Fast weighted vector inner product x*A*y with BLAS/LAPACK

Question

Is there a way to compute the weighted vector inner product xAy with vectors x and y and Matrix A using BLAS/LAPACK while avoiding additional allocations or overwriting the inputs?

I'm happy with answers that assume A is symmetric or x=y.

Answer 1

The product $x^TAy$ can be calculated as:

$\sum_{i}\sum_j x_i A_{i,j} y_j$.

Easily implemented as a double for loop. You can additionally avoid a few flops by doing either:

$\sum_{i}x_i \sum_j A_{i,j} y_j$ or

$\sum_{j}y_j\sum_i x_i A_{i,j} $.

You asked to use BLAS, but note that this is a level 2 BLAS routine, which is not actually that much faster than two for/do loops. Some exceptions:

• Your matrix is huge, meaning you can parallelize
• Your for loops violate spatial locality.
• You are using an accelerator

I am not aware of this functionality currently existing in BLAS.