BIT Numerical Mathematics

, Volume 17, Issue 4, pp 437–450 | Cite as

On the evaluation of multivariate polynomials and their derivatives

  • W. C. Rheinboldt
  • C. K. Mesztenyi
  • J. M. Fitzgerald


LetH be a polynomial inn>1 variables over the fields of real or complex numbers. An algorithm is presented here for the simultaneous evaluation ofH and its first and second (F-) derivativesH′ andH″, or of any combination ofH,H′,H″. The evaluations ofH alone or ofH andH′ together are of the same order inn andd whered is the degree ofH, while the computation ofH,H′, andH″ isd times this order. The process takes account of the sparsity pattern ofH by using a tree structure induced by the nonzero coefficients. It also allows for simultaneous operation with several polynomials with the same sparsity pattern. The data structure for the method is rather simple in nature and can be adapted easily to specific types of polynomials. Several possible implementations and their complexity are discussed.


Data Structure Computational Mathematic Complex Number Tree Structure Simultaneous Operation 
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  1. 1.
    R. H. Gallagher,Perturbation Procedures in Nonlinear Finite Element Structural Analysis, in “Computational Mechanics” (J. T. Oden, ed.) Lecture Notes in Mathematics, Vol. 461, Springer-Verlag/Germany, pp. 75–89, 1975.Google Scholar
  2. 2.
    A. D. Kerr, M. T. Soifer,The Linearization of the Prebuckling State and Its Effect on the Determined Instability Loads, J. Appl. Mech., Trans. ASME 36 (1969), 775–783.Google Scholar
  3. 3.
    D. E. Knuth,The Art of Computer Programming, Vol. 1, “Fundamental Algorithms”, Addison-Wesley Publ. Co., Reading, Mass., 1968.Google Scholar
  4. 4.
    J. T. Oden,Finite Elements of Nonlinear Continua, McGraw-Hill Book Co., New York, 1972.Google Scholar
  5. 5.
    J. M. Ortega, W. C. Rheinboldt,Iterative Solution of Nonlinear Equations in Several Variables, Academic Press, New York, 1970.Google Scholar

Copyright information

© BIT Foundations 1977

Authors and Affiliations

  • W. C. Rheinboldt
    • 1
  • C. K. Mesztenyi
    • 1
  • J. M. Fitzgerald
    • 2
  1. 1.C. Rheinboldt and Charles K. Mesztenyi Computer Science CenterUniversity of MarylandCollege ParkUSA
  2. 2.Department of MathematicsGoucher CollegeBaltimoreUSA

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