Reading through Tim Davis' book Direct Methods For Sparse Linear Systems, he says that matlab can use a supernodal Cholesky decomposition but never uses a multifrontal Cholesky decomposition. At least when using the backslash operator. By contrast, HSL MA57, used in many optimization codes, is a sparse symmetric indefinite solver that uses a multifrontal method.
I only have a very basic understanding of the differences between supernodal and multifrontal methods. So I'm curious if there are restrictions for when supernodal vs. multifrontal methods can be applied? In cases when they are both valid, are there any trade-offs between them that could lead to one being better than the other?