Efficient Evaluation Of Polynomials Research Articles

Fault tolerant systolic evaluation of polynomials and exponentials of polynomials for equispaced arguments

Many applications in digital signal processing and computer graphics, which require high speed evaluation of polynomials and exponentials of polynomials can now be implemented in hardware very efficiently because of the advances in VLSI technology. Several fast algorithms have been proposed in the recent past for the efficient evaluation of polynomials and exponentials of polynomials for equispaced arguments on uniprocessor systems. In this brief, we consider the problem of organizing this evaluation on VLSI chips in the form of systolic arrays. We use triple time redundancy and algorithm-based fault tolerance to detect and correct errors caused by transient failures in the Processing Elements (PEs) of these systolic arrays. A comparison of the space and time overheads in using these two methods for adding fault tolerance is also discussed.

Algorithm-based fault tolerant systolic evaluation of polynomials and exponentials of polynomials for equispaced arguments

Many applications in digital signal processing and computer graphics, which require high speed evaluation of polynomials and exponentials of polynomials can now be implemented in hardware very efficiently because of the advances in VLSI technology. Several fast algorithms have been proposed in the recent past for the efficient evaluation of polynomials and exponentials of polynomials for equispaced arguments on uniprocessor systems. In this paper, we consider the problem of organizing this evaluation on VLSI chips in the form of systolic arrays. We use algorithm-based fault tolerance to detect and correct errors caused by transient failures in the Processing Elements (PEs) of these systolic arrays. This approach of adding fault tolerance is shown to require minimum space and time overheads compared to the other traditional fault tolerance schemes such as space redundancy and time redundancy. The organizations presented here have the main advantage that the interconnections between the PEs are very regular and simple, and hence are very appropriate for VLSI implementation.

FAULT TOLERANT SYSTOLIC EVALUATION OF POLYNOMIALS AND EXPONENTIALS OF POLYNOMIALS FOR EQUISPACED ARGUMENTS USING TIME REDUNDANCY

Many applications which require high speed evaluation of polynomials and exponentials of polynomials can now be implemented in the hardware very efficiently because of the advances in VLSI technology. Several fast algorithms have been proposed in the recent past for the efficient evaluation of polynomials and exponentials of polynomials for equispaced arguments on uniprocessor systems. In this paper, we consider the problem of organizing this evaluation on VLSI chips in the form of systolic arrays. We present linear fault tolerant systolic arrays which can evaluate the polynomials and exponentials of polynomials of any degree for a large number of equispaced points. These organizations have the main advantage that the interconnections between the processing elements are very regular and simple, and hence are very appropriate for VLSI implementation.

A note on 'Efficient evaluation of polynomials and exponentials of polynomials for equispaced arguments' by A.H. Nuttall

A.H. Nuttall's (see ibid., vol. ASSP-35, no.10, p.1486-7, 1987) algorithm for the evaluation of a polynomial at a large number of arguments is addressed. The authors supplement Nuttall's treatment of the problem by (1) giving a relationship between the coefficients of the original polynomial and those of the equivalent one in terms of the integer variable n and (2) by deriving a formula for the computation of the initial values required for Nuttall's recursive procedure to commence. As a result of these supplements, the recursive algorithm is completely programmable and can be efficiently implemented for any order N of the polynomial. Some general observations are made on the computational complexity in recursive evaluation in contrast to direct evaluation of a polynomial.< <ETX xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">&gt;</ETX>

Efficient evaluation of polynomials and exponentials of polynomials for equispaced arguments

A kth-order polynomial can be evaluated by means of k additions and no multiplications when done in a recursive fashion at equispaced arguments. The evaluation of an exponential of a kth-order polynomial can be accomplished by k multiplications and no additions or exponentiations. Combinations of rational functions and exponentials can therefore be realized very efficiently by combining these properties.