# Hessian Matrix Calculator + Online Solver With Free Steps

A **Hessian Matrix Calculator** is used to calculate the Hessian Matrix for a multi-variable function by solving all the calculus required for the problem. This calculator comes in very handy as **Hessian Matrix** is a lengthy and hectic problem, and the calculator provides the solution at the press of a button.

## What Is a Hessian Matrix Calculator?

**A Hessian Matrix Calculator is an online calculator which is designed to provide you with solutions to your Hessian Matrix problems.**

**Hessian Matrix** is an advanced calculus problem and is used mainly in the field of **Artificial Intelligence** and **Machine Learning**.

Therefore, this **Calculator** is very useful. It has an input box for the entry of your problem and with a press of a button, it can find the solution to your problem and send it to you. Another wonderful feature of this **Calculator** is that you can use it in your browser without downloading anything.

## How to Use a Hessian Matrix Calculator?

To use the **Hessian Matrix Calculator**, you can enter a function in the input box and press the submit button, after which you will get the solution to your input function. It must be noted that this calculator can only calculate the **Hessian Matrix** for a function with a maximum of three variables.

Now, we will provide you with step-by-step instructions for using this calculator to get the best results.

### Step 1

You start by setting up a problem that you would like to find the **Hessian Matrix** for.

### Step 2

You enter the multi-variable function you would like to get the solution to in the input box.

### Step 3

To get the results, you press the **Submit** button, and it opens the solution in an interactable window.

### Step 4

Finally, you can solve more Hessian Matrix problems by entering your problem statements in the interactable window.

## How Does a Hessian Matrix Calculator Work?

A **Hessian Matrix Calculator** works by solving the second-order partial derivatives of the input function and then finding the resulting **Hessian Matrix** from them.

### Hessian Matrix

A** Hessian **or **Hessian Matrix** corresponds to the square matrix acquired from the second-order partial derivatives of a function. This matrix describes the local curves carved by a function and is used for optimizing the results obtained from such a function.

A **Hessian Matrix** is calculated only for functions with scalar constituents, which are also referred to as a **Scalar Fields**. It was originally brought forward by the German mathematician **Ludwig Otto Hesse** in the **1800s**.

### Calculate a Hessian Matrix

To calculate a **Hessian Matrix**, we first require a multi-variable function of this sort:

**f(x, y)**

It is important to note that the calculator is only functional for a maximum of three variables.

Once we have a multi-variable function, we can move forward by taking first order partial derivatives of this function:

\[\frac{\partial f(x, y)}{\partial x}, \frac{\partial f(x, y)}{\partial y}\]

Now, we carry on by taking second order partial derivatives of this function:

\[\frac{\partial^2 f(x, y)}{\partial x^2}, \frac{\partial^2 f(x, y)}{\partial y^2}, \frac{\partial^2 f(x, y)}{\partial x \partial y}, \frac{\partial^2 f(x, y)}{\partial y \partial x}\]

Finally, when we have all these four second order partial derivatives, we can calculate our Hessian Matrix by:

\[ H_f(x, y) = \bigg [ \begin{matrix} \frac{\partial^2 f(x, y)}{\partial x^2} & \frac{\partial^2 f(x, y)}{\partial x \partial y} \\ \frac{\partial^2 f(x, y)}{\partial y \partial x} & \frac{\partial^2 f(x, y)}{\partial y^2} \end{matrix} \bigg ]\]

## Solved Examples

### Example 1

Consider the given function:

\[f(x, y) = x^2y + y^2x\]

Evaluate the Hessian Matrix for this function.

### Solution

We start by solving partial derivatives for the function corresponding to both x, and y. This is given as:

\[\frac{\partial f(x, y)}{\partial x} = 2xy + y^2\]

\[\frac{\partial f(x, y)}{\partial y} = x^2 + 2yx\]

Once we have the first order partial differentials of the function, we can move forward by finding the second order differentials:

\[\frac{\partial^2 f(x, y)}{\partial x^2} = 2y\]

\[\frac{\partial^2 f(x, y)}{\partial y^2} = 2x\]

\[\frac{\partial^2 f(x, y)}{\partial x \partial y} = \frac{\partial^2 f(x, y)}{\partial y \partial x} = 2x + 2y\]

Now that we have all the second order partial differentials calculated, we can simply get our resultant Hessian Matrix:

\[ H_f(x, y) = \bigg [ \begin{matrix} \frac{\partial^2 f(x, y)}{\partial x^2} & \frac{\partial^2 f(x, y)}{\partial x \partial y} \\ \frac{\partial^2 f(x, y)}{\partial y \partial x} & \frac{\partial^2 f(x, y)}{\partial y^2} \end{matrix} \bigg ] = \bigg [ \begin{matrix}2y & 2x+2y \\ 2x+2y & 2x\end{matrix} \bigg ] \]

### Example 2

Consider the given function:

\[f(x, y) = e ^ {y \ln x}\]

Evaluate the Hessian Matrix for this function.

### Solution

We start by solving partial derivatives for the function corresponding to both x, and y. This is given as:

\[\frac{\partial f(x, y)}{\partial x} = e ^ {y \ln x} \cdot \frac{y}{x} \]

\[\frac{\partial f(x, y)}{\partial y} = e ^ {y \ln x} \cdot \ln x \]

Once we have the first order partial differentials of the function, we can move forward by finding the second order differentials:

\[\frac{\partial^2 f(x, y)}{\partial x^2} = e ^ {y \ln x} \cdot \frac{y^2}{x^2} – e ^ {y \ln x} \cdot \frac{y}{x^2} \]

\[\frac{\partial^2 f(x, y)}{\partial y^2} = e ^ {y \ln x} \cdot \ln ^2 x \]

\[\frac{\partial^2 f(x, y)}{\partial x \partial y} = \frac{\partial^2 f(x, y)}{\partial y \partial x} = e ^ {y \ln x} \cdot \frac{y}{x} \cdot \ln x +e ^ {y \ln x} \cdot \frac{1}{x} \]

Now that we have all the second-order partial differentials calculated, we can simply get our resultant Hessian Matrix:

\[ H_f(x, y) = \bigg [ \begin{matrix} \frac{\partial^2 f(x, y)}{\partial x^2} & \frac{\partial^2 f(x, y)}{\partial x \partial y} \\ \frac{\partial^2 f(x, y)}{\partial y \partial x} & \frac{\partial^2 f(x, y)}{\partial y^2} \end{matrix} \bigg ] = \bigg [ \begin{matrix}e ^ {y \ln x} \cdot \frac{y^2}{x^2} – e ^ {y \ln x} \cdot \frac{y}{x^2} & e ^ {y \ln x} \cdot \frac{y}{x} \cdot \ln x +e ^ {y \ln x} \cdot \frac{1}{x} \\ e ^ {y \ln x} \cdot \frac{y}{x} \cdot \ln x +e ^ {y \ln x} \cdot \frac{1}{x} & e ^ {y \ln x} \cdot \ln ^2 x \end{matrix} \bigg ] \]