The Gaussian function has no elementary antiderivative, and that is a theorem of Liouville. When the antiderivative does not exist, or all you have is a table of measurements, the way forward is numerical: quad, trapezoid and Simpson, from scipy.integrate, with the care each one demands.
Is the reactor temperature rising too fast? The question is a derivative, but all we have are numbers in a table. We will see finite differences, numpy.gradient and scipy.differentiate, and why shrinking the step does not always improve the derivative, because of rounding and noise.
Experimental data rarely falls on a curve. We will see how NumPy fits polynomials to noisy data, what SymPy cannot do, why extrapolating is dangerous, and why Runge’s phenomenon comes from equally spaced nodes, not from high degree.
Stored in float64, the coefficients of Wilkinson’s polynomial no longer return the roots 1 to 20: two of them become a complex pair. Before getting there, we look at roots, derivatives, integrals and division with numpy.polynomial, and at the coefficient-order trap.
From the pH of a weak acid to the volume of a real gas and the friction factor of a pipe: three equations that chemistry and engineering use every day and that only have a useful numerical solution. We solve all three with brentq and numpy.roots and rebuild the Moody chart.
Where is Halley’s Comet on any given date? The answer requires solving Kepler’s equation, which has no closed form. With SciPy’s newton and brentq, we locate the comet, make the law of areas visible, and compute the range of a projectile with air resistance.
In Kepler’s equation, the unknown appears both inside and outside a sine, and no algebra isolates it. For cases like this, SciPy approximates the root. We will see bisection and brentq, Newton’s method and why it can fail, and nonlinear systems with fsolve and root.
Most equations, integrals, and differential equations in real problems have no closed-form solution, and SymPy can’t solve them. This 13-article series shows how NumPy and SciPy tackle these problems: roots, polynomials, numerical calculus, optimization, and FFT.
Both NumPy and pandas use the concept of an axis, but with subtle differences. In NumPy, it refers to the dimension along which an operation is performed, while in pandas it indicates the direction of an operation. This article presents examples of using axis in each library.
We learn in math that there are rules for adding or multiplying matrices. But in computational math problems we sometimes have matrices or vectors that we would like to add or multiply and that do not meet the requirements. In this article we will see how the NumPy broadcasting concept is useful in these contexts and how it relates to vector quantization.
If you are venturing into data science, you have most likely seen a reshape in some code. Maybe with a mysterious value of -1 in one of the positions, as in reshape(-1, 1). But what does that mean? Why do we sometimes need to transform our data to use model methods in Scikit-Learn? That is what we will see in this article.
Monte Carlo methods are computational algorithms that rely on random samples to obtain numerical results. In this article, we will see how to obtain the value of pi with these methods.
Usain Bolt is considered by many to be the greatest sprinter of all time. In this article, we will see how to use Python to find the speed and acceleration the Jamaican reached in his races. We will also take the chance to discuss mathematical models and the care needed in choosing them and interpreting their results.
In this article we detail how to get visually elegant plots that correctly show units on their axes. We will combine the pint, numpy and matplotlib libraries.
