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### [Lapack] BLAS question

 ``` Hello Tim, Well, I think you are stucked. This is a BLAS problem and has been reported several times. There is no BLAS routine to exploit the symmetry in the output H when H is computed as H <- M^H * h * M with h Hermitian. Assume M is n-by-m and h is n-by-n. Then your optimal number of FLOPS is to compute H is 2mn^2. If you also want to compute Q then you end up with 3mn^2 operations. You can not get neither one nor the other with BLAS right now. BLAS enables you to get 4mn^2. This is a known problem and bothers lots of people. (e.g. The folks working on LDL^T for example) Here are two tricks. [1] There are two conditions for the first trick. If (a) n << m (b) h is Hermitian positive definite then (i) Cholesky of h (POTRF = n^3/3): r <- chol (h), you get r such that h = r^H*r and r upper triangular, (ii) apply r to M from the left Q <- r*M, mn^2 (iii) apply SYRK: H <- Q^H Q -> mn^2 (iv) apply Q <- r^H * Q (this is mn^2) total: 3mn^2 + n^3/3 so 3mn^2 id n<
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