A review of ideas from polynomial rootfinding

Transcription

1 A review of ideas from polynomial rootfinding Mark Richardson September 2010 Contents 1 Introduction Polynomial basics Newton Iteration Horner s Method Polynomial Evaluation Derivative evaluation Stability modifications Deflation The Fast Fourier Transform Polynomial evaluation with the FFT Polynomial multiplication and division with the FFT Problems with Newton iteration and deflation 10 5 The Lindsey-Fox method Sampling on disc of radius r A description of the method Implementation Conclusions 14 A Appendix: MATLAB code 16 A.1 Implementation of the LF method A.2 Plotting the LF roots

2 1 Introduction The problem is simply stated: given a set of n + 1 coeffcients of a degree n polynomial, {a 0, a 1,..., a n }, what are the roots? The question looks innocent enough, yet it has provoked hundreds of years of research, and literally thousands of academic papers. In spite of this, we still don t know the best general method of solving it. In this report, we shall examine some well known, and some relatively recent developments in the field. We shall predominantly be examining Newton-style techniques which generally involve obtaining some initial estimate of a root, and iterating it to convergence. The methods we shall examine will be for computational work in the monomial basis. Though care needs to be taken in certain circumstances, polynomials are in general easy to evaluate. Indeed, polynomials are particularly amenable to Newton style rootfinding techniques since their derivatives are also easy to compute. Polynomial rootfinding can be an ill-conditioned problem. That is, for some polynomials, small perturbations to the coefficients may lead to large perturbations in the roots. Consider the following equation, in an example taken from [9], for which there is a double root at z = 1. By perturbing the order one coefficient from , so that, z 2 2z + 1 = 0 z z + 1 = 0, we find that the roots have been shifted to z = & z = The difference in the roots of the perturbed equation, relative to the roots of the unperturbed equation is However, this was the effect of a relative change in the first order coefficieint of Thus, small perturbations can lead to large errors. As the example demonstrates, ill-conditioning is inherent to the problem itself, and is entirely independent of the algorithm. However, we may also be unfortunate enough to suffer from computational difficulties in addition to ill-conditioning. Take for example a degree 1000 monic 1 polynomial. If we were working in double precision and happened to suspect that a root of this polynomial were at z = 3, we would not easily be able to check this suspicion. This is because is far greater than the largest double precision number, ( ) , and we would encounter overflow. As a result of these issues, many algorithms for polynomial rootfinding are forced to restrict themselves to a certain class of polynomial. For example, the Lindsey-Fox method that we describe in Section 5 works only with the class of polynomials that have roots close to the unit disc. We begin with a standard result concerning factorisation of polynomials. 1 Monic means that the leading order coefficient is 1. 2

3 1.1 Polynomial basics Any degree n polynomial can be written P (z) = n a k z k = a 0 + a 1 z + a 2 z a n 1 z n 1 + a n z n, (1) k=0 where a = (a 0, a 1,..., a n ) is the (n + 1)-vector of polynomial coefficients. By factoring out the z-terms in sequence, the polynomial can also be written in nested form, P (z) = a 0 + z(a 1 + z(a 2 + z(... + z(a n 1 + za n )...))). (2) In factored form this polynomial is P (z) = a n D d=1 (z z d ) E d, (3) where D is the number of distinct zeros of P, and E d are positive integers with 1.2 Newton Iteration D E d = n. The basic Newton iteration for obtaining a root of f is derived from the Taylor series approximation. Given a function of a single complex variable f(z), three-times differentiable in a neighbourhood of z 0, we can write f(z) = f(z 0 ) + (z z 0 )f (z 0 ) + 1 2! (z z 0) 2 f (z 0 ) + O(z 3 ). (4) At the location of a root, f(z) = 0. Discarding the terms of O(z 2 ) and higher gives the iteration, z [k+1] = z [k] f(z[k] ) f (z [k] ). Assuming f (z 0 ) 0, if Newton s method as it is known converges to a root of the function f, then it does so quadratically. This can be shown with the following straightforward argument. Given a root z = z r, setting f(z r ) = 0, z 0 = z [k] in (4) gives, d=1 f(z [k] ) + (z r z [k] )f (z [k] ) = 1 2! (z r z [k] ) 2 f (ξ), ξ (z [k], z r ) ( ) z r z [k] f(z[k] ) = 1 f (ξ) f (z [k] ) 2 f (z [k] ) (z r z [k] ) 2 z r z [k+1] = 1 f (ξ) 2 f (z [k] ) (z r z [k] ) 2 3

