Akima presents a new method for univariate interpolation that produces a natural-looking curve when it is used for smooth curve fitting. The method is an improved version, without basic changes, of the original method included in the IMSL Library.

General properties for univariate interpolation methods are defined and their mutual compatibility is discussed to emphasize that producing a natural-looking curve needs adjustment and compromise. So, Akima’s methods do not satisfy continuity and linearity of the method and do not always preserve monotonicity or convexity. The requirement of a line segment for several collinear data points (in order to suppress excessive undulations) is extended to four or more points in the improved method (instead of three points in the original version). This improvement assures compatibility with the requirement of accuracy for a third-degree polynomial, which is the lowest-degree polynomial that can produce a good-looking curve. In addition, it allows improvement in the procedure of estimating the first derivative of the interpolating function at each data point. Thus, the first derivative of the third-degree polynomial fitted to every set of four consecutive data points that include a given point P is used as a primary estimate of the first derivative at P. A weighted mean of these four primary estimates provides the final estimate of the first derivative at P. In addition, Akima analyzes this procedure in case of an optional use of a higher-degree polynomial: generally, it reduces undulations even more, but sometimes it distorts curves that would look good otherwise.

The advantages and the conclusions are clearly demonstrated by easy-to-understand examples. The style is concise and tutorial, so no solid background is required.