Linear multistep algorithms

In contrast with one-step algorithms, a knowledge of the values of x is needed, not in one point, but in k points r ., x 2. x .k. The starting process of such an algorithm is therefore not trivial it may imply recourse to an auxiliary single-step scheme. 

If j30 = 0, the method is explicit and the computation of is straightforward. If 30 + 0, the method is implicit because an implicit algebraic equation is to be solved. Usually, two algorithms, a first one explicit and called the predictor, and a second one implicit and called the corrector, are used simultaneously. The global method is called a predictor-corrector method as, for example, the classical fourth-order Adams method, viz. 

A comprehensive discussion of algorithms, software and packages related to Gear s algorithm has been given by Byrne et al. [169]. 

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For stiff differential equations, the backward difference algorithm should be preferred to the Adams-Moulton method. The well-known code LSODE with different options was published in 1980 s by Flindmarsh for the solution of stiff differential equations with linear multistep methods. The code is very efficient, and different variations of it have been developed, for instance, a version for sparse systems (LSODEs). In the international mathematical and statistical library, the code of Hindmarsh is called IVPAG and DIVPAG. 

In [215] the author developed a three-step seventh algebraic order hybrid linear multistep method (HLMM) with three non-step points for the approximate solution special of special second order initial value problems. The author produced the main method and additional methods from the same scheme derived via interpolation and collocation techniques. The local truncation error is presented, the zero stability and the convergence and consistency properties are studied. Numerical experiments show the efficiency of the proposed algorithm. 

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