4 Taking absolute values, we have zr z [k+1] 1 2 f (ξ) f (z [k] ) z r z [k] 2. Thus, the error at the (k + 1) st step is bounded by some factor multplied by the square of the error at the k th step, and the convergence is quadratic. A similar argument shows that when the multiplicity of the root is greater than 1, Newton s method converges only linearly. In the next section, we describe a method for evaluating polynomials and their derivatives that can be used as part of Newton iteration scheme. 2 Horner s Method 2.1 Polynomial Evaluation Suppose first that we wish to evaluate the polynomial P (z) given in (1) for some z = z 0. From the nested form of P (z) in (2), we can work outwards from the innermost bracket using the following recurrence relation, known as Horner s method: b k = a k + z 0 b k+1, for k = n 1,..., 0 with b n = a n. (5) The last iterate is the value of the polynomial at z 0, b 0 = P (z 0 ). This can be simply implemented with the Matlab code in Figure 1, for a vector of randomly chosen polynomial coefficients. a = rand(1,11); nn = length(a)-1; f = a(nn+1); zz = 0.5; for k = nn:-1:1 f = a(k) + zz*f; end % P(z) = a_0 + a_1*z a_n*z^n % degree of polynomial % initialise b = b_n = a_n % value of z required % Horner iteration - P(zz) Figure 1: Matlab code for Horner evaluation of a polynomial 2.2 Derivative evaluation The intermediate b k are the values in the subsequent brackets in (2). Moreover, the last n b k coefficients are also the coefficients of the quotient polynomial Q(z) of degree n + 1 obtained by dividing P (z) by (z z 0 ). That is, if n 1 Q(z) = b k+1 z k = b 1 + b 2 z + b 3 z b n z n 1 (6) k=0 = b 1 + z(b 2 + z(b 3 + z(... + z(b n 1 + zb n )...))), 4

5 then, P (z) = (z z 0 )Q(z) + R, (7) where R = b 0 = P (z 0 ). This relationship can be verified by observing that (z z 0 )Q(z) + R = (z z 0 )(b 1 + b 2 z + b 3 z b n z n 1 ) + b 0 = (b 1 z + b 2 z b n z n ) z 0 (b 1 + b 2 z b n z n 1 ) + b 0 = (b 0 z 0 b 1 ) + (b 1 z 0 b 2 )z (b n 1 z 0 b n )z n 1 + b n z n = a 0 + a 1 z + a 2 z a n 1 z n 1 + a n z n = P (z). The quotient property enables us to easily compute the derivative P (z 0 ) at the same time as computing the function value P (z 0 ). To see this, consider differentiating (7), P (z) = (z z 0 )Q (z) + Q(z). Evaluating this expression at z = z 0, we obtain P (z 0 ) = Q(z 0 ). This suggests a very neat way of computing a function and its derivative at the same time. The Matlab code in Figure 2 achieves this. a = rand(1,11); nn = length(a)-1; f = a(nn+1); df = f; zz = 0.5; for k = nn:-1:2 f = a(k) + zz*f; df = b + zz*df; end f = a(1) + zz*b; % P(z) = a_0 + a_1*z a_n*z^n % degree of polynomial % initialise b = b_n = a_n, c = b_n % value of z required % Horner iteration steps Figure 2: Matlab code for Horner evaluation of a polynomial and its derivative 2.3 Stability modifications The recurrence relation that defines Horner s method (5) is also a first-order difference equation. An equation of this form is known to be stable for computing values of z 1, and unstable for z > 1. If using Horner s method within a Newton iteration, we should expect some of the computed values of z to lie within the unstable region of the complex plane. The instability of the Horner iteration in such cases can be controlled by writing the degree n polynomial P (z) in different nested form, 5

