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Mathematical Principles for the LF Algorithm

Module by: C. Sidney Burrus

Summary: Addendum gives the mathematical definitions and ideas used by the Lindsey-Fox algorithm

Addendum: Mathematical Principles

The mathematical principles at the core of the Lindsey-Fox algorithm for polynomial factoring are given here.

An NthNth degree polynomial is denoted by

P ( z ) = a 0 + a 1 z + a 2 z 2 + ⋯ + a N z N = ∑ n a n z n P ( z ) = a 0 + a 1 z + a 2 z 2 + ⋯ + a N z N = ∑ n a n z n (1)

or

P ( z ) = ∏ k ( z - z k ) P ( z ) = ∏ k ( z - z k ) (2)

where k=1,2,⋯,Nk=1,2,⋯,N or

P ( z ) = ∏ m ( z - z m ) M m P ( z ) = ∏ m ( z - z m ) M m (3)

where m=1,2,⋯,Qm=1,2,⋯,Q and N=∑mMmN=∑mMm. And anan is the nthnth coefficient, zkzk is the kthkth zero or root, NN is the degree of the polynomial, MmMm is the multiplicity of the mthmth zero, and QQ is number of distinct roots or zeros.

The fundamental theorem of algebra states that an NthNth degree polynomial has NN zeros.

The length-LL discrete Fourier transform (DFT) of the NN coefficients of a polynomial P(z)P(z) with L≥NL≥N are the LL equally spaced samples of the polynomial evaluated on the unit circle of the complex plane.

D F T L { a n } = P ( e 2 π i k / L ) D F T L { a n } = P ( e 2 π i k / L ) (4)

for k=0,1,2,⋯,L-1k=0,1,2,⋯,L-1

If the coefficients are multiplied by a geometric sequence, rnrn , the DFT of this modulated set of coefficients are the LL equally spaced samples of the polynomial evaluated on a circle of radius rr in the complex plane.

D F T L { r n a n } = P ( r e 2 π i k / L ) D F T L { r n a n } = P ( r e 2 π i k / L ) (5)

for k=0,1,2,⋯,L-1k=0,1,2,⋯,L-1

Using Horner's method, the number of multiplications and additions necessary to directly calculate NN equally spaced values of a degree NN polynomial on the unit circle is proportional to N2N2 . If evaluated with the DFT, it is also proportional to N2N2 . If evaluated with the FFT, it is proportional to Nlog(N)Nlog(N).

If the roots of a polynomial are at zz , the roots of the same polynomial with the sequence of coefficients reversed (“flipped"), are at 1/z1/z .

P a ' ( 1 / z ) = P a ( z ) P a ' ( 1 / z ) = P a ( z ) (6)

The “Minimum Modulus Theorem" can be stated several ways. A way most applicable to our test of the 3 node by 3 node cells is: If the minimum of an analytic function of a complex variable occurs in the interior of an open set, the minimum must in fact be a zero of the function.

If Newton's algorithm is applied to a polynomial and is started sufficiently close to a zero, it will quadratically converge to that zero if the zero is simple. If the zero is multiple, it still converges but only linearly. If Laguarre's algorithm is applied to a polynomial and is started sufficiently close to a zero, it will cubically converge to that zero if the zero is simple. If the zero is multiple, it still converges but only linearly.

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