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The lindsey-fox program can be outlined by:

Stage one: the grid search for prospective zeros

  1. Construct a polar coordinate grid on the complex plane with spacing derived from the degree of the polynomial being factored
  2. Use the FFT to evaluate the polynomial at each node along the concentric circles of the grid.
  3. Search over each 3x3 set of values for relative minima. If the center value is less than the edge values, it is a prospective zero by the Minimum Modulus Theorem of complex analysis.

Stage two: polish the prospective zeros

  1. Apply Laguerre’s algorithm to each prospective zero, correcting it to a better approximation of the “true” zero of the polynomial
  2. Test the set of polished zeros for uniqueness and discard any duplicates to give a set of candidate zeros

Stage three: Unfactor, perhaps deflate, and verify

  1. Unfactor the polished zeros i.e., reconstruct a candidate polynomial in coefficient form from the polished candidate zeros
  2. If the degree of the reconstructed polynomial is the same as that of the original polynomial and differences in their coefficients are small, the factoring is successful and finished
  3. If some zeros were missed, deflate and factor the quotient polynomial. If that does not find all of the missed zeros, deflate and factor again until all are found or until no new ones are found
  4. If deflation finds all the zeros that it can, and it still has not found them all, design a new grid with a finer spacing and return to stage one. If four restarts do not find them all and/or the reconstruction error is not small, declare failure.

Description of the three stages

Stage one is the reason this algorithm is so efficient and is what sets it apart from most other factoring algorithms. Because the FFT (fast Fourier transform) is used to evaluate the polynomial, a fast evaluation over a dense grid in the complex plane is possible. In order to use the FFT, the grid is structured in polar coordinates. In the first phase of this stage, a grid is designed with concentric circles of a particular radius intersected by a set of radial lines. The positions and spacing of the radial lines and the circles are chosen to give a grid that will hopefully separate the expected roots. Because the zeros of a polynomial with random coefficients have a fairly uniform angular distribution and are clustered close to the unit circle, it is possible to design an evaluation grid that fits the expected root density very well. In the second phase of this stage, the polynomial is evaluated at the nodes of the grid using the fast Fourier transform (FFT). It is because of the extraordinary efficiency and accuracy of the FFT that a direct evaluation is possible. In the third phase of this first stage, a search is conducted over all of the 3 by 3 node cells formed in the grid. For each 3 by 3 cell (see Figure below), if the value of the polynomial at the center node of the cell (the "x") is less than the values at all 8 of the nodes on the edges of the cell (the "o's"), the center is designated a candidate zero. This rule is based on the “Minimum Modulus Theorem” which states that if a relative minimum of the absolute value of an analytic function over an open region exists, the minimum must be a zero of the function. Finally, this set of prospective zeros is passed to the second stage. The number is usually slightly larger than the degree because some were found twice or mistakes were made. The number could be less if some zeros were missed.

Questions & Answers

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..
Even nanotechnology is pretty much all about chemistry... Its the chemistry on quantum or atomic level
Preparation and Applications of Nanomaterial for Drug Delivery
Hafiz Reply
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.
yes that's correct
I think
what is the stm
Brian Reply
is there industrial application of fullrenes. What is the method to prepare fullrene on large scale.?
industrial application...? mmm I think on the medical side as drug carrier, but you should go deeper on your research, I may be wrong
How we are making nano material?
what is a peer
What is meant by 'nano scale'?
What is STMs full form?
scanning tunneling microscope
how nano science is used for hydrophobicity
Do u think that Graphene and Fullrene fiber can be used to make Air Plane body structure the lightest and strongest. Rafiq
what is differents between GO and RGO?
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
analytical skills graphene is prepared to kill any type viruses .
Any one who tell me about Preparation and application of Nanomaterial for drug Delivery
what is Nano technology ?
Bob Reply
write examples of Nano molecule?
The nanotechnology is as new science, to scale nanometric
nanotechnology is the study, desing, synthesis, manipulation and application of materials and functional systems through control of matter at nanoscale
Is there any normative that regulates the use of silver nanoparticles?
Damian Reply
what king of growth are you checking .?
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
how did you get the value of 2000N.What calculations are needed to arrive at it
Smarajit Reply
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Source:  OpenStax, Factoring polynomials of high degree. OpenStax CNX. Apr 01, 2012 Download for free at http://cnx.org/content/col10494/1.9
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