6 P (z) = a 0 + a 1 z + a 2 z a n 1 z n 1 + a n z n = z [ ] n z 1 (z n+1 a 0 + z n+2 a 1 + z n+3 a a n 1 ) + a n = z [ ] n z 1 (... (z 1 (z 1 a 0 + a 1 ) + a 2 ) a n 1 ) + a n. (8) The corresponding recurrence relation is, β k+1 = γ 1 β k + a k+1, k = 0,..., n 1, β 0 = a 0. (9) The derivative can also be computed stably for z > 1. Differentiation of (1) gives P (z) = a 1 + 2a 2 z + 3a 3 z (n 1)a n 1 z n 2 + na n z n 1 = z [ ] n 1 a 1 z n+1 + 2a 2 z n+2 + 3a 3 z n (n 1)a n 1 z 1 + na n = z [ n 1 z ( 1... ( z ( ) ) ( ] 1 z 1 a 1 + 2a 2 + 3a n 1)an 1 ) + na n. This time, the recurrence relation is δk + 1 = γ 1 δk + (k + 1)a k+1, k = 1,..., n 1, δ 1 = a 1. (10) The function and derivative can be computed in the same loop as in Figure 3. a = rand(50,1); % P(z) = a_0 + a_1*z a_n*z^n beta = a(1); delta = a(2); % initialise beta, delta zz = 0.5; zz_rec = zz^(-1); % z and 1/z beta = zz_rec*beta + a(2); % compute first beta value for k = 2:length(a)-1 beta = zz_rec*beta + a(k+1); % P(zz) delta = zz_rec*delta + k*a(k+1); % P (zz) end f = zz^(length(a)-1)*beta; % final values df = zz^(length(a)-2)*delta; Figure 3: Matlab code for Horner deflation of a polynomial with a known root 2.4 Deflation The term deflation refers to the process of computing the polynomial Q that is obtained upon dividing a polynomial P by the factor (z z 0 ), where z 0 is a root of P. In the context of rootfinding, this can be important in ensuring that a Newton-type iteration does not converge to a root that has already been found. One strategy for computing roots is to deflate each time a root is found before restarting the process with the deflated polynomial. 6

7 Obtaining the coefficients of a deflated polynomial is a natural consequence of evaluating the polynomial using Horner s method. If z = z 0 is a root of the polynomial P (z), then clearly P (z 0 ) = 0 and thus the remainder R in (7) is zero. The quotient polynomial Q therefore contains the remaining roots of P. Achieving this is simply a case of storing the b k coefficients as we pass through the Horner evaluation algorithm. The Matlab code in Figure 4 implements this idea. a = [ ]; nn = length(a)-1; b = zeros(nn,1); b(nn) = a(nn+1); zz = -4; for k = nn:-1:2 b(k-1) = a(k) + zz*b(k); end % P(z) = a_0 + a_1*z a_n*z^n % degree of polynomial % initialise b % initialise b = b_n = a_n % known root % Horner deflation Figure 4: Matlab code for Horner deflation of a polynomial with a known root 3 The Fast Fourier Transform The Fast Fourier Transform (FFT) is a well-known technique for computing the Discrete Fourier Transform (DFT) of a set of data. The FFT became a mainstay of modern scientfic computing after a famous paper by Cooley & Tukey in Unknown to them at the time, they had actually unearthed an algorithm originally discovered by Gauss in 1805! The FFT exploits structure in the DFT matrix in order to reduce the operation count from the O(n 2 ) complexity of a standard matrix-vector multiplication to a more palatable O(n log n). Computing a DFT is necessary in all sorts of situations, and the ability to do so in an efficient manner has made a significant impact on the efficiency of modern algorithms. In this, and the following section, we describe a few of the more arcane and unexpected settings related to polynomial rootfinding in which computation of the DFT crops up. 3.1 Polynomial evaluation with the FFT Given a degree N polynomial, defined by N + 1 monomial coefficients, a DFT of the set of coefficients will evaluate the polynomial at the N + 1 roots of unity equispaced points on the unit disc. The inverse DFT will go the other way, from function values at roots of unity to polynomial coefficients. This relationship can be demonstrated by considering the definition of the DFT. For convenience, we shall work with the definition corresponding to Matlab s FFT. Given an n-vector, x, both the DFT and the FFT are defined by 7

