I'm trying to get the condition number of a multivariate function $f(a,b,c)$ to see if it is stable. I am reading the information here.
I know how to do it for a $1$-dimensional function. But for a multivariate one, I am not sure how. Also how do I even get the Jacobian because there is only 1 function here. Wouldn't that mean the Jacobian is a $1 \times 3$ matrix?
If $f(x)=f(x_1,\ldots,x_n)$ is a scalar differentiable function, then the condition number of $f$ at $x^*=(x_1^*,\ldots,x_n^*)$ is given by $$ \frac{\|J(x^*)\|}{|f(x^*)| / \|x^*\|} = \frac{\|x^*\| \|J(x^*)\|}{|f(x^*)|} $$ Assuming you choose $L_2$ norm: $$ \frac{\sqrt{ \sum_{i=1}^n (f'_{x_i})^2(x_1^*,\ldots,x_n^*) }}{|f(x_1^*,\ldots,x_n^*)| / \sqrt{\sum_{i=1}^n (x_i^*)^2}} = \frac{ \sqrt{\sum_{i=1}^n (x_i^*)^2} \; \sqrt{\sum_{i=1}^n (f'_{x_i})^2(x_1^*,\ldots,x_n^*)}}{|f(x_1^*,\ldots,x_n^*)|} $$