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| | x | | p = n | x ( n ) | p 1 / p

and finding x to minimizing this p-norm while satisfying Ax = b .

It has been shown this is equivalent to solving a least weighted norm problem for specific weights.

| | x | | p = n w ( n ) 2 | x ( n ) | 2 1 / 2

The development follows the same arguments as in the previous section but using the formula [link] , [link] derived in [link]

x = [ W T W ] - 1 A T A [ W T W ] - 1 A T - 1 b

with the weights, w ( n ) , being the diagonal of the matrix, W , in the iterative algorithm to give the minimum weighted solution norm in the same way as [link] gives the minimum weighted equation error.

A Matlab program that implements these ideas applied to our pseudoinverse problem with more unknowns than equations (case 3a) is:

% m-file IRLS2.m to find the optimal solution to Ax=b %  minimizing the L_p norm ||x||_p, using IRLS. %  Newton iterative update of solution, x, for  M < N. %  For 2<p<infty, use homotopy parameter K = 1.01 to 2 %  For 0<p<2, use K = approx 0.7 to 0.9 %  csb 10/20/2012 function x = IRLS2(A,b,p,K,KK) if nargin < 5, KK= 10;  end; if nargin < 4, K = .8;  end; if nargin < 3, p = 1.1; end; pk = 2;                                 % Initial homotopy value x  = pinv(A)*b;                         % Initial L_2 solution E = []; for k = 1:KK    if p >= 2, pk = min([p, K*pk]);      % Homotopy update of p       else pk = max([p, K*pk]); end    W  = diag(abs(x).^((2-pk)/2)+0.00001);  % norm weights for IRLS    AW = A*W;                            % applying new weights    x1 = W*AW'*((AW*AW')\b);             % Weighted L_2 solution    q  = 1/(pk-1);                       % Newton's parameter    if p >= 2, x = q*x1 + (1-q)*x; nn=p; % Newton's partial update for p>2       else x = x1; nn=1; end            % no Newton's partial update for p<2    ee = norm(x,nn);  E = [E ee];        % norm at each iteration end; plot(E)

This approach is useful in sparse signal processing and for frame representation.

The chebyshev, minimax, or L Appriximation

The Chebyshev optimization problem minimizes the maximum error:

ϵ m = max n | ϵ ( n ) |

This is particularly important in filter design. The Remez exchange algorithm applied to filter design as the Parks-McClellan algorithm is very efficient [link] . An interesting result is the limit of an | | x | | p optimization as p is the Chebyshev optimal solution. So, the Chebyshev optimal, the minimax optimal, and the L optimal are all the same [link] , [link] .

A particularly powerful theorem which characterizes a solution to Ax = b is given by Cheney [link] in Chapter 2 of his book:

  • A Characterization Theorem: For an M by N real matrix, A with M > N , every minimax solution x is a minimax solution of an appropriate N + 1 subsystem of the M equations. This optimal minimax solution will have at least N + 1 equal magnitude errors and they will be larger than any of the errors of the other equations.

This is a powerful statement saying an optimal minimax solution will have out of M , at least N + 1 maximum magnitude errors and they are the minimum size possible. What this theorem doesn't state is which of the M equations are the N + 1 appropriate ones. Cheney develops an algorithm based on this theorem which finds these equations and exactly calculates this optimal solution in a finite numberof steps. He shows how this can be combined with the minimum | | e | | p using a large p , to make an efficient solver for a minimax or Chebyshev solution.

This theorem is similar to the Alternation Theorem [link] but more general and, therefore, somewhat more difficult to implement.

The L 1 Approximation and sparsity

The sparsity optimization is to minimize the number of non-zero terms in a vector. A “pseudonorm", | | x | | 0 , is sometimes used to denote a measure of sparsity. This is not convex, so is not really a norm but the convex (in the limit) norm | | x | | 1 is close enough to the | | x | | 0 to give the same sparsity of solution [link] . Finding a sparse solution is not easy but interative reweighted least squares (IRLS) [link] , [link] , weighted norms [link] , [link] , and a somewhat recent result is called Basis Pursuit [link] , [link] are possibilities.

This approximation is often used with an underdetermined set of equations (Case 3a) to obtain a sparse solution x .

Using the IRLS algorithm to minimize the l p equation error often gives a sparse error if one exists. Using the algorithm in the illustrated Matlab program with p = 1 . 1 on the problem in Cheney [link] gives a zero error in equation 4 while using no larger p gives any zeros.

Questions & Answers

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
yes I'm doing my masters in nanotechnology, we are being studying all these domains as well..
what school?
biomolecules are e building blocks of every organics and inorganic materials.
anyone know any internet site where one can find nanotechnology papers?
Damian Reply
sciencedirect big data base
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.
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
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
characteristics of micro business
for teaching engĺish at school how nano technology help us
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
what is fullerene does it is used to make bukky balls
Devang Reply
are you nano engineer ?
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.
what is the actual application of fullerenes nowadays?
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.
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.
is Bucky paper clear?
carbon nanotubes has various application in fuel cells membrane, current research on cancer drug,and in electronics MEMS and NEMS etc
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.
Do you know which machine is used to that process?
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
how did you get the value of 2000N.What calculations are needed to arrive at it
Smarajit Reply
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Source:  OpenStax, Basic vector space methods in signal and systems theory. OpenStax CNX. Dec 19, 2012 Download for free at http://cnx.org/content/col10636/1.5
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