Graeffe's Method Research Articles

A highly parallel algorithm for root extraction

A parallel algorithm for extracting the roots of a polynomial is presented. The algorithm is based on Graeffe's method, which is rarely used in serial implementations, because it is slower than many common serial algorithms, but is particularly well suited to parallel implementation. Graeffe's method is an iterative technique, and parallelism is used to reduce the execution time per iteration. A high degree of parallelism is possible, and only simple interprocessor communication is required. For a degree-n polynomial executed on an (n+1)-processor SIMD machine, each iteration in the parallel algorithm has arithmetic complexity of approximately 2n and a communications overhead n. In general, arithmetic speedup is on the order of p/2 for a p-processor implementation.< <ETX xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">&gt;</ETX>

Algebraic complexity of computing polynomial zeros

Using the power sum techniques of Turan, we evaluate all the complex zeros of an nth degree univariate polynomial with relative errors ⩽ ϵ using O( n 2 log n ( n log n + log(1/ ϵ))) arithmetic operations. O( n log n log(1/ ϵ)) operations suffice to approximate (with relative error ⩽ ϵ) a single complex zero using Turan's method and all the zeros if they are real using Graeffe's method. In all cases n processors suffice for cn times parallel acceleration where c is a positive constant. Incorporating Turan's techniques into another, more recent algorithm gives a single complex zero with absolute error ⩽ ϵ using O(log 2 n log log(| λ 1/ ϵ|)) arithmetic parallel steps, n 2 processors (which places the problem in NC) and also gives all the complex zeros in O( n 2 log n + log log(| λ 1/ ϵ|))) arithmetic operations, where λ 1 is the absolutely largest zero. Computations with O(log( n| λ 1/ ϵ|))) binary bits support the latter estimates for the arithmetic complexity, which leads to a simple proof of the current best estimate for the Boolean circuit complexity (bit-operation complexity) of computing all the complex zeros of a polynomial, announced by A. Schönhage in 1982 but not proven yet.

Systolic designs for the root-squaring method

In this paper some systolic designs are presented for the implementation of the Graeffe root-squaring method for polynomial root solving. From a semi-systolic array, “retiming” transformations are applied to yield a purely systolic array that performs the squaring of the coefficients of an equation. The systolic array is then simulated soft-systolically in an OCCAM program listed in the Appendix. The overall design of a systolic system for the solution of equations based on the Graeffe method is also discussed.

A series expansion for the first positive zero of the Bessel functions

It is shown that the first positive zero j v , l {j_{v,l}} of the Bessel function J v ( x ) {J_v}(x) is given by \[ j v , l = 2 ( v + 1 ) 1 / 2 [ 1 + ( v + 1 ) 4 − 7 ( v + 1 ) 2 96 + 49 ( v + 1 ) 3 1152 − 8363 ( v + 1 ) 4 276480 + ⋯ ] {j_{v,l}} = 2{(v + 1)^{1/2}}\left [ {1 + \frac {{(v + 1)}}{4} - \frac {{7{{(v + 1)}^2}}}{{96}} + \frac {{49{{(v + 1)}^3}}}{{1152}} - \frac {{8363{{(v + 1)}^4}}}{{276480}} + \cdots } \right ] \] for − 1 &gt; v &gt; 0 - 1 &gt; v &gt; 0 .

Zeros of smallest modulus of functions resembling exp(z)

To determine (in various senses) the zeros of the Laplace transform of a signed mass distribution is of great importance for many problems in classical analysis and number theory. For example, if the mass consists of finitely many atoms, the transform is an exponential polynomial. This survey studies what is known when the distribution is a probability density function of small variance, and examines in what sense the zeros must have large moduli. In particular, classical results on Bessel function zeros, of Szegö on zeros of partial sums of the exponential, of I. J. Schoenberg on k‐times positive functions, and a result stemming from Graeffe′s method, are all presented from a unified probabilistic point of view.

