sensitivity analysis

root sum of squares

In general

$e_y^2 = \sum \left( \frac{\partial y}{\partial x} e_x \right)^2$

addition

$z = x + y$ $e_z^2 = \left( \frac{\partial z}{\partial x} e_x \right)^2 + \left( \frac{\partial z}{\partial y} e_y \right)^2$ $e_z^2 = e_x^2 + e_y^2$

multiplication

$z = x y$ $e_z^2 = \left( \frac{\partial z}{\partial x} e_x \right)^2 + \left( \frac{\partial z}{\partial y} e_y \right)^2$ $e_z^2 = y^2 e_x^2 + x^2 e_y^2$

general function

If we have

$f(y, x_1, x_2, ..., x_n) = 0$

which cannot be solved for $y$

$e_y^2 = \sum \left( \frac{\partial y}{\partial f} \frac{\partial f}{\partial x_i} e_{x_i} \right)^2$ $e_y^2 = \sum \left( \frac{\frac{\partial f}{\partial x_i}}{\frac{\partial f}{\partial y}} e_{x_i} \right)^2$ $e_y^2 = \frac{1}{\left( \frac{\partial f}{\partial y} \right)^2} \sum \left( \frac{\partial f}{\partial x_i} e_{x_i} \right)^2$

Charles Rymal

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sensitivity analysis

root sum of squares

addition

multiplication

general function