AlgoPy, Algorithmic Differentiation in Python algopy  documentation

AlgoPy, Algorithmic Differentiation in Python algopy documentation


Python SymPy library is created for symbolic mathematics. The SymPy project aims to become a full-featured computer algebra system while keeping the code simple to understand. Let’s see how to calculate derivatives in Python using SymPy.

When the function has a horizontal slope, and that the derivative takes on a positive value wherever the parent function has a positive slope. Let’s reproduce this result via auto-differentiation using MyGrad. Declaring a symbol is similar to saying that our function has a variable ‘x’ or simply the function depends on x. Let’s see how can we achieve this using SymPy diff() function. Symbolstakes a string of variable names separated by spaces or commas, and creates Symbols out of them.

Version Changelog¶

The differences between scalar AD and hierarchical AD have been highlighted and it has been demonstrated, at the example of the Cholesky decomposition, that it is possible to apply matrix calculus also to matrix factorizations. Luckily, addressing these challenges are relatively easy. The function evaluation and constructing the primal trace can be performed by simply running the computation in the forward direction as described above.

”’Compute the difference formula for f’ with step size h. The chain rule calculate the derivative of a composition of functions. Let’s dive into how can we actually use sympy to calculate derivatives as implied by the general differentiation rules. There are certain rules we can use to calculate the derivative of differentiable functions. This is by no means an article about the fundamentals of derivatives, it can’t be.

It aims to be an alternative to systems such as Mathematica or Maple while keeping the differentiation in python as simple as possible and easily extensible. SymPy is written entirely in Python and does not require any external libraries. So how do we go about making the integral expression for this?

Comparison operators follow the same rules as the underlying numeric types. An error estimate to calculate the derivative’s numerical value can be done by calculating the formula for the derivative in an analytical way and substituting the value at a desired point. In this post, we examine how you can calculate the value of the derivative using numerical methods in Python. A wide variety of applied problems can be solved using calculation methods that are based on mathematical principles using digital values as opposed to analytical and symbolic methods. Presently, some of the most popular Python-centric autodiff libraries include PyTorch, TensorFlow, and JAX.

We can achieve this by not modifying our previous code at all. However, since we no longer need the forward-accumulation of the gradients, we can simply modify our code to no longer compute this. We can use the nice visualization package in Python called graphviz, that is also used in the deep learning framework Keras, to graphically draw the directed acyclic graph. This requires a few more lines of code but it’s worth the effort. These unevaluated objects are useful for delaying the evaluation of the derivative, or for printing purposes. They are also used when SymPy does not know how to compute the derivative of an expression .


Up to now, I have wandwaved away the math and the deeper intuition for why and how autodiff works. By breaking down a function into elementary operations, we can leverage the differentiation rules for all the pieces to end up with the final gradient. For nested functions, we have to use the chain rule in differentiation to compute the correct gradient. We used the chain rule several times in the introduction to machine learning tutorial, but as a reminder, a function is composite if you can write it as $f(g)$. In other words, we have a function within another function.

  • The Python code below calculates the derivative of this function.
  • The purpose of AlgoPy is the evaluation of higher-order derivatives in the forward and reverse mode of Algorithmic Differentiation of functions that are implemented as Python programs.
  • The maximum likelihood estimation approach, which is widely used for this purpose, contains the underlying assumption that the likelihood function is known to follow a specified parametric probability distribution.
  • We then implemented a simple forward-mode auto-differentiation engine in Python.

When data assimilation has been used with models of carbon balance, prior or “background” errors and observation errors have largely been treated as independent and uncorrelated. Correlations between background errors have long been known to be a key aspect of data assimilation in numerical weather prediction. More recently, it has been shown that accounting for correlated observation errors in the assimilation algorithm can considerably improve data assimilation results and forecasts. The idea of including these correlations in time is new and has not been previously explored in carbon balance model data assimilation. In data assimilation, background and observation error statistics are often described by the background error covariance matrix and the observation error covariance matrix. The methods used in this paper will allow the inclusion of time correlations between many different observation types in the assimilation algorithm, meaning that previously neglected information can be accounted for.

Reverse-mode autodiff: the approach that powers modern deep learning

Equation 3 — Position as a function of time Velocity is the first derivative of position, and acceleration is the second derivative of displacement. The analytical representations are given in Equations 4 and 5, respectively. Figure 1 — Rise Over Run This definition is comparable to the first-principles definition of the derivative in differential calculus, given by Equation 2 and depicted in Figure 2.

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The calculation of the derivative is also used for gradient methods when training neural networks. The computed derivatives have a finite-precision error on par with the nominal function evaluation. There are two kinds of integrals, definite and indefinite.

