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2nd Order Taylor Series Formula inv. the Hessian Matrix, etc.  what is the x vector? 
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#1
Nov1510, 07:09 PM

P: 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 xvector, 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
Nov1610, 09:19 AM

Math
Emeritus
Sci Advisor
Thanks
PF Gold
P: 39,682

x is the vector <x a, y b>. And you need a quadratic in the second term it should be
[tex]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}[/tex] 


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