On the number of segments needed in a piecewise linear approximation

Authors:
C. L. Frenzen;Tsutomu Sasao;Jon T. Butler
Affiliations:
Department of Applied Mathematics, Naval Postgraduate School, Monterey, CA, 93943-5216, USA;Department of Computer Science and Electronics, Kyushu Institute of Technology, Iizuka, 820-8502, Japan;Department of Electrical and Computer Engineering, Naval Postgraduate School, Monterey, CA, 93943-5121, USA
Venue:
Journal of Computational and Applied Mathematics
Year:
2010

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Abstract

The introduction of high-speed circuits to realize an arithmetic function f as a piecewise linear approximation has created a need to understand how the number of segments depends on the interval a@?x@?b and the desired approximation error @e. For the case of optimum non-uniform segments, we show that the number of segments is given as s(@e)~c@e, (@e-0^+), where c=14@!"a^b|f^''(x)|dx. Experimental data shows that this approximation is close to the exact number of segments for a set of 14 benchmark functions. We also show that, if the segments have the same width (to reduce circuit complexity), then the number of segments is given by s(@e)~c@e, (@e-0^+), where c=(b-a)|f^''|"m"a"x4.