Re: Closed form solution to quadratic fit?

In article <1169666701.131894.251790@xxxxxxxxxxxxxxxxxxxxxxxxxxxx>,
Stephen_B <stephen@xxxxxxxxxxxxxx> wrote:
Is there a closed form solution to this problem? I have a set of x,y
pairs, and I would like to find the quadratic a*x^2 + b*x + c = y that
minimizes the worst case (absolute value) error between the actual data
and the quadratic approximation. I'm not trying to minimize the least
squared error, but the maximum error. Thanks.

Suppose the data points are [x_j, y_j], j=1..n.
The problem may be stated as a linear programming problem with four
variables a,b,c,t (with no sign restrictions) and 2n constraints:

minimize t
subject to
x_i^2 a + x_i b + c + t >= y_i for all i
-x_i^2 a - x_i b - c + t >= - y_i for all i

Robert Israel israel@xxxxxxxxxxx
Department of Mathematics http://www.math.ubc.ca/~israel
University of British Columbia Vancouver, BC, Canada
.

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