Accessing Algorithms in Mathematica
Whenever you use Mathematica you are accessing the world's largest collection of computational algorithms.
In each case, Mathematica automatically chooses the best algorithm to use.
Mathematica has state-of-the-art algorithms, here for factoring integers.
This generates a two-dimensional table corresponding to a matrix.
This displays the table in matrix form.
This computes the null space of the matrix.
Mathematica can solve differential equations both symbolically and numerically.
Here Mathematica solves a nonlinear differential equation numerically.
The answer is an interpolating function which implicitly represents the whole solution. Here is a parametric plot of the solution.
All of Mathematica's numerical algorithms are set up to work to whatever accuracy or precision you specify.