Как посчитать градиент функции python

With a similar procedure the forward/backward approximations used for boundaries can be derived.

Quarteroni A., Sacco R., Saleri F. (2007) Numerical Mathematics (Texts in Applied Mathematics). New York: Springer.

Durran D. R. (1999) Numerical Methods for Wave Equations in Geophysical Fluid Dynamics. New York: Springer.

Fornberg B. (1988) Generation of Finite Difference Formulas on Arbitrarily Spaced Grids, Mathematics of Computation 51, no. 184 : 699-706. PDF.

Spacing can be also specified with an array that represents the coordinates of the values F along the dimensions. For instance a uniform spacing:

Or a non uniform one:

For two dimensional arrays, the return will be two arrays ordered by axis. In this example the first array stands for the gradient in rows and the second one in columns direction:

In this example the spacing is also specified: uniform for axis=0 and non uniform for axis=1

It is possible to specify how boundaries are treated using edge_order

The axis keyword can be used to specify a subset of axes of which the gradient is calculated

Calculating gradient with NumPy

I really can not understand what numpy.gradient function does and how to use it for computation of multivariable function gradient.

For example, I have such a function:

I need to compute it’s 3-dimensional gradient (in other words, I want to compute partial derivatives with respect to all variables (q, chi, delta)).

How can I calculate this gradient using NumPy?

5 Answers 5

The problem is, that numpy can’t give you the derivatives directly and you have two options:

With NUMPY

What you essentially have to do, is to define a grid in three dimension and to evaluate the function on this grid. Afterwards you feed this table of function values to numpy.gradient to get an array with the numerical derivative for every dimension (variable).

Without NUMPY

You could also calculate the derivative yourself by using the centered difference quotient.

How to compute gradients in Tensorflow and Pytorch

Computing gradients is one of core parts in many machine learning algorithms. Fortunately, we have deep learning frameworks handle for us. This post will explain how Tensorflow and Pytorch can help us to compute gradient with an example.

Many of us has been familiar with training neural networks using TensorFlow and PyTorch. We already know how to compute gradients and use optimizers to update the weight parameters with some lines of code. This post is about to separate the part of computing gradients in these libraries to see what happens behind the code.

1. Derivatives and Gradients

In 1-dimension, the derivative of a function is defined as follows:

Generally, in multiple dimensions, gradient is the vector of partial derivatives along each dimension. Hence, a gradient has the same shape as x. And each element of the gradient will tell us what is the slope of the function f if we move in that coordinate direction.

The gradient turns out to have nice properties. It points in the direction of the greatest increase of the function. Correspondingly, the negative of gradient give us the direction of greatest decrease of the function.

2. How to evaluate gradients

One naive way to evaluate gradients in a computer is using method of finite differences, which uses the limit definition of gradient (Eq 1). Concretely, we iteratively evaluate the equation 1 for each dimension of x with a small value of h for that dimension. And it can be very slow when the size of x is large.

But thankfully, we do not have to do that. We can use calculus to compute an analytic gradient, i.e. to write down an expression for what the gradient should be.

In summary, there are 2 ways to compute gradients.

• Numerical gradients: approximate, slow, easy to write.
• Analytic gradients: exact, fast, error-prone.

In practice, we should always use analytic gradients, but check implementation with numerical gradients. This is called gradient check.

3. An example for illustration

Now, let jump to an example (from Coursera course in References). In this example, we will have some computations and use chain rule to compute gradient ourselves. We then see how PyTorch and Tensorflow can compute gradient for us.

4. PyTorch code

Implementing the code in PyTorch will give us exactly what we expect for the example above.

torch.autograd is PyTorch’s automatic differentiation engine that helps us to compute gradients.

We first create a tensor x with requires_grad=True . This signals to autograd that every operation on it should be tracked. When we call .backward() on z , autograd calculates these gradients and stores them in the tensor’s .grad attribute. Hence, we can see the gradients in x.grad .

5. Tensorflow code

In TensorFlow, optimizers are implemented using TensorFlow automatic differentiation API call Gradient Tape. This API lets us compute and track the gradient of every differentiable TensorFlow operation.

