Implicit Runge–Kutta methods have a number of desirable properties. The Gauss–Legendre methods, for example, are self-adjoint, meaning that they provide the same solution ...

The equations of motion for a free rigid body whose center of mass is at the origin are given by the following Euler equations (see [MR99]). Two quadratic first integrals of ...

NDSolve uses norms of error estimates to determine when solutions satisfy error tolerances. In nearly all cases the norm has been weighted, or scaled, such that it is less ...

Numerical integration functions. This finds a numerical approximation to the integral ∫_(0)^∞ e^-x^3 x. Here is the numerical value of the double integral ∫_(-1)^1 dx ...

Here is one way to get multiple minima: call NMinimize multiple times with different random seeds, which will cause different optimization paths to be taken. This defines a ...

As discussed in "Exact and Approximate Results", Mathematica can handle approximate real numbers with any number of digits. In general, the precision of an approximate real ...

Orthogonal polynomials. Legendre polynomials LegendreP[n,x] arise in studies of systems with three-dimensional spherical symmetry. They satisfy the differential equation ...

Complicated algebraic expressions can usually be written in many different ways. Mathematica provides a variety of functions for converting expressions from one form to ...

In a statement like x^4+x^2>0, Mathematica treats the variable x as having a definite, though unspecified, value. Sometimes, however, it is useful to be able to make ...

Structural operations on polynomials. Here is a polynomial in one variable. Expand expands out products and powers, writing the polynomial as a simple sum of terms.