Mathematica

Transpose and Flatten So now we know how to multiply matrices using Mathematica’s Dot function. This also works for higher-rank tensors (more indices) such as $m_{abc} n_{cde} = (mn)_{abde}$, which we can implement as

Mathematica is smart The take-away from this is that many clever people have worked for a very long time on optimising the Mathematica’s built-in functions, such as Dot. In particular, when Dot is called, it turns the calculation over to a set of C++ libraries for numerical linear algebra known as the Intel MKL.

Dot products The bulk of the numerical calculations that I need are basically linear algebra – matrix-matrix, vector-matrix and more exotic multiplications. All of the entries in these tensors are “machine precision”, which roughly translates to a C++ double.

Overview Calabi-Yau manifolds are special as they are Kähler manifolds which admit Ricci-flat metrics. As of the end of 2020, there are still no complete analytic expressions for such metrics on compact manifolds, other than in the somewhat trivial cases of tori.