The Numerical Factorization of a Polynomial

A number of methods can be found in the literature for the numerical factorization of a polynomial, and references to some are given in the bibliography below. Well-known examples are those of Lin and of Bairstow. Many are quadratically convergent, but most require a sufficiently close initial factorization to start with. The method of Graeffe and the qd algorithm are not ordinarily thought of as methods of factorization, but either provides, in principle, factors of the given polynomial whose zeros are zeros of equal modulus of the given polynomial. These do not require initial approximations, and the Graeffe method (but not the qd algorithm) is quadratically convergent. But neither is self-correcting, so that errors accumulate, and any approximate factor either produces may require further refinement by a method that is self-correcting. Naturally the extraction of a single zero is a special case of factorization in which one of the factors is linear, and this is to be understood throughout. A number of these methods can be related to a method that seems to have been proposed first by Sebastiao e Silva (for brevity he will hereafter be referred to as S-S) in 1941. Although Bairstow's method is a member of this class, and was published in 1914, it is only in terms of the theory set forth by S-S and further elaborated by Bauer that the relations among these methods can be properly understood. The purpose of the present note is to attempt to show in a heuristic and expository manner how some of the algorithms to which allusion has been made can be regarded as natural outgrowths of the basic S-S theory (in this connection, however, see also Stewart [26]). At the outset a theorem will be stated that is somewhat more general than the original S-S theorem. For the proof of the general theorem reference is made to Householder [15], but a proof of the S-S theorem itself will be sketched that is rather different in form from the usual one, and that can be extended, though with certain complications, to the more general theorem. This particular proof has the further advantage that the usually troublesome confluent case introduces only mild complications. A somewhat different proof has been given by Glenisson and Derwidue [12], [13] who seem to have discovered the algorithm independently. The S-S algorithm is, like the method of Bernoulli and the method of Graeffe (and of Dandelin and Lobachevsky), based on the idea of root powering, but the algorithm is different. Like these methods, it does not require an initial approximation. In one form it is quadratically convergent, like the method of Graeffe. A method that is quadratically convergent and does not require an initial approximation will be said to have Graeffe-type convergence. But the S-S algorithm, unlike the method of Graeffe, can be made error-correcting.

Remark on algorithm 256: modified Graeffe method

No abstract available.

Algorithm 256: modified Graeffe method [C2

Algorithm 256: modified Graeffe method [C2

Generalized function transformation formulae in the Lobachevsky — Graeffe method for the numerical determination of the zeros of integral and holomorphic functions

Generalized function transformation formulae in the Lobachevsky — Graeffe method for the numerical determination of the zeros of integral and holomorphic functions

Notes on the Graeffe Method of Root Squaring

Notes on the Graeffe Method of Root Squaring

On the Graeffe Method of Solution of Equations

On the Graeffe Method of Solution of Equations

On the Graeffe Method of Solution of Equations

On the Graeffe Method of Solution of Equations

Graeffe's Method and Complex Roots

Graeffe's Method and Complex Roots

On Graeffe's Method for the Numerical Solution of Algebraic Equations

On Graeffe's Method for the Numerical Solution of Algebraic Equations

On Graeffe's Method for the Numerical Solution of Algebraic Equations

On Graeffe's Method for the Numerical Solution of Algebraic Equations

On Graeffe's Method for Complex Roots of Algebraic Equations

The only really useful practical method for solving numerical algebraic equations of higher orders, possessing complex roots, is that devised by C. H. Graeffe early in the nineteenth century. When an equation with real coefficients has only one or two pairs of complex roots, the Graeffe process leads to the evaluation of these roots without great labour. If, however, the equation has a number of pairs of complex roots there is considerable difficulty in completing the solution: the moduli of the roots are found easily, but the evaluation of the arguments often leads to long and wearisome calculations. The best method that has yet been suggested for overcoming this difficulty is that by C. Runge (Praxis der Gleichungen, Sammlung Schubert). It consists in making a change in the origin of the Argand diagram by shifting it to some other point on the real axis of the original Argand plane. The new moduli and the old moduli of the complex roots can then be used as bipolar coordinates for deducing the complex roots completely: this also checks the real roots.