8 X k = M ( x m exp 2πi ) (k 1)(m 1), k = 1,..., M. (11) M m=1 Defining the M th roots of unity, ω k,m = exp ( 2πi M (k 1)), (11) can be written X k = M m=1 x m ω m 1 k,m = P M 1 (ω k,m ), k = 1,..., M. (12) This is a degree M 1 monomial defined by coefficients x k and evaluated for ω k,m. Since Matlab does not permit indexing from zero, the first coefficient is x 1 (= a 0 ). Note that it is possible to to evaluate such a polynomial for any number of roots of unity greater than or equal to M. This is made possible by zero-padding the vector of coefficients to the desired length. If for example we wished to evaluate P M 1 at 2M points on the unit circle, we would define χ k = x k for k = 1,..., M and χ k = 0 for k = M + 1,..., 2M, and compute Z k = The inverse FFT is defined by x m = 1 M M k=1 2M m=1 χ m ω m 1 k,2m, k = 1,..., 2M. ( ) 2πi X k exp (k 1)(m 1), m = 1,..., M. (13) M To verify this formula, we observe that the original coefficients x m substituting (11) into (13), x m = 1 M = 1 M [ M M k=1 M µ=1 x µ µ=1 M k=1 ( x µ exp 2πi (k 1)(µ 1) M [ ( )] k 1 2πi exp (m µ). M ) ] ( 2πi exp M can be recovered by ) (k 1)(m 1) Since m and µ are integers, the exponential term β = exp ( 2πi M (m µ)) is an M th root of unity, and therefore, β M 1 = 0. If m µ, then β 1 0 and the sum involving the square brackets is zero, M k=1 β k 1 = 1 + β + β β M 1 = βm 1 β 1 = 0. 8

9 Alternatively, if m = µ, then M k=1 βk 1 = M k=1 1 = M. Thus, the only term remaining is x m. This argument demonstrates that the DFT and its inverse are one-to-one transformations. Thus, if the FFT maps coefficients to function values, then the inverse FFT maps function values to coefficients. 3.2 Polynomial multiplication and division with the FFT An interesting corollary of this idea is that polynomial multiplication and division (deflation) can also be performed with the FFT, in O(n log n) operations. Suppose we have two polynomials; one of degree P and the other of degree Q. We know that the result of multiplying these will be a polynomial of degree P + Q. If we were to compute an P +Q+1 term DFT of each of the polynomials coefficients, this would evaluate each polynomial at the P + Q + 1 roots of unity on the unit disc. The values of the product polynomial at the same P + Q + 1 roots of unity could then be determined by point-wise multiplication of the DFT vectors. Finally, an inverse DFT of the point-wise product in Fourier-space would yield the corresponding coefficients of the product polynomial. An example of this is given in Figure 5, where a quadratic and linear polynomial are multiplied. The output shows the coefficients of the product in ascending order. a = [1 2 3]; b = [1 2]; deg = length(a)+length(b)-2; c = ifft(fft(a,deg+1).*fft(b,deg+1)) c = % coeffs in ascending order % degree of product % multiply in Fourier space Figure 5: Matlab code for polynomial multplication using the FFT Polynomial deflation with the FFT is a simple extension of this idea. We simply need to reorder the computation so that we are dividing in Fourier-space, rather than multiplying. An example of this is given in Figure 6, using the same polynomials as in the previous example. c = [ ]; a = [1 2 3]; lc = length(c); deg = lc - length(a); bb = ifft(fft(c)./fft(a,lc)); b = bb(1:deg+1) b = 1 2 % coeffs in ascending order % degree of quotient % divide in Fourier space % extract coeffcients Figure 6: Matlab code for polynomial division (deflation) using the FFT 9

