This loads packages containing some test problems and utility functions. One of the first and simplest methods for solving initial value problems was proposed by Euler: ...

The Padé approximation is a rational function that can be thought of as a generalization of a Taylor polynomial. A rational function is the ratio of polynomials. Because ...

Orthogonalize[{v_1, v_2, ...}] gives an orthonormal basis found by orthogonalizing the vectors v_i.Orthogonalize[{e_1, e_2, ...}, f] gives a basis for the e_i orthonormal ...

Mathematica provides state-of-the-art fully automated visualization of vector functions and data—suitable for representing flows, field lines and other vector fields of any ...

Sum[f, {i, i_max}] evaluates the sum \[Sum]i = 1 i_max f. Sum[f, {i, i_min, i_max}] starts with i = i_min. Sum[f, {i, i_min, i_max, di}] uses steps d i. Sum[f, {i, {i_1, i_2, ...

The differential equations that arise in practice are of two types. Here is an example of the first type. Here is an example of the second type. This equation has a symbolic ...

AlgebraicNumber[\[Theta], {c_0, c_1, ..., c_n}] represents the algebraic number in the field \[DoubleStruckCapitalQ][\[Theta]] given by c_0 + c_1 \[Theta] + ... + c_n ...

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 ...

Mathematica provides representation of algebraic numbers as Root objects. A Root object contains the minimal polynomial of the algebraic number and the root number—an integer ...

Recognize is now available as the newly added built-in Mathematica kernel function RootApproximant.