### Add notes on parabolic interpolation

parent 9ea9cf49
 ... ... @@ -26,5 +26,6 @@ print(sp.latex(sol)) q = (x_2 - x_3) * (y_2 - y_1) r = (x_2 - x_1) * (y_2 - y_3) alt = x_2 - ((x_2 - x_3) * q - (x_2 - x_1) * r) / 2 / (q - r) print(sp.latex(alt)) print(sp.simplify(sol - alt))
 ... ... @@ -2,6 +2,49 @@ title: Optimization Algorithms --- ## Parabolic Interpolation A task that comes up frequently in optimization problems is guessing a function minimum based on a parabola fit between three points. So I'll derive this technique here. First, we need a parabola that goes through three points, say \$\$(x_1, y_1)\$\$, \$\$(x_1, y_1)\$\$, and \$\$(x_3, y_3)\$\$. In the interest of generality, I'll construct it using Lagrange polynomials. \$\$ p(x) = \frac{y_1 (x - x_2) (x - x_3)}{(x_1 - x_2) (x_1 - x_3)} + \frac{y_2 (x - x_1) (x - x_3)}{(x_2 - x_1) (x_2 - x_3)} + \frac{y_3 (x - x_1) (x - x_2)}{(x_3 - x_1) (x_3 - x_2)} \$\$ Differentiating, we find that the derivative is equal to the following. \$\$ \begin{aligned} p'(x) &= \frac{y_{1} \left(x - x_{2}\right)}{\left(x_{1} - x_{2}\right) \left(x_{1} - x_{3}\right)} + \frac{y_{1} \left(x - x_{3}\right)}{\left(x_{1} - x_{2}\right) \left(x_{1} - x_{3}\right)} \\ &+ \frac{y_{2} \left(x - x_{1}\right)}{\left(- x_{1} + x_{2}\right) \left(x_{2} - x_{3}\right)} + \frac{y_{2} \left(x - x_{3}\right)}{\left(- x_{1} + x_{2}\right) \left(x_{2} - x_{3}\right)} \\ &+ \frac{y_{3} \left(x - x_{1}\right)}{\left(- x_{1} + x_{3}\right) \left(- x_{2} + x_{3}\right)} + \frac{y_{3} \left(x - x_{2}\right)}{\left(- x_{1} + x_{3}\right) \left(- x_{2} + x_{3}\right)} \end{aligned} \$\$ Setting this to zero, we find that the solution is surprisingly simple. \$\$ x_\text{min} = \frac{1}{2} \cdot \frac{x_1^2 (y_2 - y_3) + x_2^2 (y_3 - y_1) + x_3^2 (y_{1} - y_2)} {x_1 (y_2 - y_3) + x_2 (y_3 - y_1) + x_3 (y_1 - y_2)} \$\$ This can also be factored so that it only involves differences of coordinates and function values. \$\$ x_\text{min} = x_{2} - \frac{1}{2} \cdot \frac{(x_2 - x_3)^2 (y_2 - y_1) - (x_2 - x_1)^2 (y_2 - y_3)}{(x_2 - x_3) (y_2 - y_1) - (x_2 - x_1) (y_2 - y_3)} \$\$ ## Nelder-Mead TODO: Demonstrate that the Numerical Recipes formulation is equivalent to the standard formulation. ... ...
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