Polynomial Evaluation Research Articles

Hardware evaluation of mathematical functions

The paper presents a family of algorithms for evaluating the elementary mathematical functions including division, sine, cosine, tangent, arctangent, logarithm, exponential and square root. The algorithms are based on continued products and continued sums, and generate the sum on a 2-bit-at-a-time basis. The only operations required are shifting, adding, subtracting and recall of prestored constants. Consequently, the method is suitable for hardware implementation, and two designs are presented. The faster design can generate a 64-bit result in under 1 μs for most of the functions, and under 2 μS for the remainder, when using ECL technology of the F100K type. The other design is based on a pipelined structure and is approximately 40% slower and cheaper. The performance of both designs compares favourably with that of a hardwired polynomial evaluation based on Homer's scheme when using a 300 ns 64- by 64-bit multiplier. Such a design is between 35% and 130% slower than the pipelined version depending upon the function evaluated, yet only 30% cheaper.

Application of REDUCE-2 algebraic manipulation system in calibration problems of track chamber picture processing devices

The problem is considered of correcting nonlinear coordinate distortions generated by cathode ray tube scanner. Orthogonal bivariate polynomials are used to solve the computational part of this problem. The symbolic algebraic algorithm, proposed for the evaluation of orthogonal polynomials, is performed by the REDUCE-2 algebraic manipulation system.

On the application of Newbery's transformation to the Reinsch polynomial evaluation scheme

Reinsch has suggested a modification to Clenshaw's polynomial evaluation scheme which is designed to reduce the potential magnification of rounding errors, while the same objective prompted Newbery to propose treating the polynomial as a combination of two lower-degree polynomials. Here the effect of combining these two techniques is analysed and compared with alternative algorithms by means of computable error bounds which vary with the argument of the polynomial.

Addition Chain Methods for the Evaluation of Specific Polynomials

Addition chains are considered for specific polynomials. It is shown that for a wide class of polynomials the evaluation of their first n terms requires at least $n + O(n^{2/3} )$ additions. Included in this class are the first n squares, the first n cubes, $ \cdots $, the first nkth powers. The results are established by making contact with results in combinatorics.

A Theoretical Development for the Computer Generation and Display of Piecewise Polynomial Surfaces

Two algorithms for parametric piecewise polynomial evaluation and generation are described. The mathematical development of these algorithms is shown to generalize to new algorithms for obtaining curve and surface intersections and for the computer display of parametric curves and surfaces.

A note on the complexity of approximative evaluation of polynomials

A note on the complexity of approximative evaluation of polynomials

Rounding error propagation in polynomial evaluation schemes

The error propagation characteristics of the polynomial evaluation schemes of Horner, Clenshaw and Reinsch are analysed, and Newbery's transformation of the Horner method is extended to Clenshaw's algorithm. In each case computable error bounds are derived in terms of the coefficients of both the power series and Chebyshev series form of the polynomial, and are found to be of particular relevance to polynomials having equisign or strictly alternating sign coefficients or which are similar to Chebyshev polynomials. The analysis confirms the earlier hypotheses and empirical results of Newbery, but affords greater insight into how the error propagation varies with the argument of the polynomial.

On the Parallel Evaluation of Multivariate Polynomials

We prove that any multivariate polynomial P of degree d that can be computed with $C(P)$ multiplications-divisions can be computed in $O(\log d \cdot \log C(P))$ parallel steps and $O(\log d)$ parallel multiplicative steps.

A note on the evaluation of matrix polynomials

The problem of evaluating a polynomial p(x) in a matrix A arises in many applications, e.g., the Taylor approximation of eA. The 0(\sqrt{q} n^{3}) algorithm of Paterson and Stockmeyer has the drawback that it requires 0(\sqrt{q}q n^{2}) storage, where q is the degree of p and n is the dimension of A . An algorithm which greatly reduces this storage requirement without undue loss of speed is presented.

Optical computation using residue arithmetic

Using residue arithmetic it is possible to perform additions, subtractions, multiplications, and polynomial evaluation without the necessity for carry operations. Calculations can, therefore, be performed in a fully parallel manner. Several different optical methods for performing residue arithmetic operations are described. A possible combination of such methods to form a matrix vector multiplier is considered. The potential advantages of optics in performing these kinds of operations are discussed.

