2nd order approx. of barrier function

[FOIWATER]

Homework Statement

write the 2nd order taylor series for the log barrier function $$-\sum_{i=1}^{m}(b_{i}-a_{i}^{T}x)$$

Homework Equations

See Above

The Attempt at a Solution

Here is my attempt at a solution
$$\nabla f(x)=f(x_{0})-\sum_{i=1}^{m}\bigg(\dfrac{a_{i}}{b_{i}-a_{i}^{T}x}\bigg)(x-x_{0})-\dfrac{1}{2}(x-x_{0})^{T}\sum_{i=1}^{m}\bigg(\dfrac{a_{i}^{T}a_{i}}{(b_{i}-a_{i}^{T}x)^{2}}\bigg)(x-x_{0})$$

So, my problem is I've never really encountered calculating gradients and hessians for vector valued functions of only one variable (x, in this case). Am I way out to lunch?

Thanks.

 

[HallsofIvy]

Perhaps it would help if you were to say exactly what the "log barrier function" is!

 

[FOIWATER]

oh I typed it wrong

$$f(x)=-\sum_{i=1}^{m}log(b_{i}-a_{i}^{T}x)$$

But I know now that what I have above is not correct. I think the gradient might be right actually, but the hessian is certainly not.