Operations within a gradient tape scope are recorded if at least one of their variables is watched. If we watch the variable x , the tape will watch the rest of the operations that you can see below. When we call tape.gradient to compute the gradients of z with respect to x , we have the same results as before.

Conclusions

This post provided a simple example about how to compute gradients using PyTorch’s autograd and TensorFlow’s Gradient Tape. We actually use them for more complicated functions and for training deep neural networks.

The secret of differential mechanism (as I know) is from graph computation. However, the graph computation, backpropagation or the explicit implementation of these frameworks are beyond the scope of this post. You can look for more explanations in the References or their source code.

Numpy Gradient Examples using numpy.gradient() method.

Numpy is the best python module that allows you to do any mathematical calculations on your arrays. For example, you can convert NumPy array to the image, NumPy array, NumPy array to python list, and many things. But here in this tutorial, I will show you how to use the NumPy gradient with simple examples using the numpy.gradient() method.

What is Gradient?

In mathematics, Gradient is a vector that contains the partial derivatives of all variables. Like in 2- D you have a gradient of two vectors, in 3-D 3 vectors, and show on.

In NumPy, the gradient is computed using central differences in the interior and it is of first or second differences (forward or backward) at the boundaries. Let’s calculate the gradient of a function using numpy.gradient() method. But before that know the syntax of the gradient() method.

The numpy.gradient() function accepts the following important parameters.

f: The input array on which you want to calculate the gradient.

varargs: It is a list of scalars or arrays.

axis: Either 0 or 1 to do calculation row-wise or column-wise. The default value is None.

edge_order: <1, 2>, and it is optional. The gradient is calculated using N-th order accurate differences at the boundaries. The default value is 1.

Step by Step to calculate Numpy Gradient

Here you will know all the steps to compute the NumPy gradient of an array. Just follow the below steps.

Step 1: Import all the necessary libraries

Here I am using only NumPy python modules so importing it only. You can import any module in python using an import statement.

Step 2: Create a Dummy Numpy Array.

For the demonstration purpose lets the first create a NumPy array to calculate the numpy gradient. You can create a NumPy array using numpy.array() method like below. However, you can use your own dataset for gradient calculations.

Example 1: Simple Numpy Array Gradient.

First-order Differences Gradient

Second-Order Differences Gradient

Example 2: Calculation of Gradient using other NumPy arrays.

You can also calculate the gradient of a NumPy array with another NumPy array. Let’s create a second NumPy array.

Example 3: Gradient for the N-dimensional NumPy array

You can calculate the gradient for the N dimension NumPy array. The gradient will of the same dimension as the dimension array. Let’s create a two-dimensional NumPy array.

Use the code below to calculate the gradient.

The above code will return two arrays. The first one is the gradient of all the row values and the second one is the gradient along the column.

If you want to calculate row-wise then pass the axis =0 as an argument to the gradient() method and for column-wise axis =1.

These are some basic examples that show you how to calculate numpy gradient of a NumPy array. If you have also a dataset or excel file then you can read it using the pandas module and then extract data as a NumPy array.

Hope you have liked this article if you have any query or wants to know more then please contact us. We are always ready for help.

Other Examples

Numpy gradient of a function

Just like you can find the gradient for a single or multidimensional array. You can also find the NumPy gradient of a function using numdifftools python package. Suppose I have a function that returns polynomial expression on x. To find the gradient of the function I will pass the function name as an argument to the Gradient() method with the value in the square bracket.

Execute the below lines of code.

Output A gradient of a function in Python

Other Queries

Question: I have got No module named pandas error.

This error comes when you are using the pandas module in your code but you have not installed it. To remove this error just install pandas array. It will remove this import error.

You can install pandas using the pip command.

For python 2.xx version

For python 3.xx version

Similar Numpy Functions –

Numpy Random Choice : Create Random Sample Array

Find Max and Min Value of Numpy Array with its index ( 1D & 2D)

Numpy cov() method Implementation in Python : 3 Steps Only

• Total 3

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