hessian

Hessian matrix of scalar function

Syntax

• ``hessian(f,v)``
example

Description

example

````hessian(f,v)` finds the Hessian matrix of the scalar function `f` with respect to vector `v` in Cartesian coordinates.If you do not specify `v`, then `hessian(f)` finds the Hessian matrix of the scalar function `f` with respect to a vector constructed from all symbolic variables found in `f`. The order of variables in this vector is defined by `symvar`.```

Examples

Find Hessian Matrix of Scalar Function

Find the Hessian matrix of a function by using `hessian`. Then find the Hessian matrix of the same function as the Jacobian of the gradient of the function.

Find the Hessian matrix of this function of three variables:

```syms x y z f = x*y + 2*z*x; hessian(f,[x,y,z])```
```ans = [ 0, 1, 2] [ 1, 0, 0] [ 2, 0, 0]```

Alternatively, compute the Hessian matrix of this function as the Jacobian of the gradient of that function:

`jacobian(gradient(f))`
```ans = [ 0, 1, 2] [ 1, 0, 0] [ 2, 0, 0]```

Input Arguments

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`f` — Scalar functionsymbolic expression | symbolic function

Scalar function, specified as symbolic expression or symbolic function.

`v` — Vector with respect to which you find Hessian matrixsymbolic vector

Vector with respect to which you find Hessian matrix, specified as a symbolic vector. By default, `v` is a vector constructed from all symbolic variables found in `f`. The order of variables in this vector is defined by `symvar`.

If `v` is an empty symbolic object, such as `sym([])`, then `hessian` returns an empty symbolic object.

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Hessian Matrix

The Hessian matrix of f(x) is the square matrix of the second partial derivatives of f(x).

$H\left(f\right)=\left[\begin{array}{cccc}\frac{{\partial }^{2}f}{\partial {x}_{1}^{2}}& \frac{{\partial }^{2}f}{\partial {x}_{1}\partial {x}_{2}}& \cdots & \frac{{\partial }^{2}f}{\partial {x}_{1}\partial {x}_{n}}\\ \frac{{\partial }^{2}f}{\partial {x}_{2}\partial {x}_{1}}& \frac{{\partial }^{2}f}{\partial {x}_{2}^{2}}& \cdots & \frac{{\partial }^{2}f}{\partial {x}_{2}\partial {x}_{n}}\\ ⋮& ⋮& \ddots & ⋮\\ \frac{{\partial }^{2}f}{\partial {x}_{n}\partial {x}_{1}}& \frac{{\partial }^{2}f}{\partial {x}_{n}\partial {x}_{2}}& \cdots & \frac{{\partial }^{2}f}{\partial {x}_{n}^{2}}\end{array}\right]$