regRoot2

Anti-symmetric approximation of square root with discontinuous factor so that the first derivative is finite and continuous

Information

This information is part of the Modelica Standard Library maintained by the Modelica Association.

Approximates the function

y = if x ≥ 0 then sqrt(k1*x) else -sqrt(k2*abs(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 finite.
  • 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].

Typical screenshots for two different configurations are shown below. The first one with k1=k2=1:

regRoot2_a.png

and the second one with k1=1 and k2=3:

regRoot2_b.png

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

regRoot2_c.png

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

Syntax

y = regRoot2(x, x_small, k1, k2, use_yd0, yd0)

Inputs (6)

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 sqrt(k1*x) else -sqrt(k2*|x|)

k2

Default Value: 1

Type: Real

Description: y = if x>=0 then sqrt(k1*x) else -sqrt(k2*|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

Outputs (1)

y

Type: Real

Description: Ordinate value