# 2nd Order Taylor Series Formula inv. the Hessian Matrix, etc. - what is the x vector?

1. ### jaguar7

42
The formula given by my instructor for a Taylor Series approximation of the second order at point (a,b) is f(a,b) + grad(f(a,b))x + 1/2 H(f(a,b)) x

If you recognize this formula, do you know what the x vector is?

Note: x is the x-vector, and H represents the Hessian Matrix. Thanks!

The Hessian Matrix is the matrix with values [fxx, fxy, fyx, and fyy], where fxx represents the second partial derivative. Not sure the proper terminology for it...... df^2 / (dx)^2 (where d is a delta (not d) to represent partial dif.)

2. ### HallsofIvy

40,544
Staff Emeritus
Re: 2nd Order Taylor Series Formula inv. the Hessian Matrix, etc. - what is the x vec

x is the vector <x- a, y- b>. And you need a quadratic in the second term- it should be
$$f(a, b)+ \begin{bmatrix}\frac{\partial f}{\partial x}(a, b) & \frac{\partial f}{\partial y}(a, b)\end{bmatrix}\begin{bmatrix}x- a \\ y- b\end{bmatrix}+ \frac{1}{2}\begin{bmatrix}x- a & y- b\end{bmatrix}\begin{bmatrix}\frac{\partial^2 f}{\partial x^2}(a,b) & \frac{\partial^2 f}{\partial x\partial y}(a,b) \\ \frac{\partial^2 f}{\partial x\partial y}(a,b) & \frac{\partial^2 f}{\partial y^2}(a,b)\end{bmatrix}\begin{bmatrix} x- a \\ y- b\end{bmatrix}$$