Fast computation of divided differences and parallel hermite interpolation

Abstract We present parallel algorithms for fast polynomial interpolation. These algorithms can be used for constructing and evaluating polynomials interpolating the function values and its derivatives of arbitrary order (Hermite interpolation). For interpolation, the parallel arithmetic complexity is O(log2 M + log N) for large M and N, where M − 1 is the order of the highest derivative information and N is the number of distinct points used. Unlike alternate approaches which use the Lag-range representation, the algorithms described in this paper are based on the fast parallel evaluation of a closed formula for the generalized divided differences. Applications to the solution of dual Vandermonde and confluent Vandermonde systems are described. This work extends previous results in polynomial interpolation and improves the parallel time complexity of existing algorithms. 1989 Academic Press, Inc.

[1]  Fred T. Krogh,et al.  Efficient Algorithms for Polynomial Interpolation and Numerical Differentiation , 1970 .

[2]  H. T. Kung.  Fast evaluation and interpolation , 1973 .

[3]  Francis Y. L. Chin.  A Generalized Asymptotic Upper Bound on Fast Polynomial Evaluation and Interpolation , 1976, SIAM J. Comput..

[4]  Larry Rudolph,et al.  The power of parallel prefix , 1985, IEEE Transactions on Computers.

[5]  Å. Björck,et al.  Solution of Vandermonde Systems of Equations , 1970 .

[6]  Theodore Kilgore.  A characterization of the Lagrange interpolating projection with minimal Tchebycheff norm , 1978 .

[7]  John H. Reif.  Logarithmic Depth Circuits for Algebraic Functions , 1986, SIAM J. Comput..

[8]  G. Golub,et al.  The block decomposition of a vandermonde matrix and its applications , 1981 .

[9]  Alfred V. Aho,et al.  The Design and Analysis of Computer Algorithms , 1974 .

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

[11]  Nicholas J. Higham,et al.  Error analysis of the Björck-Pereyra algorithms for solving Vandermonde systems , 1987 .

[12]  Ellis Horowitz,et al.  A Fast Method for Interpolation Using Preconditioning , 1972, Information Processing Letters.

[13]  Thomas Kailath,et al.  Linear complexity parallel algorithms for linear systems of equations with recursive structure , 1987 .

[14]  Victor Pereyra,et al.  Solving confluent Vandermonde systems of Hermite type , 1971 .

[15]  J. Traub.  Iterative Methods for the Solution of Equations , 1982 .

[16]  Walter Gautschi,et al.  Optimally conditioned Vandermonde matrices , 1975 .

[17]  Nai-Kuan Tsao,et al.  On Multipoint Numerical Interpolation , 1978, TOMS.

[18]  Efstratios Gallopoulos,et al.  Fast and practical parallel polynomial interpolation , 1987 .

[19]  R. Brent,et al.  Solving Triangular Systems on a Parallel Computer , 1977 .

[20]  L. Reichel,et al.  Newton interpolation in Fejér and Chebyshev points , 1989 .

[21]  A. O. Gelʹfond.  Calculus of finite differences , 1973 .

[22]  Ambikeshwar Sharma.  Some Poised and Nonpoised Problems of Interpolation , 1972 .

[23]  H. Tal-Ezer,et al.  High degree interpolation polynomial in Newton form , 1988 .