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gams:smooth_approximations_for_max_x_0_and_min_x_0 [2007/08/10 11:07] (current)
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 +====== Smooth approximations for MAX(X,0) and MIN(X,0) ======
  
 +Q: //Do you know a smooth approximation for max(x,0), and min(x,​0)?// ​
 +
 +This comes from Prof. Ignacio Grossmann (CMU): ​
 +
 +Use the approximation ​
 +
 +     ( sqrt( sqr(x) + sqr(epsilon) )  + x ) / 2
 +
 +for max(x,0), where sqrt is the square root and sqr is the square. ​
 +
 +The error err(x) in the above approximation is maximized at 0 (the
 +point of non differentiability),​ where err(0) = epsilon. As x goes to +/-
 +infinity, err(x) goes to 0. One can reduce the maximum error to
 +epsilon/2 by using the approximation given below. This provides a
 +better approximation near the point of non smoothness but is not so
 +accurate away from this point. ​
 +
 +     ( sqrt( sqr(x) + sqr(epsilon) )  + x - epsilon ) / 2
 +
 +Because min(x,0) = -max(-x,0), you can use the above
 +approximations for min(x,0) as well. Epsilon is a small positive
 +constant.
IMPRESSUM / LEGAL NOTICEPRIVACY POLICY gams/smooth_approximations_for_max_x_0_and_min_x_0.txt ยท Last modified: 2007/08/10 11:07 (external edit)