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Approximates the function

y = if x ≥ 0 then k1*x*x else -k2*x*x, with k1, k2 > 0

in such a way that within the region -x_small ≤ x ≤ x_small, the function is described by two polynomials of third order (one in the region -x_small .. 0 and one within the region 0 .. x_small) such that

• The derivative at x=0 is non-zero (in order that the inverse of the function does not have an infinite derivative).
• The overall function is continuous with a continuous first derivative everywhere.
• If parameter use_yd0 = false, the two polynomials are constructed such that the second derivatives at x=0 are identical. If use_yd0 = true, the derivative at x=0 is explicitly provided via the additional argument yd0. If necessary, the derivative yd0 is automatically reduced in order that the polynomials are strict monotonically increasing [Fritsch and Carlson, 1980].

A typical screenshot for k1=1, k2=3 is shown in the next figure:

The (smooth, non-zero) derivative of the function with k1=1, k2=3 is shown in the next figure:

Literature

Fritsch F.N. and Carlson R.E. (1980):

Monotone piecewise cubic interpolation. SIAM J. Numerc. Anal., Vol. 17, No. 2, April 1980, pp. 238-246

y = regSquare2(x, x_small, k1, k2, use_yd0, yd0)

x	Type: Real Description: Abscissa value
x_small	Default Value: 0.01 Type: Real Description: Approximation of function for |x| <= x_small
k1	Default Value: 1 Type: Real Description: y = (if x>=0 then k1 else k2)*x*|x|
k2	Default Value: 1 Type: Real Description: y = (if x>=0 then k1 else k2)*x*|x|
use_yd0	Default Value: false Type: Boolean Description: = true, if yd0 shall be used
yd0	Default Value: 1 Type: Real Description: Desired derivative at x=0: dy/dx = yd0

y	Type: Real Description: Ordinate value