Function Mechatronix::minCubic

Function Documentation

bool Mechatronix::minCubic(real_type f0, real_type Df0, real_type l, real_type fl, real_type p, real_type fp, real_type smin, real_type smax, real_type &l_opt)

Find the minimum of a cubic passing by 3 points and initial tangent.

The cubic \( p(x) = p_0 + p'_0 x + c x^2 + d x^3 \) satisfy

\[ p(0) = p_0, \quad p'(0)= p'_0, \quad p(a) = p_a, \quad p(b) = p_b \]
The minimum is searched in the interval
\[ [s_{\min}b,s_{\max}b] \]
and thus satisfy

\[ x = \arg\min \{p(z) \,|\, z\in[s_{\min}b,s_{\max}b] \} \]
Parameters

  • f0 – the value \( f(0) \)

  • Df0 – the value \( f'(0) \)

  • l – the value \( l \)

  • fl – the value \( f(l) \)

  • p – the value \( p \)

  • fp – the value \( f(p) \)

  • smin – the value \( s_{\min} \)

  • smax – the value \( s_{\max} \)

  • l_opt – theparameter with the minimum

Returns

true if minimum found