The result of:
X' * A * X
For symmetric A has such a nice structure (Result is symmetric) that it is a pity MKL doesn't have an optimized function for it.
Quadratic Forms are very common in Machine Learning, Optimization, etc... It would benefit many users.
In case of dense, there is no single call solution as mecej4 mentioned.
However, if X and A are sparse matrices, then we added exactly this type of product to our Inspector-Executor Sparse BLAS routines. The structure is indeed quite nice. This type of product shows up a lot in multi-scale finite element methods as well where X could be a projection or elongation matrix. We call it the symmetric product with api -- mkl_sparse_sypr().
See reference documentation mkl_sparse_sypr for more details on how to use it.