The Jenkins–Traub algorithm for polynomial zeros is a fast globally convergent iterative method published in 1970 by Michael A. Jenkins and Joseph F. Traub. They gave two variants, one for general polynomials with complex coefficients, commonly known as the "CPOLY" algorithm, and a more complicated variant for the special case of polynomials with real coefficients, commonly known as the "RPOLY" algorithm. The latter is "practically a standard in black-box polynomial root-finders".
Contents
- Overview
- Root finding procedure
- Stage one no shift process
- Stage two fixed shift process
- Stage three variable shift process
- Convergence
- What gives the algorithm its power
- Analysis of the H polynomials
- Convergence orders
- Interpretation as inverse power iteration
- Real coefficients
- A connection with the shifted QR algorithm
- Software and testing
- References
This article describes the complex variant. Given a polynomial P,
with complex coefficients it computes approximations to the n zeros
The real variant follows the same pattern, but computes two roots at a time, either two real roots or a pair of conjugate complex roots. By avoiding complex arithmetic, the real variant can be faster (by a factor of 4) than the complex variant. The Jenkins–Traub algorithm has stimulated considerable research on theory and software for methods of this type.
Overview
The Jenkins–Traub algorithm calculates all of the roots of a polynomial with complex coefficients. The algorithm starts by checking the polynomial for the occurrence of very large or very small roots. If necessary, the coefficients are rescaled by a rescaling of the variable. In the algorithm proper, roots are found one by one and generally in increasing size. After each root is found, the polynomial is deflated by dividing off the corresponding linear factor. Indeed, the factorization of the polynomial into the linear factor and the remaining deflated polynomial is already a result of the root-finding procedure. The root-finding procedure has three stages that correspond to different variants of the inverse power iteration. See Jenkins and Traub. A description can also be found in Ralston and Rabinowitz p. 383. The algorithm is similar in spirit to the two-stage algorithm studied by Traub.
Root-finding procedure
Starting with the current polynomial P(X) of degree n, the smallest root of P(x) is computed. To that end, a sequence of so-called H polynomials is constructed. These polynomials are all of degree n − 1 and are supposed to converge to the factor of P(X) containing all the remaining roots. The sequence of H polynomials occurs in two variants, an unnormalized variant that allows easy theoretical insights and a normalized variant of
The construction of the H polynomials
A direct solution to this implicit equation is
where the polynomial division is exact.
Algorithmically, one would use for instance the Horner scheme or Ruffini rule to evaluate the polynomials at
Since the highest degree coefficient is obtained from P(X), the leading coefficient of
Stage one: no-shift process
For
Stage two: fixed-shift process
The shift for this stage is determined as some point close to the smallest root of the polynomial. It is quasi-randomly located on the circle with the inner root radius, which in turn is estimated as the positive solution of the equation
Since the left side is a convex function and increases monotonically from zero to infinity, this equation is easy to solve, for instance by Newton's method.
Now choose
is traced. The second stage is finished successfully if the conditions
are simultaneously met. If there was no success after some number of iterations, a different random point on the circle is tried. Typically one uses a number of 9 iterations for polynomials of moderate degree, with a doubling strategy for the case of multiple failures.
Stage three: variable-shift process
The
being the last root estimate of the second stage and
If the step size in stage three does not fall fast enough to zero, then stage two is restarted using a different random point. If this does not succeed after a small number of restarts, the number of steps in stage two is doubled.
Convergence
It can be shown that, provided L is chosen sufficiently large, sλ always converges to a root of P.
The algorithm converges for any distribution of roots, but may fail to find all roots of the polynomial. Furthermore, the convergence is slightly faster than the quadratic convergence of Newton–Raphson iteration, however, it uses at least twice as many operations per step.
What gives the algorithm its power?
Compare with the Newton–Raphson iteration
The iteration uses the given P and
is precisely a Newton–Raphson iteration performed on certain rational functions. More precisely, Newton–Raphson is being performed on a sequence of rational functions
For
is as close as desired to a first degree polynomial
where
Analysis of the H polynomials
Let
If all roots are different, then the Lagrange factors form a basis of the space of polynomials of degree at most n − 1. By analysis of the recursion procedure one finds that the H polynomials have the coordinate representation
Each Lagrange factor has leading coefficient 1, so that the leading coefficient of the H polynomials is the sum of the coefficients. The normalized H polynomials are thus
Convergence orders
If the condition
Under the condition that
one gets the aymptotic estimates for
Interpretation as inverse power iteration
All stages of the Jenkins–Traub complex algorithm may be represented as the linear algebra problem of determining the eigenvalues of a special matrix. This matrix is the coordinate representation of a linear map in the n-dimensional space of polynomials of degree n − 1 or less. The principal idea of this map is to interpret the factorization
with a root
This maps polynomials of degree at most n − 1 to polynomials of degree at most n − 1. The eigenvalues of this map are the roots of P(X), since the eigenvector equation reads
which implies that
the resulting coefficient matrix is
To this matrix the inverse power iteration is applied in the three variants of no shift, constant shift and generalized Rayleigh shift in the three stages of the algorithm. It is more efficient to perform the linear algebra operations in polynomial arithmetic and not by matrix operations, however, the properties of the inverse power iteration remain the same.
Real coefficients
The Jenkins–Traub algorithm described earlier works for polynomials with complex coefficients. The same authors also created a three-stage algorithm for polynomials with real coefficients. See Jenkins and Traub A Three-Stage Algorithm for Real Polynomials Using Quadratic Iteration. The algorithm finds either a linear or quadratic factor working completely in real arithmetic. If the complex and real algorithms are applied to the same real polynomial, the real algorithm is about four times as fast. The real algorithm always converges and the rate of convergence is greater than second order.
A connection with the shifted QR algorithm
There is a surprising connection with the shifted QR algorithm for computing matrix eigenvalues. See Dekker and Traub The shifted QR algorithm for Hermitian matrices. Again the shifts may be viewed as Newton-Raphson iteration on a sequence of rational functions converging to a first degree polynomial.
Software and testing
The software for the Jenkins–Traub algorithm was published as Jenkins and Traub Algorithm 419: Zeros of a Complex Polynomial. The software for the real algorithm was published as Jenkins Algorithm 493: Zeros of a Real Polynomial.
The methods have been extensively tested by many people. As predicted they enjoy faster than quadratic convergence for all distributions of zeros.
However, there are polynomials which can cause loss of precision as illustrated by the following example. The polynomial has all its zeros lying on two half-circles of different radii. Wilkinson recommends that it is desirable for stable deflation that smaller zeros be computed first. The second-stage shifts are chosen so that the zeros on the smaller half circle are found first. After deflation the polynomial with the zeros on the half circle is known to be ill-conditioned if the degree is large; see Wilkinson, p. 64. The original polynomial was of degree 60 and suffered severe deflation instability.