10 4 Problems with Newton iteration and deflation One strategy that we have outlined for computing the factorisation of a polynomial would be to use Newton iteration combined with a deflation scheme. That is: first compute a root by making some initial guess in the complex plane and iterating to it with Newton s method; then deflate the root to obtain some quotient polynomial that mathematically at least should contain the remaining roots. Repeating this until all the roots have been found is a typical strategy. However, the effectiveness of this technique is mitigated by two significant computational issues. Firstly, given an arbitrary starting guess in the complex plane, Newton s method is most certainly not guaranteed to converge to a root. Newton s method can often fall into limit cycles that do not converge to a root, or worse still, can become entirely unstable. Good strategies for choosing a starting guess are hard to come by and a typically imperfect one is to generate pseudo-random starting numbers using e.g. z [0] k = cos(k) + i sin(k) for k = 1, 2,... Starting with any particular z k, if Newton method either does not converge within a given number of iterations, or if any of the updated values of z become too large, it may be necessary to reset the iteration with a new starting value, z k+1. As may be expected, the complexity of the problem increases with the degree of the polynomial, and quite often this method can fail entirely. A second problem is that of introducing and accumulating errors at the deflation stage. If a root has been computed, then it is accurate to within the specified tolerance, and it is only a numerical approximation to the exact root. Therefore, when deflating the polynomial with this root, an error is introduced, leading to a perturbed quotient. These errors can accumulate, particularly over large degree polynomials where there are many stages of deflation. One may think that this is not too much of a problem, since the perturbed roots obtained from deflated polynomials will be reasonable approximations to the true roots which can be polished against the original, undeflated polynomial. While this is true to some extent, we may end up in the unfortunate situation where two of our roots are close together, and polishing of an approximated root leads to a root that has already been computed, a limit cycle, or divergence. 10

11 5 The Lindsey-Fox method The Lindsey-Fox (LF) method, described in [1], is a relatively recent innovation in the field of polynomial rootfinding. The LF method takes a novel approach, utilising the some of the FFT ideas discussed in Section 3. It claims an overall O(n 2 ) operation count, and has been reported in [1] to have accurately factored polynomials of degree as high as The polynomial rootfinding problem is in general highly ill-conditioned. The exception to this is when the roots lie close to the unit disc, and few roots of are located close together. The LF method operates in exactly this setting. This may seem like quite a restrictive condition, and indeed it is, however there are practical situations in which the method is applicable. For example, in the field of digital signal processing, very often sampling time series data can result in such polynomials. So too can approximating the roots of a periodic function by finding the roots of a Fourier polynomial. The LF method works almost exclusively with the original polynomial. Working on the assumption that the roots lie near the unit disc, the LF method uses the FFT to quickly sample concentric discs around the unit disc to check for roots. Once candidate roots have been found, they are stored and polished later against the original polynomial, using an iteration scheme. Though we shall not discuss it in this report, the LF method actually uses Laguerre iteration as a more reliable and quickly convergent alternative to Newton s method. Various checks are made, including scanning for duplicate roots. Finally, once all roots have been found, the polynomial is reconstructed from its factors, and a check is made confirming that the difference between the coeffcients of the unfactored and original polynomial is small. 5.1 Sampling on disc of radius r To sample the polynomial on concentric discs around the unit disc, the LF method uses a variation of the FFT techniques discussed in Section 3. It is a simple extension of these ideas to sample the a polynomial on a disc of arbitrary radius, r. To see this, consider the formula (12) which evaluates a degree M 1 polynomial at the (k, M) th root of unity, M P M 1 (ω k,m ) = x m ω m 1 k,m. m=1 If we instead wish to evaluate the polynomial at the same complex argument, but for some arbitrary magnitude r, we must compute P M 1 (rω k,m ). This is found indirectly, M P M 1 (rω k,m ) = x m (rω k,m ) m 1 = m=1 M m=1 11 ( xm r m 1) ω m 1 k,m.

