Hessian Matrix Calculator

Easily compute the Hessian matrix of a multivariable function. Enter your function and variables to get started.

Function (e.g., x^2*y + y^3):

Variables (e.g., x, y, z):

Separate variables with commas.

Result: Hessian Matrix

About Hessian Matrix

In mathematics, particularly in calculus, the Hessian matrix or Hessian is a square matrix of second-order partial derivatives of a scalar-valued function, or scalar field. It describes the local curvature of a function of more than one variable.

The Hessian matrix is used in optimization problems to determine the local convexity of a function. It is also used in Newton's method for finding critical points of a function. For a function $ f(x_1, x_2, ..., x_n) $, the Hessian matrix $ H(f) $ is given by:

$$ H(f) = \begin{bmatrix} \frac{\partial^2 f}{\partial x_1^2} & \frac{\partial^2 f}{\partial x_1 \partial x_2} & \cdots & \frac{\partial^2 f}{\partial x_1 \partial x_n} \\ \frac{\partial^2 f}{\partial x_2 \partial x_1} & \frac{\partial^2 f}{\partial x_2^2} & \cdots & \frac{\partial^2 f}{\partial x_2 \partial x_n} \\ \vdots & \vdots & \ddots & \vdots \\ \frac{\partial^2 f}{\partial x_n \partial x_1} & \frac{\partial^2 f}{\partial x_n \partial x_2} & \cdots & \frac{\partial^2 f}{\partial x_n^2} \end{bmatrix} $$

This calculator helps you compute this matrix for any given function and variables, making it easier to analyze the behavior of multivariable functions.

Sources: Wikipedia, MathWorld