Abstract
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.
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Rheinboldt, W.C., Mesztenyi, C.K. & Fitzgerald, J.M. On the evaluation of multivariate polynomials and their derivatives. BIT 17, 437–450 (1977). https://doi.org/10.1007/BF01933453
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DOI: https://doi.org/10.1007/BF01933453