Directly integrated into Mathematica's uniform architecture for handling lists of data is an array of highly optimized algorithms for transforming and smoothing datasets that ...

In some cases it is useful to split the differential system into subsystems and solve each subsystem using appropriate integration methods. Recombining the individual ...

DSolve[eqn, y, x] solves a differential equation for the function y, with independent variable x. DSolve[{eqn_1, eqn_2, ...}, {y_1, y_2, ...}, x] solves a list of ...

"Values for Symbols" discussed how you can use transformation rules of the form x->value to replace symbols by values. The notion of transformation rules in Mathematica is, ...

Around 1870, Marius Sophus Lie realized that many of the methods for solving differential equations could be unified using group theory. Lie symmetry methods are central to ...

Mathematica transparently works with complex variables throughout, not only numerically, but also symbolically—often relying on original results to handle intricate branch ...

The Lotka–Volterra system arises in mathematical biology and models the growth of animal species. Consider two species where Y_1(T) denotes the number of predators and Y_2(T) ...

For minimization problems for which the objective function is a sum of squares, it is often advantageous to use the special structure of the problem. Time and effort can be ...

There are many variants of quasi-Newton methods. In all of them, the idea is to base the matrix B_k in the quadratic model on an approximation of the Hessian matrix built up ...

A method like "Newton's" method chooses a step, but the validity of that step only goes as far as the Newton quadratic model for the function really reflects the function. ...