All Algebraic Functions Can Be Computed Fast
Journal of the ACM (JACM)
Convergence behavior of the Newton iteration for first order differential equations
EUROSAM '79 Proceedings of the International Symposiumon on Symbolic and Algebraic Computation
EUROCAM '82 Proceedings of the European Computer Algebra Conference on Computer Algebra
Newton's method: a great algebraic algorithm
SYMSAC '76 Proceedings of the third ACM symposium on Symbolic and algebraic computation
ISSAC '03 Proceedings of the 2003 international symposium on Symbolic and algebraic computation
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Speed of convergence of Newton-like iterations in an algebraic domain can be affected heavily by the increasing cost of each step, so much so that a quadratically convergent algorithm with complex steps may be comparable to a slower one with simple steps. This note gives two examples: solving algebraic and first-order ordinary differential equations using the MACSYMA algebraic manipulation system, demonstrating this phenomenon. The relevant programs are exhibited in the hope that they might give rise to more widespread application of these techniques.