Efficient Anti-Triangular Matrix Multiplications

[mwl]

Does anyone know of efficient software for the multiplication with an anti-triangular matrix? These are matrices whose triangular structures goes from the lower left corner to the upper right corner. Are there codes similar to the Level 3 BLAS routines for the multiplication of these matrices? They come up in light scattering computations in high dimensions.
Thanks.

 

[AlephZero]

I don't know of any codes for this, but you probably don't need to do anything more complicated than get the Level 3 BLAS code for triangular matrices from LAPACK (or from your computer supplier if you are doing high performace computing) and reverse the order of some of the loops.

 

[mwl]

Hi AlephZero,
What a great idea! Thank you so much.
mwl

 

[AlephZero]

Actually, you might not need to change the BLAS routines, if you can store some of the matrices and vectors "upside down". For example, if
$$\begin{bmatrix} a_{11} & a_{12} & a_{13} \\ a_{21} & a_{22} \\ a_{31} \end{bmatrix} \begin{bmatrix} x_1 \\ x_2 \\ x_3 \end{bmatrix} = \begin{bmatrix} y_1 \\ y_2 \\ y_3 \end{bmatrix}$$
then
$$\begin{bmatrix} a_{31} \\ a_{21} & a_{22} \\ a_{11} & a_{12} & a_{13} \end{bmatrix} \begin{bmatrix} x_1 \\ x_2 \\ x_3 \end{bmatrix} = \begin{bmatrix} y_3 \\ y_2 \\ y_1 \end{bmatrix}$$