0.1 The lindsey-fox algorithm for factoring polynomials  (Page 2/3)

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.