cied
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abeetha Reply
I start with an easy one. carbon nanotubes woven into a long filament like a string
Porter
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Porter
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Yasmin
what is the function of carbon nanotubes?
Cesar
I'm interested in nanotube
Uday
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what is nano technology
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what is system testing?
AMJAD
preparation of nanomaterial
Victor Reply
Yes, Nanotechnology has a very fast field of applications and their is always something new to do with it...
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AMJAD
what is system testing
AMJAD
what is the application of nanotechnology?
Stotaw
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
Azam
anybody can imagine what will be happen after 100 years from now in nano tech world
Prasenjit
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
Azam
name doesn't matter , whatever it will be change... I'm taking about effect on circumstances of the microscopic world
Prasenjit
how hard could it be to apply nanotechnology against viral infections such HIV or Ebola?
Damian
silver nanoparticles could handle the job?
Damian
not now but maybe in future only AgNP maybe any other nanomaterials
Azam
Hello
Uday
I'm interested in Nanotube
Uday
this technology will not going on for the long time , so I'm thinking about femtotechnology 10^-15
Prasenjit
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
the Beer law works very well for dilute solutions but fails for very high concentrations. why?
bamidele 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, Automatic generation of prime length fft programs. OpenStax CNX. Sep 09, 2009 Download for free at http://cnx.org/content/col10596/1.4
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