Make your method keep the symmetries of the problem

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Make your method keep the symmetries of the problem

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[QUICK DESCRIPTION]
Making your method keep the symmetries of the problem means that it is immune to errors that break symmetries.  This can have two important effects:
* reducing the errors in the results, and particularly eliminating entire classes of errors that have unphysical or otherwise harmful effects,
* and reducing the amount of computational work.

[PREREQUISITES]
Linear algebra, calculus.

[EXAMPLE]
For symmetric positive definite matrices, use the [[Cholesky factorization]] rather than the [[LU factorization]].

The LU factorization of a matrix is $A=LU$, while the Cholesky factorization is $A=LL^T$ where in each case $L$ is lower triangular and $U$ is upper triangular.
Using the Cholesky factorization (which preserves the symmetry of $A$) roughly halves the time to compute the factorization, and avoids the problems of swapping rows and/or columns of $A$ to preserve numerical stability.

[EXAMPLE]
Use [[w:Symplectic integrator|symplectic methods]] to solve Hamiltonian differential equations.

A Hamiltonian differential equation has the form
[maths] \frac{dz}{dt} = J\,\nabla H(z) [/maths]
where $J=\left[\begin{array}{cc}0&I\\ -I&0\end{array}\right]$.
Symplectic methods preserve the two-form $\sum_i dq_i \wedge dp_i$ where $z^T = [p^T,\, q^T]$.
Such methods also nearly preserve a "numerical energy" function (which depends on the step-size), and are much better for long-time integration of mechanical systems such as arise in celestial mechanics.

[GENERAL DISCUSSION]

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Make your method keep the symmetries of the problem

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