12 Thus, evaluation of a poynomial on a radius r can be achieved by first computing the pointwise product of the coeffcients vector (x 1, x 2,..., x M ) with a vector of ascending powers of r: (1, r,..., r M 1 ). Evaluating the FFT of this computes the required function values. The implementation is simple, and an example can be found in Figure 7. M = 11; x = rand(m,1); r = 0.9; vals_ud = fft(x.*(r.^(0:m-1) )); % polynomial and radius % values Figure 7: Matlab code for FFT evaluation of a polynomial on a disc of radius r. 5.2 A description of the method The main idea behind the Lindsey-Fox method is to sample the polynomial over a circular mesh around the unit disc. If the grid is sufficiently fine, then deductions about the likely locations of the roots can be made by using a corollary of the maximum modulus theorem. Figure 8(a) shows an example of the type of search grid used. Note that it is only necesary to search in one half of the complex plane, since any root that is found will have an associated complex conjugate which can be trivially obtained by negating the imaginary component. (a) (b) Figure 8: The LF search grid around the lower half of the unit disc Three concentric discs are sampled simultaneously. Each of the function values corresponding to the red crosses in Figure 8(a) are examined in turn. The absolute value of the function at the central node is compared to the absolute value of the function at each of the surrounding nodes, as in Figure 8(b). If this absolute value is less than the surrounding values, then the node is likely to be in the vicinity of a root. Once a particular set of concentric discs have been scanned, and any potential root candidates identified, the process restarts. The function values on a fourth concentric disc (inside the interior disc, say) are sampled, the previous middle disc becomes the new exterior disc, and the old interior disc becomes the new middle disc. 12

13 Naturally, the number of initial approximations discovered increases as the search grid becomes finer. A finer search grid not only leads to more initial candidate roots, but these guesses are likely to be better initial approximations. The polishing step is therefore much more likely to end up converging the correct root, rather than to one that has already been determined or worse, falling into a limit cycle, or diverging. Moreover, since most, if not all of the roots are initially approximated with the grid search, there is no need for sequential deflation steps and the accumulated error they introduce. However, it is entirely possible that the grid search will miss some roots in the initial search. If this happens, deflation is then necessary. Often, the number of roots missed by the grid search will be a relatively small number less than 100, say. Then, the companion matrix method can be used to find approximation to the remaining roots, which can then be polished with a suitable iterative scheme. 5.3 Implementation We provide an implementation of the kernel of the method in Appendix A.1. The code performs a search over just three concentric discs, but can easily be adapted to cycle over several radii. We also omit the iteration step. In Appendix A.2, we give some code which plots the intitial guesses found by the LF search method against the roots computed by the companion matrix method. Figure 9 shows sample output for two degree 99 polynomials. Figure 9: LF search approximations to the roots The red circles are roots computed with the companion matrix, and the blue crosses are the LF search approximations, before iteration. Note that many but nowhere near all of the roots have been approximated. This is because the code is only checking three discs. If more discs are added and the search grid made correspondingly finer, a majority, and perhaps all, of the roots are likely to be found. 13

