Polynomial roots

Polynomial root finding, as in the previous section, has some technical pitfalls that one would like to avoid. It is easier to write reliable software for matrix diagonalization (QMOBAS, TMOBAS) than it is for polynomial root finding hence, diagonalization is the method of choice for Huckel calculations. 

The polynomial root-solving program POLRT was used in the examples below. It is from the IBM Scientific Subroutines. A listing of the source program is given in Table 10.1. POLRT is vyy fast, even on a personal computer. 

Computer simulation was carried out on an IBM 370/155 computer, using double-precision calculation and the double-precision version of the Bairstow method polynomial roots subroutine in the IBM scientific subroutines package. For the monomer-dimer case, the quadratic can be and was solved exactly—e.g., see Ref. 3. Estimates of propagated error are provided in the tables where needed. The analysis given here reduces to that given previously (3) when the system concerned is a monomer-dimer system. 

Let us look at the process of finding polynomial roots as an input to output process ... 

We start with a short history of the polynomial-root finding problem that will explain the eminent role of matrices for numerical computations by example. 

The study of polynomial roots literally lies at the origin of modern Numerical Analysis. 

MATLAB s polynomial-roots finder roots does not handle repeated or clustered roots very well, but otherwise it is the best 0(n3) root finder available. Note that an operations count of 0(nP) for an algorithm signifies that the algorithm performs K n additions and multiplications (for some algorithm specific constants K and j, but depending on n) to obtain its output from n input data. Most of the polynomial-root finders of the last century unfortunately were even slower 0(n4) algorithms and all in all much too slow and inaccurate. 

To illustrate we first verify the identical behavior of the MATLAB QR based polynomial-root finder roots and MATLAB s QR based matrix eigenvalue finder eig for p s companion matrix P = C(p) First we define p by its coefficient vector in MATLAB s workspace, then we invoke the MATLAB polynomial-root finder roots, followed by its matrix eigenvalue finder eig on the companion matrix ofp. Finally we display the companion matrix P of p. As an example we use p(x) = x3 — 2.x2 + 4 here and represent p by its coefficient vector [1 -2 0 4] in the following line of MATLAB commands. 

In order to call newtonpoly (start, n) implicitly on a command line with success, the two implicit inputs start and n must have been declared previously and must be available in the current workspace. Of course, one can also call newtonpoly9 explicitly by entering newtonpoly (21,42), for example, on the command line if one wants to see the list of 41 Newton iterates for the same polynomial-root problem, starting from start = xo = 21. 

MATLAB s 0(n3) polynomial-root finder roots, used for the same polynomialp, encounters different problems and computes 4 complex conjugate root pairs instead. These lie on a small radius circle around the ninefold root 2. As input for roots, we represent our polynomial (x — 2)9 of degree 9 in extended form by its coefficient vector [1 -18. .. 2304 -512],... 

Only the earlier mentioned faster 0(n2) polynomial-root finder pzero discovers the ninefold real root 2 of p correctly see the Resources appendix for a quote of the literature for pzero and the folder pzero on our CD for the actual MATLAB code of pzero. 

Prior knowledge may in fact be used even though LPSVD is a linear procedure. The number of spectral components K may be known exactly, thus guiding the truncation of the SVD. Furthermore, the frequencies of the spectral components may be known, so that the correct roots can be readily picked (or replaced with the theoretical values) during the polynomial rooting.118... 

With the introduction of the HSVD method, the difficult step of polynomial rooting was avoided. Comparisons of LPSVD and HSVD generally show that the results are very similar, with only minor differences in the parameters of the smallest spectral components93 (Fig. 21). Incorporation of the Lanczos... 

LPSVD T Y N N N Linear prediction and polynomial rooting Good... 

Note that in the cases k > 6 there appears at least one polynomial root which is numerically smaller than 1. The form of the general solution (B.42) therefore implies that the values sn then do not converge to a finite value, as n tends towards infinity. This is in accord with the known instability of BDF for k > 6 [187],... 

Convergence is slower for higher k, as is also implied by the values of the polynomial roots in Table B.l. 

Except in the case of polynomial root-finding and complex roots, there are more efficient and easier-to-implement algorithms. 

Keeping only the terms up to the second order in and using the basic properties of determinants and polynomial roots we find that the first-order terms vanish and the energy reads E = —Aso -I- (1/2) Yijtj U(Rij), where... 

In addition to FindRoot, there is also NSolve [4], which is more suited to finding the roots of polynomials. Following the Mathematica system, polynomial root finding is based on the Jenkins-Traub algorithm [8]. 

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