*(English)*Zbl 1198.65142

Summary: Many scientific and engineering problems are described using Ordinary Differential Equations (ODEs), where the analytic solution is unknown. Much research has been done by the scientific community on developing numerical methods which can provide an approximate solution of the original ODE. In this work, two approaches have been considered based on BDF and Piecewise-linearized Methods. The approach based on BDF methods uses a Chord-Shamanskii iteration for computing the nonlinear system which is obtained when the BDF schema is used. Two approaches based on piecewise-linearized methods have also been considered. These approaches are based on a theorem proved in this paper which allows to compute the approximate solution at each time step by means of a block-oriented method based on diagonal Padé approximations. The difference between these implementations is in using or not using the scale and squaring technique.

Five algorithms based on these approaches are developed.

##### MSC:

65L99 | Numerical methods for ODE |

34-04 | Machine computation, programs (ordinary differential equations) |

##### Keywords:

ordinary differential equation (ODE); initial value problem (IVP); backward differentiation formula (BDF) method; piecewise-linearized method; diagonal Padé approximation;##### References:

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