Linear implicit differentiation formulas of variable step and order

An algorithm is presented for solving nonlinear ordinary differential equations that generates a broad class of implicit linear differentiation formulas. Specific interest is concerned in one type of formula which for constant timestep reduces to the well-known Gear formulas. Although these formulas are mathematically fully equivalent to the BDF formulas as presented by Brayton et al. [1], their construction is quite different. Instead of using previously calculated function values, we employ predictions extrapolated from these values to set up and evaluate the differentiation formula. A recursive relation for these predictions is derived in order to simplify their calculation in the next timestep. As predictions are needed for order and error control, our algorithm appears to be more efficient as to the number of arithmetic operations than Brayton's algorithm, as well as more general and systematic. Moreover, a change of the order can be accomplished without extra work. Due to the available predictions, an interpolation to determine function values at intermediate time instants, as for instance required in plotting procedures, can be performed in a fast way.