A tutorial on numerical differentiation in Python

Don’t forget that these returned expressions are SymPy expressions on which we can use solve(), subs(), expand() and other similar functions. Implemented the forward mode of automatic differentiation with the help of dual numbers using Python. Equation 6 — Example Function of Time Obtain the analytical first derivative of Equation 6 using the differentiation rules linked above.

The synchronous variant of our method outperformed an existing multi-GPU implementation in terms of accuracy while running at a comparable execution time. Oftentimes, input values of functions are specified in the form of an argument-value pair, which in large data arrays, can be significantly data-intensive to process. Fortunately, many problems are much easier to solve if you use the derivative of a function, helping across different fields like economics, image processing, marketing analysis, etc. Python is a popular programming language for scientific computing , .

Then when accumulating the gradient in the reverse direction, we have all the required information. Briefly, in the previous part, we learned about dual numbers and explored their relationship to the derivative, and how we can exploit that to compute gradients. We then implemented a simple forward-mode auto-differentiation engine in Python.

derivative 0.5.3

We discuss the reasons for this alternative approach and explain the underlying idea. Examples illustrate how AlgoPy can be used from a user’s point of view. And this is all that is required to find the derivative of a function in Python. So, below we will find the derivative of the function, x4 + 7×3 + 8.


In principle 3 and 4 differ only by who does the work, the computer or the programmer. 3 is preferred over 4 due to consistency, scalability, and laziness. Automatic derivatives are very cool, aren’t prone to numeric errors, but do require some additional libraries . This is the most robust but also the most sophisticated/difficult to set up choice. If you’re fine restricting yourself to numpy syntax then Theano might be a good choice.

The goal of this package is to provide some common numerical differentiation techniques that showcase improvements that can be made on finite differences when data is noisy. This article by no means was a course about derivatives or how can we solve derivatives in Python but an article about how can we leverage python SymPy packages to perform differentiation on functions. Derivatives are awesome and you should definitely get the idea behind it as they play a crucial role in Machine learning and beyond.

This section covers how to do basic calculus tasks such as derivatives, integrals, limits, and series expansions in SymPy. If you are not familiar with the math of any part of this section, you may safely skip it. Savitzky-Galoy derivatives of any polynomial order with independent left and right window parameters.

To evaluate an unevaluated derivative, use the doit() method. The first parameter of the diff function should be the function you want to take the derivative of. The second parameter should be the variable you are taking the derivative with respect to. This way, dydx will be computed using central differences and will have the same length as y, unlike numpy.diff, which uses forward differences and will return (n-1) size vector. Just for the sake of completeness, you can also do differentiation by integration (see Cauchy’s integral formula), it is implemented e.g. in mpmath . When I said “symbolic differentiation” I intended to imply that the process was handled by a computer.

It’s possible scipy is calling numpy incorrectly, but very unlikely. See if np.interp() works – it may provide a more helpful error if not. Is there an easy way to do finite differences in numpy without implementing it yourself?


We present one academic and two industrial application examples (biochemical process/Diesel-oxidation catalysis process) where we achieve speedups that range between 10 and 100. In addition to our core results, we also describe an efficient adjoint approach for the treatment of differential algebraic equations and present adjoint formulas for constrained least-squares problems. Efforts to implement variational data assimilation routines with functional ecology models and land surface models have been limited, with sequential and Markov chain Monte Carlo data assimilation methods being prevalent.

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We propose a method for an efficient optimization of experimental designs, using a combination of discrete adjoint computations, Taylor arithmetic and matrix calculus. We demonstrate that the advantageous complexity results are not only of theoretical nature, but lead to significant speedups in practice as well. With our implementation we are very close to the theoretical bound of the cheap gradient principle.

It has a clear syntax, a large standard library and there exist many packages useful for scientific computing. The de facto standard for array and matrix manipulations is provided by the package NumPy and thus many scientific programs in Python make use of it. In consequence, concepts such as broadcasting, slicing, element-wise operations and numerical linear algebra functions are used on a regular basis. In addition to scipy differentiate, you can also use analytical differentiation in Python. The SymPy package allows you to perform calculations of an analytical form of a derivative.

So we are able to make functions and compute their derivatives, but how do we use these functions? We need to be able to plug a value into these equations and get a solution. Post your problem as a new question and link to it here. Providing an example that causes your error to occur will probably be needed.

Gist 3 — Numerically Solve Equation 3Figure 3 plots the numerical differentiation results and the analytical solutions for velocity and acceleration. The Python code in Gist1 evaluates the numerical derivative of any function by applying the theory presented above. A practical example of numerical differentiation is solving a kinematical problem. Kinematics describes the motion of a body without considering the forces that cause them to move. Numerical differentiation is finding the numerical value of a function’s derivative at a given point.