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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.

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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.

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  • $\begingroup$ What's an accelerator? $\endgroup$
    – Bananach
    Oct 19, 2023 at 19:49
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    $\begingroup$ mostly GPU, but it could also be some other specialized piece of hardware. $\endgroup$ Oct 19, 2023 at 20:00

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