On interval predictor-corrector methods

Abstract

One can approximate numerically the solution of the initial value problem using single or multistep methods. Linear multistep methods are used very often, especially combinations of explicit and implicit methods. In floating-point arithmetic from an explicit method (a predictor), we can get the first approximation to the solution obtained from an implicit method (a corrector). We can do the same with interval multistep methods. Realizing such interval methods in floating-point interval arithmetic, we compute solutions in the form of intervals which contain all possible errors. In this paper, we propose interval predictor-corrector methods based on conventional Adams-Bashforth-Moulton and Nyström-Milne-Simpson methods. In numerical examples, these methods are compared with interval methods of RungeKutta type and methods based on high-order Taylor series. It appears that the presented methods yield comparable approximations to the solutions.

Development of different load flow methods for solving large‐scale ill‐conditioned systems

“…Alternatively, an explicit method can be employed instead; this methodology is known as predictor-corrector. 26 Therefore, an explicit method is used to compute an approximate value of the following point (b x kþ1 ð Þ ); then, the implicit method is run to correct it. When an explicit Adams-Bashforth method is used as predictor and an implicit Adams-Moulton method is used as corrector, the approach is called Adams-Bashforth-Moulton.…”

“…Frequently, an iterative technique is employed to address the problem of computing f ( x ( k +1) ) without knowing the value of x ( k +1) ; however, this technique is totally inefficient since it needs to nest two iterative techniques into the algorithm. Alternatively, an explicit method can be employed instead; this methodology is known as predictor‐corrector . Therefore, an explicit method is used to compute an approximate value of the following point ( ); then, the implicit method is run to correct it.…”

Interval methods of Adams-Bashforth type with variable step sizes

“…We can distinguish three main kinds of interval methods for solving the initial value problem: methods based on high-order Taylor series (see, e.g., [1,2,6,15,33,34]), explicit and implicit methods of Runge-Kutta type ( [9,10,22,28,30,37]), and explicit and implicit multistep methods ( [9, 17-19, 22, 25, 26]), including interval predictor-corrector methods [27]. In interval methods based on Runge-Kutta and in interval multistep methods considered so far, a constant step size has been used.…”

Interval versions of Milne’s multistep methods

“…The last ones concern interval methods based on conventional methods of Adams-Bashforth, Adams-Moulton, Nyström, and Milne-Simpson types. These methods have been considered and compared by us, especially in [30]. Although we usually apply such methods of low orders, it appears that for a suitable choice of step sizes the obtained enclosures are as good as enclosures obtained by methods based on high-order Taylor series.…”

“…Although we usually apply such methods of low orders, it appears that for a suitable choice of step sizes the obtained enclosures are as good as enclosures obtained by methods based on high-order Taylor series. Moreover, explicit interval multistep method (of AdamsBashforth and Nyström types) can be used as predictors for implicit interval methods (of Adams-Moulton and Milne-Simpson types)-see [30] for details. From our analysis, it appears that the explicit multistep methods of Nyström type are a little bit better than those of Adams-Bashforth type-the widths of intervals are smaller.…”