14 6 Conclusions Whilst the Lindsey-Fox method does present a fresh perspective on the subject of polynomial rootfinding, care needs to be taken with regard to the claims made by the engineers of the algorithm regarding its reliability and robustness. That is not to say that the recommendation of this report that the LF method is unreliable. However, potential users should be aware that as far as the author is aware the robustness of the method has not so far been tested in a rigorous comparative setting. It is important to try and put things in perspective. In particular, the methods described in this report represent a very small cross-section of the vas literature available on polynomial rootfinding. Indeed, a recent book by McNamee [7] on the subject included a reference list with more than 8000 entries! For each of the methods described, it is easy to come up with an example that will cause it to fail. This indeed is the perennial problem faced by rootfinding algorithms. Quite apart from the inherent ill-conditioning of many problems, it is notoriously difficult to design rootfinding algorithms that are both robust and stable. In the few cases where this is achieved, the cost is often high operation count and computational complexity. One class of techniques that has been omitted from our discussion so far falls into this category. The companion matrix method is behind the the Matlab roots() command. A particular matrix is constructed from the coefficients of the polynomial and the eigenvalues of this matrix turn out to be exactly the roots of the polynomial. Stable algorithms exist to compute eigenvalues, and typically, an iterative scheme such as the QR-algorithm is used. Relying as it does on matrix multiplication, the QR method tallies up an O(n 3 ) operation count. More recently, attempts have been made recently to utilise structure in the companion matrix in order enabling the QR method to converge in O(n 2 ) operations [8]. When the rootfinding problem is restricted to finding roots on a region of the real axis, rather than in a region of the complex plane, the situation is somewhat nicer. Here, several successful attempts have been made to retain the robustness of the companion-matrix method whilst reducing the operation count to O(n 2 ). In particular, both Boyd in 2002 [4] and Battles in his 2004 Oxford DPhil Thesis [5] described how using a Chebyshev basis can lead to savings by allowing recursive subdivision of the approximation interval. This technique ensures that the size of the eigenvalue problem to be solved is never too large and is in fact the method used for computing roots in Chebfun [6]. 14

15 References [1] Sitton, G; Burrus, SC; Fox, JW; Trietel, S Factoring very high degree polynomials IEEE Signal Processing Magazine, Vol. 20(6) pp (2003) [2] Burrus, SC Horner s Method for Evaluating and Deflating Polynomials Unpublished notes: [3] Higham, NJ Accuracy and Stability of Numerical Algorithms SIAM, Second edition (2002) [4] Boyd, JP Computing zeros on a real interval through Chebyshev expansion and polynomial rootfinding SIAM Journal on Numerical Analysis 40 pp (2002) [5] Battles, Z Numerical Linear Algebra for Continuous Functions DPhil thesis, Oxford University Computing Laboratory (2004) [6] Chebfun website [7] McNamee, JM Numerical Methods for Roots of Polynomials Elsevier, First edition (2007) [8] Van Barel, M; Vandebril, R; Van Dooren, P Implicit double shift QR-algorithm for companion matrices Technical report, Department of Computer Science, K.U.Leuven (2008) [9] Datta, BN Numerical Linear Algebra and Applications SIAM, Second edition (2010) 15

16 A Appendix: MATLAB code A.1 Implementation of the LF method N = 100; c = rand(n,1); % c_1 + c_2 x c_n x^{n-1} mm = 200; % nodes in lower half plane if mm < N/2+1, error( require mm >= N/2 +1 ) else M = 2*(mm-1); end rr = [ ]; % radii of three concentric discs exps = (0:N-1) ; % exponential factors ffta = fft(c.*(rr(1).^exps),m); % evaluate polynomial on the rings fftb = fft(c.*(rr(2).^exps),m); fftc = fft(c.*(rr(3).^exps),m); cra = abs([ffta(end) ; ffta(1:mm+1)]); % append end terms and take abs val crb = abs([fftb(end) ; fftb(1:mm+1)]); crc = abs([fftc(end) ; fftc(1:mm+1)]); k = 2:mm+1; Bk = crb(k); % indices of middle-disc without end values % value function on the middle-ring % locate the local minima locs = (Bk < cra(k-1)) & (Bk < cra(k)) & (Bk < cra(k+1))... & (Bk < crc(k-1)) & (Bk < crc(k)) & (Bk < crc(k+1))... & (Bk < crb(k-1)) & (Bk < crb(k+1)); indx = find(locs); lx = length(indx); lf_rts=[]; if (lx) theta = -2*pi/M*(indx-1); lf_rts = exp(1i*theta); end % indices of local minima % number of minima found % convert indices into complex numbers A.2 Plotting the LF roots actual_rts = roots(flipud(c)); % compute roots with companion matrix plot(lf_rts, * ), hold on % plot LF approximations, plot(conj(lf_rts), * ) % companion matrix roots plot(actual_rts, or ) % and the three concentric discs uc = exp(1i*linspace(0,2*pi,200) ); plot(uc*rr, k ), hold off axis equal, axis off 16