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
[tex]\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}[/tex]
then
[tex]\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}[/tex]