Accurate simple zeros of polynomials in floating point arithmetic

Accurate simple zeros of polynomials in floating point arithmetic

In the paper, we examine the local behavior of Newton’s method in floating point arithmetic for the computation of a simple zero of a polynomial assuming that an good initial approximation is available. We allow an extended precision (twice the working precision) in the computation of the residual. We prove that, for a sufficient number of iterations, the zero is as accurate as if computed in t...

Accurate Polynomial Evaluation in Floating Point Arithmetic

One of the three main processes associated with polynomials is evaluation; the two other ones being interpolation and root finding. Higham [1, chap. 5] devotes an entire chapter to polynomials and more especially to polynomial evaluation. The small backward error the Horner scheme introduce when evaluated in floating point arithmetic justifies its practical interest. It is well known that the c...

Synthesizing Accurate Formulas for the Floating-Point Arithmetic

Many critical embedded systems perform floating-point computations yet their accuracy is difficult to assert and strongly depending on how formulas are written in programs. In this article, we focus on the synthesis of accurate formulas mathematically equal to the original formulas occurring in source codes. In general, an expression may be rewritten in many ways using associativity, distributi...

Accurate Floating Point Arithmetic through Hardware Error-Free Transformations

This paper presents a hardware approach to performing accurate floating point addition and multiplication using the idea of errorfree transformations. Specialized iterative algorithms are implemented for computing arbitrarily accurate sums and dot products. The results of a Xilinx Virtex 6 implementation are given, area and performance are compared against standard floating point units and it i...

Floating-Point Arithmetic