Hessian Calculator
Calculate the Hessian matrix and analyze critical points of scalar functions.
Scalar Function f(x, y)
Point of Evaluation
Hessian Matrix Formula
H = [d^2f/dx^2 d^2f/dxdy]
[d^2f/dydx d^2f/dy^2]
Hessian Determinant
3.000000
f(1, 1) = 3.000000
Hessian Matrix
| 2.0000 | 1.0000 |
| 1.0000 | 2.0000 |
detDeterminant
3.000000
trTrace
4.000000
Critical Point Analysis
Local Minimum
Not a critical point (gradient != 0)
Matrix Properties
Positive Definite: Yes
Negative Definite: No
About the Hessian
Second Derivative Test
At a critical point: If det(H) > 0 and d^2f/dx^2 > 0, its a local minimum. If det(H) > 0 and d^2f/dx^2 < 0, its a local maximum. If det(H) < 0, its a saddle point.
Applications
- Optimization and machine learning
- Newton's method for optimization
- Curvature analysis of surfaces
- Second-order Taylor approximation