Accurate quotient-difference algorithm: Error analysis, improvements and applications

The compensated quotient-difference (Compqd) algorithm is proposed along with some applications. The main motivation is based on the fact that the standard quotient-difference (qd) algorithm can be numerically unstable. The Compqd algorithm is obtained by applying error-free transformations to improve the traditional qd algorithm. We study in detail the error analysis of the qd and Compqd algorithms and we introduce new condition numbers so that the relative forward rounding error bounds can be derived directly. Our numerical experiments illustrate that the Compqd algorithm is much more accurate than the qd algorithm, relegating the influence of the condition numbers up to second order in the rounding unit of the computer. Three applications of the new algorithm in the obtention of continued fractions and in pole and zero detection are shown.

[1]  Lizhi Cheng,et al.  Accurate Evaluation of Polynomials in Legendre Basis , 2014, J. Appl. Math..

[2]  B. Parlett,et al.  From qd to LR, or, how were the qd and LR algorithms discovered? , 2011 .

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

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

[5]  Philippe Langlois,et al.  More Instruction Level Parallelism Explains the Actual Efficiency of Compensated Algorithms , 2007 .

[6]  Philippe Langlois,et al.  How to Ensure a Faithful Polynomial Evaluation with the Compensated Horner Algorithm , 2007, 18th IEEE Symposium on Computer Arithmetic (ARITH '07).

[7]  Lisa Lorentzen,et al.  Padé approximation and continued fractions , 2010 .

[8]  J. Hadamard,et al.  Essai sur l'étude des fonctions données par leur développement de Taylor , 1892 .

[9]  Hassane Allouche,et al.  Reliable root detection with the qd-algorithm: When Bernoulli, Hadamard and Rutishauser cooperate , 2010 .

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

[11]  G. Szegő.  Polynomials orthogonal on the unit circle , 1939 .

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

[13]  Donald E. Knuth.  The Art of Computer Programming 2 / Seminumerical Algorithms , 1971 .

[14]  Xiangke Liao,et al.  Accurate evaluation of a polynomial in Chebyshev form , 2011, Appl. Math. Comput..

[15]  Annie A. M. Cuyt.  Floating-point versus Symbolic Computations in theQD-algorithm , 1997, J. Symb. Comput..

[16]  Siegfried M. Rump,et al.  Accurate Floating-Point Summation Part I: Faithful Rounding , 2008, SIAM J. Sci. Comput..

[17]  Donald Ervin Knuth,et al.  The Art of Computer Programming, Volume II: Seminumerical Algorithms , 1970 .

[18]  Nicolas Louvet,et al.  Algorithmes compensés en arithmétique flottante : précision, validation, performances. (Compensated algorithms in floating point arithmetic : accuracy, validation, performances) , 2007 .

[19]  Philippe Langlois,et al.  Algorithms for accurate, validated and fast polynomial evaluation , 2009 .

[20]  Henry C. Thacher,et al.  Applied and Computational Complex Analysis. , 1988 .

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

[22]  Peter Henrici,et al.  Finding zeros of a polynomial by the Q-D algorithm , 1965, CACM.

[23]  Xiaoye S. Li,et al.  Algorithms for quad-double precision floating point arithmetic , 2000, Proceedings 15th IEEE Symposium on Computer Arithmetic. ARITH-15 2001.

[24]  Lizhi Cheng,et al.  Accurate evaluation of a polynomial and its derivative in Bernstein form , 2010, Comput. Math. Appl..

[25]  Siegfried M. Rump,et al.  Verification methods: rigorous results using floating-point arithmetic , 2010, Acta Numerica.

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

[27]  Heinz Rutishauser.  Anwendungen des Quotienten-Differenzen-Algorithmus , 1954 .

[28]  H. Rutishauser.  Der Quotienten-Differenzen-Algorithmus , 1954 .

[29]  A. C. Aitken.  XXV.—On Bernoulli's Numerical Solution of Algebraic Equations , 1927 .

[30]  B. Parlett.  What Hadamard Missed , 1996 .

[31]  Jim Euchner.  Design , 2014, Catalysis from A to Z.

[32]  A. C. Aitken.  XII.—Further Numerical Studies in Algebraic Equations and Matrices , 1932 .

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

[34]  W. J. Thron,et al.  Continued Fractions: Analytic Theory and Applications , 1984 .

[35]  Annie A. M. Cuyt,et al.  Handbook of Continued Fractions for Special Functions , 2008 .

[36]  Siegfried M. Rump,et al.  Accurate Floating-Point Summation Part II: Sign, K-Fold Faithful and Rounding to Nearest , 2008, SIAM J. Sci. Comput..

[37]  C. Lanczos.  An iteration method for the solution of the eigenvalue problem of linear differential and integral operators , 1950 .

[38]  Peter W. Markstein,et al.  IA-64 and elementary functions - speed and precision , 2000 .

[39]  B. Parlett,et al.  Accurate singular values and differential qd algorithms , 1994 .