Automatic codes for differential equations can be inadequate when the solutions have discontinuities.If the user provides an external mdicator for discontinuities (e.g., a switching function whose sign changes mdicate discontinuities), a code can be more efficient.A technique for detection and location of a discontinmty ts dtscussed which can be implemented when such indicators are not practical.It estimates the order and magmtude of the discontinuity and hence the stepsize which keeps the error under control while stepping over the discontinuity.

MultistepRunge-Kutta-like formulas which enable a multmtep method to start or restart at a high order after lust one Runge-Kutta (RK) step are presented.These formulas greatly improve the efficiency of mnltistep methods in a situation m which they were previously out performed by RK methods--i e., m which there are problems with frequent dlscontinmties or sudden large increases in derivatives, all of which cause an automatic program to reduce order and step size suddenly.

No abstract available.

Two new techniques are presented for reducing the number of depth calculations in hidden surface elimination. Two new algorithms using the techniques are compared with three existing algorithms and it is shown by examples that the new techniques reduce the number of multiplications involved in the depth calculations. A technique for increasing the parallelism of operations is also presented. This allows the calculation to be done more rapidly in hardware and is particularly useful for generating line drawings rather than the usual TV raster scan images in the common raster-scan hidden surface algorithms.

An interactive computer graphics program was used to search for stiffly stable methods of orders seven through fifteen.Previously, methods of orders up to six were known.Methods of orders seven and eight were found in this search.The evidence gathered for the class of methods which requires a minimum of previous values to be kept indicates that no stiffly stable methods of this type exist for orders eight through fifteen.

This paper describes how the assembly program, now being written for the new University of Illinois digital computer ILLIAC 2, optimizes the computation and compilation of the address field. The compilation is necessary because the address field may contain general expressions which involve the contents of modifiers (index registers) and are therefore only computable at execution time rather than at assembly time. The optimization is advisable because some computation can be done at assembly time; the remainder must be done at execution time, but may be done in either of the two arithmetic units available in ILLIAC 2. Expressions involving modifiers additively or subtractively were first introduced into this assembly language to enable the programmer to make simple use of the multiple indexing facilities available in hardware. Since it had been found desirable in other assemblers to allow multiplication (and division) of symbolic addresses and integers, and since, in this assembler, symbolic addresses may be defined as representing either numbers or the contents of modifiers, it was decided to allow general expressions of both numbers and modifiers. A simple algorithm is described which, in a single scan of the address field, reorganizes the calculation in order to assign each of the operations in the address field in a near optimum manner to the appropriate one of the three calculating units—the assembly computer, the floating-point accumulator (FAC) and the address accumulator (AAC).