Linear algebra

Scientific programming languages provide facilities to work with linear algebra. For example, the following Julia program solves a system of linear equations:

A = rand(20, 20)  # A is a 20x20 matrix
b = rand(20)      # b is a 20-element vector
x = A\b           # x is the solution to A*x = b

Working with large vectors and matrices is a key feature of these languages, as linear algebra lays the foundation to mathematical optimization, which in turn enables major applications such as deep learning.

Mathematical optimization

In a scientific programming language, we can compute function optima with a syntax close to mathematical language. For instance, the following Julia code finds the minimum of the polynomial ${\displaystyle P(x,y)=x^{2}-3xy+5y^{2}-7y+3}$.

using Optim

P(x,y) = x^2 - 3x*y + 5y^2 - 7y + 3

z₀ = [ 0.0
       0.0 ]     # starting point for optimization algorithm

optimize(z -> P(z...), z₀, Newton();
         autodiff = :forward)

In this example, Newton's method for minimizing is used. Modern scientific programming languages will use automatic differentiation to compute the gradients and Hessians of the function given as input; cf. differentiable programming. Here, automatic forward differentiation has been chosen for that task. Older scientific programming languages such as the venerable Fortran would require the programmer to pass, next to the function to be optimized, a function that computes the gradient, and a function that computes the Hessian.

With more knowledge of the function to be minimized, more efficient algorithms can be used. For instance, convex optimization provides faster computations when the function is convex, quadratic programming provides faster computations when the function is at most quadratic in its variables, and linear programming when the function is at most linear.