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On the other hand, + is a third-order approximation, so the difference between + and + can be used to adapt the step size. The FSAL—first same as last—property is that the stage value k 4 {\displaystyle k_{4}} in one step equals k 1 {\displaystyle k_{1}} in the next step; thus, only three function evaluations are needed per step.
Flory–Stockmayer theory is a theory governing the cross-linking and gelation of step-growth polymers. [1] The Flory–Stockmayer theory represents an advancement from the Carothers equation, allowing for the identification of the gel point for polymer synthesis not at stoichiometric balance. [1]
The substitutions of Euler can be generalized by allowing the use of imaginary numbers. For example, in the integral +, the substitution + = + can be used. Extensions to the complex numbers allows us to use every type of Euler substitution regardless of the coefficients on the quadratic.
For example, consider the ordinary differential equation ′ = + The Euler method for solving this equation uses the finite difference quotient (+) ′ to approximate the differential equation by first substituting it for u'(x) then applying a little algebra (multiplying both sides by h, and then adding u(x) to both sides) to get (+) + (() +).
The first Dahlquist barrier states that a zero-stable and linear q-step multistep method cannot attain an order of convergence greater than q + 1 if q is odd and greater than q + 2 if q is even. If the method is also explicit, then it cannot attain an order greater than q ( Hairer, Nørsett & Wanner 1993 , Thm III.3.5).
The simplest case refers to the formation of a strictly linear polymer by the reaction (usually by condensation) of two monomers in equimolar quantities. An example is the synthesis of nylon-6,6 whose formula is [−NH−(CH 2) 6 −NH−CO−(CH 2) 4 −CO−] n from one mole of hexamethylenediamine, H 2 N(CH 2) 6 NH 2, and one mole of adipic acid, HOOC−(CH 2) 4 −COOH.
The method has been discovered and forgotten many times, dating back to Newton's Principiae, [1] as recalled by Richard Feynman in his Feynman Lectures (Vol. 1, Sec. 9.6) [2] In modern times, the method was rediscovered in a 1956 preprint by René De Vogelaere that, although never formally published, influenced subsequent work on higher-order symplectic methods.
The idea to combine the bisection method with the secant method goes back to Dekker (1969).. Suppose that we want to solve the equation f(x) = 0.As with the bisection method, we need to initialize Dekker's method with two points, say a 0 and b 0, such that f(a 0) and f(b 0) have opposite signs.