Compensated Horner scheme in complex floating point arithmetic

Several different techniques and softwares intend to improve the accuracy of results computed in a fixed finite precision. Here we focus on a method to improve the accuracy of polynomial evaluation via Horner’s scheme. Such an algorithm exists for polynomials with real floating point coefficients. In this paper, we provide a new algorithm which deals with polynomials with complex floating point coefficients. We show that the computed result is as accurate as if computed in twice the working precision. The algorithm is simple since it only requires addition, subtraction and multiplication of floating point numbers in the same working precision as the given data. Such an algorithm can be useful for example to compute zeros of polynomial by Newton-like methods.

[1]  Jean-Michel Muller,et al.  Some functions computable with a fused-mac , 2005, 17th IEEE Symposium on Computer Arithmetic (ARITH'05).

[2]  Peter Linz,et al.  Accurate floating-point summation , 1970, CACM.

[3]  Stef Graillat,et al.  Error-free transformations in real and complex floating point arithmetic , 2007 .

[4]  Siegfried M. Rump,et al.  Verification of Positive Definiteness , 2006 .

[5]  Yves Nievergelt,et al.  Scalar fused multiply-add instructions produce floating-point matrix arithmetic provably accurate to the penultimate digit , 2003, TOMS.

[6]  Siegfried M. Rump,et al.  Accurate Sum and Dot Product , 2005, SIAM J. Sci. Comput..

[7]  James Hardy Wilkinson,et al.  Rounding errors in algebraic processes , 1964, IFIP Congress.

[8]  Nicholas J. Higham,et al.  INVERSE PROBLEMS NEWSLETTER , 1991 .

[9]  Donald E. Knuth,et al.  The art of computer programming. Vol.2: Seminumerical algorithms , 1981 .

[10]  Mei Han An,et al.  accuracy and stability of numerical algorithms , 1991 .

[11]  T. J. Dekker,et al.  A floating-point technique for extending the available precision , 1971 .

[12]  James Demmel,et al.  Design, implementation and testing of extended and mixed precision BLAS , 2000, TOMS.

[13]  S. Oishi,et al.  ACCURATE FLOATING-POINT SUMMATION , 2005 .

[14]  Richard P. Brent,et al.  Error bounds on complex floating-point multiplication , 2007, Math. Comput..

[15]  James Demmel,et al.  Accurate and Efficient Floating Point Summation , 2003, SIAM J. Sci. Comput..

[16]  Philippe Langlois,et al.  Compensated Horner Scheme , 2005, Algebraic and Numerical Algorithms and Computer-assisted Proofs.