Stable evaluation of polynomials in time 𝑙𝑜𝑔𝑛

An algorithm is investigated which evaluates real polynomials of degree n in time log ⁡ n \log n at asymptotically minimum costs. The algorithm is considerably stable with respect to round-off.

On the calculation of the acyclic polynomial

Graphical methods are developed for recursive evaluation of the acyclic polynomial. Analytical formulas of the acyclic polynomials for several specific series of graphs are given. Mathematical properties of the derivatives of the acyclic polynomial are given.

A note on the signs of truncated Chebyshev polynomials

When one or more of the lowest degree terms are removed from the power series form of a Chebyshev polynomialTn(x), the resulting polynomial is found to be of constant sign, a fact which is of importance when analysing the stability of polynomial evaluation schemes.

The fast Fourier transform by polynomial evaluation

A description, leading to a program, for Fast Fourier Transform is given by means of the relationship to polynomial evaluation.

Evaluation of polynomials with super-preconditioning

Certain questions concerning the arithmetic complexity of univariate polynomial evaluation are considered. The principal technical results show that there exist polynomials f, g, and h with h = fg, such that h requires substantially fewer arithmetic operations than either f or g. However, if the coefficients of f are algebraically independent, then any h = fg is as hard to evaluate as f. The question of the relative complexities of f and fg is viewed as a special case of the following question: given an operator Δ which maps polynomials to sets of polynomials, what savings in arithmetic operations is achievable by evaluating some polynomial h ϵ Δ( f) rather than f? Observations and open questions concerning several operators are discussed.

A New Approach to Error Arithmetic

By modification of the standard definition of relative error, a form of error arithmetic is developed that is well suited to floating-point computations. Rules are given for conversion from interval analysis to the new approach, and vice versa, both for real and complex variables. Illustrative applications include accumulation of products, quotients, sums and inner products, and the evaluation of polynomials. The paper also includes some new error bounds for basic operations in floating-point arithmetic.

On the evaluation of multivariate polynomials and their derivatives

LetH be a polynomial inn>1 variables over the fields of real or complex numbers. An algorithm is presented here for the simultaneous evaluation ofH and its first and second (F-) derivativesH′ andH″, or of any combination ofH,H′,H″. The evaluations ofH alone or ofH andH′ together are of the same order inn andd whered is the degree ofH, while the computation ofH,H′, andH″ isd times this order. The process takes account of the sparsity pattern ofH by using a tree structure induced by the nonzero coefficients. It also allows for simultaneous operation with several polynomials with the same sparsity pattern. The data structure for the method is rather simple in nature and can be adapted easily to specific types of polynomials. Several possible implementations and their complexity are discussed.

A General Hardware-Oriented Method for Evaluation of Functions and Computations in a Digital Computer

A parallel computational method, amenable for efficient hardware-level implementation, is described. It provides a simple and fast algorithm for the evaluation of polynomials, certain rational functions and arithmetic expressions, solving a class of systems of linear equations, or performing the basic arithmetic operations in a fixed-point number representation system. The time required to perform the computation is of the order of m carry-free addition operations, m being the number of digits in the solution. In particular, the method is suitable for fast evaluation of mathematical functions in hardware.

Complexity measures and hierarchies for the evaluation of integers and polynomials

The complexity of evaluating integers and polynomials is studied. A new model is proposed for studying such complexities. This model differs from previous models by requiring the construction of constant to be used in the computation. This construction is given a cost which is dependent upon the size of the constant. Previous models used a uniform cost, of either 0 or 1, for operations involving constants. Using this model, proper hierarchies are shown to exist for both integers and polynomials with respect to evaluation cost. Furthmore, it is shown that almost all integers (polynomials) are as difficult to evaluate as the hardest integer (polynomial). These results remain true even if the underlying basis of binary operations which the algorithm performs are varied.

A Generalized Asymptotic Upper Bound on Fast Polynomial Evaluation and Interpolation

It is shown in this paper that the evaluation and interpolation problems corresponding to a set of points, $\{ x_i \} _{i = 0}^{n - 1} $, with $(c_i - 1)$ higher derivatives at each $x_i $ such that $\sum _{i = 1}^{n - 1} c_i = N$, can be solved in $O([N\log N][(\log n) + 1])$ steps.l This upper bound matches perfectly with the known upper bounds of the two extreme cases, which are $O(N\log ^2 N)$ and $O(N\log N)$ steps when $n = N$ and $n = 1$, respectively.