An algorithm for piecewise linear approximations

JC Pleasant - Proceedings of the 18th annual Southeast regional …, 1980 - dl.acm.org
JC Pleasant
Proceedings of the 18th annual Southeast regional conference, 1980dl.acm.org
An algorithm is described for approximating a function F (x) on a finite interval [a, b] whose
second derivative is of constant sign on (a, b) by a continuous piecewise linear function, with
any desired accuracy. Given a positive number ε, the algorithm finds a continuous piecewise
linear functionL (x)= mi x+ bi, xi-l≤×≤ xi, i= 1, 2..., nwhere a= xo< xl<...< xn= b, such thatmax
{| L (x)-F (x)|: xi-l≤ x≤ xi}≅= εfor i= 1, 2,..., nl, and| L (x)-F (x)|≤ εfor xn-l≤×≤ xn. In contrast
to a method described by Phillips (1968), the derivative of F (x) is not used in the calculation …
An algorithm is described for approximating a function F(x) on a finite interval [a,b] whose second derivative is of constant sign on (a,b) by a continuous piecewise linear function, with any desired accuracy. Given a positive number ε, the algorithm finds a continuous piecewise linear functionL(x) = mi x + bi, xi-l ≤ × ≤ xi,i = 1,2 ...,nwhere a = xo < xl < ... < xn = b, such thatmax {|L(x) - F(x)|: xi-l ≤ x ≤ xi} ≈ = εfor i = 1,2,...,n-l, and|L(x) - F(x)| ≤ εfor xn-l ≤ × ≤ xn. In contrast to a method described by Phillips (1968), the derivative of F(x) is not used in the calculation of L(x). A computer implementation of the algorithm is discussed and an example of its use is provided.
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